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Suppose that $f$ is a quadratic polynomial and $g$ is a cubic polynomial, and both $f$ and $g$ have a leading coefficient of $1$. What is the maximum degree of the polynomial $(f(x))^3 - (g(x))^2 + f(x) - 1$?
Since $f$ has a degree of $2$, the degree of $(f(x))^3$ is $6$. Also, since $g$ has a degree of $3$, the degree of $(g(x))^2$ is $6$. Furthermore, as $f$ and $g$ both have a leading coefficient of $1$, then $(f(x))^3$ and $(g(x))^2$ both have a leading coefficient of $1$. Hence, when subtracting $(f(x))^3 - (g(x))^2$, the leading terms cancel, and so $(f(x))^3 - (g(x))^2$ has a maximum degree of $5$. We can see that a degree of $\boxed{5}$ can be achieved by taking $f(x) = x^2 + x$ and $g(x) = x^3$, for example. | Math Dataset |
Maximizing throughput gain via resource allocation in D2D communications
Yucheng Wu1,
Xiaocui Liu1,
Xiang He1,
Qiong Yu1 &
Weiyang Xu1
By reusing the cellular resources, device-to-device (D2D) communication is becoming a very promising technology that greatly enhances the spectrum utilization. To harvest the benefits that D2D communications can offer, efficient resource allocation strategy is required to guarantee the demands of quality of service (QoS) for both cellular and D2D users. This paper proposes a resource allocation scheme to alleviate the performance deterioration of the D2D communications with spectrum reuse. To maximize the overall throughput gain, the proposed scheme is designed to reduce the rate loss of cellular users and improve the rate of D2D users simultaneously in a two-step manner. Specifically, it first calculates the reuse gain for a single D2D pair and a single cellular user. Next, a maximum weight bipartite matching is further proposed to select the reuse pair to maximize the overall network throughput gain. Numerical results demonstrate that the proposed resource allocation scheme can significantly improve the network throughput performance with average user rate guaranteed.
The device-to-device (D2D) communication is widely recognized as one of the key technology of the evolving 5G architecture due to the enhanced cellular spectrum utilization [1]. In the D2D scenario, the terminals can communicate directly with one another without the base station (BS) [2]. Therefore, the end-to-end latency can be decreased; also, the area spectral efficiency can be improved simultaneously. Therefore, the network is able to accommodate more users [3, 4].
It is worth noting that D2D communications rely on the reuse of cellular spectrum resources; thus, the performance of the cellular system will be subject to the interference incurred as a consequence. This key problem has drawn much attention from both the academic and industrial fields. In references, methods in [4–7] suggest to mitigate the interference that cellular users suffer by either limiting the D2D user's transmit power or choosing the D2D users only in the interference limited area. However, the two approaches mentioned above cannot fully enhance the performance of D2D communications.
On the other hand, the motivation of works in [8–11] is to increase the network throughput. In [8], a single D2D pair is allowed to reuse a single cellular user's resource to maximize the throughput, and also, a closed expression of the optimal power allocation is given. In [9], the overall network throughput is maximized via reusing cellular users' resources by multiple D2D pairs where the optimization problem is solved in three steps, i.e., access control, power allocation, and channel allocation. Moreover, the literatures in [10, 11] still consider the resource allocation with the goal of maximizing the throughput while taking the throughput gain as the access control criterion. Unfortunately, none of the above studies take into account the performance loss of cellular users incurred by the spectrum reuse. In [12], the authors propose a power management scheme for an adjacent femtocell network and formulate a non-convex optimization problem in order to maximize the capacity under the power constraints. The joint uplink subchannel and power allocation in cognitive small cells using cooperative Nash bargaining game theory is investigated in [13], where the cross-tier interference mitigation, minimum outage probability requirement, imperfect CSI, and fairness are considered. In [14], the authors propose an iterative gradient user association and power allocation approach with attention to load balance constraints, energy harvesting by base stations, user quality of service requirements, energy efficiency, and cross-tier interference limits. More recently, [15] analyzes the characteristics of optimal joint power control and D2D matching strategy, based on which an energy-efficient iterative algorithm for D2D communications is proposed.
For the future evolution of cellular networks, it is significant to maintain the quality of service (QoS) of both cellular and D2D users. To this end, this paper proposes a resource allocation algorithm that maximizes the throughput gain while reducing the rate loss of cellular users and increasing the rate of D2D users at the same time. It is demonstrated that the resource allocation in this study can be modeled as a mixed integer nonlinear programming (MINLP) optimization problem. To find a tractable solution, the original MINLP problem is decomposed into two subproblems, where the optimal solutions are able to be obtained in a two-step manner without reducing the feasible domain. Specifically, the first subproblem is to obtain the maximum reuse gain when a single D2D user shares a single cellular user's resource and determine whether it is eligible for spectrum reuse. Moreover, the second subproblem determines the best pairing between D2D and cellular users and finally maximizes the overall network throughput.
The rest of the paper is organized as follows. The system model and optimization problem description are given in Section 2. Then, in Section 3, the optimal resource allocation algorithm is investigated in detail. Numerical results are presented in Section 4 to demonstrate the performance of the proposed scheme. Finally, Section 5 concludes this paper.
System model and problem formulation
Introduction of system model
In the time-division duplexing (TDD) system, D2D users are enabled to access time-frequency resources of the cellular networks. As a result, both D2D and cellular users are subject to the interference from each other. As shown in Fig. 1, the receiver of the D2D pair D1 is interfered by the cellular user C1, also the BS is interfered by the transmitter of the D2D pair D1. Assuming that there are N available orthogonal frequency resource blocks (RB) in one cell, the BS allocates resources to N cellular users with the traditional algorithm. Here, we assume that the number of cellular users is fixed, the case of varying numbers can refer to [16]. Let C={1,2,…,N} and D={1,2,…,M} denote the sets of cellular and D2D users, respectively. Furthermore, only one or zero D2D user is allowed to share the same RB with the cellular user n. At last, it is assumed that the BS has the knowledge of the channel state information (CSI), which is kept constant during the coherence time and changes independently in different coherence intervals. Furthermore, the proposed algorithm is based on a generic model in device-to-device and cellular hybrid network, which could be applicable in content sharing, gaming, connectivity extension, traffic offloading, disaster relief, etc.
The illustration of interference between D2D and cellular users
The path loss model in [5] is employed in this paper. Specifically, the path gain between the terminals i and j (j=1 represents the BS) can be modeled as:
$$ g_{i,j} = K\beta_{i,j}\eta_{i,j}d_{i,j}^{-\alpha} $$
where K is a system-related constant, β i,j represents the multipath gain of the link between terminals i and j, which follows the exponential distribution, η i,j denotes the shadow gain of the link, following the logarithm distribution, d i,j indicates the distance of the link, and α indicates the path-loss factor. In order to distinguish different links in the system model, we adopt the following rules: D m,m indicates the D2D link m and the corresponding path gain is \(g_{D_{m,m}}\), C n,B indicates the link between cellular user n and BS and the path gain is represented as \(g_{C_{n,B}}\), D m,B denotes the link between the transmitter of D2D pair m to BS and the path gain is \(g_{D_{m,B}}\), and C n,m represents the link between the cellular user n to the receiver of the D2D pair m, whereas the path gain is expressed as \(g_{C_{n,m}}\).
Our study aims to maximize the throughput gain while reducing the rate loss of cellular users. First, it is necessary to measure the rate loss of cellular users, which can be expressed as the difference between the rate of cellular users with or without the interference caused by D2D users under the same transmit power constraint. Thus, the rate loss can be formulated as follows:
$$ R_{\text{loss}} = {\log_{2}}\left(1+\xi_{n}^{\text{no}}\right)-{\log_{2}}\left({1+\xi_{n}}\right) $$
where R loss is the rate loss and ξ n and \(\xi _{n}^{\text {no}}\) indicate the signal to interference plus noise ratio (SINR) in the case of sharing resources with D2D users or not, respectively. The expressions of ξ n and \(\xi _{n}^{\text {no}}\) are
$$ \begin{aligned} \xi_{n} &= \frac{P_{n}g_{C_{n,B}}}{\sigma^{2} + P_{m}g_{D_{m,B}}},&\forall n \in C\\ \xi_{n}^{\text{no}} &= \frac{P_{n}g_{C_{n,B}}}{\sigma^{2}},&\forall n \in C \end{aligned} $$
where P n and P m indicate the transmission power of the cellular user and D2D user, separately, and σ 2 denotes the variance of the additive white Gaussian noise (AWGN).
On the other hand, this study is proposed to minimize the rate loss of the cellular user while ensuring the rate gain of the D2D user. Specifically, when the same time-frequency resource is shared between a single D2D pair and a cellular user, the optimization problem can be described as:
$$ \begin{aligned} &\mathop{\max}\limits_{{P_{n}},{P_{m}}} {\log_{2}}\left({1+{\xi_{m}}}\right)\\ &\mathop{\min}\limits_{{P_{n}},{P_{m}}} {\log_{2}}\left({1+\xi_{n}^{\text{no}}}\right)-{\log_{2}}\left({1+{\xi_{n}}}\right) \end{aligned} $$
where ξ m is the SINR obtained after D2D users access the cellular resources, i.e., \(\xi _{m} = P_{m}g_{Dm,m}/\left (\sigma ^{2} + P_{n}g_{C_{n,m}}\right)\).
Equation (4) is a multi-objective optimization problem, where the non-inferior solution can be solved by the weighted evaluation function method [17]. We use the linear weighting method to construct the evaluation function, which can be written as follows:
$$ \begin{aligned} &\mathop{\max}\limits_{{P_{n}},{P_{m}}} {\lambda_{1}}\left({{{\log }_{2}}\left({1 + {\xi_{m}}} \right)} \right)\\ &\quad+ {\lambda_{2}}\left({{{\log }_{2}}\left({1 + \xi_{n}^{{\rm{no}}}} \right) - {{\log }_{2}}\left({1 + {\xi_{n}}} \right)} \right) \end{aligned} $$
The target function should be converted to the maximum problem without any bias. Let λ 1=1 and λ 2=−1, so that the evaluation function is obtained. The purpose of this function is to maximize the system throughput gain. A suboptimal solution could be given to the original optimization problem, thus maximizing the system throughput gain can take into account the performance gain of the D2D user and at the same time the performance loss of the cellular user.
Considering that there are multiple D2D links and cellular links in the cell, the optimization problem is described as:
$$ {\begin{aligned} &\mathop {\max }\limits_{{x_{m,n}},{P_{n}},{P_{m}}} \sum\limits_{m \in {D_{m}}} \sum\limits_{n \in {C_{n}}} \big\{\log_{2}\left({1 + {\xi_{n}}} \right) + x_{m,n}\log_{2}\left(1+\xi_{m} \right)\\ &\quad- \log_{2}\left(1 + \xi_{n}^{\text{no}} \right)\big\}\\ s.t. \qquad &{\xi_{n}} = \frac{{{P_{n}}{g_{{C_{n,B}}}}}}{{{\sigma^{2}} + {P_{m}}{g_{{D_{m,B}}}}}} \ge {\xi_{n,\min }};\\ &{\xi_{m}} = \frac{{{P_{m}}{g_{{D_{m,m}}}}}}{{{\sigma^{2}} + {P_{n}}{g_{{C_{n,m}}}}}} \ge {\xi_{m,\min }};\\ &\sum\limits_{m} {{x_{m,n}}} \le 1,\;{x_{m,n}} \in \left\{ {0,1} \right\};\\ &\sum\limits_{n} {{x_{m,n}}} \le 1,\;{x_{m,n}} \in \left\{ {0,1} \right\};\qquad \forall n \in C;\;\forall m \in {D_{A}}\\ &0 \le {P_{n}} \le {P_{n,\max }};\\ &0 \le {P_{m}} \le {P_{m,\max }};\qquad \qquad \qquad~ \forall n \in C;\;\forall m \in {D_{A}} \end{aligned}} $$
where D A (D A ∈D) represents the subset of D2D users that can access the cellular network, ξ n,min and ξ m,min are the minimum SINR requirements for cellular users and D2D users, respectively, and ξ m represents the SINR of D2D user with interference caused by the cellular user. According to the expressions of ξ n and \(\xi _{n}^{\text {no}}\), it can be found that when ξ n ≥ξ n,min, it is straightforward to derive that \(\xi _{n}^{\text {no}} \ge {\xi _{n,\min }}\), thus the constraint of \(\xi _{n}^{\text {no}} \ge {\xi _{n,\min }}\) is not required. x m,n is the identifier of the resource reuse, i.e., when the D2D user m reuses the resource of cellular user n, then x m,n =1; otherwise, x m,n =0. Since the optimization problem contains the integer variable x m,n and the objective function is nonlinear, it can be considered as a mixed integer nonlinear programming problem which is difficult to directly obtain the optimal solution. Alternatively, this optimization procedure can be decomposed into two subproblems without changing the feasible domain of the original problem. After that, the corresponding optimal solutions to subproblems are obtained separately. The next section will present the detailed description of solving the optimization problem.
Resource allocation for throughput gain maximization
Two subproblems are obtained from the original mixed integer nonlinear programming problem to facilitate the optimization procedure. The first subproblem is to solve the maximum reuse gain of a single D2D user when reusing a single cellular user's resource and determine whether it is eligible to share the spectrum. The second subproblem determines the best pairings that maximize the overall network throughput gain, when multiple D2D users reuse multiple cellular users' resources.
Joint access control and power allocation based on multiplexing gain
In order to maximize the overall network throughput gain, it is necessary to determine the subset of D2D users that can access the cellular network. First, we need to establish the optimal objective function for maximizing the throughput gain with constraints of QoS and transmit power. Then, by solving the objective function, the optimal power allocation and maximum throughput gain can be obtained. Finally, we can obtain the subset of D2D users D A by judging whether the maximum throughput gain is greater than zero.
From the analysis of the Section 2, it can be found that Eq. (6) becomes the optimization problem of maximizing the throughput gain when λ 1=1 and λ 2=−1. Accordingly, the expression is
$$ { \begin{aligned} &\left({P_{n}^{*},P_{m}^{*}} \right)\mathop { = \arg \max }\limits_{{P_{n}},{P_{m}}}\big\{ {\log_{2}}\left({1 + {\xi_{n}}} \right) + {\log_{2}}\left({1 + {\xi_{m}}} \right)\\ &\quad- {\log_{2}}\left({1 + \xi_{n}^{\text{no}}} \right)\big\}\\ s.t.\quad &{\xi_{n}} = \frac{{{P_{n}}{g_{{C_{n,B}}}}}}{{{\sigma^{2}} + {P_{m}}{g_{{D_{m,B}}}}}} \ge {\xi_{n,\min }};\\ &{\xi_{m}} = \frac{{{P_{m}}{g_{{D_{m,m}}}}}}{{{\sigma^{2}} + {P_{n}}{g_{{C_{n,m}}}}}} \ge {\xi_{m,\min }};\\ &0 \le {P_{n}} \le {P_{n,\max }};\\ &0 \le {P_{m}} \le {P_{m,\max }} \end{aligned}} $$
where \(\xi _{n}^{\text {no}} = P_{n}g_{C_{n,B}}/\sigma ^{2}\).
Obviously, Eq. (7) is a nonlinear programming problem. When the equal sign of QoS constraint is established, it can be converted to a function of P n and P m separately, i.e.,
$$ \begin{aligned} {l_{c}}:~{P_{n}} &= \frac{{{g_{{D_{m,B}}}}{\xi_{n,\min }}\;}}{{{g_{{C_{n,B}}}}}}{P_{m}}+\frac{{{\xi_{n,\min }}{\sigma^{2}}}}{{{g_{{C_{n,B}}}}}}\\ l_{d}:~{P_{n}} &= \frac{{{g_{{D_{m,m}}}}\;}}{{{g_{{C_{n,m}}}}{\xi_{m,\min }}}}{P_{m}}-\frac{{{\sigma^{2}}}}{{{g_{{C_{n,m}}}}}} \end{aligned} $$
where l c represents the QoS constraint of cellular user n and the power allocation which is larger than l c can satisfy cellular user n's QoS. Whereas l d represents the QoS constraint of D2D user m, the power allocation which is smaller than l d can satisfy D2D user m's QoS. In addition, the power allocation should follow the maximum and minimum power constraint of cellular users and D2D users. Thus, the area enclosed by straight lines l c and l d and power constraints is the feasible solution range of power allocation, which is the represented by Γ and shown as the shadow area in Fig. 2. In order to solve the nonlinear programming problem of Eq. (7), we need to apply the following conclusion:
The admissible area according to power and QoS constraints. a The scope of solution 1. b The scope of solution 2. c The scope of solution 3
The power distribution exists at the lower boundary of the feasible solution domain, i.e., a straight line \({P_{n}}{g_{{C_{n,B}}}} = {\xi _{n,\min }}\left ({{\sigma ^{2}} + {P_{m}}{g_{{D_{m,B}}}}} \right)\).
First, the following relationship can be obtained with straightforward mathematical manipulations:
$$ \begin{aligned} f\left({P_{n}},{P_{m}}\right) &= g\left({P_{n}},{P_{m}}\right) + h\left({P_{n}},{P_{m}}\right)\\ g\left({P_{n}},{P_{m}}\right) &= {\log_{2}}\left({1 + {\xi_{m}}} \right)\\ h\left({P_{n}},{P_{m}}\right) &= {\log_{2}}\left({1 + {\xi_{n}}} \right) - {\log_{2}}\left({1 + \xi_{n}^{{\rm{no}}}} \right) \end{aligned} $$
Evidently, g(P n ,P m ) is a monotonically decreasing function of P n , and it can be proved that h(P n ,P m ) is a monotonically increasing function with respect to P n . Consequently, we can get \({\xi _{n}} < \xi _{n}^{\text {no}}\) for any P m ≠0. Furthermore, let us define
$$ \begin{aligned} h\left(\kappa {P_{n}},{P_{m}}\right) - h\left({P_{n}},{P_{m}}\right) &= {\log_{2}}\left({\frac{{1 + \kappa {\xi_{n}}}}{{1 + \kappa \xi_{n}^{{\rm{no}}}}}} \right)\\ &\quad- {\log_{2}}\left({\frac{{1 + {\xi_{n}}}}{{1 + \xi_{n}^{\text{no}}}}} \right) \end{aligned} $$
For any κ>1, we can have the following:
$$ \left({\frac{{1 + \kappa {\xi_{n}}}}{{1 + \kappa \xi_{n}^{\text{no}}}}} \right) < \left({\frac{{1 + {\xi_{n}}}}{{1 + \xi_{n}^{\text{no}}}}} \right) $$
Also, it comes to
$$ \begin{aligned} &h\left(\kappa {P_{n}},{P_{m}}\right) < h\left({P_{n}},{P_{m}}\right)\\ &g\left(\kappa {P_{n}},{P_{m}}\right) < g\left({P_{n}},{P_{m}}\right) \end{aligned} $$
Finally, we arrive at the conclusion f(k P n ,P m )<f(P n ,P m ). Thus, for any P m ∈Γ, f(P n ,P m ) is a monotonically decreasing function with respect to P n ; thus, the optimal solution corresponds to the lower boundary of constraint domain Γ, i.e., \(P_{n}g_{C_{n,B}}={\xi _{n,\min }}{\sigma ^{2}} + {P_{m}}{g_{Dm,B}}\). Therefore, the power distribution exists at the lower boundary of the feasible solution domain, and the theorem is proved. □
When applying Theorem 1 to the original optimization problem, the feasible solution range can be reduced to the lower boundary. As a result, the original optimization problem can be transformed into the following equation:
$$ {P_{m}} = \frac{{{P_{n}}{g_{{C_{n,B}}}} - {\xi_{n,\min }}{\sigma^{2}}}}{{{\xi_{n,\min }}{g_{{D_{m,B}}}}}} $$
It is necessary to point out that a constant after conversion of log2(1+ξ m,min), which does not affect the solution to the problem, can be safely removed in Eq. (13).
Consequently, the original optimization problem is converted to
$$ \begin{aligned} P_{n}^{*}&= \mathop{\max}\limits_{{P_{n}}} \left({{{\log }_{2}}\left({1 + \frac{{\left({{P_{n}}{g_{{C_{n,B}}}} - {\xi_{n,\min }}{\sigma^{2}}} \right){g_{{D_{m,m}}}}}}{{\left({{\sigma^{2}} + {P_{n}}{g_{{C_{n,m}}}}} \right){\xi_{n,\min }}{g_{{D_{m,B}}}}}}} \right) } \right.\\ &\quad-\left. {{{\log }_{2}}\left({1 + \frac{{{P_{n}}{g_{{C_{n,B}}}}}}{{{\sigma^{2}}}}\;} \right)} \right) \end{aligned} $$
Let us define
$$ {\begin{aligned} Q\left({{P_{n}}} \right) = {\left.{\left({1 + \frac{{\left({{P_{n}}{g_{{C_{n,B}}}} - {\xi_{n,\min }}{\sigma^{2}}} \right){g_{{D_{m,m}}}}}}{{\left({{\sigma^{2}} + {P_{n}}{g_{{C_{n,m}}}}} \right){\xi_{n,\min }}{g_{{D_{m,B}}}}}}} \right)} \right/ {\left({1 + \frac{{{P_{n}}{g_{{C_{n,B}}}}}}{{{\sigma^{2}}}}\;} \right)}} \end{aligned}} $$
Thus, we have the partial derivative as
$$ \frac{{\partial Q}}{{\partial {P_{n}}}} = \frac{{ - ACP_{n}^{2} - 2BC{P_{n}} + AE - DB}}{F} $$
$$\begin{aligned} A &= {\sigma^{2}}\left({{\xi_{n,\min }}{g_{{C_{n,m}}}}{g_{{D_{m,B}}}} + {g_{{D_{m,m}}}}{g_{{C_{n,B}}}}} \right)\\ B &= {\sigma^{2}}{\sigma^{2}}{\xi_{n,\min }}\left({{g_{{D_{m,B}}}} - {g_{{D_{m,m}}}}} \right)\\ C &= {\xi_{n,\min }}{g_{{C_{n,m}}}}{g_{{D_{m,B}}}}\\ D &= {\sigma^{2}}{\xi_{n,\min }}{g_{{D_{m,B}}}}\left({{g_{{C_{n,m}}}} + {g_{{C_{n,B}}}}} \right)\\ E &= {\sigma^{2}}{\sigma^{2}}{\xi_{n,\min }}{g_{{D_{m,B}}}}\\ F &= {\left({CP_{n}^{2} + D{P_{n}} + E} \right)^{2}} \end{aligned} $$
Let us further define \(-ACP_{n}^{2} - 2BC{P_{n}} + AE - DB = 0\), then the extreme point of Q(P n ) can be calculated by
$$ P_{n}^{{\Delta}} = \frac{{2BC \pm \sqrt {\left({2BC} \right)^{2} + 4AC\left(AE - DB\right)} }}{{ - 2AC}} $$
One of the poles is negative and another is positive; thus, in the range P n ∈(0,P n,max], the positive solution \(P_{n}^{{\Delta }}\) could be its extreme points. Due to the fact that 4A C(A E−D B)≪(2B C)2, then the extreme value of the solution can be simplified as:
$$ \begin{aligned} P_{n}^{{\Delta}} &= \frac{{2BC + \sqrt {\left({2BC} \right){}^{2} + 4AC\left(AE - DB\right)} }}{{ - 2AC}}\\ &\approx \frac{{ - 2{\sigma^{2}}{\xi_{n,\min }}\left({{g_{{D_{m,B}}}} - {P_{m,\max }}{g_{{D_{m,m}}}}} \right)}}{{{\xi_{n,\min }}{g_{{C_{n,m}}}}{g_{{D_{m,B}}}} + {g_{{D_{m,m}}}}{g_{{C_{n,B}}}}}} \end{aligned} $$
If there is an extreme value in the feasible solution range, the maximum value is the extreme value. If there is no extreme value in the feasible solution range, the maximum value is the boundary. Therefore, the solution to the optimal power distribution is:
$$ {{P_{n}^{*} = \left\{ \begin{aligned} &P_{n}^{{\Delta}},&{P_{a}} \le P_{n}^{{\Delta}} \le {P_{b}}\\ &{P_{a}},&P_{n}^{{\Delta}} \le {P_{a}}\\ &{P_{b}},&{P_{b}} \le P_{n}^{{\Delta}} \end{aligned} \right.; \newline \qquad P_{m}^{*} = \frac{{P_{n}^{*}{g_{{C_{n,B}}}} - {\xi_{n,\min }}{\sigma^{2}}}}{{{\xi_{n,\min }}{g_{{D_{m,B}}}}}}}} $$
where \(P_{a} = \frac {{{\sigma ^{2}}\left ({{\xi _{n,\min }}{g_{{D_{m,m}}}} + {\xi _{n,\min }}{\xi _{m,\min }}{g_{{D_{m,B}}}}} \right)}}{{{g_{{C_{n,B}}}}{g_{{D_{m,m}}}} - {\xi _{n,\min }}{\xi _{m,\min }}{g_{{D_{m,B}}}}{g_{{C_{n,m}}}}}}\) is obtained by using (8), \(P_{n}^{{\Delta }}\) is the extremum obtained by (19), and P b is the intersection solution of the feasible solution boundary, which is different from the change of the feasible solution range. Interesting remarks can be obtained as follows.
When the feasible solution domain is shown as in the case of Fig. 2a, c, the range of P n is from P a to P b , where \(P_{b} = \xi _{n,\min }\left (\sigma ^{2}+g_{D_{m,B}}\right)/g_{C_{n,B}}\)
When the feasible solution domain is shown as in the case of Fig. 2b, the range of P n is from P a to P n,max
The optimal power distribution pair \(\left (P^{*}_{n},P^{*}_{m}\right)\) can be obtained by Eq. (19). It is necessary to confirm whether the multiplexed pair can bring the throughput gain. Hence, \(\left (P^{*}_{n},P^{*}_{m}\right)\) will be substituted into Eq. (21) to obtain \(R_{n,m}^{\text {Gain}}\), which can be defined as
$$ R_{n,m}^{\text{Gain}} = {\log_{2}}\left({1 + {\xi_{n}}} \right) + {\log_{2}}\left({1 + {\xi_{m}}} \right) - {\log_{2}}\left({1 + \xi_{n}^{\text{no}}} \right) $$
If \(R_{n,m}^{\text {Gain}}\) is greater than zero, it comes to the conclusion that D2D user m is actually qualified to reuse the resource of cellular user n.
Multiple D2D users multiplex multiple cellular users' resources
After the D2D user set D A and the maximum reuse gain of the reused pair are obtained, it is next required to determine the best pairing which could maximize the overall network throughput gain when multiple D2D users reuse multiple cellular users' resources. The problem can be modeled as the weighted matching of the weighted bipartite graphs in graph theory, which is represented as follows:
$$ \begin{aligned} &\mathop{\max}\limits_{{x_{m,n}}} \sum\limits_{n \in {{C_{m}',}}m \in D'} {{x_{n,m}}R_{n,m}^{\text{Gain}}}\\ s.t.\quad &\sum\limits_{m} {{x_{m,n}}} \le 1,~{x_{m,n}} \in \left\{ {0,1} \right\},~\forall m \in {D_{A}}\\ &\sum\limits_{n} {{x_{m,n}}} \le 1,~{x_{m,n}} \in \left\{ {0,1} \right\},~\forall n \in C \end{aligned} $$
where D ′ represents the set of accessible D2D users and C m′ denotes the set of cellular users of which the resources that D2D users m can reuse. Figure 3 shows the bipartite graph optimal matching problem of Eq. (21) with D A ={1,2,…,M 1}, where M 1 is the maximum number of D2D pairs allowed to access the cellular spectrum. When the D2D user m reuses the resource of the cellular user n, it establishes a connection and takes \(R_{n,m}^{\text {Gain}}\) as a weight.
Weighted bipartite graph for D2D users and cellular users
The solution to the above problem can be solved by the Kuhn-Munkres algorithm in [18], and the details is beyond the scope of this paper. The pseudo-code of the maximizing throughput gain via resource allocation is summarized in Algorithm 1.
Simulation results and discussion
In order to verify the performance of our scheme, the resource allocation algorithm based on maximized system throughput proposed in [5] is used as the benchmark. The throughput gain in [5] is defined as the maximum throughput increase after the introduction of D2D, shown as follows:
$$ R_{n,m}^{\text{Gain}} = R_{n,m}^{\max} - R_{n}^{\max } $$
The throughput gain in this paper is defined as the gain obtained by the power distribution according to the maximum throughput gain:
$$ R_{n,m}^{\text{Gain}} = R_{n,m}^{\max, \rm gain} $$
In order to compare with fairness, the rate gain in the reference [5] is modified. The formula is as follows:
$$ R_{n,m}^{\text{Gain}} = R_{n,m}^{\max} - R_{n}^{{\rm{no}}} $$
where \(R_{n}^{\text {no}}\) is the rate of cellular user without interference under the same transmit power constraint.
The simulation parameters are listed in Table 1. Specifically, the multipath fading follows the exponential distribution, and shadow fading follows a log-normal distribution.
Figure 4 shows the relationship between the system throughput gain and the number of D2D pairs for a cell multiplexing with the D2D users. MT denotes the maximum throughput algorithm in [5]. It can be observed that the throughput gain increases when the number of D2D users increases. As the maximum transmission distance of D2D users increases, the throughput gain reduces consequently. However, the gain of the proposed algorithm is significantly higher than that of the MT scheme, since the purpose of optimization taken in this paper is to maximize the throughput gain.
The relationship between the access rate and the number of D2D pairs
Figure 5 demonstrates the access rate, which is defined as the ratio of the actual number of access D2D users to the total number of D2D users. The access rates of the two algorithms decrease as the D2D user increases and decreases with the maximum transmission distance of the D2D user. In this paper, the access rate based on the multiplexing gain access control is slightly lower than the MT algorithm. The reason behind is that the access control is based on the throughput gain, and to ensure the access quality, D2D users who can not bring the gain are not permitted to access the cellular spectrum.
The relationship between the total rate loss of cellular users and the number of D2D pairs
The total rate loss of cellular users is shown in Fig. 6. The rate loss of both the proposed algorithm and MT is independent of the distance of D2D users and the number of D2D pairs. The total rate loss of cellular users of the proposed allocation is much lower than that of the MT algorithm, which reduces the cost of resource sharing for cellular users.
The relationship between the system throughput gain and the number of D2D pairs
Figure 7 shows the cumulative distribution function (CDF) of average rate loss of cellular users. It is observed from this figure that the rate loss of cellular users using this method is much lower than that of MT algorithm and the loss range is more concentrated. Therefore, one can come to the conclusion that the proposed algorithm reduces not only the rate loss, but also the cost of spectrum sharing between the cellular and D2D users.
The CDF of cell user's rate loss
Aiming at reducing the performance loss caused by the reuse of cellular resources by D2D users, the concept of reuse cost is proposed to measure the rate loss of cellular users. The multi-objective optimization problem of maximizing the gain of D2D users and minimizing the loss of cellular users is established and transformed into single-objective optimization problem by constructing evaluation function. To solve the optimization problem, the original mixed integer nonlinear programming is divided into two sub-problems, and the optimal solution of the sub-problems is given. The simulation results show that the proposed algorithm can maximize the throughput gain and reduce the rate of loss of cellular users while ensuring the QoS requirements of D2D users and cellular users.
In this study, we assume the perfect CSI while the channel estimation can never be error-free in practice [19]. Therefore, the effect of imperfect CSI on the resource allocation scheme in D2D communication is worth studying further.
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The authors would like to thank Dr. Wenjiang Feng of Chongqing University for providing the code of the maximum throughput algorithm.
This work is supported by the National Natural Science Foundation of China under Grant 61201177.
All data are fully available without restriction.
Chongqing University, No. 174 Shazhengjie, Shapingba, Chongqing, 400044, China
Yucheng Wu
, Xiaocui Liu
, Xiang He
, Qiong Yu
& Weiyang Xu
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YW, XL, and XH contributed to the main idea, designed and implemented the algorithms, and drafted the manuscript. QY and WX designed and carried out the simulation and analyzed the results. All authors read and approved the final manuscript.
Correspondence to Yucheng Wu.
Wu, Y., Liu, X., He, X. et al. Maximizing throughput gain via resource allocation in D2D communications. J Wireless Com Network 2017, 220 (2017) doi:10.1186/s13638-017-1007-z
D2D communications
Throughput gain
Rate loss | CommonCrawl |
\begin{document}
\title{ extsc{Algorithmic Extensions of Dirac's Theorem}
\begin{abstract} In 1952, Dirac proved the following theorem about long cycles in graphs with large minimum vertex degrees: Every $n$-vertex $2$-connected graph $G$ with minimum vertex degree $\delta\geq 2$ contains a cycle with at least $\min\{2\delta,n\}$ vertices. In particular, if $\delta\geq n/2$, then $G$ is Hamiltonian. The proof of Dirac's theorem is constructive, and it yields an algorithm computing the corresponding cycle in polynomial time. The combinatorial bound of Dirac's theorem is tight in the following sense. There are 2-connected graphs that do not contain cycles of length more than $2\delta+1$. Also, there are non-Hamiltonian graphs with all vertices but one of degree at least $n/2$. This prompts naturally to the following algorithmic questions. For $k\geq 1$, \begin{itemize} \item[(A)] How difficult is to decide whether a 2-connected graph contains a cycle of length at least $\min\{2\delta+k,n\}$? \item[(B)] How difficult is to decide whether a graph $G$ is Hamiltonian, when
at least $n - k$ vertices of $G$ are of degrees at least $n/2-k$?
\end{itemize}
The first question was asked by Fomin, Golovach, Lokshtanov, Panolan, Saurabh, and
Zehavi. The second question is due to Jansen, Kozma, and Nederlof. Even for a very special case of $k=1$, the existence of a polynomial-time algorithm deciding whether $G$ contains a cycle of length at least $\min\{2\delta+1,n\}$ was open. We resolve both questions by proving
the following algorithmic generalization of Dirac's theorem: If all but $k$ vertices of a $2$-connected graph $G$ are of degree at least $\delta$, then deciding whether $G$ has a cycle of length at least $\min\{2\delta +k, n\}$ can be done in time $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$.
The proof of the algorithmic generalization of Dirac's theorem builds on new graph-theoretical results that are interesting on their own. \end{abstract}
\tableofcontents
\section{Introduction}\label{sec:intro}
The fundamental theorem of Dirac from 1952 guarantees the existence of a Hamiltonian cycle in a graph with a large minimum vertex degree.
\begin{theorem}[Dirac~{\cite[Theorem~3]{Dirac52}}] \label{thm:diracs}
If every vertex of an $n$-vertex graph $G$ is of degree at least $n/2$, then $G$ is Hamiltonian, that is, contains a Hamiltonian cycle.
\end{theorem}
\Cref{thm:diracs} follows from a more general statement of Dirac about long cycles in a graph.
\begin{theorem}[Dirac~{\cite[Theorem~4]{Dirac52}}]\label{thm:circum} Every $n$-vertex $2$-connected graph $G$ with minimum vertex degree $\delta(G)\geq 2$, contains a cycle with at least $\min\{2\delta(G),n\}$ vertices. \end{theorem}
Both Dirac's theorems were the first instances of results that developed into one of the core areas in Extremal Graph Theory. One of the main questions in this research domain is to establish vertex degree characterization of Hamiltonian graphs and conditions enforcing long paths or cycles in graphs. The (very) incomplete list of results in this area includes the classical theorems of Erd{\H{o}}s and Gallai \cite{ErdosG59}, Ore~\cite{ore60}, Bondy and Chv\'{a}tal~\cite{bondyC76}, P\'{o}sa \cite{posa62},
Meyniel \cite{Meyniel73}, and Bollob\'{a}s and Brightwell \cite{bollobasB93}, see also the Wikipedia entry on the Hamiltonian path.\footnote{\url{https://en.wikipedia.org/wiki/Hamiltonian_path}} The chapters of Bondy \cite{MR1373656} and Bollob\'{a}s \cite{MR1373679} in the Handbook of Combinatorics, as well as Chapter~3 in the Extremal Graph Theory book \cite{MR506522} provide excellent introduction to this important part of graph theory. The survey of Li \cite{Li13survey} is a comprehensive (but a bit outdated) overview of the area. After almost 70 years, the field remains active, see for example the very recent proof of the Woodall's conjecture by Li and Nung \cite{MR4140611}.
Computing long cycles and paths is also an important topic in parameterized complexity.
It served as a test-bed for developing several fundamental algorithmic techniques including the color coding of Alon, Yuster and Zwick \cite{AlonYZ95}, the algebraic approaches of Koutis and Williams \cite{Koutis08,Williams09}, matroids-based methods \cite{FominLS14}, and the determinants-sum technique of Bj{\"{o}}rklund from his FOCS 2010 Test of Time Award paper \cite{DBLP:journals/siamcomp/Bjorklund14}. We refer to \cite{FominK13}, \cite{KoutisW16}, and \cite[Chapter~10]{cygan2015parameterized} for an overview of algorithmic ideas and techniques developed for computing long paths and cycles in graphs.
Despite the tremendous progress in graph-theoretical and algorithmic studies of longest cycles, all the developed tools do not answer the following natural and ``innocent'' question. By \Cref{thm:circum}, deciding whether a $2$-connected graph $G$ contains a cycle of length at least $\min\{2\delta(G),n\}$ can be trivially done in polynomial time by checking degrees of all vertices in $G$.
\begin{tcolorbox}[colback=green!5!white,colframe=blue!40!black]
\textbf{Question~1:} Is there a polynomial time algorithm to decide whether a $2$-connected graph $G$ contains a cycle of length at least
$\min\{2\delta(G) +1,n\}$?
\end{tcolorbox} The methods developed in the extremal Hamiltonian graph theory do not answer this question. The combinatorial bound in \Cref{thm:circum} is known to be sharp; that is, there exist graphs that have no cycles of length at least $\min\{2\delta(G)+1,n\}$. Since the extremal graph theory studies the existence of a cycle under certain conditions, such type of questions are beyond its applicability. The techniques of parameterized algorithms do not seem to be much of use here either. Such algorithms compute a cycle of length at least $k$ in time $2^{\mathcal{O}(k)} \cdot n^{\mathcal{O}(1)}$, which in our case is $2^{\mathcal{O}(\delta(G))}\cdot n^{\mathcal{O}(1)}$. Hence when $\delta(G)$ is, for example, at least $n^{1/100}$, these algorithms do not run in polynomial time.
Similarly, the existing methods do not answer the question about another ``tiny algorithmic step'' from Dirac's theorem, what happens when all vertices of $G$ but one are of large degree?
\begin{tcolorbox}[colback=green!5!white,colframe=blue!40!black] \textbf{Question~2:} Let $v$ be a vertex of the minimum degree of a $2$-connected graph $G$.
Is there a polynomial time algorithm to decide whether $G$ contains a cycle of length at least
$\min\{2\delta(G-v),n\}$?
\end{tcolorbox} (We denote by $G-v$ the induced subgraph of $G$ obtained by removing vertex $v$.) Note that graph $G-v$ is not necessarily 2-connected and we cannot apply \Cref{thm:circum} to it.
The incapability of existing techniques to answer Questions 1 and 2 was the primary motivation for our work. We answer both questions affirmatively and in a much more general way. Our result implies that in polynomial time one can decide whether $G$ contains a cycle of length at least $2\delta(G-B)+k$ for $B\subseteq V(G)$ and $k\geq 0$ as long as $k+ |B| \in \mathcal{O}(\log{n})$. (We denote by $G-B$ the induced subgraph of $G$ obtained by removing vertices of $B$.) To state our result more precisely, we define the following problem.
\defparproblema{\pname{Long Dirac Cycle}\xspace} {Graph $G$ with vertex set $B\subseteq V(G)$ and integer $k\ge 0$.}
{$k+|B|$}
{Decide whether $G$ contains a cycle of length at least $\min\{2\delta(G-B), |V(G)|-|B|\}+k$. }
In the definition of \pname{Long Dirac Cycle}\xspace we use the minimum of two values for the following reason. The question whether an $n$-vertex graph
$G$ contains a cycle of length at least $ 2\delta(G-B)+k$ is meaningful only for $\delta(G-B)\leq n/2$. Indeed, for $\delta(G-B)>n/2$, $G$ does not contain a cycle of length at least $2\delta(G-B)+k>n$. However, even when $\delta(G-B)>n/2$, deciding whether $G$ is Hamiltonian, is still very intriguing. By taking the minimum of the two values, we capture both interesting situations.
The main result of the paper is the following theorem providing an algorithmic generalization of Dirac's theorem.
\begin{theorem}[\textbf{Main Theorem}]\label{theorem:main}On an $n$-vertex 2-connected graph $G$,
\pname{Long Dirac Cycle}\xspace is solvable in time $2^{\mathcal{O} (k+|B|)} \cdot n^{\mathcal{O} (1)}$. \end{theorem}
In other words, \pname{Long Dirac Cycle}\xspace is fixed-parameter tractable parameterized by $k+|B|$ and the dependence on the parameters is single-exponential.
This dependence is asymptotically optimal up to the Exponential Time Hypothesis (ETH) of Impagliazzo, Paturi, and Zane~\cite{ImpagliazzoPZ01}. Solving \pname{Long Dirac Cycle}\xspace in time $2^{o (k)} \cdot n^{\mathcal{O} (1)}$ even with $B = \emptyset$ yields recognizing in time $2^{o(n)}$ whether a graph is Hamiltonian. A subexponential algorithm deciding Hamiltonicity would fail ETH. In \Cref{thm:hard_on_B} we show that solving \pname{Long Dirac Cycle}\xspace in time $2^{o (|B|)} \cdot n^{\mathcal{O} (1)}$ even for $k = 1$ would contradict ETH as well. It is also \ensuremath{\operatorClassNP}-complete to decide whether a $2$-connected graph $G$ has a cycle of length at least $(2+\varepsilon)\delta(G)$ for any $\varepsilon> 0$ (\Cref{thm:tightness}).
The 2-connectivity requirement in the statement of the theorem is important---without it \pname{Long Dirac Cycle}\xspace is already \ensuremath{\operatorClassNP}-complete for $k=|B|=0$. Indeed, for an $n$-vertex graph $G$ construct a graph $H$ by attaching to each vertex of $G$ a clique of size $n/2$. Then $H$ has a cycle of length at least $2\delta (H)\geq n$ if and only if $G$ is Hamiltonian. However, when instead of a cycle we are looking for a long path, the 2-connectivity requirement could be omitted. More precisely, consider the following problem.
\defparproblema{\pname{Long Dirac Path}\xspace} {Graph $G$ with vertex set $B\subseteq V(G)$ and integer $k\ge 0$.}
{$k+|B|$}
{Decide whether $G$ contains a path of length at least $\min\{2\delta(G-B), |V(G)|-|B|-1\}+k$. }
Theorem~\ref{theorem:main} yields the following.
\begin{corollary}\label{theorem:main_path}
On a connected $n$-vertex graph $G$,
\pname{Long Dirac Path}\xspace is solvable in time $2^{\mathcal{O} (k+|B|)} \cdot n^{\mathcal{O} (1)}$. \end{corollary}
Indeed, when $G$ is connected, the graph $G+v$, obtained by adding a vertex $v$ and making it adjacent to all vertices of the graph, is $2$-connected. The minimum vertex degree of $G+v$ is equal to $\delta(G)+1$, and $G$ has a path of length at least $t$ if and only if $G+v$ has a cycle of length at least $t+2$.
\Cref{theorem:main} answers several open questions from the literature. Fomin, Golovach, Lokshtanov, Panolan, Saurabh and Zehavi in~\cite{fomin_et_al:LIPIcs:2019:11168} asked about
the parameterized complexity of problems (with parameter $k$) where for a given (2-connected) graph $G$ and $k\geq 1$, the task is to check whether $G$ has a path (cycle) with at least $2\delta(G)+k$ vertices. By \Cref{theorem:main} and Corollary~\ref{theorem:main_path} (the case $B=\emptyset$), both problems are fixed-parameter tractable.
Jansen, Kozma, and Nederlof in \cite{DBLP:conf/wg/Jansen0N19}
conjectured that if
at least $n - k$ vertices of graph $G$ are of degree at least $n/2-k$, then deciding whether $G$ contains a Hamiltonian cycle
can be done in time $2^{\mathcal{O} (k)} \cdot n^{\mathcal{O} (1)}$. \Cref{theorem:main} resolves this conjecture. Indeed, if $G$ is Hamiltonian, it is $2$-connected. Then let $B$, $|B|\leq k$, be the set of vertices such that every vertex from $V(G)\setminus B$ is of degree (in $G$) at least $n/2-k$. Then $\delta(G-B)\geq n/2 -k - |B|\geq n/2 -2k$. If $n-|B|\leq n-2k-2|B|$, we put $k'=|B|$, otherwise we put $k'=n-2\delta(G-B)$. Note that because $2\delta(G-B)\geq n -4k$, in both cases we have that $k'\leq 4k$. Also by the choice of $k'$,
$\min\{2\delta(G-B), n-|B|\}+k'=n$ and hence
$G$ has a cycle of length at least $\min\{2\delta(G-B), n-|B|\}+k'$
if and only if $G$ is Hamiltonian.
By \Cref{theorem:main}, deciding whether $G$ has a cycle of length at least
$\min\{2\delta(G-B),n-|B|\}+k'$ can be done in time $2^{\mathcal{O} (k'+|B|)} \cdot n^{\mathcal{O} (1)}=2^{\mathcal{O} (k)} \cdot n^{\mathcal{O} (1)}$. Interestingly, while the conjecture of Jansen, Kozma and Nederlof follows from the statement of \Cref{theorem:main}, to prove the theorem, we need to resolve this conjecture directly.
We state \Cref{theorem:main} for the decision variant of the problem. However, the proof is constructive and the corresponding cycle can be found within the same running time. Note that standard self-reduction arguments are not applicable here because deleting or contracting edges could change the minimum vertex degree.
\noindent\textbf{Related work.} Until very recently, graph-theoretical and algorithmic studies of the longest paths and cycles coexisted in parallel universes without almost any visible interaction. In 1992, H\"{a}ggkvist \cite{Hagvist92}, as a corollary of his structural theorem, provided an algorithm that decides in time $n^{\mathcal{O}(k)}$ whether a graph with the minimum vertex degree at least $n/2-k$ is Hamiltonian.
In 2019, Jansen, Kozma, and Nederlof in \cite{DBLP:conf/wg/Jansen0N19} gave two algorithms of running times $2^{\mathcal{O} (k)} \cdot n^{\mathcal{O} (1)}$ that decide whether the input graph $G$ is Hamiltonian when either the minimum vertex degree of $G$ is at least $n/2-k$ or at least $n-k$ vertices of $G$ are of degree at least $n/2$. The first result of Jansen, Kozma, and Nederlof strongly improves the algorithm of H\"{a}ggkvist.
However, the methods they use, like the structural theorem of H\"{a}ggkvist \cite{Hagvist92}, are specific for Hamiltonicity and are not applicable for the more general
problem of computing the longest cycle. Second, their parameterized algorithms work only in one of the scenarios: either when all vertices are of degree at least $n/2-k$ or when at least $n-k$ vertices are of degree at least $n/2$. Whether both scenarios could be combined, that is, the existence of a parameterized algorithm deciding Hamiltonicity when $n-k$ vertices are of degree at least $n/2-k$, was left open.
Fomin, Golovach, Lokshtanov, Panolan, Saurabh and Zehavi in~\cite{fomin_et_al:LIPIcs:2019:11168} gave an algorithm that in time $2^{\mathcal{O} (k)} \cdot n^{\mathcal{O} (1)}$ decides whether a $2$-connected graph $G$ contains a cycle of length at least $d+k$, where $d$ is the degeneracy of $G$. Since the minimum vertex degree $\delta(G)$ does not exceed the degeneracy of $G$, this result also implies an algorithm for finding a cycle of length at least $\delta(G)+k$ in $2$-connected graphs.
None of the works \cite{DBLP:conf/wg/Jansen0N19} and \cite{fomin_et_al:LIPIcs:2019:11168} could be used to address Questions~1 and~2, the very special cases of \Cref{theorem:main}.
More generally, \Cref{theorem:main} fits into a popular trend in parameterized complexity called
``above guarantee'' parameterization. The general idea of this paradigm is that the natural parameterization of, say, a maximization problem by the solution size is not satisfactory if there is a lower bound for the solution size that is sufficiently large. For example, there always exists a satisfying assignment that satisfies half of the clauses or there is always a max-cut containing at least half the edges. Thus nontrivial solutions occur only for the values of the parameter that are above the lower bound. This indicates that for such cases, it is more natural to parameterize the problem by the difference of the solution size and the bound. Since the work of Mahajan and Raman~\cite{MahajanR99} on \textsc{Max Sat} and \textsc{Max Cut}, the above guarantee approach was
successfully applied to various problems, see e.g.~\cite{AlonGKSY10,CrowstonJMPRS13,GargP16,DBLP:journals/mst/GutinKLM11,GutinIMY12,GutinP16,GutinRSY07,LokshtanovNRRS14,MahajanRS09}. In particular, \cite{BezakovaCDF17} and \cite{fomin_et_al:LIPIcs:2020:11724} study the longest path above the shortest $s,t$-path and the girth of a graph.
\section{Preliminaries}\label{section:prelim}
\section{Overview of the proof}\label{section:overview} The original proof of Dirac is not constructive because it does not provide any procedure for constructing a cycle of length at least $2\delta(G)$. There are algorithmic proofs of Dirac's theorem; see, e.g., the thesis of Locke \cite{locke1983extremal}. The idea of Locke's proof that also provides a polynomial-time algorithm for constructing a cycle of length at least $2\delta(G)$ is to start from some cycle and to grow by inserting new vertices and short paths. Thanks to the conditions on the graph's degrees, such a procedure always constructs a cycle of the required length. On a very general level, our proof of Theorem~\ref{theorem:main} uses the same strategy. For an instance $(G,B,k)$ of \pname{Long Dirac Cycle}\xspace, we try to grow a cycle iteratively. However, enlarging the cycle by ``elementary'' improvements could get stuck with a cycle of length significantly smaller than
$\min\{2\delta(G-B), |V(G)|-|B|\}+k$. It appears that the cycles that cannot be improved by ``elementary'' operations induce a very particular structure in a graph. These structural theorems play a crucial role in our algorithm.
The main technical contribution is the new graph decomposition that we call Dirac decomposition\xspace. The formal definition is given in Section~\ref{sec:bananas}. Dirac decomposition\xspace
is defined for a cycle $C$ in $G$. Let $C$ be a cycle of length less than $2\delta(G-B) +k$. Informally, the components of Dirac decomposition are connected components in $G-V(C)$. (For an intuitive description of the decomposition, we will assume that $B=\emptyset$. Handling vertices of $B$ requires more technicalities---we have to refine the graph and work with its refinement.) Since $G$ is $2$-connected, we can reach $C$ by a path starting in such a component in $G$. One of the essential properties of Dirac decomposition\xspace is a limited number of vertices in $V(C)$ that have neighbors outside of $C$. In fact, we can choose two short paths $P_1$ and $P_2$ in $C$ (and short means that their total length is of order $k$) such that all connections between connected components of $G-V(C)$ and $C$ go through $V(P_1)\cup V(P_2)$. The second important property is that each connected component of $G-(V(P_1)\cup V(P_2))$ is connected with $P_i$ in $G$ in a very restricted way: The maximum matching size between its vertex set and the vertex set of $P_i$ is at most one.
Dirac decomposition\xspace appears to be very useful for algorithmic purposes. For a cycle $C$ and a vertex set $B$, given a Dirac decomposition\xspace for $C$ and $B$, in time $2^{\mathcal{O} (k+|B|)} \cdot n^{\mathcal{O} (1)}$ we either solve the problem or succeed in enlarging $C$
(Theorem~\ref{thm:cyclebanana}). We also provide an algorithm that either constructs a Dirac decomposition\xspace in polynomial time or obtains additional structural information that again can be used either to solve the problem or to enlarge the cycle. More precisely, first, we need to eliminate the ``extremal'' cases. When $\delta(G-B)\in \mathcal{O}(k)$, the classical result of Alon, Yuster, and Zwick \cite{AlonYZ95} solves the problem in time
$2^{\mathcal{O} (k+|B|)} \cdot n^{\mathcal{O} (1)}$. Another extremal case is when $|B|\le k$ and $\delta(G - B) \ge \frac{n}{2}-k$. In that case, for solving \pname{Long Dirac Cycle}\xspace, we have to decide in time $2^{\mathcal{\mathcal{O}}(k)}\cdot n^{\mathcal{O}(1)}$ whether $G$ is almost Hamiltonian, i.e., a cycle in $G$ that cover all but $\mathcal{O}(k)$ vertices. The existence of such an algorithm for Hamiltonian cycles was conjectured in
\cite{DBLP:conf/wg/Jansen0N19} and Theorem~\ref{theorem:hamiltonian} settles this conjecture. We give an overview of the proof of Theorem~\ref{theorem:hamiltonian} later in this section. If we are in none of the extremal cases, then (Lemma~\ref{lemma:main_cycle_lemma}) in polynomial time we can either (a) enlarge the cycle $C$, or (b) compute a vertex cover of $G-B$ of size at most $\delta(G-B)+2k$, or (c) compute a Dirac decomposition\xspace. In cases (a) and (c), we can proceed iteratively. For the case (b) we give an algorithm that solves the problem in time $2^{\mathcal{O}(k+|B|)}\cdotn^{\mathcal{O}(1)}$ (Theorem~\ref{thmVCad}).
The most critical and challenging component of the proof is Theorem~\ref{thm:cyclebanana} about algorithmic properties of Dirac decomposition\xspace.
We use the properties of Dirac decomposition\xspace to show that an enlargement of a cycle $C$ of length at most $2\delta(G-B)+k-1$ can be done in a very particular way. By an extension of Dirac's existential theorem, Theorem~\ref{thm:relaxed_long_cycle}, we can assume that $C$ is of length at least $2\delta(G-B)$. The most interesting and not-trivial situation that could occur is that for some vertices $x\in V(P_1)$ and $y\in V(P_2)$, we replace the shortest $(x,y)$-path in $C$ by a detour with a particular property. This detour leaves $x$, moves to a vertex $s$ of some $2$-connected component of $G-V(C)$, visits some vertices in this component, leaves it from a vertex $t$, and goes to vertex $y$. Since the length of the longest $(x,y)$-path in $C$ is at least $\delta(G-B)$, to decide whether such a detour exists, it is sufficient to solve the following problem.
For vertices $s,t$ of a $2$-connected graph $G$, decide whether $G$ contains an $(s,t)$-path of length at least $\delta(G-B)+k$. We give an algorithm solving this problem in time $2^{\mathcal{O}(k+|B|)}\cdot n^{\mathcal{O}(1)}$ (Theorem~\ref{thmEG}). The combinatorial bound that an $(s,t)$-path of length $\delta(G)$ always exists if $G$ is $2$-connected, is the classical theorem of Erd{\H{o}}s and Gallai \cite[Theorem~1.16]{ErdosG59}. Because of that, we name the problem of computing an $(s,t)$-path of length at least $\delta(G-B)+k$ by the \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace problem. \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace is an interesting problem on its own, and to prove Theorem~\ref{thmEG}, we use another structural result which we call Erd{\H {o}}s-Gallai decomposition\xspace. Similar to Dirac decomposition\xspace, this decomposition is very useful from the algorithmic perspective. In Section~\ref{sec:erdos-gallaiPath}, we define this decomposition, provide efficient algorithms for constructing it, and use it to solve \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace. Another interesting component of the solution to \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace is the algorithm for computing the longest cycle passing through two specified vertices (Theorem~\ref{thmTLDP}). We are not aware of the previous work in parameterized algorithms on this natural problem.
Figure~\ref{fig:proofsketch} displays the most important steps of the proof and the dependencies between them. In the remaining part of this section, we highlight the ideas behind each of the auxiliary steps (Theorems~\ref{thmTLDP}, \ref{thmEG}, \ref{thmVCad}, and~\ref{theorem:hamiltonian}) in the proofs of Theorem~\ref{theorem:main} and Theorem~\ref{thm:cyclebanana}.
\begin{figure}
\caption{The main steps and connections in the proof of Theorem~\ref{theorem:main}.}
\label{fig:proofsketch}
\end{figure}
The first auxiliary problem whose solution we use in the proof of Theorem~\ref{thmEG} is the following.
\defparproblema{\pname{Long $(s,t)$-Cycle}\xspace} {Graph $G$ with two vertices $s,t\in V(G)$ and integer $k \ge 0$.} {$k$} {Decide whether there is a cycle in $G$ of length at least $k$ that passes through $s$ and $t$. }
When $s\neq t$, an equivalent formulation is to decide whether $G$ contains two internally disjoint $(s,t)$-paths of total length at least $k$.
In Section~\ref{sec:tldp} we prove the following theorem. \begin{theorem}\label{thmTLDP} \pname{Long $(s,t)$-Cycle}\xspace is solvable in time
$2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$. \end{theorem} While the first idea to design an algorithm claimed in Theorem~\ref{thmTLDP} would be to use the color coding technique of Alon, Yuster and Zwick \cite{AlonYZ95}, this idea does not work directly. The reason is that color coding can be used only to find in the claimed running time the cycle whose length is of order of $k$. However, it is quite possible that the lengths of all solutions are much larger than $k$; in such situation color coding cannot be applied directly. Our approach in proving Theorem~\ref{thmTLDP} builds on ideas from~\cite{DBLP:journals/ipl/FominLPSZ18,Zehavi16}, where a parameterized algorithms for finding a directed $(s,t)$-path and a directed cycle of length at least $k$ were developed.
The main idea of the proof is the following. First, we use color coding to verify whether the considered instance has a solution composed by two $(s,t)$-paths of total length at most $3k$. If the instance has a solution, we return it and stop. Otherwise, we conclude that the total length of the paths of every solution is at least $3k+1$. This allows to use structural properties of paths. Let $P_1$ and $P_2$ be the $(s,t)$-paths of a solution of minimum total length. Then there are vertices $x_1$ and $x_2$ on $P_1$ and $P_2$, respectively, such that (i) the total length of the $(s,x_1)$-subpath $P_1'$ of $P_1$ and the $(s,x_2)$-subpath $P_2'$ of $P_2$ is exactly $k$, (ii) either $x_1=s$ or the length of the $(x_1,t)$-subpath $P_1''$ of $P_1$ is at least $k$, and, symmetrically, (iii) either $x_2=s$ or the length of the $(x_2,t)$-subpath $P_2''$ of $P_2$ is at least $k$. Then $P_1''$ and $P_2''$ are internally disjoint paths that are shortest disjoint paths avoiding $V(P_1')\cup V(P_2')\setminus\{x_1,x_2\}$. We use the method of random separation to distinguish the following three sets: $V(P_1')\cup V(P_2)\setminus\{x_1,x_2\}$, the last $\min\{k,|V(P_1)|-2\}$ internal vertices of $P_1''$, and the last $\min\{k,|V(P_2)|-2\}$ internal vertices of $P_2''$. This allows to highlight the crucial parts of the shortest solution and then find a solution.
The second problem whose solution we use in the proof of Theorem ~\ref{theorem:main}, comes from another classical theorem due to Erd{\H{o}}s and Gallai from \cite[Theorem~1.16]{ErdosG59}, see also \cite{locke1985generalization}.
For every pair of vertices $s,t$ of a 2-connected graph $G$, there is a path of length at least $\min_{v\in V(G)\setminus \{s,t\}}\deg v$. The proof of this result is constructive, and it implies a polynomial time algorithm that finds such a path. We define \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace as follows.
\defparproblema{\pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace}
{Graph $G$ with vertex set $B\subseteq V(G)$, two vertices $s,t\in V(G)$ and integer $k \ge 0$.}
{$k+|B|$}
{Decide whether $G$ contains an $(s,t)$-path of length at least $\delta(G-B)+k$.
}
In Section~\ref{sec:erdos-gallaiPath}, we prove the following theorem. This theorem plays an important role in the proof of Theorem~\ref{thm:cyclebanana}. \begin{theorem}\label{thmEG}
\pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace is solvable in time
$2^{\mathcal{O}(k+|B|)}\cdot n^{\mathcal{O}(1)}$
on $2$-connected graphs. \end{theorem}
Similar to \pname{Long Dirac Path}\xspace, the requirement that the input graph is $2$-connected is important. It is easy to prove that \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace is NP-complete for $k=|B|=0$ when the input graph is not $2$-connected.
To prove Theorem~\ref{thmEG}, we apply the following strategy. We take an $(s,t)$-path $P$ and try to extend it as much as possible. The principal tool in enlarging the path
$P$ is Corollary~\ref{thm:relaxed_st_path}, which is an extension of the theorem of Erd{\H{o}}s and Gallai that takes into account the vertices of $B$. In the extremal case, when we cannot extend the path anymore, we obtain a graph decomposition whose properties become useful from the algorithmic perspective. We call this decomposition by the name of Erd{\H {o}}s-Gallai decomposition\xspace and prove that, in that case, the graph can be decomposed in a very particular way. Roughly speaking, after a certain refinement of the graph, the $(s,t)$-path $P$ consists of a prefix $P_1$ and a suffix $P_2$ with the following properties. These parts of the path are sufficiently far from each other in $P$. Moreover, all components of the graph $G-V(P_1\cup P_2)$, we call them Erd{\H {o}}s-Gallai component\xspace, are connected to $P_1$ and $P_2$ in a very restricted way. Such a graph-theoretical insight helps us to characterize how a long $(s,t)$-path traverses through an Erd{\H {o}}s-Gallai component\xspace. This property allows us to design the recursive algorithm that proves the theorem.
The next auxiliary result required for proof of Theorem~\ref{theorem:main}, concerns \pname{Long Dirac Cycle}\xspace parameterized by the vertex cover of a graph. It is well-known, see e.g., \cite{CyganFKLMPPS15}, that a longest path in a graph $G$ could be found in time $2^{\mathcal{O}(t)}n^{\mathcal{O}(1)}$, where $t$ is the size of the minimum vertex cover of $G$. However, we need a much more refined result for the proof of the main theorem, where the parameter is not just the size of the vertex cover, but the difference between that size and $\delta(G - B)$. We define the following parameterized problem.
\defparproblema{\textsc{\pname{Long Dirac Cycle}\xspace\ / Vertex Cover Above Degree}} {Graph $G$ with vertex set $B\subseteq V(G)$, vertex cover $S$ of $G$ of size $\delta(G-B)+p$ and integer $k\ge 0$.}
{$p+|B|$} {Decide whether $G$ contains a cycle of length at least $2\delta(G-B)+k$. }
Section~\ref{sec:vcalgo} is devoted to the proof of the following theorem, which we need for both Theorem~\ref{theorem:hamiltonian} and Theorem~\ref{theorem:main}.
\begin{theorem}\label{thmVCad}
\textsc{\pname{Long Dirac Cycle}\xspace\ / Vertex Cover Above Degree} is solvable in $2^{\mathcal{O}(p+|B|)}\cdot n^{\mathcal{O}(1)}$ running time.
\end{theorem}
To prove Theorem~\ref{thmVCad}, we establish the new structural result, Lemma~\ref{thm:path_cover}. The lemma reduces the crucial case of the problem about the long cycle to a particular path cover problem. This equivalence becomes very handy because we can use color-coding to compute the particular path cover, and thus by the lemma, to compute a long path. In spirit, Lemma~\ref{thm:path_cover} is close to the classical theorem of Nash-Williams~\cite{NashWil71}, stating that a 2-connected graph $G$ with $\delta(G) \ge (n + 2)/3$ is either Hamiltonian or contains an independent set of size $\delta(G) + 1$. An extension of this theorem is due to H\"{a}ggkvist \cite{Hagvist92}, which was used by Jansen, Kozma and Nederlof \cite{DBLP:conf/wg/Jansen0N19} in their algorithm for Hamiltonicity below Dirac's condition. In our case, we cannot use the structural theorem of H\"{a}ggkvist as a black box, and build on the new graph-theoretic lemma instead.
The last ingredient we need to prove Theorem~\ref{theorem:main}, is its special case when the minimum degree of $\delta(G - B)$ is nearly $\frac{n}{2}$.
Specifically, the problem is defined as follows.
\defparproblema{\textsc{Almost Hamiltonian Dirac Cycle}}
{Graph $G$, integer $k\ge 0$ and vertex set $B \subset V(G)$, such that $|B| \le k$ and $\delta(G - B) \ge \frac{n}{2}-k$.} {$k$} {Find the longest cycle in $G$. } Observe that for a 2-connected graph, Theorem~\ref{thm:relaxed_long_cycle} always gives a cycle of length $2\delta(G - B) \ge n - 2k$. Thus it is more natural to state the problem in the form above, as the length of the longest cycle is necessarily between $n - 2k$ and $n$, which is at most $2 \delta(G - B) + 2k$. In other words, we look for an almost Hamiltonian cycle, in a sense that it does not cover only $\mathcal{O}(k)$ vertices. Now we state our result for \textsc{Almost Hamiltonian Cycle} that we prove in Section~\ref{sec:HamCycles}.
\begin{restatable}{theorem}{theoremhamiltonian}
\label{theorem:hamiltonian}
Let $G$ be a given 2-connected graph on $n$ vertices and let $k$ be a given integer.
Let $B\subseteq V(G)$ be such that $|B|\le k$ and $\delta(G - B) \ge \frac{n}{2}-k$.
There is a $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$ running time algorithm that finds the longest cycle in $G$.
\end{restatable}
The key obstacle for proving the theorem is the low-degree set $B$, since for empty $B$, we could simply apply the Nash-Williams theorem~\cite{NashWil71} and obtain either a Hamiltonian cycle or an independent set of size $\delta(G) + 1$, and in the latter case use our result for \textsc{\pname{Long Dirac Cycle}\xspace\ / Vertex Cover Above Degree}. Assume there exists a Hamiltonian cycle in $G$ (for almost Hamiltonian cycles the algorithm is similar), it induces a certain path cover of the vertices of $B$, where the endpoints of paths belong to $V(G)\setminus B$, and their total length is $\mathcal{O}(k)$. Such a path cover can be found by color-coding and dynamic programming in time $2^{\mathcal{O}(k)} n ^{\mathcal{O}(1)}$. Now either the rest of the graph is not 2-connected, and we have a $\mathcal{O}(k)$-sized separator, or we can apply the Nash-Williams theorem and obtain a cycle covering everything except the path cover, or a large independent set. The latter case is dealt with by Theorem~\ref{thmVCad}, and for the case of the small separator we design a special algorithm that leverages the fact that the resulting components are very dense. So the main case is when the graph splits into a long cycle and the path cover. Now we crucially use that the paths in the path cover start and end outside of $B$, thus the endpoints of a path have high degree, each of them sees roughly half of the vertices of the long cycle. This makes it ``hard'' to not be able to insert the path somewhere in the cycle and make it longer. However, this last intuitive idea is achieved by a very intricate case analysis that constitutes the most of technical difficulty of the proof.
Also, in some of the cases, we cannot make the cycle longer nor conclude that it is impossible, but instead we are able to find either a small separator or a large independent set. Again, we settle these cases by using the respective specialized algorithms.
\section{Preliminaries and classical theorems}\label{section:prelim}
\noindent\textbf{Graph notation.} Most of the graph notation that we use here are standard and are compatible with the notation used in the textbook of Diestel~\cite{Diestel}. Graphs in this paper are finite and undirected. The vertex set of a graph $G$ is denoted by $V(G)$ and the edge set of $G$ is denoted by $E(G)$. We use shorthands $n=|V(G)|$ and $m=|E(G)|$. An edge of an undirected graph with endpoints $u$ and $v$ is denoted by $uv$.
Graph $H$ is a {\em{subgraph}} of graph $G$ if $V(H)\subseteq V(G)$ and $E(H)\subseteq E(G)$. For a subset $S\subseteq V(G)$, the subgraph of $G$ {\em{induced}} by $S$ is denoted by $G[S]$; its vertex set is $S$ and its edge set consists of all the edges of $E(G)$ that have both endpoints in $S$. For $B\subseteq V(G)$, we use $G-B$ to denote the graph $G[V\setminus B]$, and for $F\subseteq E(G)$ by $G-F$ we denote the graph $(V(G),E(G)\setminus F)$. We also write $G- v$ instead of $G- \{v\}$.
For graph $G$ and edge $uv\in E(G)$, by {\em{contracting}} edge $uv$ we mean the following operation. We remove $u$ and $v$ from the graph, introduce a new vertex $w_{uv}$, and connect it to all the vertices $u$ or $v$ were adjacent to. The \emph{neighborhood} of a vertex $v$ in $G$ is
$N_G(v)=\{u\in V~|~ uv\in E(G)\}$ and the \emph{closed neighborhood} of $v$ is $N_G[v]=N_G(v)\cup \{v\}$. For a vertex set $S\subseteq V$, we define $N_G[S]=\bigcup_{v \in S} N[v]$ and $N_G(S)=N_G[S]\setminus S$. We denote by $\deg_G(v)$ the \emph{degree} of a vertex $v$ in graph $G$, which is just the number of edges incident with $v$. We may omit indices if the graph under consideration is clear from the context. We use $\delta(G)$ for minimum vertex degree of graph $G$.
A {\em{path}} $P$ in a graph is a nonempty sequence of vertices $v_0,\ldots,v_{k}$ such that for every $i=0,\ldots,k-1$ we have $v_iv_{i+1}\in E(G)$ and $v_i\neq v_j$ for all $i\neq j$. Vertices $v_0$ and $v_k$ are the \emph{endpoints} of path $P$ and $v_1, \dots, v_{k-1}$ are \emph{internal}. If $P=v_0v_1\dots v_k$ is a path, then the graph obtained from $P$ by adding edge $x_k x_0$ is a cycle. The {\em{length}} of a path or cycle is equal to the cardinality of its edge set. The \emph{distance} between vertices $u$ and $v$ in a graph $G$ is the shortest length of a path between $u$ and $v$. For vertices $s,t\in V(G)$, an $(s,t)$-path is a path with the first vertex $s$ and the last vertex $t$.
A \emph{Hamiltonian path (cycle)} in a graph $G$ is a path (cycle) passing through all the vertices of $G$. Two paths $P$ and $Q$ are internally disjoint if
every internal vertex of one path is not a vertex of the other path, that is, $P$ and $Q$ may only share their endpoints.
The \emph{concatenation} of internally vertex-disjoint paths $P= v_0,\ldots,v_{k}$ and $Q=v_{k}, v_{k+1}, \ldots, v_\ell$ is $PQ= v_0,\ldots,v_{k}, v_{k+1}, \ldots, v_\ell$. Note that $PQ$ is a path if $v_0\neq v_\ell$ and is a cycle if $v_0= v_\ell$.
An \emph{arc} in a cycle $C$ is a path whose all edges belong to $C$. A \emph{chord} of a cycle $C$ is a path connecting two non-adjacent vertices of $C$ that is internally vertex-disjoint with $C$.
An undirected graph $G$ is connected if for every pair $u,v$ of its vertices there is a path between $u$ and $v$. A vertex set $X\subseteq V(G)$ is \emph{connected} if the subgraph $G[X]$ is connected. A connected component of $G$ is the subgraph induced by a maximal connected vertex subset of $G$. A connected graph $G$ with at least three vertices is \emph{2-connected} if for every $v\in V(G)$, $G-v$ is connected. Similarly, a vertex set $X\subseteq V(G)$ is \emph{2-connected} if the subgraph $G[X]$ is 2-connected. A \emph{block} of $G$ is the subgraph induced by a maximal \emph{2-connected} subset. A vertex $v$ is a \emph{cut-vertex} if it belongs to at least two blocks. All other vertices of a block are \emph{inner} vertices. Blocks in a graph form a forest structure (viewing each block as a vertex of the forest and two blocks are adjacent if they share a cut-vertex). The blocks corresponding to the leaves of the block-forest, are referred as \emph{leaf-blocks}. A connected component is \emph{separable} if it contains a cut-vertex, or equivalently, if it is not 2-connected.
A \emph{vertex cover} $X$ of a graph $G$ is a subset of the vertex set $V(G)$ such that $X$ covers the edge set $E(G)$, i.e., every edge of $G$ has at least one endpoint in $X$. An \emph{independent set} $I$ in a graph $G$ is a subset of the vertex set $V(G)$ such that the vertices of $I$ are pairwise nonadjacent. A \emph{path cover} of a graph $G$ is a family of disjoint paths in $G$ such that every vertex of $G$ belongs to some of these paths.
\noindent\textbf{Classical results.}
Besides Dirac's theorem from \cite{Dirac52}, already stated as \Cref{thm:diracs}, we use the result that guarantees a long path between two fixed vertices of a $2$-connected graph. Its different versions can be found throughout the works of Locke \cite{locke1983extremal,locke1985generalization}. The version below is from the paper of Egawa, Glas, and Locke \cite[Lemma 5]{Egawa1991}.
\begin{lemma}[Egawa, Glas, and Locke \cite{Egawa1991}]\label{lemma:path_by_large_degree}
Let $G$ be a $2$-connected graph with at least $4$ vertices, and let $s,t \in V(G)$ be a pair of vertices in $G$, and let $d$ be an integer.
If all vertices in $G$, except $s$, $t$ and one other vertex, have degree at least $d$, then there exists an $(s,t)$-path of length at least $d$ in $G$. \end{lemma}
In addition, we rely on several other classical theorems. In some parts of the proof we use one more result from the Dirac's work~\cite{Dirac52}.
\begin{theorem}[Dirac \cite{Dirac52}]\label{prop:cycle_delta}
Every graph $G$ contains a cycle of of length at least $\delta(G)+1$. \end{theorem}
We remark that \Cref{lemma:path_by_large_degree} and \Cref{prop:cycle_delta} are constructive in the following sense. Their proofs can be turned into polynomial time algorithms producing an $(s,t)$-path of length at least $d$ and a pa cycle of of length at least $\delta(G)+1$.
In 1976, Bondy and Chv\'{a}tal~\cite{bondyC76} proved the following generalization of \Cref{thm:diracs} that we use in Section~\ref{sec:vcalgo}. Let $G$ be an $n$-vertex graph. The \emph{closure} ${\sf cl}(G)$ of $G$ is the graph obtained from $G$ by iteratively making two distinct vertices $u$ and $v$ adjacent whenever the sum of their degrees is at least $n$. Note that if $\deg_G(u)+\deg_G(v)\geq n$ for all pairs of vertices $u$ and $v$, then ${\sf cl}(G)=K_n$. In particular, the closure of a $n$-vertex graph satisfying the conditions of \Cref{thm:diracs} is $K_n$.
\begin{theorem}[Bondy and Chv\'{a}tal \cite{bondyC76}]\label{thm:bh} A graph $G$ has a Hamiltonian cycle if and only if ${\sf cl}(G)$ has a Hamiltonian cycle. \end{theorem}
We remark that the proof of \Cref{thm:bh} yields a polynomial time algorithm constructing from a cycle with an added edge a cycle in a graph without the new edge. Thus by repeating this argument for every added edge, we obtain a polynomial time algorithm for constructing a Hamiltonian cycle in $G$ from a Hamiltonian cycle in ${\sf cl}(G)$.
Another classical result that we require is the well-known Menger's theorem.
\begin{theorem}[Menger's theorem, \cite{menger1927allgemeinen,goring2000short}]
Let $G$ be a graph and $A, B\subseteq V(G)$ be two subsets of its vertices.
Let $s$ be the minimum number of vertices separating $A$ and $B$ in $G$.
There are $s$ vertex-disjoint paths going from $A$ to $B$. \end{theorem}
Throughout the paper we are mostly working with $2$-connected graphs, so we just need the following corollary of the Menger's theorem.
\begin{corollary}
Let $G$ be a $2$-connected graphs and $A,B\subseteq V(G)$ be two subsets of its vertices such that $|A|,|B|\ge 2$.
There exist two vertex-disjoint paths going from $A$ to $B$ in $G$. \end{corollary}
Finally, we will make use of a strengthening of Dirac's theorem due to Nash-Williams \cite{NashWil71}. We state it in the form following Jansen, Kozma and Nederlof \cite{DBLP:conf/wg/Jansen0N19}, where the following algorithmic statement is proven. \begin{theorem}[Nash-Williams \cite{NashWil71, DBLP:conf/wg/Jansen0N19}]
\label{proposition:cycle_or_is}
Let $G$ be a $2$-connected graph with $n$ vertices, with $\delta(G) \ge (n + 2)/3$. Then, we can find in $G$, in time $\mathcal{O}(n^3)$, either a Hamiltonian cycle, or an independent set of size $\delta(G) + 1$. \end{theorem}
\noindent\textbf{Parameterized algorithms.} We will use several results from parameterized complexity as black boxes. Let us recall that \pname{Longest Cycle}\xspace (\pname{Longest Path}\xspace) are the problems where for given graph $G$ and integer $k$, the task is to decide whether $G$ has a cycle (a path) of length at least $k$. In \textsc{Long $(s,t)$-Path}, for $s,t\in V(G)$, the task is to decide whether an $(s,t)$-path of length at leas $k$ exists. The first algorithms for \pname{Longest Cycle}\xspace and \pname{Longest Path}\xspace of running time $2^{\mathcal{O}(k)}\cdotn^{\mathcal{O}(1)}$ are due to Alon, Yuster and Zwick \cite{AlonYZ95}. The fastest known randomized algorithm for \pname{Longest Path}\xspace\ on undirected graph is due to Bj{\"{o}}rklund, Husfeldt, Kaski and Koivisto~\cite{BjHuKK10} and runs in time $1.657^k \cdot n^{\mathcal{O}(1)} $. Tsur gave the fastest known deterministic algorithm for the problem running in time $2.554^k \cdot n^{\mathcal{O}(1)}$~\cite{Tsur19b}. For \pname{Longest Cycle}\xspace, the current fastest randomized algorithm running in time $4^k\cdot n^{\mathcal{O}(1)}$ is due to Zehavi~\cite{Zehavi16} and the best deterministic algorithm runs in time $4.884^k\cdot n^{\mathcal{O}(1)}$~\cite{DBLP:journals/ipl/FominLPSZ18}. For \textsc{Long $(s,t)$-Path} the best known running time is $4.884^k\cdot n^{\mathcal{O}(1)}$~\cite{DBLP:journals/ipl/FominLPSZ18}.
\begin{theorem}[\cite{AlonYZ95},\cite{DBLP:journals/ipl/FominLPSZ18}]\label{prop:longest_cycle}
\pname{Longest Path}\xspace, \pname{Longest Cycle}\xspace, and \textsc{Long $(s,t)$-Path} admit algorithms with running time $2^{\mathcal{O}(k)}\cdotn^{\mathcal{O}(1)}$. \end{theorem}
We also use the following algorithms of Jansen, Kozma, and Nederlof \cite{DBLP:conf/wg/Jansen0N19}. \begin{theorem}[\cite{DBLP:conf/wg/Jansen0N19}]\label{theorem:JansenKN}
If a graph $G$ has at least $n-k$ vertices of degree at least $\frac{n}{2}$ or if a graph $G$ has $\delta(G)\ge\frac{n}{2}-k$, a Hamiltonian cycle in $G$ can be found in time $2^{\mathcal{O}(k)}\cdotn^{\mathcal{O}(1)}$. \end{theorem}
\section{Generalized theorems}\label{section:generaltheorems} The classical theorems of Dirac and Erd{\H{o}}s-Gallai provide bounds on the length of cycles and paths in terms of vertex degrees in graph $G$. In our algorithmic extension of Dirac's theorem, we deal with a more general problem when the cycle's length is bounded by vertex degrees of graph $G-B$. In our algorithm we use generalizations of
these classical results stated for vertex degrees in graph $G-B$. These generalizations are simple and most likely they are known as a folklore. However we could not find them in the literature and prove them here for completeness.
The first theorem is the generalization of Dirac's theorem (Theorem~\ref{thm:circum}): Theorem~\ref{thm:circum} is its special case with $B=\emptyset$.
\begin{theorem}\label{thm:relaxed_long_cycle}
Let $G$ be a $2$-connected $n$-vertex graph.
For any $B \subseteq V(G)$ there exists a simple cycle in $G$ of length at least $\min\{n-|B|,2\delta(G- B)\}$. Moreover, there is a polynomial time algorithm constructing a cycle of such length.
\end{theorem}
\begin{proof}
We assume that $0 < |B| \le n-1$ and $\delta(G-B)>1$, other cases are trivial.
Consider graph $G-B$.
It consists of one or more connected components.
If $G$ has at least one connected component, say $H$, that is $2$-connected and is of size at least $2\delta(G-B)$, then a cycle of length at least $\min(|V(H)|, 2\delta(H))\ge 2\delta(G-B)$ can be found inside $H$ by Theorem~\ref{thm:circum}.
Now assume that each connected component in $G-B$ either contains a cut-vertex or consists of less than $2\delta(G-B)$ vertices.
Assume that there are at least two connected components in $G-B$, say $H_1$ and $H_2$.
Since $\delta(G-B)>1$, both $H_1$ and $H_2$ consist of at least three vertices.
If $H_1$ contains a cut vertex, take one of its leaf-blocks, say $L_1$, and put $S_1=L_1$.
Note that all vertices but one are of degree at least $\delta(G- B)$ in $L_1$.
As $\delta(G-B)>1$, $L_1$ consists of at least three vertices and is $2$-connected.
If $H_1$ is $2$-connected, put $S_1=H_1$.
Find $S_2$ in the same way for $H_2$.
By Menger's theorem, there are two vertex-disjoint paths from $V(S_1)$ to $V(S_2)$.
Thus, there are two distinct vertices $u_1, v_1 \in V(S_1)$ that are connected correspondingly with $u_2, v_2 \in V(S_2)$ with two vertex-disjoint paths.
Note that the total length of these paths is at least four, since the are no edges $u_1 u_2$ and $v_1 v_2$ in $G$.
By Lemma~\ref{lemma:path_by_large_degree}, there is a path of length at least $\delta(G- B)$ between $u_1$ and $v_1$ in $S_1$ if $|V(S_1)|\ge 4$.
If $|V(S_1)|<4$, then $S_1$ is a cycle on three vertices, so there also exists a path of length at least $2\ge |V(S_1)|-1\ge \delta(G-B)$ between $u_1$ and $v_1$ in $S_1$.
Analogously, there is a path of length at least $\delta(G-B)$ between $u_2$ and $v_2$ in $S_2$.
Combine these two paths and the two paths outside and obtain a cycle of length at least $2\delta(G- B)+4$ in $G$.
Now assume that there is exactly one connected component in $G-B$ of size $n-|B|$.
Note that it consists of at least three vertices as $\delta(G-B)>1$.
If it is $2$-connected, then its size is less than $2\delta(G-B)$.
Hence, the desired cycle is obtained automatically by Theorem~\ref{thm:diracs}.
If it is not $2$-connected, take any of its cut vertices and add it to $B$ to obtain $B'$.
Note that $\delta(G- B')\ge \delta(G- B)-1$.
Now
$G- B'$ consists of at least two connected components, so apply the discussion above for this case and obtain a cycle of length at least $2\delta(G- B')+4\ge 2\delta(G- B)+2$.
The proof is constructive and all its steps (computing 2-connected components, finding a cut-vertex, computing two vertex-disjoint paths, etc.) are implementable in polynomial time.
\end{proof}
The similar theorem for paths can be now derived.
\begin{theorem}\label{thm:relaxed_long_path}
Let $G$ be a connected $n$-vertex graph.
For any $B\subseteq V(G)$ there exists a simple path in $G$ of length at least $\min\{n-|B|-1, 2\delta(G-B)\}$. Moreover, there is a polynomial time algorithm constructing a path of such length.
\end{theorem}
\begin{proof}
Construct graph $G'$ from $G$ by adding to it a universal vertex, that is the vertex adjacent to all vertices of $G$.
Note that $\delta(G'-B)=\delta(G-B)+1$ and $G'$ consists of $n+1$ vertices.
Also, $G'$ is $2$-connected, since $G$ is connected.
Thus, by Theorem~\ref{thm:relaxed_long_cycle}, $G'$ contains a simple cycle of length at least $\min\{n+1-|B|, 2\delta(G-B)+2\}$.
If this cycle does not contain the universal vertex, this cycle is contained in $G$ as well, and we automatically obtain a path of length at least $\min\{n+1-|B|, 2\delta(G-B)+2\}-1$ in $G$.
If the cycle contains the universal vertex, remove this vertex from the cycle.
Since this vertex is incident with two edges of the cycle, we obtain a path of length at least $\min\{n+1-|B|, 2\delta(G-B)+2\}-2$ in $G$. Again, the construction can be easily turned into a polynomial time algorithm.
\end{proof} The following Corollary generalizes the theorem of Erd{\H{o}}s and Gallai from \cite[Theorem~1.16]{ErdosG59}.
\begin{corollary}\label{thm:relaxed_st_path}
Let $G$ be a $2$-connected graph and let $s, t$ be a pair of distinct vertices in $G$.
For any $B \subseteq V(G)$ there exists a path of length at least $\delta(G- B)$ between $s$ and $t$ in $G$. Moreover, there is a polynomial time algorithm constructing a cycle of such length.
\end{corollary}
\begin{proof}
Suppose that $n-|B|\ge2\delta(G- B)$.
Use Theorem~\ref{thm:relaxed_long_cycle} to find a cycle of length at least $2\delta(G-B)$.
By Menger's theorem, there are two vertex-disjoint paths from $\{s,t\}$ to this cycle in $G$.
Take these paths and the longer arc of the cycle and obtain a path of length at least $\delta(G- B)$ between $s$ and $t$.
Consider the case when $n-|B|< 2\delta(G- B)$, so $\delta(G- B)\ge (|V(G- B)|+1)/2$.
If $n-|B|\le 3$, then $\delta(G- B)\le 2$, so it is enough to find any path of length two between $s$ and $t$.
If $n-|B|\ge 4$, then $G- B$ is $2$-connected, as it contains a Hamiltonian cycle by the classical Dirac's theorem.
Apply Menger's theorem to $G$ and find two vertex-disjoint paths from $\{s,t\}$ to $V(G)\setminus B$.
Let these paths be a path going from $s$ to $s'\in V(G-B)$ and from $t$ to $t'\in V(G-B)$.
By Lemma~\ref{lemma:path_by_large_degree}, there is a path of length at least $\delta(G- B)$ between $s'$ and $t'$ in $G- B$.
Combine this path with the paths from $s$ to $s'$ and from $t$ to $t'$.
This yields a path of length at least $\delta(G- B)$ between $s$ and $t$ in $G$.
\end{proof}
\begin{comment}
The cases for $n\le 3$ are trivial, so we assume that $n>3$.
We also assume that $n-|B|\ge 3$ and $\delta(G-B)\ge 2$, otherwise we are looking for a path of length at most two, and connected graph with more than three vertices always contains it.
If $\delta(G-B)=2$, then it is enough to find a path of length at least three.
The only type of connected graphs on $n>3$ vertices that contain no path of length three are star graphs (other graphs have matchings consisting of two edges, which can be combined into a path of length at least three).
All vertices of star graphs except one have degree one, so $\delta(G-B)=2$ cannot hold for any $B\subseteq V(G)$.
Hence, $G$ is not a star graph and contains a path of length at least three.
Thus, we can assume that $\delta(G-B)\ge 3$.
Assume that $\gbref{G}{B}$ is $2$-connected.
Note that $\gbref{G}{B}$ is equivalent to $G-B'$ for some $B'\subseteq B$.
Apply Theorem~\ref{thm:relaxed_long_cycle} to $G-B'$ and $B\setminus B'$.
This yields a cycle of length at least $\min\{|V(G-B')|-|B\setminus B'|, 2\delta((G-B')-(B\setminus B'))\}=\min\{n-|B|, 2\delta(G-B)\}$.
Delete an arbitrary edge of this cycle and obtain a path of the required length.
It is left to consider the case when $\gbref{G}{B}$ contains a cut vertex, say $c$.
There are at least two leaf blocks in $\gbref{G}{B}$.
Denote these leaf-blocks by $L_1$ and $L_2$ and their cut vertices by $c_1$ and $c_2$ respectively.
For each $i$, all vertices in $L_i$, except $c_i$, have the same degree in $L_i$ and $\gbref{G}{B}$.
Since $L_i$ contains at least one inner vertex from $V(G)\setminus B$, $\delta(L_i-(B\cup \{c_i\}))\ge \delta(G-B)-1$.
Hence, $|V(L_i-(B\cup\{c_i\}))|\ge \delta(G-B)$.
We claim that $L_i$ contains a path of length at least $\delta(G-B)$ starting in $c_i$.
If $|V(L_i-(B\cup \{c_i\}))|\le 2\delta(G-B)-2$, then $L_i-(B\cup\{c_i\})$ contains a hamiltonian cycle by the classical Dirac's theorem.
That is, $L_i$ contains a cycle of length $|V(L_i-(B\cup\{c_i\}))|\ge \delta(G-B)$ not containing $c_i$.
Since $L_i$ is connected, there is a path of length at least one connecting $c_i$ with this cycle.
This yields a path of length at least $\delta(G-B)$ starting from $c_i$ in $L_i$.
Otherwise, $|V(L_i-(B\cup \{c_i\}))|>2\delta(G-B)-2$.
By Theorem~\ref{thm:relaxed_long_cycle} applied to $L_i$ and $B\cup\{c_i\}$, there is a cycle of length at least $\min\{|V(L_i)|-|B\cup\{c_i\}|, 2\delta(G-B)-2\}=2\delta(G-B)-2 \ge \delta(G-B)+1$ in $L_i$.
There is a path (possibly of zero length) connecting $c_i$ with the cycle, so there is a path of length at least $\delta(G-B)$ starting from $c_i$ in $L_i$.
Combine the path starting in $c_1$ with the path starting in $c_2$ with any path between $c_1$ and $c_2$ (possibly of zero length).
The obtained path is of length at least $2\delta(G-B)$.
The proof is complete.
\end{comment}
\newcommand\claimqed{
$\lrcorner$} \section{Long $(s,t)$-Cycle}\label{sec:tldp} In this section we give an FPT algorithm that finds a cycle of length at least $k$ passing through designated terminal vertices $s$ and $t$. When the length of such cycle is of order $\mathcal{O}(k)$, then the classical methods like color-coding solve the problem. The difficulty is that the length of the cycle can be arbitrarily bigger than $k$. For that case we build on the approach from ~\cite{DBLP:journals/ipl/FominLPSZ18} that was used to design an algorithm for a longest $(s,t)$-path.
Now we are ready to prove Theorem~\ref{thmTLDP}. We restate it here.
\noindent {\bf Theorem~\ref{thmTLDP}.} {\it The \pname{Long $(s,t)$-Cycle}\xspace problem is solvable in $\mathcal{O}((2e)^{3k}\cdot mn)$ time by a randomized Monte Carlo algorithm and in $(2e)^{3k}k^{\mathcal{O}(\log k)}\cdot mn\log n$ deterministic time. }
\begin{proof} Let $(G,s,t,k)$ be an instance of \pname{Long $(s,t)$-Cycle}\xspace. Clearly, we can assume that $G$ is connected, because if $s$ and $t$ are in distinct connected components, then we have a trivial no-instance, and if $s$ and $t$ are in the same connected component of a disconnected graph, then we can consider the problem on the component containing $s$ and $t$ instead of $G$. To avoid additional case analysis, we assume that $s\neq t$. Otherwise, if $s=t$, we can do the following. If $k\leq 3$, then to solve the problem, it is sufficient to check whether $G$ has a cycle containing $s$ and this easily can be done in linear time. If $k\geq 4$, then we apply the algorithm from ~\cite{DBLP:journals/ipl/FominLPSZ18}. To be able to do it formally, we create a new vertex $t'$ that is a false tween of $s$ and then check whether the obtained graph has an $(s,t')$-path of length at least $k$. Fomin, Lokshtanov, Panolan, Saurabh and Zehavi~\cite{DBLP:journals/ipl/FominLPSZ18} do not state explicitly the dependency of their algorithm on the graph size. However, it can be seen that the running times of the randomized and deterministic variants of their algorithm are dominated by $\mathcal{O}((2e)^{3k}\cdot mn)$ and $(2e)^{3k}k^{\mathcal{O}(\log k)}\cdot mn\log n$, respectively.
We also assume that $k\geq 4$. If $k\leq 3$, then to solve the problem, it is sufficient to find any two internally disjoint $(s,t)$-paths, and this can be done by the standard flow techniques (see, e.g., the recent textbook~\cite{Williamson19}) in time $\mathcal{O}(n+m)$, because we are looking for a flow of volume 2.
The algorithm works in two stages. First we try to find two internally vertex-disjoint $(s,t)$-paths of total length $\ell$ for $\ell\in\{k,\ldots,3k\}$. If such paths are found, they form the required cycle, so we stop. Otherwise, we proceed to Stage 2, where we assume that the long $(s,t)$-cycle, if it exists, is longer that $3k$.
\noindent\textbf{Stage~1.} First, we check whether there are two internally disjoint $(s,t)$-paths of total length $\ell$ for some $\ell\in\{k,\ldots,3k\}$. For this, we apply the classical color-coding technique of Alon, Yuster, and Zwick~\cite{AlonYZ95}. Here the arguments are standard and
we only sketch how to solve the decision version of the problem. The algorithm may be easily modified to construct the paths. We describe a randomized Monte Carlo algorithm and explain how to derandomize it in the concluding part of the theorem proof.
We color the vertices of $G$ uniformly at random by $3k$ colors $\{1,\ldots,3k\}$. We say that two $(s,t)$-paths $P_1$ and $P_2$ form a \emph{colorful solution} if the vertices of each of the paths have distinct colors and the colors of the internal vertices of $P_1$ are distinct from the colors of the internal vertices of $P_2$. (Clearly, the colors of $s$ and $t$ are the same in both paths.) In other words, in the $(s,t)$-cycle formed by $P_1$ and $P_2$ all vertices are colored in different colors.
We find a colorful solution by dynamic programming. Denote by $c(x)$ the color of a vertex $x\in V(G)$, and let $p=c(s)$ and $q=c(t)$.
If $p=q$, then there is no colorful solution. Suppose that $p\neq q$. For a vertex $x\in V(G)$ and a nonempty set of colors $X\subseteq\{1,\ldots,3k\}$, define
$\alpha(x,X)={\tt true}$ if there is an $(s,x)$-path $P$ with $|X|$ vertices that are colored by distinct colors from $X$, and we set $\alpha(x,X)={\tt false}$ otherwise. The values of $\alpha(x,X)$ are computed for all $x\in V(G)$ and all $X\subseteq \{1,\ldots,3k\}$ starting from sets of size one.
For every $x\in V(G)$ and every $i\in\{1,\ldots,3k\}$, we define \begin{equation}\label{eq:dp-cc-one} \alpha(x,\{i\})= \begin{cases} {\tt true}&\mbox{if }x=s\text{ and }i=p\\ {\tt false}&\mbox{otherwise}. \end{cases} \end{equation}
Assume that $|X|\geq 2$ and the table of values of $\alpha(x,Y)$ is constructed for all $x\in V(G)$ and all $Y\subseteq\{1,\ldots,3k\}$ such that $|Y|<|X|$. Then for $x\in V(V)$, we set \begin{equation}\label{eq:dp-cc-two} \alpha(x,X)= \begin{cases} \bigvee\limits_{y\in N_G(x)}\alpha(y,X\setminus\{c(x)\})&\mbox{if }c(x)\in X\\ {\tt false}&\mbox{if }c(x)\notin X. \end{cases} \end{equation} By exactly the same arguments as for the color-coding algorithm for \pname{Longest Path}\xspace (see, e.g.,~\cite[Chapter~5]{CyganFKLMPPS15}), we obtain that (\ref{eq:dp-cc-one}) and (\ref{eq:dp-cc-two}) allow to compute the table of values of $\alpha(x,X)$ for all $x\in V(G)$ and all nonempty $X\subseteq\{1,\ldots,3k\}$ in time $\mathcal{O}(2^{3k}\cdot mn)$, because we compute $\alpha(x,X)$ for $n$ vertices $x$ and $2^{3k}-1$ sets $X$, and to compute $\alpha(x,X)$ using (\ref{eq:dp-cc-two}), we consider the neighbors of $x$.
To complete the description of the algorithm that verifies the existence of a colorful solution, we observe that such a solution exists if and only if there are $X,Y\subseteq \{1,\ldots,3k\}$ such that $X\cap Y=\{p,q\}$, $|X|+|Y|\geq k+2$, and $\alpha(t,X)=\alpha(t,Y)={\tt true}$.
Hence, it takes time $\mathcal{O}(2^{3k}\cdot mn)$ to decide whether there is a colorful solution. If there is a colorful solution, $(G,s,t,k)$ is a yes-instance of \pname{Long $(s,t)$-Cycle}\xspace. However, the absence of a colorful solution does not imply that we have a no-instance.
Assume that there are two internally disjoint $(s,t)$-paths $P_1$ and $P_2$ in $G$ whose total length is between $k$ and $3k$. That is, $k\leq |V(P_1)\cup V(P_2)|\leq 3k$. Then the probability that all vertices of $V(P_1)\cup V(P_2)$ are colored by distinct colors is at least $\frac{(3k)!}{(3k)^{3k}}\geq e^{-3k}$. The probability that there is no colorful solution is at most $1-e^{3k}$. Therefore, by trying to find a colorful solution for $N=\lceil e^{3k}\rceil$ random colorings, we either conclude that we have a yes-instance, or return no-answer with the mistake probability at most $(1-e^{3k})^N\leq e^{-1}$. This gives us a Monte Carlo algorithm with running time $\mathcal{O}((2e)^{3k}\cdot mn)$.
\noindent\textbf{Stage~2.} From now, we assume that we failed to solve the problem at Stage~1. This means that each solution is an $(s,t)$-cycle of length $3k+1$. As in Stage~1, we find
two disjoint $(s,t)$-paths of total length at least $3k+1$. This is done by generalizing the technique of Fomin, Lokshtanov, Panolan, Saurabh and Zehavi from~\cite{DBLP:journals/ipl/FominLPSZ18} for finding an $(s,t)$-path of length at least $k$.
Now instead of color-coding, we use the technique of random separation~\cite{CaiCC06}.
The main step of our procedure for Stage~2 is given in Algorithm~\ref{alg:step}.
\begin{algorithm}[h] Color the vertices of $V(G)\setminus\{s,t\}$ uniformly at random by three colors $1, 2,$ and $3$, and denote by $X_1,X_2,X_3$ the vertices colored by the corresponding colors.\\ \For{$i=1,2,3$} { \ForEach{$v\in X_i$ at distance $k$ from $t$ in $G_i=G[X_i\cup\{t\}]$} { Find a shortest $(v,t)$-path $P$ in $G_i$\; Find an $(s,v)$-path $P_1$ and an $(s,t)$-path $P_2$, such that $P_1$ and $P_2$ are
internally disjoint and both these paths avoid internal vertices of $P$\; \tcc*[h]{that is, $P_1$ and $P_2$ are paths in $G-(V(P)\setminus\{v,t\})$}\\ \If{ such paths $P_1$ and $P_2$ exist} {\Return{the paths $P_1'=P_1P$ and $P_2$}\; {\bf quit}. } }
} \caption{Main step of Stage~2.}\label{alg:step} \end{algorithm}
Due to the conditions that
$P_1$ does not contain internal vertices of $P$, avoids $t$, and is internally disjoint with $P_2$,
we have that
the concatenation $P_1'$ of $P_1$ and $P$ is a path. Moreover, $P_1'$ and $P_2$ are internally vertex disjoint $(s,t)$-paths. Since the length of $P$ is $k$, the length of $P_1'$ is at least $k$. We conclude that if the algorithm returns $P_1'$ and $P_2$, then these paths form a required $(s,t)$-cycle of length at least $k$. The algorithm runs in $\mathcal{O}(n+m)$ time, as $P_1$ and $P_2$ can be found (if they exist) by the standard flow algorithm, see e.g., ~\cite{Williamson19}.
However, the proof that the algorithm finds a solution in a yes-instance with a reasonable probability is non-trivial. It follows from the following lemma.
\begin{lemma}\label{cl:probability} If $(G,s,t,k)$ is a yes-instance of \pname{Long $(s,t)$-Cycle}\xspace, then the described algorithm finds a solution with probability at least $\frac{2}{3^{3k-1}}$. \end{lemma}
\begin{proof}[Proof of Lemma~\ref{cl:probability}] Suppose that $(G,s,t,k)$ is a yes-instance. Then there are two internally disjoint $(s,t)$-paths $P_1$ and $P_2$ with total length at least $k$. We assume that the total length of paths $P_1$ and $P_2$ is minimum. Recall that the total length of $P_1$ and $P_2$ is at least $3k+1$. We assume that the vertices of $P_1$ and $P_2$ are ordered in the path order starting with $s$. Thus whenever we refer to the first or the last vertices of the paths, these vertices respect this ordering. We follow the same convention for all $(x,y)$-paths, that is, we order the vertices starting from $x$. We consider two cases.
\noindent {\bf Case~1.} \emph{The shortest path among $P_1$ and $P_2$ is of length at most $k$. } Without loss of generality,
we assume that the length of $P_2$ is at most $k$. Then the length of $P_1$ is at least $2k+1$. Denote by $A$ the set of the first $k-1$ internal vertices of $P_1$, by $B$ the set of the last $k$ internal vertices of $P_1$, and by $C$ the set of internal vertices of $P_2$. Because the vertices of $V(G)\setminus \{s,t\}$ are colored uniformly at random, with probability at least $\frac{3!}{3^{|A|}\cdot 3^{|B|}\cdot 3^{|C|}}\geq \frac{2}{3^{3k-1}}$ \begin{itemize} \item[(i)] vertices of each of the sets $A$, $B$, and $C$ receive the same colors, and \item[(ii)] vertices of distinct sets are of distinct colors. \end{itemize} We show that if (i) and (ii) holds, then the algorithm finds a solution. For further analysis, we assume that $A\subseteq X_1$, $B\subseteq X_2$, and $C\subseteq X_3$.
Let $v$ be the internal vertex of $P_1$ at distance $k$ from $t$ in the path. Then $v\in B\subseteq X_2$. Denote by $P_1'$ the $(s,v)$-subpath of $P_1$. Let $P$ be an arbitrary shortest $(v,t)$-path in $G_2=G[X_2\cup\{t\}]$. Notice that $P$ is internally disjoint with $P_2$, because $V(P_2)\setminus \{s,t\}\subseteq X_3$, and $X_2\cap X_3=\emptyset$. We claim that $P_1'$ and $P$ are internally disjoint.
Targeting towards a contradiction, assume that $V(P_1')\cap V(P)\setminus\{v\}\neq\emptyset$. Let $u$ be the first vertex of $P_1'$ that is in $V(P)$. Let $Q_1$ be the $(s,u)$-subpath of $P_1'$ and let $Q$ be the $(u,t)$-subpath of $P$. Recall that the first $k-1$ internal vertices of $P_1$ are in $A\subseteq X_1$. This implies that $A\subseteq V(Q_1)$. Threfore, the length of $Q_1$ is at least $k$. Let $\hat{P}_1=Q_1Q$. We obtain that $\hat{P}_1$ is an $(s,t)$-path of length at least $k$ that is internally disjoint with $P_2$. Hence, $\hat{P}_1$ and $P_2$ form a solution. However, the length of $\hat{P}_1$ is less than the length of $P_1$, contradicting the condition that $P_1$ and $P_2$ form a solution of minimum total length. Hence $P_1'$ and $P$ are internally disjoint.
We have that $\hat{P}_1=P_1'P$ is a path internally disjoint with $P_2$. We also have that $A\subseteq V(P_1)$ and, therefore, the length of $\hat{P}_1$ is at least $k$. Thus $\hat{P}_1$ and $P_2$ is a solution. By the construction of $\hat{P}_1$, the length of $\hat{P}_1$ is at most the length of $P_1$. Since $P_1$ and $P_2$ compose a solution of minimum total length, the length of $P_1$ is the same as the length of $\hat{P}_1$. Hence, $v$ is at distance $k$ from $v$ in $G_3$.
Summarizing, there is a vertex $v$ at distance $k$ from $t$ in $G_2$ such that for any shortest $(v,t)$-path $P$ in graph $G_2$, in graph $G-(V(P)\setminus\{v,t\})$ there exist an $(s,v)$-path $P_1$ and an $(s,t)$-path $P_2$ that are internally disjoint. In this case our algorithm finds a solution.
\noindent {\bf Case~2.} \emph{The length of each of the paths $P_1$ and $P_2$ is at least $k+1$.} Let $B$ be the last $k$ internal vertices of $P_1$ and let $C$ be the last $k$ internal vertices of $P_2$. Since each of the paths $P_1$ and $P_2$ is of length at least $k+1$ because the total length of both paths is at least $3k+1$, we conclude the following. For some positive integers $k_1$ and $k_2$ such that $k_1+k_2=k-1$,
the first $k_1$ internal vertices of $P_1$ are not in $B$, and the first $k_2$ internal vertices of $P_2$ are not in $C$. Denote by $A_1$ the first $k_1$ internal vertices of $P_1$ and by $A_2$ the first $k_2$ internal vertices of $P_2$. We set $A=A_1\cup A_2$. Thus $|A|=k-1$ and the sets $A$, $B$ and $C$ are pairwise disjoint. Since the vertices of $V(G)\setminus \{s,t\}$ are colored uniformly at random, we have that with probability $\frac{3!}{3^{|A|}\cdot 3^{|B|}\cdot 3^{|C|}}\geq \frac{2}{3^{3k-1}}$ \begin{itemize} \item[(i)] vertices of each of the sets $A$, $B$, and $C$ are of the same color, and \item[(ii)] vertices of distinct sets are of different colors. \end{itemize} As in Case~1, we show that if a coloring satisfies (i) and (ii), then the algorithm finds a solution. Without loss of generality, we assume that $A\subseteq X_1$, $B\subseteq X_2$, and $C\subseteq X_3$.
Let $v_1$ be the internal vertex of $P_1$ at distance $k$ from $t$ in $P_1$, and let $v_2$ be the internal vertex of $P_2$ at distance $k$ from $t$ in $P_2$. Note that $v_1\in B\subseteq X_2$ and $v_2\in C\subseteq X_3$. Denote by $P_1'$ the $(s,v_1)$-subpath of $P_1$ and by $P_1''$ the $(v_1,t)$-subpath of $P_1$. Similarly, we define $P_2'$ as the $(s,v_2)$-subpath of $P_2$ and $P_2''$ as the $(v_2,t)$-subpath of $P_2$.
We prove the following claim. \begin{claim}\label{cl:oneoftwo} At least one the following options holds. \begin{itemize} \item Either for every shortest $(v_1,t)$-path $Q_1$ in $G_2=G[X_2\cup\{t\}]$, paths $Q_1$ and $P_2$ are internally disjoint, \item or for every shortest $(v_2,t)$-path $Q_2$ in $G_3=G[X_3\cup\{t\}]$, paths $Q_2$ and $P_1$ are internally disjoint. \end{itemize} \end{claim}
\noindent \emph{Proof of Claim~\ref{cl:oneoftwo}.} The proof is by contradiction. Assume that there is a shortest $(v_1,t)$-path $Q_1$ in $G_2$ and a shortest $(v_2,t)$-path $Q_2$ in $G_3$ such that neither $Q_1$ and $P_2$ are internally disjoint, nor $Q_2$ and $P_1$ are internally disjoint. See Figure~\ref{fig:paths}.
\begin{figure}
\caption{Structure of the paths $P_1$, $P_2$, $Q_1$, and $Q_2$. a) A dashed line shows path $P_1$, a dotted line indicates $P_2$, and solid lines are used for $Q_1$ and $Q_2$. b) Solid lines indicate paths $R_1$, $R_2$, $S_1$, and $S_2$. The thin lines are used for $R_1$ and $R_2$, while the thick lines for $S_1$ and $S_2$. The choice of $w$ is demonstrated in c).}
\label{fig:paths}
\end{figure}
Notice that $Q_1$ and $Q_2$ are internally disjoint since they are paths in $G_2$ and $G_3$ respectively, and $t$ is the unique common vertex of these graphs. Let $u_1$ be the vertex of $V(Q_1)\cap V(P_2)$ distinct from $t$ that is at the minimum distance from $t$ in $Q_1$. Similarly, let $u_2$ be the vertex of $V(Q_2)\cap V(P_1)$ distinct from $t$ that is at the minimum distance from $t$ in $Q_2$. The choice of $u_1$ and $u_2$ is shown in Figure~\ref{fig:paths} (a). Because $V(P_2'')\subseteq X_3$ and $V(P_1'')\subseteq X_2$, we have that $u_1\in V(P_2')$ and $u_2\in V(P_1')$. Denote by $R_1$ the $(s,u_2)$-subpath of $P_1$ and by $R_2$ the $(s,u_1)$-subpath of $P_2$. Let also $S_1$ be the $(u_1,t)$-subpath of $Q_1$ and $S_2$ be the $(u_2,t)$-subpath of $Q_2$. The construction of these paths is shown in Figure~\ref{fig:paths} (b).
We claim that paths $S_1$ and $R_1$ have no common vertices. For the sake of contradiction, let $V(S_1)\cap V(R_1)\neq \emptyset$ and assume that $w$ is the first vertex of $R_1$ in $V(S_1)$ (see Figure~\ref{fig:paths} (c)). Since $R_1$ and $R_2$ are internally vertex disjoint, $w$ is an internal vertex of $S_1$. By the choice of $u_1$, there is no internal vertex of $S_1$ that belongs to $P_2$.
Hence, the concatenation $\hat{P}_1$ of the $(s,w)$-subpath of $R_1$ and the $(w,t)$-subpath of $S_1$, gives a path that is internally vertex disjoint with $P_2$. Observe also that $A_1\subseteq V(\hat{P}_1)$, because $w\in B$ and the first $k_1$ internal vertices of $P_1$ are in $A_1$. Therefore, the total length of paths $\hat{P}_1$ and $P_2$ is at least $k$. However, the length of the $(s,w)$-subpath of $R_1$ is less than the length of $P_1'$ and the length of the $(w,t)$-subpath of $S_1$ is less than $k$. Therefore, the length of
$\hat{P}_1$ is less than the length of $P_1$, contradicting the choice of $P_1$ and $P_2$.
This proves that
$S_1$ and $R_1$ have no common vertices. By the same arguments, $S_2$ and $R_2$ also have no common vertices.
Consider $(s,t)$-paths $\hat{P}_1=R_2S_1$ and $\hat{P}_2=R_1S_2$. Since paths $S_1$ and $R_1$ do not intersect and paths $S_2$ and $R_2$ also do not intersect, we have that paths $\hat{P}_1$ and $\hat{P}_2$ are internally disjoint.
Because $A_1\subseteq V(\hat{P}_2)$ and $A_2\subseteq V(\hat{P}_1)$, the total length of paths $\hat{P}_1$ and $\hat{P}_2$ is at least $k$.
However, because the length of $P_1''$ is less than the length of $S_1$ and because the length of $P_2''$ is less than the length of $S_2$, the total length of $\hat{P}_1$ and $\hat{P}_2$ is less than the total length of $P_1$ and $P_2$. This contradict the choice of $P_1$ and $P_2$ and proves the claim.
\claimqed
By Claim~\ref{cl:oneoftwo}, without loss of generality, we assume that for every shortest $(v_1,t)$-path $Q_1$ in $G_2$, paths $Q_1$ and $P_2$ are internally disjoint.
Now we repeat the arguments from Case~1. We observe that every shortest $(v_1,t)$-path $Q_1$ in $G_2$ is internally disjoint with $P_1'$. Indeed, if this is not the case, we can select the first vertex $u$ of $P_1'$ that is in $Q_1$. Then by replacing $P_1$ by the concatenation of the $(s,u)$-subpath of $P_1'$ and the $(u,t)$-subpath of $Q_1$, we obtain a solution with a shorter total length. But this contradicts the choice of $P_1$ and $P_2$. Since $Q_1$ and $P_1'$ are internally vertex disjoint, we have that the cycle formed by paths $\hat{P}_1=P_1'Q_1$ and $P_2$ is a solution. This implies that $Q_1$ and $P_1''$ have the same length. Therefore $v_1$ is at distance $k$ from $t$.
We conclude that there is $v_1$ at distance $k$ from $t$ in $G_2$ such that for every shortest $(v,t)$-path $Q_1$ in $G_2$, there are an $(s,v_1)$-path $P_1'$ and an $(s,t)$-path $P_2$ in $G-(V(P)\setminus\{v,t\})$ that are internally disjoint. Then the algorithm finds a solution. This concludes Case~2 and the proof of Lemma~\ref{cl:probability}.
\end{proof}
By Lemma~\ref{cl:probability}, if we iterate Algorithm~\ref{alg:step} $3^{3k-1}/2$ times, then we either find a solution, or return the no-answer with the error probability at most $(1-\frac{2}{3^{3k-1}})^{3^{3k-1}/2}\leq e^{-1}$.
Thus we have a Monte Carlo algorithm with false negatives that runs in time $\mathcal{O}(3^{3k}\cdot (n+m))$.
\noindent\textbf{Derandomization.}
For the Monte Carlo algorithm that we use in the first stage (finding a short cycle), derandomization uses the standard technique. We replace random colorings by functions from the \emph{$(n,3k)$-perfect hash family} of functions of size $e^{3k}k^{\mathcal{O}(\log k)}\cdot \log n$ that can be constructed in time $e^{3k}k^{\mathcal{O}(\log k)}\cdot n\log n$ by the results of Naor, Schulman, and Srinivasan~\cite{NaorSS95} (we refer to~\cite[Chapter~5]{CyganFKLMPPS15} for the detailed introduction to the technique). This allows us to check in $(2e)^{3k}k^{\mathcal{O}(\log k)}\cdot mn\log n$ deterministic time whether there are two internally vertex disjoint $(s,t)$-paths in $G$ whose total length is at least $k$ but at most $3k$.
To derandomize the algorithm from the second stage that uses random separation, we have to do an extra work.
This is because commonly random separation is used to distinguish two sets~ \cite[Chapter~5]{CyganFKLMPPS15}. In our algorithm we distinguish three sets; derandomization here is slightly different and is based on Lemma~\ref{lem:derand}. Lemma~\ref{lem:derand} could be a folklore, but we did not find it in the literature and prove it here for completeness.
Let $k$ and $n$ be positive integers. An \emph{$(n,k)$-universal} set is a family $\mathcal{U}$ of subsets of $\{1,\ldots,n\}$ such that for any $S\subseteq \{1,\ldots,n\}$ of size $k$, the family $\{A\cap S\mid A\in \mathcal{U}\}$ contains all $2^k$ subsets of $S$. We use the following result of Naor, Schulman, and Srinivasan~\cite{NaorSS95}.
\begin{proposition}[\cite{NaorSS95}]\label{prop:derand} For any $n, k\geq 1$, one can construct an $(n,k)$-universal set of size $2^kk^{\mathcal{O}(\log k)}\cdot\log n$ in time $2^kk^{\mathcal{O}(\log k)}\cdot n\log n$. \end{proposition}
Using Proposition~\ref{prop:derand}, we prove the following lemma.
\begin{lemma}\label{lem:derand} For an $n$-element set $\Omega$ and a positive $k$, there is a family of functions $\mathcal{F}_{n,k}$ mapping $\Omega$ to $\{1,2,3\}$ of size $2^{5k}k^{\mathcal{O}(\log k)}\cdot (\log n)^2$ such that for every triple of disjoint nonempty sets $A_1,A_2,A_3\subseteq \Omega$, each of size at most $k$, there is $f\in\mathcal{F}_{n,k}$ with the property that \begin{itemize} \item $f(x)=f(y)$ if $x,y\in A_i$ for some $i\in\{1,2,3\}$, \item $f(x)\neq f(y)$ if $x\in A_i$ and $y\in A_j$ for distinct $i,j\in\{1,2,3\}$. \end{itemize} Moreover, $\mathcal{F}_{n,k}$ can be constructed in $2^{5k}k^{\mathcal{O}(\log k)}\cdot n^2\log n$ time.
\end{lemma}
\begin{proof} If $n\leq 3k$, then we define $\mathcal{F}_{n,k}$ to be the family of all at most $3^{3k}$ mappings $f\colon \Omega\rightarrow \{1,2,3\}$. Hence, from now we assume that $n\geq 3k$. Let $\Omega=\{\omega_1,\ldots,\omega_n\}$.
We apply Proposition~\ref{prop:derand} to construct the following family of universal sets. We construct an $(n,3k)$ universal set $\mathcal{U}^{(1)}$. Then for every positive $p\leq n$, we construct an $(p,2k)$-universal set $\mathcal{U}^{(2)}_p$. Then $\mathcal{F}_{n,k}$ is constructed as follows. For every $U=\{i_1,\ldots,i_p\}\in \mathcal{U}^{(1)}$ and every $W=\{j_1,\ldots,j_q\}\in \mathcal{U}^{(2)}_p$, we construct $f\colon \Omega\rightarrow \{1,2,3\}$ such that for every $h\in\{1,\ldots,n\}$, $$ f(\omega_h)= \begin{cases} 1&\mbox{if }h\notin \{i_1,\ldots,i_p\}\\ 2&\mbox{if }h\in\{i_1,\ldots,i_p\}\setminus\{i_{j_1},\ldots,i_{j_q}\}\\ 3&\mbox{if }h\in\{i_{j_1},\ldots,i_{j_q}\}. \end{cases} $$
To see that $\mathcal{F}_{n,k}$ satisfies the required property, consider arbitrary disjoint sets $A_1,A_2,A_3\subseteq \Omega$ of size at most $k$. We assume without loss of generality that each $A_i$ is of size exactly $k$ (otherwise, we can complement the sets by adding elements of $\Omega$ that are outside these sets). Let $A_i=\{\omega_{i_1^i},\ldots,\omega_{i_k^i}\}$ for $i\in\{1,2,3\}$. Let $S=\{i_1^1,\ldots,i_k^1\}\cup \{i_1^2,\ldots,i_k^2\}\cup\{i_1^3,\ldots,i_k^3\}$. By definition, the $(n,3k)$-universal set $\mathcal{U}^{(1)}$, contains a set $X$ such that
$S\cap X=\{i_1^2,\ldots,i_k^2\}\cup\{i_1^3,\ldots,i_k^3\}$. Let $p=|X|$ and assume that $X=\{j_1,\ldots,j_p\}$. Again by definition, the $(p,2k)$-universal set $\mathcal{U}_p^{(2)}$ contains a set $Z$ such that for every $s\in Z$, $j_s\neq i_1^2,\ldots,i_k^2$, and for every $t\in\{1,\ldots,k\}$, there is $s\in Z$ such that $j_s=i_t^2$. This implies that for $f\in \mathcal{F}_{n,k}$ constructed for $X\in \mathcal{U}^{(1)}$ and $Y\in\mathcal{U}_p^{(2)}$, $f(x)=i$ if $x\in A_i$ for $i\in\{1,2,3\}$. Therefore, $f$ distinguishes the sets $A_1,A_2,A_3$.
By Proposition~\ref{prop:derand}, $|\mathcal{U}^{(1)}|=2^{3k}k^{\mathcal{O}(\log k)}\cdot\log n$ and $|\mathcal{U}^{(2)}_p|=2^{2k}k^{\mathcal{O}(\log k)}\cdot\log n$. Therefore,
$|\mathcal{F}_{n,k}|\leq 2^{5k}k^{\mathcal{O}(\log k)}(\log n)^2$. By Proposition~\ref{prop:derand}, the universal sets can be constructed in time $2^{3k}k^{\mathcal{O}(\log k)}\cdot n^2\log n$. Then we construct $\mathcal{F}_{n,k}$ in time $2^{5k}k^{\mathcal{O}(\log k)}\cdot n(\log n)^2$. \end{proof}
To derandomize our algorithm, we apply Lemma~\ref{lem:derand}. Notice that the only property of random colorings that we use in the algorithm is that with sufficiently high probability the sets $A$, $B$, and $C$ defined in the proof of Lemma~\ref{cl:probability} are colored by distinct colors. The sets $A$, $B$, and $C$ have sizes at most $k$, and they are subsets of $V(G)\setminus\{s,t\}$. This implies that the random colorings can be replaced by functions of the family $\mathcal{F}_{n-2,k}$ for $\Omega=V(G)\setminus\{s,t\}$. Since Algorithm~\ref{alg:step} runs in $\mathcal{O}(n+m)$ time, the running time is $2^{5k}k^{\mathcal{O}(\log k)}\cdot (n+m)(\log n)^2$. Taking into account the time for constructing $\mathcal{F}_{n-2,k}$, we conclude that
the problem can be solved in $2^{5k}k^{\log k}\cdot mn\log n$ deterministic time.
Recall that in the first stage of our algorithm, we try to find two internally disjoint $(s,t)$-paths of total length $\ell$ for some $\ell\in\{k,\ldots,3k\}$, and this can be done in
$(2e)^{3k}\cdot mn$ randomized and $(2e)^{3k}k^{\mathcal{O}(\log k)}\cdot mn\log n$ deterministic time. Since $(2e)^{3}\geq 2^{5}\geq 3^{3}$ and $nm\geq n(n-1)$ as $G$ is assumed to be connected, we obtain that the running time of the first stage dominates the running time of the second.
We conclude that the problem can be solved in
$(2e)^{3k}\cdot mn$ randomized and $(2e)^{3k}k^{\mathcal{O}(\log k)}\cdot mn\log n$ deterministic time. It is plausible that the running time for the first stage can be improved by making use of more sophisticated techniques for \pname{Longest Path}\xspace and \pname{Longest Cycle}\xspace (see, e.g.,~\cite{FominLS14,Zehavi16}) but such an improvement goes beyond the scope of our paper. \end{proof}
\section{\pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace}\label{sec:erdos-gallaiPath}
In this section we prove Theorem~\ref{thmEG}:
\emph{\pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace is solvable in time
$2^{\mathcal{O}(k+|B|)}\cdot n^{\mathcal{O}(1)}$.}
The proof of the theorem relies on the structural properties of graphs with a long path. The notions of \emph{Erd{\H {o}}s-Gallai decomposition\xspace} and \emph{Erd{\H {o}}s-Gallai component\xspace} are crucial here. We prove several combinatorial and algorithmic properties of Erd{\H {o}}s-Gallai decomposition\xspace, and then apply the obtained properties in the proof of \Cref{thmEG}.
\subsection{Erd{\H {o}}s-Gallai decompositions and structures}
We need to introduce the operation of \emph{$B$-refinements}. The intuition behind this operation is the following. In our proof, we will be using the following rerouting strategy. Suppose we have an $(s,t)$-path $P$, and we want to construct a longer path by rerouting some parts of $P$ through a connected component $H$ of $G-V(P)$. If $H$ is 2-connected, we can try to apply \Cref{thm:circum}
to argue that such an enlargement of $P$ is possible. However, when $H$ is not $2$-connected, we want to eliminate some ``insignificant'' parts of $H$. While in the refinement we contract some of the edges inside $H$, all edges between $H$ and the remaining part of the graph remain.
\begin{definition}[\textbf{$B$-refinement of $H$}]
Let $H$ be a connected subgraph of a graph $G$ and $B\subset V(G)$.
The \emph{$B$-refinement} of $H$, denoted by $\gbref{B}{H}$, is the graph obtained by the following process.
Start with $\gbref{B}{H}:=G$.
While ${H}$ is not $2$-connected and contains a leaf-block
with all inner vertices from $B$, contract all edges in $H$ from this leaf-block to its cut-vertex.
\end{definition}
In other words, $\gbref{B}{H}$ is obtained from $G$ by repeatedly contracting edges of $H$ from the leaf-blocks of $H$ whose inner vertices are from $B$. Note that in $B$-refinement only edges with both endpoints in $B$ can be contracted. We also say that $\gbref{B}{H}$ is obtained from $G$ by applying $B$-refinement to $H$.
\begin{comment}
\todo[inline]{Check if we use the proposition below anywhere at all. If yes, then maybe rewrite the first paragraph of the proof. }
$B$-refinement of a graph also defines a notion of separability.
\begin{proposition}\label{prop:separabil}
Let $G$ be a connected graph and $B$ be a vertex set such that $|V(G)\setminus B|>2$.
Then $\gbref{G}{B}$ is $2$-connected if and only if $G$ contains no cut-vertex separating two vertices in $V(G)\setminus B$ from each other.
\end{proposition}
\begin{proof}
Suppose that $\gbref{G}{B}$ is $2$-connected, so $\gbref{G}{B}-v$ is connected for any $v \in V(\gbref{G}{B})$.
As $\gbref{G}{B}$ is a subgraph of $G$ and $V(G)\setminus B\subseteq \gbref{G}{B}$, all vertices in $V(G)\setminus B$ are contained in the same connected component in $G-v$ for any $v \in V(G)$.
Equivalently, there is no cut-vertex in $G$ separating a pair of vertices in $V(G)\setminus B$.
Suppose that $\gbref{G}{B}$ is not $2$-connected. Since $|V(G)\setminus B|>2$, we have that
$\gbref{G}{B}$ contains at least three vertices. Thus it has at least two leaf-blocks.
Each of these leaf-blocks contains an inner vertex from $V(G)\setminus B$, denote these vertices by $s$ and $t$.
Each path from $s$ and $t$ path through cut vertices of their leaf-blocks. Thus, there is a vertex $c \in V(\gbref{G}{B})$ that lies on any path from $s$ to $t$.
Note that edges in $E(G)\setminus E(\gbref{G}{B})$ do not connect any two vertices in $V(\gbref{G}{B})$ with each other.
Hence, all paths between $s$ and $t$ in $G$ are contained in $\gbref{G}{B}$ as well.
It follows that $c$ separates $s$ from $t$ in $G$.
\end{proof}
\end{comment}
\begin{figure}\label{fig:bananapath}
\end{figure}
We are ready to introduce the primary tool for solving \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace. This structure arises in the extremal cases when we cannot enlarge an $(s,t)$-path by local replacement used in the proof of the Erd{\H {o}}s-Gallai's theorem. This is where the name we use for the decomposition comes from.
\begin{definition}[\textbf{Erd{\H {o}}s-Gallai decomposition\xspace and Erd{\H {o}}s-Gallai component\xspace}] Let $P$ be a path in a $2$-connected graph $G$ and let $B\subseteq V(G)$. We say that two disjoint paths $P_1$ and $P_2$ in $G$ induce \emph{an Erd{\H {o}}s-Gallai decomposition\xspace for $P$ and $B$} in $G$ if
\begin{itemize}
\item
Path $P$ is of the form $P=P_1 {P'}P_2$, where the inner path ${P'}$ has at least $\delta(G- B)$ edges.
\item
Let $G'$ be the graph obtained from $G$ by applying $B$-refinement to every connected component $H$ of $G- V(P_1 \cup P_2)$, except those components $H$ with $V(H)\subseteq B$. Note that no edges of the paths $P_1$ and $P_2$ are contracted.
There are at least two connected components $H'$ in $G'-V(P_1\cup P_2)$ with $V(H')\not\subseteq B$. For every such connected component $H'$ holds $|V(H')|\ge 3$ and one of the following.
\begin{enumerate}[label=(R\arabic*)]
\item\label{enum:tunnel_path_bic} $H'$ is $2$-connected and the maximum size of a matching in $G'$ between $V(H')$ and $V(P_1)$ is one, and between $V(H')$ and $V(P_2)$ is also one;
\item\label{enum:tunnel_path_cut_left} $H'$ is not 2-connected, exactly one vertex of $P_1$ has neighbors in $H'$, that is
$|N_{G'}(V(H'))\cap V(P_1)|=1$, and no inner vertex from a leaf-block of $H'$ has a neighbor in $P_2$;
\item\label{enum:tunnel_path_cut_right} The same as \ref{enum:tunnel_path_cut_left}, but with $P_1$ and $P_2$ interchanged. That is, $H'$ is not 2-connected,
$|N_{G'}(V(H'))\cap V(P_2)|=1$, and no inner vertex from a leaf-block of $H'$ has a neighbor in $P_1$.
\end{enumerate}
\end{itemize} The set of \emph{Erd{\H {o}}s-Gallai component\xspace}s for an Erd{\H {o}}s-Gallai decomposition\xspace
is defined as follows.
First, for each component $H'$ of type \ref{enum:tunnel_path_bic}, $H'$ is an Erd{\H {o}}s-Gallai component\xspace of the Erd{\H {o}}s-Gallai decomposition\xspace.
Second, for each $H'$ of type \ref{enum:tunnel_path_cut_left}, or of type \ref{enum:tunnel_path_cut_right}, all its leaf-blocks are also Erd{\H {o}}s-Gallai component\xspace{s} of the Erd{\H {o}}s-Gallai decomposition\xspace. The example of an Erd{\H {o}}s-Gallai decomposition\xspace is given in Figure~\ref{fig:bananapath}. \end{definition}
The following lemma provides a polynomial time algorithm that either finds a long path in the given graph or constructs an Erd{\H {o}}s-Gallai decomposition\xspace.
\begin{lemma}\label{lemma:st_path_or_tunnel}
Let $G$ be a $2$-connected graph with two distinct vertices $s$ and $t$, $B\subseteq V(G)$ be a subset of vertices such that $s,t\in B$, and $k>0$ be an integer such that $4k+2|B|+4\le\delta(G-B)$.
There is a polynomial time algorithm that
\begin{itemize}
\item either outputs an $(s,t)$-path $P$ of length at least $\delta(G-B)+k$,
\item or outputs an $(s,t)$-path $P$ with $V(P)\cup B =V(G)$,
\item or outputs an $(s,t)$-path $P$ with paths $P_1, P_2$ that induce an Erd{\H {o}}s-Gallai decomposition\xspace for $P$ and $B$ in $G$.
\end{itemize} \end{lemma} \begin{proof}
By Corollary~\ref{thm:relaxed_st_path}, an $(s,t)$-path $P$ of length at least $\delta(G-B)$ can be found in polynomial time.
If the length of $P$ is at least $\delta(G-B)+k$, we output it and stop. Otherwise, we try to make $P$ longer by replacing some of its parts with paths in $G- V(P)$.
We first contract some edges of $G$ in a way similar to the definition of Erd{\H {o}}s-Gallai decomposition\xspace{s}. For each connected component $H$ of $G-V(P)$ such that $V(H)$ is not in $B$, we perform $B$-refinement of $H$. That is, while $H$ is not 2-connected and has a leaf-block with all inner vertices from $B$, we contract all edges of this leaf-block. We denote the resulting graph by $G'$. Note that $G'$ still contains $P$ as a subgraph and that $\delta(G'-B)\ge\delta(G-B)$.
If we find an $(s,t)$-path that is longer than $P$ in $G'$, this path can be easily transformed into a path of the same or greater length in $G$.
Moreover, if we find paths $P_1$ and $P_2$ that induce an Erd{\H {o}}s-Gallai decomposition\xspace for $P$ in $G'$, then $P_1, P_2$ induce an Erd{\H {o}}s-Gallai decomposition\xspace for $P$ in graph $G$ as well.
Thus, from now on, we proceed with the graph $G'$.
We start with the trivial case. If $V(G')\setminus P\subseteq B$, then $V(P)\cup B=V(G)$.
Hence, the algorithm just outputs $P$ and stops.
From now on we assume that
$(V(G')\setminus B) \setminus V(P)\neq\emptyset$.
Let $H'$ be a connected component in $G'-V(P)$ that contains at least one vertex in $V(G')\setminus B$.
We consider several cases. The first case is a trivial case when $H'-B$ contains at most two vertices.
The second case corresponds to {Erd{\H {o}}s-Gallai component\xspace}s of type
\ref{enum:tunnel_path_bic}, while the third case to {Erd{\H {o}}s-Gallai component\xspace}s of types \ref{enum:tunnel_path_cut_left} and \ref{enum:tunnel_path_cut_right}.
If we find out that $P$ can be enlarged, we replace $P$ with the longer path in $G$ and start trying to make it longer again.
Throughout the proof and all its claims, we consider that $P$ cannot be made longer with the replacement operation.
\noindent
\textbf{Case 1:} \emph{$H'-B$ contains at most two vertices.}
In this case, each vertex in $V(H'- B)$ has at least $\delta(G'-B)-2$ neighbors in $P$.
If the length of $P$ is less than $2\delta(G-B)-4\ge \delta(G-B)+4k$, then each vertex in $V(H'-B)$ has two consecutive vertices in $P$ as neighbors.
Hence, any such vertex can be inserted in $P$ between such two neighbors, so the length of $P$ increases by one.
\textbf{Conclusion of Case 1.} Either $H'-B$ consists of at least three vertices, or the length of $P$ can be increased (in polynomial time).
\noindent
\textbf{Case 2:} \emph{$H'$ is $2$-connected.} We start with the following claim.
\begin{claim}\label{claim:case2-connected}
If there is a matching of size at least three between $V(H')$ and $V(P)$ in $G'$, then the length of $P$ can be enlarged in polynomial time.
\end{claim}
\noindent \emph{Proof of Claim~\ref{claim:case2-connected}.}
As $\delta(G'-B)\ge \delta(G-B)$,
$2\delta(G'-B)-2>\delta(G-B)+4k+2|B|>\delta(G-B)+k$.
So we assume that the length of $P$ is at most $2\delta(G'-B)-1$.
Let $u_1v_1, u_2v_2, u_3v_3$ be a matching in $G'$ such that $u_1, u_2, u_3 \in V(H')$ and $v_1, v_2, v_3 \in V(P)$.
If no vertex in $V(H'-B)$ has a neighbor in $P$, then $\delta(H'-B)\ge \delta(G'-B)$.
By Corollary~\ref{thm:relaxed_st_path}, there is a path of length at least $\delta(G'-B)$ between any pair of vertices in $H'$.
Because the length of $P$ is at most $2\delta(G'-B)-1<2\delta(G'-B)+4$, at least for one pair $\{v_i,v_j\}$, $i\neq j$,
the distance
between $v_i $ and $v_j$ in $P$ is less than $\delta(G'-B)+2$. Then we replace the $(v_i,v_j)$-subpath in $P$ with the path $v_i u_i \leadsto u_j v_j$, where $u_i \leadsto u_j$ is a path between $u_i$ and $u_j$ in $H'$ of length at least
$\delta(G'-B)$. The length of $v_i u_i \leadsto u_j v_j$ is at least $\delta(G'-B)+2$ and hence we can enlarge $P$.
Now we assume that there is at least one vertex $w\in V(H'-B)$ with a neighbor in $P$. We can assume that in the matching $u_1v_1, u_2v_2, u_3v_3$, one of the vertices $u_i=w$. (If all $u_i\in B$, we just replace $u_1$ with $w$.) Vertex $w$ has at least $\max\{1,\delta(G'-B)-\delta(H'-B)\}$ neighbors in $P$. Let $S$ be the set of neighbors of $u_1, u_2, u_3$ in $P$, that is, $S:=(N_{G'}(u_1)\cup N_{G'}(u_2) \cup N_{G'}(u_3))\cap V(P)$.
Then the size of $S$ is at least $\max\{\delta(G'-B)-\delta(H'-B),3\}$. Let $s_1,s_2, \dots, s_{|S|}$, be the order of vertices from $S$ in the path $P$. If the length of one of the subpaths $s_i, s_{i+1}$, $i\in \{1, \dots, |S|-1\}$, of $P$ is $1$, we can enlarge $P$ by replacing $s_i, s_{i+1}$ with an $(s_i, s_{i+1})$-path of length at least 2 going through $V(H')$.
Moreover, at least two of these subpaths go between neighbors of $u_i$ and $u_j$ for distinct $i$ and $j$. If one of these paths, say between $s_\ell$ and $s_{\ell+1}$, is of length less than $\delta(H'-B)+2$, we
can increase $P$ by replacing it with path $s_\ell u_i \leadsto u_j s_{\ell+1} $, where $u_i \leadsto u_j$ is a path between $u_i$ and $u_j$ in $H'$ of length at least $\delta(H'-B)$. This means that if we cannot enlarge $P$, then the length of $P$ is at least $2(|S|-3)+ 2(\delta(H'-B)+2)\geq 2(\delta(G'-B)-\delta(H'-B)-3)+ 2(\delta(H'-B)+2)=2\delta(G'-B)-2>\delta(G-B)+k$.
\claimqed
By the claim and the fact that $G$ is $2$-connected, we can assume that the maximum size of a matching between $V(H')$ and $V(P)$ in $G'$ is exactly two.
\begin{claim}\label{claim:eg_path_bic_degree}
There is a path of length at least $\delta(G'-B)-2$ between any pair of vertices in $H'$.
\end{claim}
\begin{claimproof}
Let $h_1v_1, h_2v_2$ be the edges of the maximum matching between $V(H')$ and $V(P)$ in $G'$, where $h_1, h_2\in V(H')$, $v_1, v_2 \in V(P)$. Note that no vertex in $V(H')\setminus\{h_1,h_2\}$ has neighbours in $V(P)\setminus \{v_1,v_2\}$.
If $h_1$ and $h_2$ have no neighbours other than $v_1$ and $v_2$ in $V(P)$, then, trivially, $N_G(V(H'))\cap V(P)=\{v_1,v_2\}$, so $\delta(H'-B)\ge \delta(G'-B)-2$.
Without loss of generality, we now assume that $h_1$ has a neighbour $v_3 \in V(P)\setminus \{v_1,v_2\}$. Then no vertex in $V(H')\setminus \{h_1,h_2\}$ can have $v_1$ as a neighbour. Analagously, if $h_2$ has a neighbour other than $v_1$ or $v_2$, no vertex in $V(H')\setminus\{h_1,h_2\}$ can have $v_2$ as a neighbour. Hence, if $N_G(h_i)\not\subseteq \{v_1,v_2\}$ for both $i=1$ and $i=2$, then $\delta(H'-(B\cup \{h_1,h_2\}))\ge \delta(G'-B)-2$.
We now assume that $h_2$ has no neighbours other than $v_1$ and $v_2$ in $V(P)$. If $h_2$ is adjacent to $v_1$, then no vertex in $V(H')\setminus \{h_1, h_2\}$ can be adjacent to $v_2$, as we would obtain a matching $h_1v_3$, $h_2v_1$, $h_3v_2$ of size at least three. Hence, if $h_2v_1 \in E(G')$, then $\delta(H'-(B\cup\{h_1,h_2\}))\ge \delta(G'-B)-2$. If $h_2$ is not adjacent to $v_1$, then all vertices in $V(H')\setminus\{h_1\}$ only can have $v_2$ as a neighbour, so $\delta(H'-(B\cup\{h_1\}))\ge \delta(G'-(B\cup\{h_1\}))-1\ge \delta(G'-B)-2$.
It is left to apply \Cref{thm:relaxed_st_path} to all of the cases.
\end{claimproof}
Hence, if there is a matching between $V(H')$ and two vertices on $P$ that are closer than $\delta(G'- B)$ to each other and $P$ can be made longer.
Suppose that we have a matching between $V(H')$ and $V(P)$ with endpoints $h_1,h_2 \in V(H')$ and $v_1, v_2 \in V(P)$, where $v_1$ is closer to $s$ on $P$ than $v_2$.
Let $a_1$ denote the distance from $s$ to $v_1$ on $P$ and $a_2$ denote the distance from $v_2$ to $s$ on $P$.
Then, if $|V(P)|+1-(a_1+a_2)<\delta(G'-B)$, $P$ can be enlarged using the long $(h_1,h_2)$-path of length at least $\delta(G'-B)-2$ in $H'$.
Otherwise, $a_1+a_2\le |V(P)|+1-\delta(G'-B)$.
In particular, $a_1,a_2 \le |V(P)|+1-\delta(G'-B)$.
Thus, $v_1$ is within the first $|V(P)|+2-\delta(G'-B)$ vertices of $P$ and $v_2$ is within the last $|V(P)|+2-\delta(G'-B)$ vertices of $P$.
\textbf{Conclusion of Case 2.} If $H'$ is $2$-connected, then either $P$ can be made longer or the following holds.
The size of the maximum matching between $H'$ and $P$ is exactly 2. Moreover, for any maximum matching between $H'$ and $P$, the endpoint of one edge of the matching is one of the first $|V(P)|-\delta(G'- B)+2$ vertices of $P$ and one is within the last $|V(P)|-\delta(G'- B)+2$ vertices of $P$.
\noindent
\textbf{Case 3:} \emph{$H'$ is not $2$-connected.} Let $L$ be a leaf-block $L$ of $H'$ and let $c$ be
the cut-vertex of the leaf-block $L$.
Note that $V(L)\setminus B\setminus \{c\}$ is not empty and $\delta(L- (B\cup \{c\}))\ge \delta(H'-B)-1$.
Assume first that there is a matching of size three between $V(L)$ and $V(P)$ in $G'$.
Similar to Case 2, then there is a vertex in $V(L-(B\cup \{c\}))$ with at least $\delta(G'-B)-\delta(L-(B\cup \{c\}))$ neighbors in $V(P)$. In this case, since the length of $P$ is at most $\delta(G-B)+k< 2(\delta(G'-B)-\delta(L-(B\cup \{c\}))-1)+2\delta(L-(B\cup\{c\}))$, we can reroute a part of $P$ through $H'$ and thus make it longer.
Now we may assume that the maximum matching size between $V(L)$ and $V(P)$ in $G'$ is at most two. Again, similar to Case 2 and \Cref{claim:eg_path_bic_degree} we derive that $\delta(L- (B'\cup\{c\}))\ge \delta(G'-(B\cup\{c\}))-2\ge \delta(G'-B)-3$ for some $B'\supseteq B$.
Hence, there is a path of length at least $\delta(G'-B)-3$ between any pair of vertices in $L$ by \Cref{thm:relaxed_st_path}.
It follows that there is a path of length at least $\delta(G'-B)-3$ between an inner vertex of a leaf-block of $H'$ and any other vertex in $H'$.
For each leaf-block in $H'$, there is at least one inner vertex that has at least one neighbor in $P$, otherwise $G'$ is not $2$-connected.
Suppose first that there are two inner vertices of two distinct leaf-blocks in $H'$ that have two distinct neighbors in $V(P)$. There is always path between these two inner vertices of length at least $2(\delta(G'-B)-3)$: we can find two paths in each leaf-block starting in the cut vertex and ending in an inner vertex of length at least $\delta(G'-B)-3$.
Since the length of $P$ is at most $\delta(G-B)+k\leq 2(\delta(G'- B)-3)+2$, the subpath of $P$ between their neighbours is shorter than if than the path between them through $H'$.
So we can enlarge $P$ by using this path.
Note that if there are at least two vertices $V(P)$ having at least one inner leaf-block vertex of $H'$ as a neighbour, then we can always pick two inner vertices as described in the previous paragraph.
Hence, if $P$ cannot be made longer, there is exactly one vertex $v \in V(P)$ that is connected to inner vertices of the leaf-blocks of $H'$.
Then, in fact, $\delta(L-(B\cup\{c\}))\ge \delta(G'-(B\cup\{c\}))-1\ge \delta(G'-B)-2$ for each leaf-block $L$ of $H$ with cut vertex $c$.
The following claim holds.
\begin{claim}\label{claim:eg_path_sep_degree}
There is a path of length at least $\delta(G'-B)-2$ between any inner vertex of a leaf-block and any other vertex of $H'$.
\end{claim}
Since $G$ is $2$-connected, there is at least one other vertex $u \in V(P)$ that has neighbors in $V(H')$.
If the distance between $u$ and $v$ on $P$ is less than $(\delta(G'- B)-2)+2$, then $P$ can be made longer.
As there is a path of length at least $\delta(G'- B)-2$ between their neighbours in $H$.
Hence, $H'$ can only have neighbors among the first and among the last $|V(P)|+2-\delta(G'- B)$ vertices of $P$ analogously to Case 2.
\textbf{Conclusion of Case 3.}
If $H'$ contains at least three vertices and is not 2-connected, then
either $P$ can be made longer, or the following properties hold.
All inner vertices of its leaf-blocks that have neighbors in $V(P)$ have exactly one neighbor on $P$, and this neighbour is the same for all inner vertices.
This neighbour vertex is within the first (or the last) $|V(P)|+3-\delta(G'- B)$ vertices of $P$.
All other neighbours of $V(H')$ on $P$ are, oppositely, within the last (or the first) $|V(P)|+2-\delta(G'- B)$ vertices of $P$.
\noindent
\textbf{Constructing Erd{\H {o}}s-Gallai decomposition\xspace.} We use the structural properties of $G'$ to construct an Erd{\H {o}}s-Gallai decomposition\xspace in graph $G'$, and hence in $G$.
We start from an $(s,t)$-path $P$ in $G'$ and try to increase its length by applying one of the algorithms from Cases 1--3. Assume that we cannot increase the length of $P$ anymore. Then we have a path $P$ and every connected component $H'$ of $G'-V(P)$ should satisfy the properties summarized in the conclusion of
Case~2 or Case~3. We show that in this case we either could construct in polynomial time a new path $P$ of length at least $\delta(G- B)+k$, or to construct an Erd{\H {o}}s-Gallai decomposition\xspace.
Then each $H'$ has neighbors within the first $k+2$ vertices of $P$ and within the last $k+2$ vertices of $P$.
Denote by $P_1$ the shortest subpath of $P$ starting in $s$ that contains all starting neighbors (that is, neighbours that are closer to $s$ than to $t$ in $P$) among all possible components $H'$.
Analogously, denote by $P_2$ the shortest subpath of $P$ ending in $t$ that contains all ending neighbors (that is, neighbours that are closer to $t$ than to $s$ in $P$) among all possible $H'$. Thus $P=P_1 P' P_2$.
\begin{claim}\label{claim:p'_long}
The length of $P'$ is at least $\delta(G-B)-k$.
\end{claim}
\begin{claimproof}
We know that $|V(P_1)|,|V(P_2)|\le |V(P)|-\delta(G'- B)+2$.
The length of each of $P_1$ and $P_2$ is at most $|V(P)|-\delta(G'-B)+1$, so the length of $P'$ is at least \[(|V(P)|+1)-2(|V(P)|-\delta(G'-B)+1)=2\delta(G'-B)-|V(P)|>\delta(G-B)-k.\]
\end{claimproof}
Denote by $s'$ and $t'$ the endpoints of $P'$, so $P_1$ and $P_2$ are the $(s,s')$-subpath and the $(t',t)$-subpath of $P$ respectively.
The following claim is rather useful.
\begin{claim}\label{claim:p'_no_neighbours}
There is a connected component $H'$ in $G'-(V(P_1)\cup V(P_2))$ with $V(H')\setminus B=V(P'-\{s',t'\})\setminus B$.
\end{claim}
\begin{claimproof}
We actually need to show that each vertex $v \in V(P'-\{s',t'\})$ can only have neighbours in $V(P)$ or $B$.
Suppose that there is $v\in V(P'-\{s',t'\})$ with a neighbour $u \in V(G')\setminus V(P)\setminus B$.
Then $u$ is in some connected component $H''$ of $G'-V(P)$ with $|V(H''-B)|\ge |\{u\}|>0$.
Note that then $H''$ has a vertex with a neighbour in $P$ that is not in $V(P_1)\cup V(P_2)$.
This contradicts the choice of $P_1$ and $P_2$.
\end{claimproof}
We now show that the length of $P'$ can be actually assumed to be at least $\delta(G-B)$, as agrees with the definition of Erd{\H {o}}s-Gallai decomposition\xspace. This strengthens \Cref{claim:p'_long}.
\begin{claim}\label{claim:distP1} If the distance between $P_1$ and $P_2$ in $P$ is less than $\delta(G- B)$, then $G'$ contains an $(s,t)$-path of length at least $\delta(G-B)+k$. Moreover, this path can be computed in polynomial time. \end{claim}
\begin{claimproof}
Suppose that the distance between $P_1$ and $P_2$ in $P$ is less than $\delta(G- B)$.
Equivalently, $|V(P'-\{s',t'\})|<\delta(G- B)-1$.
Vertices in $P'-\{s',t'\}$ are adjacent in $G'$ only to vertices in $B$ and vertices from $V(P)$ by \Cref{claim:p'_no_neighbours}.
Hence, each vertex in $V((P'-\{s',t'\})- B)$ has at least three neighbors in $V(P_1)\cup V(P_2)$, as it has at most $|V(P'-\{s',t'\})|-1\le\delta(G- B)-3$ neighbors in $V(P'-\{s',t'\})$.
Consider the first vertex in $P'$ that is not in $B$ and is at distance at least $\delta(G- B)/2$ from the start of $P'$. Denote this vertex by $v$.
Note that the length of the $(s',v)$-subpath of $P'$ is at most $\delta(G-B)/2+|B|$.
By \Cref{claim:p'_long}, the distance from $v$ to the last vertex of $P'$, i.e.\ the length of the $(v,t')$-subpath of $P'$, is
at least $(|V(P')|-1)-(\delta(G- B)/2+|B|)\ge \delta(G-B)-k-\delta(G-B)/2-|B|=\delta(G-B)/2-k-|B|$ .
Hence, the distance from $v$ to each of the endpoints of $P'$ is at least $\delta(G-B)/2-k-|B|$. Vertex $v$ has at least two neighbors in $V(P_1)$ or in $V(P_2)$, as it has at least three neighbours in $V(P_1)\cup V(P_2)$.
Without loss of generality, assume that it has two neighbors in $P_1$.
\begin{figure}
\caption{Construction of a long path in $G'$ when $P'$ is shorter than $\delta(G-B)$.}
\label{fig:eg_path_p'_short}
\end{figure}
One of its neighbors, say $u_1$, is different from $s'$.
Construct an $(s,t)$-path as follows. Start from $s$, move to $u_1$ along $P_1$, then from $u_1$ to $v$, then follow the path $P'$ backwards from $v$ to $s'$. By the construction of
$P_1$, there is at least one component $H'$ in $G'- V(P)$ that is connected with $s'$. Thus from
$s'$ we enter $H'$, and follow a path of length at least $\delta(G'-B)-2$ in $H'$ (such path always exists by either \Cref{claim:eg_path_bic_degree} or \Cref{claim:eg_path_sep_degree}) to reach some vertex in $P_2$.
We complete the construction of the path by
following along $P_2$ to $t$ (see \Cref{fig:eg_path_p'_short}).
The length of the constructed path is at least $$\underbrace{1}_{s\leadsto u_1\leadsto v}+\underbrace{\delta(G-B)/2-k-|B|}_{v\leadsto s'}+\underbrace{1+(\delta(G-B)-2)+1)}_\text{$s'\leadsto t$ through $H'$},$$ which equals $\frac{3}{2}\delta(G-B)-k-|B|+1>\delta(G-B)+k$.
\end{claimproof}
By the claim, if the length of $P'$ is less than $\delta(G- B)$, then we find in polynomial time the desired path and stop.
Otherwise, the distance between $P_1$ and $P_2$ in $P$ is at least $\delta(G- B)$, hence $|V(P_1)|+|V(P_2)|\le k+1$ as $|V(P)|\le \delta(G-B)+k$.
\begin{claim}\label{claim:p'_is_bic}
The connected component $H'$ from \Cref{claim:p'_no_neighbours} is of type \ref{enum:tunnel_path_bic} in $G'-(V(P_1) \cup V(P_2))$, or a path of length at least $\delta(G-B)+k$ in $G'$ can be found in polynomial time.
\end{claim}
\begin{claimproof}
We first show that $H'$ is $2$-connected after $B$-refinements are applied to it.
Denote the component $H'$ with applied $B$-refinements by $H''$ and assume that $G'=\gbref{B}{H'}$.
If $H''$ is not $2$-connected, then it contains at least two leaf-blocks, as $|V(P'-\{s',t'\})|\ge \delta(G-B)-1>2$.
Since $\delta(H'-B)\ge \delta(G-(B\cup(V(P_1)\cup V(P_2))))\ge \delta(G-B)-k-1$, each leaf-block of $H''$ should contain at least $\delta(G-B)-k$ vertices outside $B$.
Hence, $H'-B$ consists of at least $2(\delta(G-B)-k)-1\ge \delta(G-B)+k\ge |V(P)|$ vertices.
This is not possible since $V(H'-B)\subseteq V(P'-\{s',t'\})$ and $|V(P'-\{s',t'\})|<|V(P)|$.
It is left to show that the matching conditions of type \ref{enum:tunnel_path_bic} are also satisfied.
Assume that these conditions do not hold.
Without loss of generality, assume that the maximum matching size between $V(H'')$ and $V(P_1)$ is at least two in $\gbref{B}{H'}$.
Then there are two edges $v_1 h_1, v_2 h_2 \in E(G')$ with $v_1,v_2 \in V(P_1)$ and $h_1, h_2 \in V(H'')$.
Without loss of generality, we assume that $v_1$ is closer to $s$ on $P$ than $h_2$.
In particular, $v_1 \neq s'$.
As $H''$ is $2$-connected, then by \Cref{thm:relaxed_st_path}, it contains a path of length at least $\delta(H''-B)=\delta(H'-B)\ge \delta(G-B)-k-1$ between $h_1$ and $h_2$.
As discussed above in the proof of \Cref{claim:p'_long} (see \Cref{fig:eg_path_p'_short}), there is a path connecting $s'$ with some vertex in $P_2$ going through a component $H$ in $G-V(P)$.
Hence, there is an $(s',t)$-path of length at least $\delta(G-B)$ that does not have common vertices with $H''$. Then we concatenate the following paths.
Take the $(s,v_1)$-subpath of $P_1$, proceed further with the edge $v_1h_1$ and the $(h_1,h_2)$-path inside $H''$, then with the edge $h_2v_2$ and the $(v_2,s')$-subpath of $P_1$.
Finish with the $(s',t)$-path.
The obtained path is an $(s,t)$-path of length at least $$\underbrace{1}_{s\leadsto h_1}+\underbrace{\delta(G-B)-k-1}_{h_1\leadsto h_2}+\underbrace{1}_{h_2\leadsto s'}+\underbrace{\delta(G-B)}_{s'\leadsto t},$$
which equals to $2\delta(G-B)-k+1>\delta(G-B)+k$.
Thus, if $H''$ is not of type \ref{enum:tunnel_path_bic}, then we can find a long path in $G'$ in polynomial time.
\end{claimproof}
Note that every connected component in $G'-V(P_1 \cup P_2)$ corresponds either to Case 2, or to Case 3, or to \Cref{claim:p'_is_bic}, or is fully contained in $B$.
A connected component from Case 2 or \Cref{claim:p'_is_bic} corresponds to \ref{enum:tunnel_path_bic}-type connected components of {Erd{\H {o}}s-Gallai decomposition\xspace}s.
The connected components from Case 3 correspond to \ref{enum:tunnel_path_cut_left}-type and \ref{enum:tunnel_path_cut_right}-type connected components depending on whether the vertex $v$ is from $V(P_1)$ or from $V(P_2)$.
Thus, $P_1$ and $P_2$ induce an Erd{\H {o}}s-Gallai decomposition\xspace for $P$ and $B$ in $G'$, and hence in $G$.
\end{proof}
The following proposition about long paths inside Erd{\H {o}}s-Gallai component\xspace{s} is clear from the proof of \Cref{lemma:st_path_or_tunnel}.
\begin{proposition}
For any Erd{\H {o}}s-Gallai component\xspace of any Erd{\H {o}}s-Gallai decomposition\xspace in $G$ for $B\subseteq V(G)$, there is a path of length at least $\delta(G-B)-2$ between any pair of vertices of this Erd{\H {o}}s-Gallai component\xspace. \end{proposition}
We start to establish the properties of Erd{\H {o}}s-Gallai component\xspace{s} that will be exploited by the algorithm. To state the first property, we need the following definition. \begin{definition} We say that a path $P$ \emph{enters} a subgraph $H$, if at least one edge of $H$ is also an edge of $P$. \end{definition} Informally, the property is the following. Consider an Erd{\H {o}}s-Gallai component\xspace $M$ for some Erd{\H {o}}s-Gallai decomposition\xspace and consider also an $(s,t)$-path $P'$. Path $P'$ can hit some vertices of $M$. However, if $P'$ enters $M$, then all vertices of $H$ hit by $P$, that is, all common vertices of $P$ and $M$, appear consecutively in $P'$.
\begin{lemma}\label{lemma:st_path_banana_consecutive}
Let $G$ be a $2$-connected graph, $B\subseteq V(G)$, $P$ be an $(s,t)$-path in $G$. Let paths $P_1, P_2$ induce an Erd{\H {o}}s-Gallai decomposition\xspace for $P$ and $B$ in $G$. Let also $G'$ be the graph obtained after $B$-refinements of connected components of
$G- V(P_1 \cup P_2)$,
and let $M$ be an Erd{\H {o}}s-Gallai component\xspace. Then for every $(s,t)$-path $P'$ in $G'$, if $P' $ enters $M$, then all vertices of $M\cap V(P')$ appear consecutively in $P'$. \end{lemma} \begin{proof} Targeting towards a contradiction, assume that the statement of the lemma does not hold. Then there is an $(s,t)$-path $P'$ that contains at least one edge of $M$, but vertices of $M$ does not appear consecutively in $P'$.
That is, there are vertices $v_1, v_2 \in V(M)$, $v_1\neq v_2$, such that $P'$ is of the form
$s, \dots, v_1, \dots, x,\dots, v_2, \dots, t$, where $x\not \in V(M)$. No internal vertex of
the $(s,v_1)$-subpath and the $(v_2,t)$-subpath of $P'$ belongs to $V(M)$.
Moreover, the $(v_1, v_2)$-subpath of $P'$ contains at least one edge of $M$ and at least one edge outside of $M$.
Let $G'$ be the graph obtained from $G$ after applying all possible $B$-refinements.
According to the definition of Erd{\H {o}}s-Gallai component\xspace, $M$ can be one of the following three types. Either it is a connected component of
$G'- V(P_1 \cup P_2)$ (this corresponds to type \ref{enum:tunnel_path_bic}), or it is a leaf-block of a connected component of
$G'- V(P_1 \cup P_2)$ (this corresponds to types \ref{enum:tunnel_path_cut_left} and \ref{enum:tunnel_path_cut_right}).
Therefore, we consider three cases.
\noindent
\textbf{Case 1.}
Suppose that $M$ is an Erd{\H {o}}s-Gallai component\xspace of type \ref{enum:tunnel_path_bic}. That is, $M$ is a connected component of $G'- V(P_1 \cup P_2)$ and also $M$ is 2-connected.
Consider the $(s,v_1)$-subpath of $P'$ in $G$. Since $s\in V(P_1)$, there exists vertex
$w_1$ that is the last vertex on this subpath that is from $V(P_1\cup P_2)$.
Then the subpath is of form $s\leadsto w_1 \leadsto v_1$, where all inner vertices of the subpath $w_1\leadsto v_1$ are from $V(H)\setminus V(M)$, where $H$ is the connected component $M$ before the $B$-refinements. But after the $B$-refinement of $H$ in $G$, all inner edges of this path are contracted. Then in $G'$ this $(w_1,v_1)$-subpath consists of just single edge $w_1 v_1$.
Analogously, consider the $(v_2, t)$-subpath of $P'$ and let $w_2$ be the first vertex from $V(P_1)\cup V(P_2)$ on this subpath.
The $(v_2, w_2)$-subpath goes only through vertices in $V(H)\setminus V(M)$ in $G$ and turns into the edge between $v_2$ and $w_2$ in $G'$.
The last subpath to consider is the $(v_1,v_2)$-subpath of $P'$.
It goes between vertices in $M$ and contains at least one edge outside $M$; hence it should contain at least one vertex in $V(P_1)\cup V(P_2)$.
Let $u$ be the first vertex on this subpath that is from $V(P_1)\cup V(P_2)$.
Then either the $(v_1,u)$-subpath or the $(u,v_2)$-subpath contains an edge of $M$.
First, suppose that the $(v_1,u)$-subpath contains an edge of $M$.
Denote by $v_3$ the last vertex from $V(M)$ on this subpath.
Then $v_3\neq v_1$ and the $(v_3,u)$-subpath contains only vertices in $V(H)\setminus V(M)$ as internal vertices.
Hence, in this case there is an edge between $v_3$ and $w_3=u$ in $G'$.
Now for the case when the $(u,v_2)$-subpath contains an edge of $M$.
Denote by $v_3$ the first vertex on this subpath that is from $M$.
Then $v_3\neq v_2$ and the $(u,v_3)$-subpath does not contain vertices of $M$ as internal vertices.
Denote by $w_3$ the last vertex in $V(P_1)\cup V(P_2)$ on this subpath.
We obtain a path between $w_3$ and $v_3$ that goes only through $V(H)\setminus V(M)$ in $G$, so there is an edge between $w_3$ and $v_3$ in $G'$.
\noindent\textbf{Conclusion of Case 1.}
If $M$ is an Erd{\H {o}}s-Gallai component\xspace corresponding to a connected component of type \ref{enum:tunnel_path_bic}, then there is a matching $v_1w_1, v_2w_2, v_3w_3$ of size three between $V(M)$ and $V(P_1)\cup V(P_2)$ in $G'$.
Hence, there is a matching between $V(M)$ and $V(P_i)$ of size two for some $i\in\{1,2\}$.
This contradicts to the corresponding condition \ref{enum:tunnel_path_bic} of {Erd{\H {o}}s-Gallai decomposition\xspace}s; hence Case~1 cannot occur.
\noindent\textbf{Case 2.}
Now suppose that there is a type \ref{enum:tunnel_path_cut_left} connected component $H$ in $G-(V(P_1)\cup V(P_2))$ such that $M$ is a leaf-block of the component obtained after some edges of $H$ were contracted in the process of $B$-refinement $\gbref{B}{H}$.
Denote
the cut-vertex of this leaf-block $M$ by $c$. We will refer to all remaining vertices of $M$ as to \emph{inner} vertices.
By the definition of \ref{enum:tunnel_path_cut_left}-type components, $N_{G'}(V(M))\cap V(P_1)=\{w\}$ for some $w \in V(P_1)$.
Again consider the $(s,v_1)$-subpath, the $(v_1, v_2)$-subpath, and the $(v_2,t)$-subpath of $P'$.
The $(v_1, v_2)$-subpath contains a vertex $u \in V(P_1)\cup V(P_2)$ as internal vertex, so we can also break it into $(v_1,u)$-subpath and $(u,v_2)$-subpath.
Note that at least two of these four subpaths do not contain $c$.
Each of these subpaths is an $(x,y)$-path for $x\in V(P_1)\cup V(P_2)$ and $y\in V(H)$.
We claim that if such $(x,y)$-path does not contain $c$, then it contains $w$.
Suppose that an $(x,y)$-path does not contain $c$, so $y$ is an inner vertex of $M$.
This path does not contain $c$, and to reach $y$ it should reach some inner vertex of $M$ from the outside, since the path starts in $V(P_1)\cup V(P_2)$.
Hence, this path should contain $w$. Otherwise there is an edge between $V(P_2)$ and some inner vertex of $M$ in $G'$, which contradicts the property \ref{enum:tunnel_path_cut_left}.
Thus at least two of the four subpaths contain $w$.
The only possible option for this is when $u=w$ and both $(v_1,u)$-subpath and $(u,v_2)$-subpath do not contain $c$.
Then both $(s,v_1)$-subpath and $(v_2,t)$-subpath do not contain $w$, since $w$ can appear only once in $P'$.
Both of them reach an inner vertex of $M$ from the outside of $M$.
If a path reaches an inner vertex of $M$ and avoids $w$, then it should contain the cut-vertex $c$.
Therefore, the $(s,v_1)$-subpath and the $(v_2,t)$-subpath both contain $c$.
This is contradiction, since these two paths are vertex-disjoint.
\noindent\textbf{Conclusion of Case 2.}
If $M$ is an Erd{\H {o}}s-Gallai component\xspace corresponding to a connected component of type \ref{enum:tunnel_path_cut_left}, then $P'$ necessarily contains an edge between $V(P_2)$ and an inner vertex of $M$.
This contradicts the definition of \ref{enum:cycle_tunnel_path_cut_left}-type components.
\noindent\textbf{Case 3.} The case when $M$ is an Erd{\H {o}}s-Gallai component\xspace of type \ref{enum:tunnel_path_cut_right} is symmetrical.
In each of the three cases we obtained a contradiction with one of the properties of an Erd{\H {o}}s-Gallai decomposition\xspace. This completes the proof. \end{proof}
In order to proceed further with the structural properties of Erd{\H {o}}s-Gallai decomposition\xspace{s}, we need the following definition and lemma.
\begin{definition}[\textbf{$B$-leaf-block separator}]\label{definition:b_leaf_block_sep}
Let $H$ be a connected graph that is not $2$-connected and $B$ be a subset of its vertices.
Let $I$ be the set of inner vertices of all leaf-blocks of $H$.
We say that $S \subseteq V(H)\setminus I$ is a \emph{$B$-leaf-block separator} of $H$, if $S$ separates at least one vertex in $V(H)\setminus (I\cup B)$ from $I$ in $H$. \end{definition} \begin{lemma}\label{lemma:separator_in_non_2c}
Let $H$ be a connected graph with at least one cut-vertex and let $B$ be a subset of its vertices.
Let $S$ be a $B$-leaf-block separator of $H$.
Then for any vertex $v$ that is not an inner vertex of a leaf-block of $H$, there is a cut-vertex $c$ of a leaf-block of $H$ and a $(c,v)$-path of length at least $\frac{1}{2}\left(\delta(H-B)-|S|\right)$ in $H$. \end{lemma} \begin{proof}
We assume that $\delta(H-B)> |S|$, since the other case is trivial.
Consider graph $H-(B\cup S)$.
We know that there is at least one connected component in this graph that does not contain any vertex from $I$ and contains at least one vertex not in $B$.
Denote this connected component by $T$.
We know that $\delta(T)\ge \delta(H-(B\cup S))>1$.
By \Cref{prop:cycle_delta}, $T$ contains a cycle $C$ of length at least $\delta(T)+1$.
We know that $C$ is fully contained in some non-leaf-block of $H$.
Denote this block by $K$.
Now let $v$ be a vertex in $V(H)\setminus I$ given from the lemma statement.
It is easy to see that we can always choose the vertex $c$ in a way that any $(c,v)$-path contains at least one edge of $K$.
Take such vertex and an arbitrary $(c,v)$-path.
Edges of $K$ induce a subpath of non-zero length in this path.
Let $x,y$ be the endpoints of this subpath.
We know that $x\neq y$.
We need the following claim.
\begin{claim}\label{lemma:biconnected_cycle_to_any_path}
If a $2$-connected graph contains a cycle on $k$ vertices, then it contains a path of length at least $\lceil \frac{k}{2} \rceil$ between any pair of vertices. \end{claim}
\noindent \emph{Proof of Claim~\ref{lemma:biconnected_cycle_to_any_path}.}
Take two distinct vertices $s$, $t$.
To show that there is a path between $s$ and $t$ of length at least $\lceil \frac{k}{2} \rceil$, we apply Menger's theorem to $\{s,t\}$ and the vertex set of the cycle of length $k$.
This gives two vertex-disjoint paths going from $s$ and $t$ to two vertices $s'$ and $t'$ on the cycle.
Take the longer arc between $s'$ and $t'$ on the cycle and combine it with the two paths. The resulting path is of length at least $\lceil \frac{k}{2} \rceil$.
\claimqed
By Claim~\ref{lemma:biconnected_cycle_to_any_path}, there is a path of length at least $\delta(T)/2>\frac{1}{2}(\delta(H-B)-|S|)$ between $x$ and $y$ in $K$.
Replace the subpath of the initial $(c,v)$-path with this subpath.
This yields a $(c,v)$-path of desired length. \end{proof}
In Lemma~\ref{lemma:st_path_banana_consecutive}, we proved that if a path enters an Erd{\H {o}}s-Gallai component\xspace, then after leaving it, it cannot come back. The following lemma guarantees, that if we have a yes-instance, then there is a solution path that enters at least one Erd{\H {o}}s-Gallai component\xspace. \begin{lemma}\label{lemma:st_path_edge_of_banana}
Let $G$ be a graph, $B\subseteq V(G)$ be a subset of its vertices and $P_1, P_2$ induce an Erd{\H {o}}s-Gallai decomposition\xspace for an $(s,t)$-path $P$ in $G$ of length less than $\delta(G-B)+k$.
Let $k$ be an integer such that $5k+4|B|+6 < \delta(G-B)$.
If there exists an $(s,t)$-path of length at least $\delta(G-B)+k$ in $G$, then there exists $(s,t)$-path of length at least $\delta(G-B)+k$ in $G$ that enters an Erd{\H {o}}s-Gallai component\xspace. \end{lemma} \begin{proof}
Since the length of $P$ is less than $\delta(G-B)+k$, we may assume that $|V(P_1)\cup V(P_2)|<k+2$.
Assume that there is an $(s,t)$-path $P'$ of length at least $\delta(G-B)+k$ in $G$ that contains no edge of an Erd{\H {o}}s-Gallai component\xspace.
We show that there exists an $(s,t)$-path of length at least $\delta(G-B)+k$ that enters some Erd{\H {o}}s-Gallai component\xspace.
The path $P'$ path can contain only edges with endpoints in $V(P_1)\cup V(P_2) \cup B$ or edges of non-leaf-blocks of \ref{enum:tunnel_path_cut_left}- or \ref{enum:tunnel_path_cut_right}-type components.
All other edges are edges of Erd{\H {o}}s-Gallai component\xspace{s}.
There are at most $2|V(P_1)\cup V(P_2)\cup B|\le 2(k+2+|B|)$ edges in $P'$ that have endpoints in the corresponding set.
Hence, $P'$ contains at least $\delta(G-B)+k-2(k+2+|B|)=\delta(G-B)-k-4-2|B|$ edges that lie inside non-leaf-blocks of separable components of the Erd{\H {o}}s-Gallai decomposition\xspace .
Let $u$ be the vertex on $P'$ such that the $(s,u)$-subpath of $P'$ is of length exactly $k+1$.
Denote this subpath by $P'_1$.
Analogously, let $v$ be the vertex on $P'$ such that the $(v,t)$-subpath of $P'$ is of length exactly $k$, and denote this subpath by $P'_2$.
Note that $P'_1$ and $P'_2$ are on a distance at least $\delta(G-B)-k>0$ from each other on $P'$.
The $(u,v)$-subpath of $P'$ consists of at least $(\delta(G-B)-k-4-2|B|)-2k> 2|B|$ edges of the non-leaf-blocks.
Hence, at least one non-leaf-block edge in $P'$ is not incident to any vertex in $B$.
Let $H$ be the connected component in $G-(V(P_1)\cup V(P_2))$ that contains this edge and let $H'$ be its $B$-refinement.
The graph $H'$ contains at least one edge of the $(u,v)$-subpath of $P'$, and none of these edges are incident to an inner vertex of its leaf-blocks.
We note that the whole path $P'$ cannot go through any inner leaf-block vertex of $H'$.
Suppose that this is not true and it contains such vertex.
Since it does not contain any leaf-block edge, this path should enter and leave this inner vertex from the outside of $H$.
And the only way to enter a \ref{enum:tunnel_path_cut_left}-type or a \ref{enum:tunnel_path_cut_right}-type component of the Erd{\H {o}}s-Gallai decomposition\xspace is to go from the only vertex of $V(P_1)$ or of $V(P_2)$ correspondingly.
Thus, this vertex of either $V(P_1)$ or $V(P_2)$ is contained twice on the path, and that is not possible.
Consider now the graph $H'-(V(P'_1)\cup V(P'_2))$.
\textbf{Case 1.}
Suppose that there is a connected component in this graph that contains an inner vertex of a leaf-block of $H'$ and some vertex of the $(u,v)$-subpath of $P'$ simultaneously.
Denote this leaf-block by $L$ and the vertex of the $(u,v)$-subpath by $w$.
Note that all paths between $w$ and vertices of $L$ go through the cut-vertex of $L$.
If there are multiple choices of $w$ for $L$, choose the one which is the closest to the cut-vertex of $L$.
As $P'$ does not contain any inner leaf-block vertex of $H'$, the connected component of $w$ in $H'-(V(P'_1)\cup V(P'_2))$ contains the whole leaf-block $L$.
Hence, there is a path connecting $w$ with any inner vertex of $L$.
Choose any inner vertex of $L$ that is connected to $V(P_1)$ (if $H'$ is \ref{enum:tunnel_path_cut_left}-type) or to $V(P_2)$ (if $H'$ is \ref{enum:tunnel_path_cut_right}-type).
Denote this vertex by $z$.
Since $L$ is a Erd{\H {o}}s-Gallai component\xspace, there is a path in $L$ of length at least $\delta(G-B)-2$ connecting $z$ with the cut-vertex of $L$, so there is a $(w,z)$-path of length at least $\delta(G-B)-2$ in $H'$.
Note that the only common vertex of this path and $P'$ is the vertex $w$.
We also know that the $(s,w)$-subpath and the $(w,t)$-subpath of $P'$ are of length at least $k+1$, since $w$ is not in $V(P'_1)\cup V(P'_2)$.
Now prolong the $(w,z)$-path in $G$ by going outside $H'$ from $z$ to the vertex from $V(P_1)$ or $V(P_2)$ depending on the type of $H$, and finally go from this vertex to $t$ following the initial path $P$.
We obtain a $(w,t)$-path $Q$ that has at least two common vertices with $P'$.
\begin{figure}\label{fig:lemma_banana_edge}
\end{figure}
Denote by $x$ the second vertex on this $(w,t)$-path that is common with $P'$.
Denote the $(w,x)$-subpath of $Q$ by $Q'$.
The path $Q'$ is of length at least $\delta(G-B)-1$ and contains at least one Erd{\H {o}}s-Gallai component\xspace edge, since it contains the $(w,z)$-path as a proper subpath.
Suppose that $x$ is a part of the $(s,w)$-subpath of $P'$.
Then consider constructing the following path (see \Cref{fig:lemma_banana_edge}).
Take the $(s,x)$-subpath of $P'$, then go following the path $Q'$ from $x$ to $w$, and finish with the $(w,t)$-subpath of $P'$.
The constructed path is of length at least $0+(\delta(G-B)-1)+(k+1)\ge \delta(G-B)+k$, so we are done.
If $x$ is not a part of the $(s,w)$-subpath of $P'$, then it is a part of the $(w,t)$-subpath of $P'$.
Then the required path is combined of the $(s,w)$-subpath of $P'$, then of $Q'$ and of the $(x,t)$-subpath of $P'$.
Its total length is at least $(k+1)+(\delta(G-B)-1)+0\ge \delta(G-B)+k$.
\textbf{Case 2.}
It is left to consider the case when $V(P'_1)\cup V(P'_2)$ separates all inner leaf-block vertices from all vertices of the $(u,v)$-subpath of $P'$ that are from $V(H')$.
Note that at least one vertex of the $(u,v)$-subpath is from $V(H')\setminus B$.
Denote the set of all inner leaf-block vertices in $H'$ by $I$.
We know that $V(P'_1)\cup V(P'_2)$ separates at least one vertex in $V(H')\setminus B$ from $I$.
Apply Lemma~\ref{lemma:separator_in_non_2c} to $H'$ and $S=V(P'_1)\cup V(P'_2)$.
Suppose that $H'$ is of type \ref{enum:tunnel_path_cut_left}.
Then take any vertex in $V(H')$ that is connected with $V(P_2)$ by an edge (after the edge contractions).
Denote this vertex by $y$.
We know that $y$ is not in $I$, so there is a $(c,y)$-path of length at least $\frac{1}{2}(\delta(H'-B)-|S|)$ in $H'$ for cut-vertex $c$ of some leaf-block $L$ in $H'$.
This leaf-block has at least one inner vertex that is connected to $V(P_1)$ by an edge.
Denote such vertex by $x$.
There is a path of length at least $\delta(G-B)-3$ between $x$ and $c$ inside $L$.
Combine this $(x,c)$-path with the $(c,v)$-path and obtain a path of length at least $1+(\delta(G-B)-3)+(\frac{1}{2}(\delta(H'-B)-|S|))+1$ between $V(P_1)$ and $V(P_2)$.
This path does not intersect internally with $P_1$ or $P_2$.
Hence, there is an $(s,t)$-path in $G$ of length at least $\delta(G-B)+\frac{1}{2}\delta(H'-B)-\frac{1}{2}|S|-1$.
We know that $\delta(H'-B)\ge \delta(G-(B\cup V(P_1)\cup V(P_2)))\ge \delta(G-B)-k-1$.
Thus, our $(s,t)$-path is of length at least $\delta(G-B)+\frac{1}{2}((\delta(G-B)-k-1)-(2(k+1))-2)=\delta(G-B)+\frac{1}{2}(\delta(G-B)-3k-5)\ge \delta(G-B)+k$.
This path contains an edge of the leaf-block $L$, which is an Erd{\H {o}}s-Gallai component\xspace edge, so we are done.
When $H'$ is of type \ref{enum:tunnel_path_cut_right}, the proof is symmetrical.
The proof of the lemma is complete. \end{proof}
We are now ready to formulate a very crucial lemma of this section. It serves as a basic tool for applying recursion in Erd{\H {o}}s-Gallai decomposition\xspace in the algorithm for \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace. Basically, it provides a way to search for a long part of the $(s,t)$-path inside an Erd{\H {o}}s-Gallai component\xspace wrapped up in a $2$-connected subgraph of $G$.
\begin{lemma}\label{lemma:st_path_banana_to_2_connected}
Let paths $P_1,P_2$ induce an Erd{\H {o}}s-Gallai decomposition\xspace for an $(s,t)$-path $P$ and $B\subseteq V(G)$ in graph $G$.
Let $M$ be an Erd{\H {o}}s-Gallai component\xspace in $G$. Then there is a polynomial time algorithm that outputs a $2$-connected subgraph $K$ of $G$ and two vertices $s', t' \in V(K)$, such that every
$(s,t)$-path $P'$ in $G$ that enters $M$, the following hold
\begin{enumerate}
\item $V(K)\setminus B=(V(M)\cup\{s',t'\})\setminus B$;
\item $P'[V(K)]$ is an $(s',t')$-subpath of $P'$ and an $(s',t')$-path in $K$;
\item $\delta(K-(B\cup\{s',t'\})) \ge \delta(G-(B\cup\{s',t'\}))$;
\end{enumerate} \end{lemma}
\begin{proof}
We consider several cases depending on the type of the connected component $H$ of $G-(V(P_1)\cup V(P_2))$ that contains $M$.
We start with the simpler case, when $H'=\gbref{H}{B}$ is separable.
By $G'$ we as usual denote the graph $G$ where all edges corresponding to $B$-refinements of the Erd{\H {o}}s-Gallai decomposition\xspace induced by $P_1,P_2$ are applied.
\noindent\textbf{Case 1}.
The component $H$ is of type \ref{enum:tunnel_path_cut_left} (type \ref{enum:tunnel_path_cut_right} is symmetrical as ususal).
Then $M$ is some leaf-block of $H'$.
If an $(s,t)$-path $P'$ enters $M$, then $P'[V(M)\cap V(P')]$ is a path in $L$ by Lemma~\ref{lemma:st_path_banana_consecutive}.
Moreover, we know that this path starts in an inner vertex of $M$ and ends in the cut-vertex of $M$.
Denote these two vertices by $v$ and $c$ respectively.
Also, $N_G(H)=\{w\}$ for some $w\in V(P_1)$, and by definition of \ref{enum:tunnel_path_cut_left}-type connected components, $P'$ contains a $(w,v)$-subpath going (in either direction) internally only through vertices in $V(H)\cap B$.
There are two cases of how $K$ should be constructed.
If there is a single and the only inner vertex $v \in V(M)$ such that there is an edge between $w$ and $v$ in $G'$, then any path $P'$ that enters $M$ contains a path between $v$ and $c$ as a subpath.
Thus, put $K:=M$ and $s':=v$, $t':=c$.
Clearly, $K,s',t'$ satisfy all three conditions in the lemma statement.
The other case is when there are at least two inner vertices in $M$ that are neighbors to $w$ in $G'$.
We cannot put $s'$ equal to any vertex of $M$, because we cannot be sure that $P'$ passes through a concrete inner vertex.
But we are sure that $P'$ passes through $w$.
Construct $K$ in the following way.
Denote by $B'$ the set of vertices in $B$ that are reachable from $V(M)\setminus\{c\}$ in $H-\{c\}$.
Then put $K:={G[V(M)\cup B' \cup \{w\}]}$, $s'=w$, $t'=c$.
Note that $K$ is an induced subgraph of $G$ and is $2$-connected as $G'[V(M)\cup \{w\}]$ is $2$-connected.
The first and the last two conditions in the lemma statement are satisfied, and we claim that the second one is satisfied as well.
We already know that $P'$ contains a $(w,c)$-subpath.
This subpath goes from $w$ to an inner vertex of $M$ through the vertices in $B'$, and then follows a path inside $M$.
Hence, this subpath is contained in $K$.
It is left to show that no vertex from $V(M)\cup B'\cup \{w\}$ can appear in $P'$ outside of the $(w,c)$-subpath.
We do it by contradiction.
Assume that there is such vertex $v \in V(M)\cup B' \cup \{w\}$.
If $v \in V(M)$, then $v \neq c$, hence $v$ is an inner vertex of $M$.
Then $P'$ should contain a $(s,v)$-subpath or a $(v,t)$-subpath that does not go through $w$ nor $c$, but $\{w,c\}$ separates $V(M)$ from $V(P_1)\cup V(P_2)$.
Thus, $v \in B'$.
Then there exists either $(s,v)$-subpath or $(v,t)$-subpath in $P'$ that does not contain $w$ and any vertex from $V(M)$.
Hence, this subpath connects $v$ with some vertex $u \in V(P_2)$ and goes only through $B'$.
We know that after the edge contractions for $G'$ the vertex $v$ becomes identified with an inner vertex of $M$, so there is an edge between $u$ and this inner vertex.
This is not possible by the definition of type \ref{enum:tunnel_path_cut_left} connected components.
We obtain a contradiction.
\noindent\textbf{Case 2.}
$H$ is of type \ref{enum:tunnel_path_bic}, so $H'$ is $2$-connected and $M=H'$.
We know that the maximum matching size between $V(P_i)$ and $V(M)$ in $G'$ is exactly one for each $i\in\{1,2\}$.
For each $i$, it splits into two possible options: either $|N_{G'}(V(P_i))\cap V(M)|=1$ or $|N_{G'}(V(M))\cap V(P_i)|=\{w_i\}$, where $w_i$ has at least two neighbors in $V(M)$ in $G'$.
We now consider several subcases of Case 2 depending on the combinations of these options.
If for each $i\in \{1,2\}$, $|N_{G'}(V(P_i))\cap V(M)|=1=\{v_i\}$ for some $v_i\in V(M)$, then, an $(s,t)$-path $P'$ can enter or leave $M$ only through the vertices $v_1$ and $v_2$.
Note that $v_1\neq v_2$, since $\{v_1,v_2\}$ separates $V(M)$ from the rest of the graph in $G$.
Thus, if $P'$ enters $M$, then it necessarily contains a $(v_1, v_2)$-subpath inside $M$.
By Lemma~\ref{lemma:st_path_banana_consecutive}, we have that $P'[V(P')\cap V(M)]$ is exactly the $(v_1, v_2)$-subpath inside $M$.
Thus, it is enough to put $K:=M$ and $s':=v_1$, $t':=v_2$.
The first two and the last conditions of the lemma are satisfied for this choice of $K, s'$ and $t'$.
Also, no vertex in $V(M)\setminus \{v_1,v_2\}$ has neighbors outside $V(H)$ in $G$, so $\delta(K-(B\cup \{s',t'\}))=\delta(M- \{v_1,v_2\}])\ge \delta(G-(B\cup\{s',t'\}))$, and the third condition is also satisfied.
The other case is when for each $i \in \{1,2\}$, $|N_{G'}(V(M))\cap V(P_i)|=\{w_i\}$, where $w_i$ has at least two neighbors in $V(M)$ in $G'$.
It is easy to see that to enter or leave any vertex of $H$ in $G$, an $(s,t)$-path $P'$ should go through $w_1$ and $w_2$.
Since $G$ is $2$-connected, $w_1\neq w_2$ and $P'$ contains a $(w_1, w_2)$-subpath going internally only through vertices in $V(H)$.
Put $K:=G[V(H)\cup \{w_1,w_2\}]$, $s':=w_1$, $t':=w_2$.
Clearly, $K$ is $2$-connected because $G$ is $2$-connected, $\{w_1,w_2\}$ separates $V(H)$ from the rest of $G$, and degrees of $w_1$ and $w_2$ in $K$ are at least two.
We need to show that the second condition is satisfied as well.
If it is not satisfied, then $P'[V(K)]$ consists of at least two disjoint paths.
We know that one of these paths is the $(w_1,w_2)$-subpath.
Hence, the other one contains at least one vertex from $V(H)$ but does not contain $w_1$ or $w_2$.
This is not possible since $\{w_1,w_2\}$ separates $V(H)$ from the rest of the graph.
Thus, the first two and the last condition are satisfied.
It is easy to see that the third condition is satisfied as well, because vertices in $V(H)$ have no outside neighbors apart from $s'$ and $t'$ in $G$.
It is left to consider the case when $N_{G'}(V(M))\cap V(P_1)=\{w_1\}$, where $w_1$ has at least two neighbors in $V(M)$ in $G'$, and $N_{G'}(V(P_2))\cap V(M)=\{v_2\}$ (the case when $1$ and $2$ are interchanged is symmetrical).
This is the most non-clear case.
We know that if $P'$ enters $M$, then it should pass through both $w_1$ and $v_2$.
Moreover, the $(w_1,v_2)$-subpath of $P'$ goes internally only through $V(H)$.
Let $B'$ be the set of vertices reachable from $v_2$ by the edges in $E(H)\setminus E(H')$.
The difficulty beyond choosing $K$ in this case is to satisfy the second condition.
We split on two cases.
Assume that there is no edge between $w_1$ and $v_2$ in $G'$.
Then put $K:=G[(V(H)\cup \{w_1\})\setminus B']$, $s':=w_1$ and $t':=v_2$.
Clearly, $K$ is equal to $G[V(H)\cup \{w_1\}]$ with applied $B$-refinements so it is $2$-connected.
The first, the third and the fourth condition of the lemma are satisfied by the arguments similar to the cases considered above.
It is left to show that the second condition is satisfied.
Suppose that $P'[V(K)]$ contains a path different from the $(w_1,v_2)$-subpath.
Then there is at least one vertex $u \in V(K)\setminus V(M)$ that is not on this subpath.
Then $P'$ should contain either an $(s,u)$-subpath or an $(t,u)$-subpath that does not go through $w_1$ or $v_2$.
Moreover, this subpath does not contain any vertex of $V(M)$ by Lemma~\ref{lemma:st_path_banana_consecutive}.
Denote by $x$ the last vertex on this supbath that is not from $V(K)$.
Then $P'$ contains an $(x,u)$-subpath, where $x \in V(P_1)\cup V(P_2)$
After the $B$-refinement of $H$, this path yields an edge in $G'$ between $x$ and $y$ for some $y \in V(M)$.
This is only possible when either $x=w_1$ or $y=v_2$.
The case $x=w_1$ is not possible because the $(x,u)$-subpath does not contain $w_1$.
Hence, it should be the case that $y=v_2$.
Then $u$ is a vertex reachable from $v_2$ in $H$ outside $V(M)$.
That is, $u \in B'$.
Hence, $u \notin V(K)$.
We obtain a contradiction, so all four conditions are satisfied.
It is left to consider the case when there is an edge between $w_1$ and $v_2$ in $G'$.
It is clear that in this case the graph $G[V(H)\cup \{w_1\}]$ is $2$-connected.
Unfortunately, we cannot put $K$ equal to this graph because this might break the second condition of the lemma.
We already know, however, that the graph $K:=G[(V(H)\cup \{w_1\})\setminus B']$ with $s':=w_1$ and $t':=v_2$ would satisfy all conditions of the lemma except, possibly the first.
Thus, there are two cases.
When the graph $G[(V(H)\cup \{w_1\})\setminus B']$ is $2$-connected, then consider $K,s'$ and $t'$ similarly to the case when there is no edge between $w_1$ and $v_2$.
Otherwise, the graph $G[(V(H) \cup \{w_1\})\setminus B']$ is not $2$-connected.
Then $w_1$ is connected to exactly two vertices from $V(M)$ in $G'$.
One of these two vertices is $v_2$.
The other one we denote by $v_1$.
Then $P'$ necessarily contains a $(v_1,v_2)$-subpath inside $M$.
Then it is sufficient to put $K:=M$, $s':=v_1$, $t':=v_2$, as $M$ is $2$-connected.
The proof is complete. \end{proof}
\subsection{Algorithm for \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace}\label{subsec:algorithmstpath}
We are almost set to proceed with the proof of Theorem~\ref{thmEG}. The algorithm is based on Lemmata
~\ref{lemma:st_path_or_tunnel}, \ref{lemma:st_path_banana_consecutive}, \ref{lemma:separator_in_non_2c}, and \ref {lemma:st_path_banana_to_2_connected} on properties of Erd{\H {o}}s-Gallai decomposition\xspace{s}. For the proof of the correctness of the algorithm, we will need one more lemma.
\begin{lemma}\label{lemma:almost_ham_path}
Let $G$ be a $2$-connected graph with $B\subseteq V(G)$ such that $\frac{6}{5}\delta(G-B)\ge |V(G)|$ and $\delta(G-B)\ge 4|B|$.
Then for any pair of distinct vertices $s,t\in V(G)$, the longest $(s,t)$-path in $G$ contains all vertices from $V(G-B)$. \end{lemma} \begin{proof}
The proof is by contradiction.
Suppose that there is an $(s,t)$-path $P$ in $G$ such that the length of $P$ is maximum possible, but there is $v \in V(G)\setminus B$ with $v \notin V(P)$.
By Corollary~\ref{thm:relaxed_st_path}, the length of $P$ is at least $\delta(G-B)$.
Hence, $|V(P-B)|> \delta(G-B)-|B|$.
Then $v$ has at most $|V(G-B)|-|V(P-B)|<\frac{1}{5}\delta(G-B)+|B|$ neighbors outside $V(P)$.
Hence, $v$ has more than $\frac{4}{5}\delta(G-B)-|B|$ neighbors from $V(P)$.
Note that $v$ should not have any two consecutive vertices in $P$ as neighbors, otherwise $P$ can be made longer.
Hence, $2(\frac{4}{5}\delta(G-B)-|B|)<|V(P)|$.
Equivalently, $|V(P)|>|V(G)|+\frac{3}{5}\delta(G-B)-2|B|\ge |V(G)|$.
This is a contradiction. \end{proof}
For reader's convenience, we restate Theorem~\ref{thmEG} here.
\noindent\textbf{Theorem~\ref{thmEG}.} \emph{\pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace is solvable in $2^{\mathcal{O}(k+|B|)}\cdot n^{\mathcal{O}(1)}$ running time on $2$-connected graphs.}
\begin{proof}
The recursive algorithm is presented in Algorithm~\ref{alg:long_eg_st_path}.
Note that this algorithm requires that $s,t\in B$ in the given input instance.
Any instance can be reduced to instance with this restriction by adding $s,t$ into $B$ and increasing $k$ by at most two.
This changes the parameters by a constant value and does not significantly affect the running time of the algorithm.
Also, this algorithm does not just determine whether the given instance is a yes-instance.
If the given instance is a no-instance, the algorithm also outputs the maximum length of an $(s,t)$-path in $G$ in the form $\delta(G-B)+x$, where $x\ge 0$ and $x < k$.
Note that algorithm actually also finds a path of such length, and it possible to change it so the path is in the output of the algorithm.
We now go through the lines of the algorithm to explain its correctness.
\SetNlSty{}{}{}
\let\oldnl\nl
\newcommand\nonl{
\renewcommand{\nl}{\let\nl\oldnl}}
\begin{algorithm}[!h]
\SetKwFunction{LongestPath}{longest\_path}
\SetKwFunction{LongSTPath}{long\_st\_path}
\SetKwFunction{LongErdosSTPath}{long\_eg\_st\_path}
\SetKwFunction{LongSTCycle}{long\_st\_cycle}
\SetKwFunction{HamPath}{hamiltonian\_path}
\Indm\nonl\LongErdosSTPath{$G,B,s,t,k$}
\Indp
\KwIn{an instance $(G,B,s,t,k)$ of \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace, where $G$ is $2$-connected and $s,t\in B$}
\KwResult{$k$, if $(G,B,s,t,k)$ is a yes-instance, or an integer $x$, such that the maximum length of an $(s,t)$-path in $G$ is $\delta(G-B)+x$.}
\If{$k=0$}{
\Return{$k$}\;
}
\If{$5k+5|B|+6\ge\delta(G-B)$}{
$x\longleftarrow k$\;
\While{\LongSTPath$(G,s,t,\delta(G-B)+x)$ is \textsc{No}}{
$x \longleftarrow x-1$\;
}
\Return $x$\;
}
\uIf{the algorithm of Lemma~\ref{lemma:st_path_or_tunnel} applied to $G,B,s,t,k$ returns $P$ with $P_1, P_2$}{
$x\longleftarrow 0$\;
\ForEach{Erd{\H {o}}s-Gallai component\xspace $M$ of the Erd{\H {o}}s-Gallai decomposition\xspace of $(G,B)$ induced by $P_1, P_2$\label{alg:line:st_path_bananas}}{
$K,s',t' \longleftarrow $ result of Lemma~\ref{lemma:st_path_banana_to_2_connected} applied to $G,B,P_1,P_2$ and $M$\;
$x'\longleftarrow$ \LongErdosSTPath $(K,B\cup\{s',t'\},s',t',k)$\;
$H \longleftarrow (V(G)\cup \{a,b\}, (E(G)\setminus E(G[T])) \cup \{as,at,bs', bt'\})$\;
$r \longleftarrow \max\{(\delta(G-B)+k)-(\delta(K-(B\cup \{s',t'\}))+x'),0\}$\;
\While{\LongSTCycle$(H,a,b,r+4)$ is \textsc{No}}{$r \longleftarrow r-1$\;}
$x \longleftarrow \max\{x, (\delta(K-(B\cup \{s',t'\}))+x'+r)-\delta(G-B)\}$\;
}
\Return $x$\;
}\uElseIf{it returns $P$ with $V(P)\cup B=V(G)$\label{alg:line:ham_path}}{
\ForEach{$B'\subseteq B\setminus\{s,t\}$}{
$H \longleftarrow (V(G-B')\cup \{s',t'\}, E(G-B')\cup \{s's, tt'\})$\;
\If{\HamPath$(H)$}{
$x \longleftarrow \max\{x, |V(H)|-1-\delta(G-B)\}$\;
}
}
\Return $\min\{x,k\}$\;
}\Else{
\Return {$k$}\;
}
\caption{Recursive algorithm solving \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace on $2$-connected graphs.}
\label{alg:long_eg_st_path}
\end{algorithm}
The first two conditional operators handle the most trivial cases of the problem.
The first conditional operator is for the case $k=0$, which corresponds to trivial yes-instances by Corollary~\ref{thm:relaxed_st_path}.
The second operator ensures that parameters $k$ and $|B|$ are small enough compared to $\delta(G-B)$ to apply results discussed earlier in this section.
If they are not, the algorithm just employs the algorithm from \Cref{prop:longest_cycle}
for \textsc{Long $(s,t)$-Path}, which works in $2^{\mathcal{O}(\delta(G-B)+k)}\cdotn^{\mathcal{O}(1)}=2^{\mathcal{O}(k+|B|)}\cdotn^{\mathcal{O}(1)}$.
When the third conditional operator is reached, Lemma~\ref{lemma:st_path_or_tunnel} can indeed be applied to the input instance.
Thus, in polynomial time either an $(s,t)$-path of length at least $\delta(G-B)+k$ is found, or an $(s,t)$-path $P$ with $V(P)\cup B=V(G)$ is found, or an $(s,t)$-path $P$ and two paths $P_1, P_2$ are found.
The paths $P_1$ and $P_2$ induce an Erd{\H {o}}s-Gallai decomposition\xspace for $P$ in $(G,B)$.
If the path of length at least $\delta(G-B)+k$ is found, our algorithm correctly decides that the given instance is a yes-instance and stops.
Otherwise, it enters the third conditional operator body.
The conditional operator in line \ref{alg:line:ham_path} checks that we should deal with the case covered by Lemma~\ref{lemma:almost_ham_path}.
We shall now explain this in detail.
Suppose that we enter the conditional operator body, i.e., \ $V(P)\cup B=V(G)$.
Since the length of $P$ is at most $\delta(G-B)+k-1$, we get that $\delta(G-B)+k+|B|\ge |V(G)|$.
Since this operator can be reached only if $5(k+|B|)\le |V(G)|$, we can apply Lemma~\ref{lemma:almost_ham_path} to our instance.
We can now look for an $(s,t)$-path that contains all vertices in $V(G-B)$.
Clearly, any such path is a hamiltonian path in the graph $G-B'$ for some $B'\subseteq B$ with $s,t\notin B'$.
To achieve that any hamiltonian path in $G-B'$ corresponds to an $(s,t)$-path, we add two additional vertices $s', t'$ of degree one to obtain the graph $H$.
Moreover, in the graph $H$ all vertices have degree at least $\delta(G-B)$, except, probably, at most $|B|+2$ vertices.
We know that $2\delta(G-B)>|V(H)|$, so we can apply one of the two FPT-algorithms from \Cref{theorem:JansenKN}
for solving \textsc{Hamiltonian Path} in $H$.
This algorithm runs in $2^{\mathcal{O}(|B|+2)}\cdotn^{\mathcal{O}(1)}$ time.
Thus, the longest path in $G-B$ is found by the algorithm for the correct choice of $B'$.
We now move to the most crucial part of the algorithm.
This part deals with Erd{\H {o}}s-Gallai component\xspace{s} of the Erd{\H {o}}s-Gallai decomposition\xspace induced by $P_1$ and $P_2$.
We note that when line~\ref{alg:line:st_path_bananas} of the algorithm is reached, there are at least two distinct Erd{\H {o}}s-Gallai component\xspace{s} of the Erd{\H {o}}s-Gallai decomposition\xspace of $(G,B)$ by definition.
By Lemma~\ref{lemma:st_path_edge_of_banana} and Lemma~\ref{lemma:st_path_banana_to_2_connected}, if the given instance is a yes-instance, there is an Erd{\H {o}}s-Gallai component\xspace that contains a long subpath of the desired path.
Let $M$ be an Erd{\H {o}}s-Gallai component\xspace fixed by the foreach cycle.
Lemma~\ref{lemma:st_path_banana_to_2_connected} applied to this Erd{\H {o}}s-Gallai component\xspace yields a triple $K,s',t'$.
The following lines of the algorithm focus on finding maximum $x'$ such that there is an $(s',t')$-path in $K$ of length at least $\delta(K-(B\cup\{s',t'\}))+x'$.
We know that such path in $K$ exists for $x'=0$ by Corollary~\ref{thm:relaxed_st_path}.
We shall analyze the running time of this recursion later in this proof.
Note that any $(s',t')$-path in $K$ can be expanded to an $(s,t)$-path in $G$ using at least $p:=|\{s',t'\}\setminus\{s,t\}|$ edges.
Also, $s,t\in B$, so $|B\cup \{s',t'\}|\le |B|+p$.
Hence, if there is an $(s',t')$-path in $K$ of length at least $\delta(K-(B\cup\{s',t'\}))+x'\ge \delta(G-(B\cup \{s',t'\}))+x'\ge \delta(G-B)-p+x'$, there is a path of length at least $\delta(G-B)+x'$ in $G$.
It follows that if $x' \ge k$ then the algorithm can safely decide that the given instance is a yes-instance.
Otherwise, the maximum possible $x'<k$ is found and it is left for the algorithm to expand the $(s',t')$-path in $K$ to an $(s,t)$-path in $G$.
That is, it needs to find two vertex-disjoint paths of sufficient total length going from ${s',t'}$ to ${s,t}$ in $G$.
An additional restriction for these paths is that they should not contain any edge of $M$.
Since the sufficient total length is bounded by $k+2$, we can safely employ the algorithm for \pname{Long $(s,t)$-Cycle}\xspace, Theorem~\ref{thmTLDP} from Section~\ref{sec:tldp}, running in $2^{\mathcal{O}(k)}\cdotn^{\mathcal{O}(1)}$ time.
The correctness of the algorithm is now clear and we move to analyze the recursion running time.
We know that without the recursive call, the algorithm runs in $2^{\mathcal{O}(k+|B|)}\cdotn^{\mathcal{O}(1)}$ time.
For convenience, we write this running time bound in the form $2^{\mathcal{O}(k+|B|)}\cdot(n-2)^{\mathcal{O}(1)}$.
Note that this is possible since $n>2$ for any $2$-connected graph $G$.
Thus, we can already assume that if the algorithm runs without making recursive calls, it runs in $2^{c_1(k+|B|)}\cdot(n-2)^{c_2}$ time, where $c_1,c_2\ge 1$ are constant integers given by the non-recursive subroutine.
Since the recursive call is made when the graph contains at least two Erd{\H {o}}s-Gallai component\xspace{s}, it is always made to an instance with the smaller number of vertices.
We will now prove that our algorithm runs in $2^{c_1(k+|B|)}\cdot (n-2)^{c_2+1}$ time by induction on $n$.
The base of our induction are instances for which no recursive calls are made.
Consider an instance for which at least two recursive calls are made.
We want to prove that the algorithm running time $2^{\mathcal{O}(k+|B|)}\cdotn^{\mathcal{O}(1)}$.
First note that the parameter $k+|B|$ does not increase in a recursive call, because $|(B\cup \{s',t'\})\cap V(K)|\le |B\cap V(G)|$.
Let $q\ge 2$ be the number of Erd{\H {o}}s-Gallai component\xspace{s} in $G$.
For $i\in [q]$, denote by $K_i, s'_i, t'_i$ the triple given by Lemma~\ref{lemma:st_path_banana_to_2_connected} for the $i$-{th} Erd{\H {o}}s-Gallai component\xspace of $G$.
Denote also $n_i:=|V(K_i)|$.
The running time of the algorithm for the instance given by $K_i$ is at most $2^{c_1(k+|B|)}\cdot (n_i-2)^{c_2+1}$ by induction.
Note that all $q$ sets $V(K_i)\setminus\{s'_i, t'_i\}$ are pairwise disjoint.
Also, none of these sets contains $s$ or $t$.
Hence, $\sum_{i=1}^{q} (n_i-2)\le n-2$.
We now want to upper-bound the sum $\sum_{i=1}^q (n_i-2)^{c_2+1}$.
\begin{proposition}
Let $a_1, a_2, \ldots, a_q$ be a sequence of $q\ge 2$ positive integers with $\sum_{i=1}^q a_i=n$.
Let $x>1$ be an integer.
Then $\sum_{i=1}^q a_i^x \le (n-1)^x+1 \le n^x - n^{x-1}$.
\end{proposition}
\begin{proof}
First, we show that the maximum of the sum $\sum_{i=1}^q a_i^x$ is achieved with $q=2$, $a_1=n-1$, $a_2=1$, if the sum $\sum_{i=1}^q a_i=n$ is fixed.
To show that the maximum cannot be achieved with $q>2$, it is enough to see that replacing $a_{q-1}$ and $a_q$ with $a_{q-1}+a_q$ yields a greater total sum, as $(a_{q-1}+a_q)^x>a_{q-1}^x+a_q^x$.
We know that the maximum is achieved with $a_1^x+a_2^x$ for some positive integers $a_1, a_2$ with $a_1+a_2=n$.
Without loss of generality, we can assume that $a_1\ge a_2$.
Suppose that $a_2>1$.
Consider replacing $a_1$ with $a_1+1$ and $a_2$ with $a_2-1$.
We need to show that the total sum does not decrease, i.e., \ $(a_1+1)^x+(a_2-1)^x\ge a_1^x+a_2^x$, or $(a_1+1)^x-a_1^x\ge a_2^x-(a_2-1)^x$.
Rewrite the left and the right part to obtain
\[\sum_{i=0}^x \binom{x}{i}a_1^i-a_1^x\ge a_2^x-\sum_{i=0}^x\binom{x}{i}a_2^i(-1)^{x-i}.\] Then \[\sum_{i=0}^{x-1}\binom{x}{i}a_1^i\ge -\sum_{i=0}^{x-1}\binom{x}{i}a_2^i(-1)^{x-i},\] and
\[\sum_{i=0}^{x-1}\binom{x}{i}(a_1^i+a_2^i(-1)^{x-i})\ge 0.\]
Each summand of the sum in the last inequality is non-negative since $a_1^i\ge a_2^i$ for any $i\ge 0$.
Thus, the initial inequality holds and we can replace $(a_1,a_2)$ with $(a_1+1,a_2-1)$ if $a_2>1$ so the total sum does not decrease.
Hence, the maximum is achieved with $a_1=n-1$ and $a_2=1$.
It is left to show that $(n-1)^x+1\le n^x-n^{x-1}$.
We rewrite it as
\[1\le (n-1)\cdot (n^{x-1}-(n-1)^{x-1}),\]
which holds as $n>1$ and $x>1$.
The proof is complete.
\end{proof}
With this proposition, we have that the running time of the algorithm is upper-bounded by
\begin{multline*}
2^{c_1(k+|B|)}\cdot (n-2)^{c_2}+2^{c_1(k+|B|)}\cdot \sum_{i=1}^q n_i^{c_2+1}\le 2^{c_1(k+|B|)}\cdot ((n-2)^{c_2}+((n-2)^{c_2+1}-(n-2)^{c_2})),
\end{multline*}
so the induction hypothesis holds.
This concludes the proof.
\end{proof}
\section{Algorithm for small vertex covers}\label{sec:vcalgo}
In this section we prove Theorem~\ref{thmVCad} stating that \textsc{\pname{Long Dirac Cycle}\xspace\ / Vertex Cover Above Degree} is solvable in $2^{\mathcal{O}(p+|B|)}\cdot n^{\mathcal{O}(1)}$ running time. Recall that the task of this problem is, given a graph $G$, a subset of vertices $B$, a vertex cover $S$ of $G$ of size $\delta(G-B)+p$ and a nonnegative integer $k$, decide whether $G$ has a cycle of length at least $2\delta(G-B)+k$. We start by assembling combinatorial results about paths and vertex covers, which we later use in the algorithm.
\begin{comment} \todo[inline]{The theorem below. Is it used anywhere? and if used, downgrade it to lemma} \begin{theorem}
Let $G$ be a graph with vertex cover $S$, such that $|S|=d+p$, where $d=\min_{v\in I}\deg_G(v)$ for $I=V(G)\setminus S$.
Let $k$ be the length of the longest cycle in $G$. Then for every set
$X\subseteq I$ such that $|X|=\lfloor\frac{k}{2}\rfloor-2p$,
there exists a cycle of length $k$ in $G$ containing all vertices of $X$. \end{theorem} \begin{proof}
Let $C$ be the longest cycle in $G$ containing the maximum number of vertices from $X$.
Suppose that $C$ does not contain all vertices of $X$. Let $S'=V(C)\cap S$. Since vertices $I=V(G)\setminus S$ form an independent set, we have that all vertices of $C$ outside $S'$ are pairwise nonadjacent. Hence $k \le 2|S'|$.
Consider a vertex $u \in X \setminus V(C)$. Vertex $u$ has at least $d$ neighbors in $S$; therefore
there are at most $p$ non-neighbors of $u$ in $S$, and thus in $S'$.
Let $v_1, v_2, \ldots, v_{|S'|}, v_{|S'|+1}$ be the vertices in $S'$ in the order they appear in $C$, and $v_{|S'|+1}=v_1$.
Assume that for some $i\in\{1,\dots, |S'|\}$, vertices $v_i$ and $v_{i+1}$ are adjacent to $u$. Then
$v_i$ and $v_{i+1}$ cannot be adjacent in $C$. Indeed, if that was the case, then
by inserting $u$ between $v_i$ and $v_{i+1}$ in $C$, we construct a path of length $k+1$. This contradicts the assumption that $C$ is the longest path in $G$. Therefore, if $v_i$ and $v_{i+1}$ are adjacent to $u$, then there is exactly one vertex from $I$ between $v_i$ and $v_{i+1}$ in $C$.
At least $|S'|-p$ vertices among $v_1, v_2, \ldots, v_{|S'|}$ are adjacent to $u$.
We try to insert $u$ in $C$ by taking a pair $v_i, v_{i+1}$ of neighbors of $u$ that have a vertex from $ w\in I\setminus X$ between them in $C$. That is, we want to switch $u$ with $w$. We claim that there is always a possibility to switch.
There are $|S'|$ pairs $v_i$, $v_{i+1}$. Each vertex of $S'$ that is not adjacent to $u$, ``eliminates'' at most two pairs where $u$ could be inserted. In addition, a vertex from $X\cap C$ could prohibit at most one pair. Since
\[|S'|-2p-|X\cap C|\ge |S'|-2p-|X|+1\ge \frac{k}{2}-2p-|X|+1\ge 1,\]
at least for one pair $v_i$, $v_{i+1}$ we can switch $u$ with the vertex $w$ from $ I\setminus X$ that is between $v_i$ and $v_{i+1}$
in $C$. This contradicts the selection of $C$ being a cycle of length $k$ with the maximum number of vertices from $X$.
\end{proof} \end{comment} The following lemma provides conditions when a part of long cycle $C$ can be rerouted through any sufficiently large independent set.
\begin{lemma}\label{lemma:cycle_contains_x}
Let $G$ be a graph with a given subset of vertices $B$ and a vertex cover $S$ such that $S\supseteq B$ and $|S|=\delta(G-B)+p$ for some $p\geq 0$.
Let $k$
be a non-negative integer and let $X\subseteq I=V(G)\setminus S$ be such that $|X|=\delta(G-B)-3p$.
If $G$ has a cycle $C$ of length $2\delta(G-B)+k$, then it also has a cycle $C'$ such that
\begin{itemize}
\item
The length of $C'$ is $2\delta(G-B)+k$,
\item $C'$ contains all vertices of $X$, and
\item $V(C)\cap S=V(C')\cap S$.
\end{itemize} \end{lemma} \begin{proof}
Because $S$ is a vertex cover, the length of any cycle in $G$ does not exceed $2|S|$. Hence if $G$ contains a cycle of length $2\delta(G-B)+k$, we have that $k\le 2p$.
Suppose that $G$ contains a cycle $C$ of length $2\delta(G-B)+k$.
Among all cycles of length $2\delta(G-B)+k$, we select a cycle $C'$ such that $V(C)\cap S=V(C')\cap S$
and, subject to that,
with the maximum number of vertices from $X$. We claim that all vertices of $X$ are in $C'$.
Targeting towards a contradiction, assume that there is a vertex $x\in X$ that is not in $C'$. Let $S'=S\cap V(C')$.
Note that $|S'|\ge \frac{|C'|}{2}= \delta(G-B)+\frac{k}{2}$.
Because $S$ is a vertex cover, all neighbors of $x$ are in $S$. Then $x$ has at least $\delta(G-B)$ neighbors in $S$ and, therefore, all but $p$ vertices of $S'$ are adjacent to $x$.
Let $v_1, v_2, \ldots, v_{|S'|}$ be the vertices of $S'$ in the order they appear on the cycle $C'$.
Note that for each $i\in\{1,\dots, |S'|\}$, vertices $v_{i}$ and $v_{i+1}$ (and $v_{|S'|}$, $v_1$) are either adjacent vertices in $C'$, or there exists exactly one vertex from $I$ that is between them in $C'$. We want to show that there exists at least one pair $\{v_{i},v_{i+1}\}$ such that both $v_{i}$ and $v_{i+1}$ are adjacent to $x$ and a vertex $u\in I\setminus X$ is between $v_{i}$ and $v_{i+1}$ in $C'$. If such a pair exists, then by swapping $u$ and $x$ in $C'$, we would obtain a cycle that has a larger number of vertices from $X$ leading to a contradiction.
There are at least $\delta(G-B)+k-p$ vertices from $I$ in $C'$, so there are at least $\delta(G-B)+k-p$ pairs $\{v_i, v_{i+1}\}$ that have a vertex from $I$ between them.
We know that at most $p$ vertices in $S'$ are not adjacent to $x$.
Since each vertex in $S'$ is a member of at most two pairs, vertex $x$ is adjacent to all but $2p$ such pairs $\{v_i, v_{i+1}\}$.
Suppose that $C'$ already contains $t\geq 0$ vertices from $X$. Note that by our assumption, $t<|X|=\delta(G-B)-3p$, thus $\delta(G-B)+k-p-2p-t>0$. Therefore, at least one pair of vertices $\{v_i, v_{i+1}\}$ is adjacent to $x$ and $v_i u v_{i+1}$ is a subpath of $C'$ for some $u\in I\setminus X$. Therefore, by rerouting $C'$ through $x$, instead of $u$, we construct a cycle $C''$ of length
$2\delta(G-B)+k$, such that $V(C)\cap S=V(C'')\cap S$, and $C''$ containing $t+1$ vertices of $X$. But by our assumption, cycle $C'$ contains the maximum number of vertices $t$ from $X$. We achieved the contradiction that concludes the proof of the lemma. \end{proof}
We will need the following two simple facts about the number of vertices in a vertex cover that have a small amount of neighbors outside the vertex cover.
\begin{lemma}\label{thm:many_vertices_with_large_degree}
Let $G$ be a graph, $S\subseteq V(G)$ be a vertex cover of $G$, and let $I=V(G)\setminus S\neq\emptyset$.
Let $d=\min_{v \in I}\deg_G(v)$, $b=|S|-d \ge 0$, and
$\beta=\frac{d}{|I|}$. Then for any
$\alpha\in (0, \frac{1}{\beta})$,
the number of vertices in $S$ having less that $\alpha d$ neighbors in $I$ is strictly less than $\frac{b}{1-\alpha\beta}$. \end{lemma} \begin{proof}
Let $s$ be the number of vertices in $S$ with less than $\alpha d$ neighbors in $I$.
On one hand, the number of edges between $I$ and $S$ is at least $d|I|$.
On the other hand, it is less than $\alpha d s + (|S|-s)|I|=\alpha ds + (d+b-s)|I|$.
Hence, \[d|I|<\alpha ds+(d+b-s)|I|.\] This is equivalent to
\[d<\frac{\alpha ds}{|I|}+d+b-s.\]
Thus
\[b>s\cdot \left(1-\alpha \cdot \frac{d}{|I|}\right)=s\cdot\left(1-\alpha\beta\right),\] and we conclude that
$s < \frac{b}{1-\alpha\beta}$.
\end{proof}
\begin{lemma}\label{lemma:many_vertices_with_neighbors_in_x}
Let $G$ be a graph with $B\subseteq V(G)$ and a vertex cover $S\supseteq B$ with $|S|=\delta(G-B)+p$, where $0<p< \delta(G-B)/8$.
Then for any $X\subseteq V(G)\setminus S$ with $|X|\geq\delta(G-B)-3p$, at most $2p$ vertices in $S$ have less than $2p$ neighbors in $X$. \end{lemma} \begin{proof}
Consider the graph $G[S\cup X]$.
Apply Lemma~\ref{thm:many_vertices_with_large_degree} to this graph with $I=X$.
Clearly, $d\ge \delta(G-B)>8p$, because $B\subseteq S$ and $p< \delta(G-B)/8$. Therefore, $b\leq p$.
Because $p< \delta(G-B)/8$, we also have that
$\beta= d/|X|\le |S|/(\delta(G-B)-3p)=(\delta(G-B)+p)/(\delta(G-B)-3p)\le 1+4p/(\delta(G-B)-3p)< \frac{9}{5}$. Pick $\alpha=\frac{1}{4}$, so $\alpha\beta<\frac{9}{20}$.
By Lemma~\ref{thm:many_vertices_with_large_degree}, at most $p/(1-\alpha \beta)<\frac{20}{11}p<2p$ vertices in $S$ have less than $\alpha d > \frac{1}{4} \cdot 8p = 2p$ neighbors in $I=X$. \end{proof}
The following structural lemma provides necessary and sufficient conditions for the existence of a long cycle in graph crossing specified subsets of the vertex cover and the independent set. These conditions can be checked in FPT time (Lemma~\ref{lemma:path_cover_dp}), and both lemmata are the crucial components in the proof of Theorem~\ref{thmVCad}.
\begin{lemma}\label{thm:path_cover}
Let $G$ be a graph, $B\subseteq V(G)$, and let $p> 0$ be an integer such that $p<\delta(G-B)/8$. Assume that $G$ has a vertex cover $S$ such that
$|S|=\delta(G-B)+p$ and $B\subseteq S$.
Let $k\ge 0$ be an integer and let $X \subseteq I=V(G)\setminus S$ such that $|X|\geq \delta(G-B)-3p$.
Let $A \subseteq S$ be the set of vertices of $S$ with at least $p+1$ neighbors in $X$, and let $Z=S\setminus A$.
Then there is a cycle of length $2\delta(G-B)+k$ in $G$ containing all vertices in $X \cup Z$ if and only if there is a set $Y\subseteq I$ and a path cover $\mathcal{P}$ of $G[S \cup Y]$, such that:
\begin{itemize}
\item[(i)] $\mathcal{P}$ consists of $|S|+k-2q-|Y|$ paths, where $\frac{k}{2} \le q \le p$,
\item[(ii)] $\mathcal{P}$ contains no path with an endpoint in $Z$ or $Y$,
\item[(iii)] At least
$p-q$ of paths in $\mathcal{P}$ are paths of length $0$, that is, covering a single vertex of $A$,
\item[(iv)] $|Y|\le 2|Z|$,
\item[(v)] $|X\cup Y| \le \delta(G-B)+k-q$, and
\item[(vi)] $|I| \ge \delta(G-B)+k-q$.
\end{itemize} \end{lemma} \begin{proof}
Let $C$ be a cycle of length $2\delta(G-B)+k$ in $G$ containing all vertices from $X\cup Z$.
Define $Y\subseteq I$ to be the set of the vertices of $C$ in $I$ having neighbors in $Z$ in the cycle.
Clearly, $|Y|\leq 2|Z|$ satisfying (iv).
Let $S'=S\cap V(C)$ and define $q:=|S'|-\delta(G-B)$.
Note that $q \ge \frac{k}{2}$ because $2|S'|\ge |C|$, and that $q \le p$ because $|S'|\le |S|$. Hence, the conditions for $q$ in (i) are satisfied. Since $X\cup Y \subseteq V(C)$ and $| V(C)\setminus (X\cup Y)|\ge|S'|=\delta(G-B)+q$, we have that $|X\cup Y|\le \delta(G-B)+k-q$ and (v) holds. Notice that
$|I|\geq |V(C)\cap I|=|V(C)|-|S'|=(2\delta(G-B)+k)-(q+\delta(G-B))=\delta(G-B)+k-q$ and (vi) is fulfilled.
Because $p<\delta(G-B)/8$ and $|X|\geq \delta(G-B)-3p$, $|Z|<2p$ by Lemma~\ref{lemma:many_vertices_with_neighbors_in_x}. Then $|Y|\leq 2|Z|< 4p$ and, therefore, $|S'|+|Y|< \delta(G-B)+q+4p\leq \delta(G-B)+5p$. Since $C$ has $2\delta(G-B)+k\geq 2\delta(G-B)$ vertices and $\delta(G-B)\geq 8p$, we obtain that $|S'|+|Y|< |C|$. This means that $C[S'\cup Y]$ is a proper subgraph of $C$, that is, the union of disjoint paths. Consider the path cover $\mathcal{P}'$ of $S'\cup Y$ which is produced by $C$, that is, $\mathcal{P}'$ is the set of paths that are connected components of $C[S'\cup Y]$ (see Figure~\ref{fig:pathcover}).
It consists of $|C|-|S'|-|Y|$ paths, as each vertex from $V(C)\setminus (S'\cup Y)$ on $C$ is a neighbor to exactly two endpoints in the path cover produced by $C$. Note also that the endpoints of each path of $\mathcal{P}'$ are in $A$.
This path cover still does not cover vertices in $S\setminus S'$, so we add $|S|-|S'|=p-q$ paths of zero length covering each vertex from $S\setminus S' \subseteq A$; this satisfies (iii).
The obtained path cover $\mathcal{P}$ is a path cover of $S\cup Y$ consisting of exactly $|C|+|S|-2|S'|-|Y|=|S|+k-2q-|Y|$ paths implying (i). Since all the path in $\mathcal{P}'$ have their endpoints in $A$ and each or $p-q$ trivial paths is a vertex of $A$, we obtain that (ii) is fulfilled. We conclude that $\mathcal{P}$ satisfies conditions (i)--(vi) in the statement of the lemma.
\begin{figure}
\caption{Illustration of how a cycle forms a path cover $\mathcal{P}$ from Lemma~\ref{thm:path_cover}.
Edges belonging to different paths are colored with different colors.
Vertices in $A$ that have no incident colored edge are covered by zero-length paths in $\mathcal{P}$.}
\label{fig:pathcover}
\end{figure}
We now prove the opposite direction.
Let $Y\subseteq I$ and let $\mathcal{P}$ be a path cover of $G[S\cup Y]$ satisfying conditions (i)--(vi) of the lemma. In particular, $|\mathcal{P}|=|S|+k-2q-|Y|$, where $\frac{k}{2}\leq q\leq p$, by (i).
We show that there exists a cycle of length $2\delta(G-B)+k$ containing all vertices of $X\cup Z$.
We remove from $\mathcal{P}$ arbitrary $p-q$ zero length paths covering single vertices of $A$ using (iii).
The obtained set of paths $\mathcal{P}'$ consists of $|S|-p-q+k-|Y|=\delta(G-B)+k-q-|Y|$ paths and covers $G[S'\cup Y]$, where $S'\subseteq S$ and $|S'|=|S|-(p-q)=\delta(G-B)+q$.
Now choose an arbitrary subset $I'$ of $I$ of size $\delta(G-B)+k-q$ containing all vertices from $X\cup Y$ that exists due to (v) and (vi).
We consider $H=G[S'\cup I']$. Notice that $|V(H)|=|S'|+|I'|=2\delta(G-B)+k$.
We claim that graph $H$ contains a Hamiltonian cycle. Clearly, this suffices for the proof, because the length of such a cycle is $2\delta(G-B)+k$ and it contains all the vertices of $X\cup Z$ as required.
Let $H'$ be the graph obtained from $H$ be the deletion of edges $e\in H[S']$ that are not included in the paths of $\mathcal{P}'$. It is straightforward to see that it is sufficient to show that $H'$ has a Hamiltonian cycle. By construction, $H'[S']$ is the union of paths that are subpaths of the elements of $\mathcal{P}'$. By (ii), no path of $\mathcal{P}'$ has an endpoint in $Y$.
Since $\mathcal{P}'$ consists of paths with endpoints in $S'$ covering $S'\cup Y$ and there is no edges between vertices in $Y$, removal of each vertex from $Y$ breaks one path into two.
Hence, the number of disjoint paths forming $H'[S']$ is exactly $|\mathcal{P}'|-|Y|=\delta(G)+k-q-|Y|+|Y|=|I'|$.
This implies that the graph $H''$ obtained from $H'$ by making every pair of distinct vertices of $I'$ adjacent has a Hamiltonian cycle if and only if the same holds for $H'$, because no Hamiltonian cycle of $H''$ cannot contain an edge $uv$ with $u,v\in Y$. Otherwise, such a cycle would
cover
$S'$ by less than $|I'|$ paths.
Now the degree of each vertex from $I'$ in $H''$ is at least $\delta(G-B)-(p-q)+|I'|-1=2\delta(G-B)+k-p-1$.
Take a vertex $v\in S'\setminus Z$. Then $v\in A$ and, by the definition of $A$, $v$ has at least $p+1$ neighbors in $X\subseteq I'$. Hence it has at least $p+1$ neighbors in $H''$.
Therefore, the sum of vertex degrees of a vertex from $S'\setminus Z$ and a vertex from $I'$ in $H''$, is at least $2\delta(G-B)+k=|V(G')|$. We construct $H'''$ from $H''$ by making adjacent every pair of vertices $u$ and $v$ with $u\in I'$ and $v\in S'\setminus Z$. Theorem~\ref{thm:bh} implies that $H'''$ has a Hamiltonian cycle if and only if $H''$ has a Hamiltonian cycle.
Finally, we construct a Hamiltonian cycle in $H'''$ using the paths of $\mathcal{P}'$. For this, recall that each path of $\mathcal{P}'$ has its endpoints in $A$ by (ii).
Notice that there are exactly $|I'\setminus Y|=\delta(G-B)+k-q-|Y|$ vertices of $I'$ that are not covered by the paths. Since the number of paths in $\mathcal{P}'$ is $\delta(G-B)+k-q-|Y|$ and every endpoint of a path is adjacent to every vertex of $I'\setminus Y$, it is straightforward to see that we can construct a Hamiltonian cycle joining the paths of $\mathcal{P}'$ via the vertices of $I'\setminus Y$.
Thus we conclude that $H'''$ has a Hamiltonian cycle. This implies that $H$ has a Hamiltonian cycle and competes the proof.
\end{proof}
By Lemma~\ref{thm:path_cover}, to find a cycle of length $2\delta(G-B)+k$ in $G$ containing all vertices in $C \cup Z$, it suffices to identify a path cover $\mathcal{P}$. Such a path cover can be computed by making use of color-coding. More precisely.
\begin{lemma}\label{lemma:path_cover_dp}
Given $G, B, S, k,$ and $X,A,Z$ defined in the same way as in Lemma~\ref{thm:path_cover}, the existence of $Y$ and a path cover $\mathcal{P}$ of $G[S\cup Y]$ satisfying (i)--(vi)
can be determined in $2^{\mathcal{O}(p)}\cdotn^{\mathcal{O}(1)}$ running time. \end{lemma}
\begin{proof}
Because $p<\delta(G-B)/8$ and $|X|\geq \delta(G-B)-3p$, $|Z|<2p$ by Lemma~\ref{lemma:many_vertices_with_neighbors_in_x}. Then we are looking for $Y\subseteq I$ with $|Y|\leq 2|Z|< 4p$ by (iv). Also by (i), $\frac{k}{2}\leq q\leq p$.
We assume without loss of generality that $q$ and the cardinality $r$ of $Y$ are fixed, as an algorithm can iterate over all $\mathcal{O}(p^2)$ possible pairs of these values in an outer loop. We also assume that (vi) holds for the given value of $q$.
The algorithm is now to find a set of disjoint paths $\mathcal{P}$ covering all vertices in $S$ and a set $Y\subseteq I$ of size $r$.
Since Lemma~\ref{thm:path_cover} requires an upper bound (v) on $|X\cup Y|$, we will aim to
maximize
$|X\cap Y|$, i.e.\ the number of vertices from $X$ used by the paths of $\mathcal{P}$.
As the paths of $\mathcal{P}$ cover exactly $|S|+|Y|$ vertices and their number is exactly $|S|+k-2q-|Y|$ by (i), the total length of these paths is exactly $2|Y|+2q-k\leq 10p$.
This allows us to deal with a bounded number of paths of positive length.
By (ii), there is no path in $\mathcal{P}$ with an endpoint in $Z\cup Y$.
In particular, this means that all paths of zero length are vertices in $A$ and the endpoint of nontrivial paths are in $A$.
Each nontrivial path has exactly two endpoints in $A$. Then, because the total number of path $\mathcal{P}$ is $|S|+k-2q-|Y|$, the number of nontrivial paths $t$ is at most $|A|-|\mathcal{P}|=|A|- (|S|+k-2q-|Y|)=|Y|-|Z|+2q-k\leq 6p$. Note also that because $|\mathcal{P}|=|S|+k-2q-|Y|$, the nontrivial paths should cover exactly $s=|S|+k-2q-|Y|-t$ vertices of $A$ and they should leave uncovered at least $p-q$ vertices of $A$ to satisfy (iii). Clearly, $s\leq 20p$, because the total length of the nontrivial paths is at most $10p$.
Thus, our task is reduced to deciding whether there is a set $Y\subseteq I$ of size $r\leq 4p$
and
a family of $t\leq |Y|-|Z|+2q-k\leq 6p$ disjoint nontrivial paths $\mathcal{P}'$
such that
\begin{itemize}
\item[(a)] the endpoints of the paths of $\mathcal{P}'$ are in $A$,
\item[(b)] the paths cover the vertices of $Y$ and exactly
$s=|S|+k-2q-|Y|-t\leq 20p$
vertices of $A$, and they leave uncovered at least $p-q$ vertices of $A$,
\item[(c)] subject to (a)--(b), $|Y\cap X|$ is maximum.
\end{itemize}
The color-coding technique of Alon, Yuster, and Zwick~\cite{AlonYZ95} is a standard tool for solving problems of this type. Since the approach is standard (see, e.g, the book~\cite[Chapter~5]{cygan2015parameterized}), we only briefly sketch the algorithm. In the same way as in the proof of Theorem~\ref{thmTLDP}, we give a sketch of a randomized Monte Carlo algorithm and then explain how it can be derandomized.
For each positive integer $t\leq |Y|-|Z|+2q-k$, we verify whether there are $Y$ and $\mathcal{P}'$ satisfying (a) and (b) and find the maximum size of $|X\cap Y|$. After iterating over all possible values of $t$, the algorithm returns a solution that gives the maximum value of $X\cap Y$. For a given $t$, we compute $s=|S|+k-2q-|Y|-t$ and verify whether $|A|-s\geq p-q$. We discard the current choice of $t$ if $|A|-s< p-q$.
From now we assume that the value of $t$ is fixed and $|A|-s\geq p-q$.
We use the following randomized procedure.
We color the vertices of $I$ by $r=|Y|$ distinct colors uniformly at random and then the vertices of $A$ are colored uniformly at random with another set of $s$ distinct colors.
We also assume that the vertices of $Z$ are colored as well by pairwise distinct colors that are different from the colors used for $I$ and $A$. We denote by $C_I$, $C_A$, and $C_Z$ the sets of colors used to color $I$, $A$, and $Z$, respectively. Let also $C=C_I\cup C_A\cup C_Z$. Clearly, $|C|=\mathcal{O}(p)$.
We say that $Y\subseteq I$ and a set of disjoint nontrivial paths $\mathcal{P}'$ satisfying (a) and (b) is a \emph{coloful} solution if the vertices of the paths are colored by distinct colors.
The main steps of our algorithm either finds the maximum $|X\cap Y|$ for a colorful solution or reports that a colorful solution does not exist.
For a set of colors $R\subseteq C$, denote by $\alpha(R)$ the maximum number of vertices of $X$ that can be covered by a nontrivial path $P$ with $|R|$ vertices such that their the endpoint are in $A$ and the vertices of $P$ are colored by distinct colors from $R$; we assume that $\alpha(R)=-\infty$ if such a path does not exist. We observe that for every $R\subseteq C$, the value of $\alpha(R)$ can be computed in $2^{\mathcal{O}(p)}\cdot n^{\mathcal{O}(1)}$ time by a straightforward modification of the standard dynamic programming algorithm for finding a colorful $|R|$-path (see~\cite{AlonYZ95} and~\cite[Chapter~5]{cygan2015parameterized}). It is easy to incorporate the condition that the endpoits are in $A$. To maximize the number of vertices of $X$ used by a path, we can assume that the vertices of $X$ are of weight one and the vertices of $V(G)\setminus X$ are given zero weights. Then we use the variant of the algorithm that finds a colorful path of maximum weight. From now, we assume that we are given the table of values of $\alpha(R)$ for all $R\subseteq C$. Note that this table of size $2^{\mathcal{O}(p)}$ can be constructed in $2^{\mathcal{O}(p)}\cdot n^{\mathcal{O}(1)}$ time.
Let $R\subseteq C$, and $\ell\leq t$ be a positive integer. Denote by $\beta(R,\ell)$ the maximum number of vertices of $X$ that can be covered by exactly $\ell$ nontrivial path with $|R|$ vertices in total such that their endpoint are in $A$ and the vertices of the paths are colored by distinct colors from $R$; we assume that $\beta(R,\ell)=-\infty$ if such paths do not exist; in particular $\beta(R,\ell)=-\infty$ if $|R|\leq 1$. It is straightforward to see that $\beta(R,1)=\alpha(R)$ for every $R\subseteq C$. To compute $\beta(R,\ell)$ for $\ell>1$, we use the following straightforward recurrence for $|R|\geq 2$. \begin{equation}\label{eq:rec-beta} \beta(R,\ell)=\max\{\alpha(R')+\beta(R\setminus R',\ell-1)\mid \emptyset\neq R'\subset R\}. \end{equation}
We use (\ref{eq:rec-beta}) to compute the table of values of $\beta(R,t)$ for all nonempty $R\subseteq C$. Because $|C|=\mathcal{O}(p)$, computing the table can be done in $2^{\mathcal{O}(p)}\cdot n^{\mathcal{O}(1)}$ time.
By the choice of $C_I$, $C_A$ and $C_Z$, we have that $\beta(C,t)$ is the maximum number of vertices of $X$ that can be covered by a colorful solution, and $\beta(C,t)=-\infty$ if there is no colorful solution.
To obtain an optimum (non-colorful) solution, we define $N=\lceil e^{s+t}\rceil\geq \frac{r^r\cdot s^s}{r!\cdot s!}$ and iterate the randomized procedure $N$ times. Then the algorithm returns a solution that gives the maximum value $|X\cap Y|$ over all coloful solution or reports that there is no solution if the algorithm fails to find a colorful solution in every iteration.
Suppose that $Y\subseteq I$ of size $r$ and $\mathcal{P}'$ of size $t$ satisfy (a) and (b) and provide the maximum value of $|X\cap Y|$. Then with probability at least $\frac{r!}{r^r}$, the vertices of $Y$ are colored by distinct colors from $C_I$ by a random coring. Similarly, with probability at least $\frac{s!}{s^s}$, the $s$ vertices of $A$ covered by the paths of $\mathcal{P}'$ are colored by distinct colors of $C_A$. Then with probability at least $\frac{r!\cdot s!}{r^r\cdot s^s}$, the vertices of the paths of $\mathcal{P}'$ are colored by distinct colors. Respectively, the probability that this does not holds, that is, there are at least two vertices of the same color, is at most $(1-\frac{r!\cdot s!}{r^r\cdot s^s})$. By the choice of $N$, we obtain that the probability that for every iteration, at least two vertices of paths of $\mathcal{P}$ have the same color, is at most $(1-\frac{r!\cdot s!}{r^r\cdot s^s})^N\leq e^{-1}$. Thus, the probability that the randomized algorithm fails to return an optimum solution is at most $e^{-1}<1$.
To evaluate the running time, recall that the tables of values of $\alpha(\cdot)$ and $\beta(\cdot,t)$ can be computed in $2^{\mathcal{O}(p)}\cdot n^{\mathcal{O}(1)}$ time. Since $r<4p$ and $s\leq 20p$, $N=2^{\mathcal{O}(p)}$ and, therefore, the total running time is $2^{\mathcal{O}(p)}\cdot n^{\mathcal{O}(1)}$.
To derandomize the algorithm, we use the standard technique (see~\cite{AlonYZ95} and~\cite[Chapter~5]{cygan2015parameterized}). For given $r$ and $s$, we construct the $(|I|,r)$ and $(|A|,s)$-perfect hash families
of the functions $\mathcal{F}_I$ and $\mathcal{F}_A$, respectively, of sizes $e^{r}r^{\mathcal{O}(\log k)}\cdot \log |I|$ and $e^{s}s^{\mathcal{O}(\log s)}\cdot \log |A|$, respectively, using the results of Naor, Schulman, and Srinivasan~\cite{NaorSS95}. These families can be constructed in time $2^{\mathcal{O}(p)}\cdot n\log n$. Then we replace the random colorings of $I$ and $A$ by the functions from $\mathcal{F}_I$ and $\mathcal{F}_A$, respectively, and iterate the main step over all these functions. This gives deterministic $2^{\mathcal{O}(p)}\cdot n^{\mathcal{O}(1)}$ running time.
To conclude the proof, note that algorithms finds the maximum possible size of $|X\cap Y|$ for $Y\subseteq I$ of size $r$ such that $S\cap Y$ can be covered by a set of paths $\mathcal{P}$ satisfying conditions (i)--(iv) and (vi) of Lemma~\ref{lemma:many_vertices_with_neighbors_in_x}. To verify (v), it is sufficient to check additionally whether $|X\cup Y|\leq \delta(G-B)+k-q$, by the maximality of $|X\cap Y|$. This concludes the proof. \end{proof}
Everything is settled for the proof of Theorem~\ref{thmVCad}. For convenience, we restate the theorem here.
\noindent\textbf{Theorem~\ref{thmVCad}}. \emph{\textsc{\pname{Long Dirac Cycle}\xspace\ / Vertex Cover Above Degree} is solvable in $2^{\mathcal{O}(p+|B|)}\cdot n^{\mathcal{O}(1)}$ running time.}
\begin{proof}
Let $(G, B, S, k)$ be a given instance of the problem. We assume without loss of generality that $B \subseteq S$; otherwise we can set $S:=S\cup B$ and $p:=p+|B\setminus S|$,
which increases $p$ by at most $|B|$. Let $I=V(G)\setminus S$.
Note that $G$ has no cycle longer than $2|S|\le 2\delta(G-B)+2p$.
In particular, if $k>2p$, then the given instance is a no-instance. Therefore, we can assume that $k\leq 2p$.
If $\delta(G-B)\leq 8p$, then $2\delta(G-B)+k\leq 18p$ and one can verify whether $G$ has a cycle of length $2\delta(G-B)+k$ in $2^{\mathcal{O}(p)}\cdot n^{\mathcal{O}(1)}$ time using, e.g., the algorithm given by Zehavi~\cite{Zehavi16}.
From now on, we assume that $\delta(G-B)>8p$. It is also convenient to assume that our aim is to verify the existence of a cycle of length \emph{exactly} $2\delta(G-B)+k$; for this we iterate over all possible values of the parameter from the initial given value of $k$ and $2p$.
Also, if $p=0$, then $k=0$ and each vertex in $I$ is adjacent to all vertices in $S=\delta(G-B)$.
Then $G$ contains all edges between $S$ and $I$, so a cycle of length at least $2\delta(G-B)=2|S|$ exists in $G$ if and only if $|S|\geq |I|$ and $|S|\geq 2$.
Thus, we can now assume that $p>0$.
If $|I| < \delta(G-B)+k-p$, then $(G, B, S, k)$ is a no-instance. Hence, we can assume that this is not the case.
Our algorithm chooses an arbitrary $X \subseteq I$ of size $\delta(G-B)-3p$.
By Lemma~\ref{lemma:cycle_contains_x}, the algorithm can now look for a cycle of length $2\delta(G-B)+k$ in $G$ containing all vertices from $X$.
Then we partition $S$ into two subsets $A$ and $Z$.
The subset $A$ consists of all vertices in $S$ that have at least $p+1$ neighbors in $X$.
The subset $Z$ consists of all other vertices in $S$. The running time of the procedure computing $Z$ is clearly polynomial.
By Lemma~\ref{lemma:many_vertices_with_neighbors_in_x}, the cardinality of $Z$ is at most $2p$.
Before we can apply Lemmata~\ref{thm:path_cover} and~\ref{lemma:path_cover_dp}, we need to ensure that the cycle we are looking for contains \emph{all} vertices from $Z$.
To achieve that, we allow our algorithm to brute-force over all $2^{|Z|}=2^{\mathcal{O}(p)}$ options of how the cycle intersects $Z$.
When an option is fixed, consider deleting from $G$ all vertices of $Z$ outside the fixed intersection.
This can change the value of $p$, as $p=|S|-\delta(G-B)$, and both $|S|$ and $\delta(G-B)$ may change after the deletion.
As a consequence, the equality $|X|=\delta(G-B)-3p$ could no longer hold, so we need to change $X$ correspondingly.
Rewrite $\delta(G-B)-3p=4\delta(G-B)-3|S|$.
Note that the removal of a single vertex of $Z$ from $G$ always decreases $|S|$ by one and can decrease $\delta(G-B)$ by at most one.
Hence, the value $\delta(G-B)-3p$ can only increase.
Thus, after the deletion, to ensure $|X|=\delta(G-B)-3p$, we add some vertices from $I$ to $X$.
By Lemma~\ref{lemma:cycle_contains_x}, the choice of these vertices can be arbitrary and we can be sure that there is a cycle containing $X$ while its intersection with $S$ remains the same.
Each vertex in $A$ still has at least $p+1$ neighbors in $X$.
Since $X$ now can containin some new vertices from $I$, a vertex in $Z$ may have at least $p+1$ neighbors in $X$.
If such a vertex exists, we simply move it from $Z$ to $A$.
Observe that the value of the parameter $p$ may be only decreased and the deletion does not violate the property $\delta(G-B)>8p$.
Note that the deletion operation discussed above also can imply an increment in $k$ as $\delta(G-B)$ can decrease.
This is safe as Lemma~\ref{lemma:cycle_contains_x} does not depend on the value of $k$ other than for estimating the length of the cycle.
After the intersection of the cycle with $Z$ is fixed and all vertices from $Z$ outside it are deleted from $G$, the algorithm finally employs the routine from Lemma~\ref{lemma:path_cover_dp} to find the path cover from Lemma~\ref{thm:path_cover}, hence to find the cycle.
The total running time of the algorithm (under the assumption that $B\subseteq S$) is proportional to the number of sets $Z$, which is $2^{\mathcal{O}(p)}$, times the time required to compute the path cover for each of the sets, which is $2^{\mathcal{O}(p)}\cdot n^{\mathcal{O}(1)}$ by Lemma~\ref{lemma:path_cover_dp}. Hence the total running time is
$2^{\mathcal{O}(p)}\cdot n^{\mathcal{O}(1)}$. Taking into account that to ensure the assumption that $B\subseteq S$ we may increase the initial value of $p$ by at most $|B|$, we conclude that the algorithm runs in $2^{\mathcal{O}(p+|B|)}\cdot n^{\mathcal{O}(1)}$ time.
\end{proof}
\section{Finding almost Hamiltonian cycles} \label{sec:HamCycles}
This section is dedicated to the proof of Theorem~\ref{theorem:hamiltonian}. To recall, the theorem states that given a graph $G$ with a set $B \subset V(G)$ and a parameter $k$ such that $|B| \le k$ and
$\delta(G - B) \ge \frac{n}{2} - k$, in time $2^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$ we can find the longest cycle in $G$. Before we move on to prove the theorem itself, we show how to deal with the special case where there is a small separator in the graph, as it is an important subroutine in the main algorithm. Another key ingredient to the proof of Theorem~\ref{theorem:hamiltonian} is our \textsc{\pname{Long Dirac Cycle}\xspace\ / Vertex Cover Above Degree} result, presented in Section~\ref{sec:vcalgo}.
\subsection{Small separator lemma}
We show an algorithm for \textsc{Almost Hamiltonian Dirac Cycle} when there is a small (i.e. of size $\mathcal{O}(k)$) separator $B$ in $G$. Intuitively, the presense of a small separator makes the problem easier in the following sense. Each component of $G - B$ still has high minimal degree, slightly less than $\frac{n}{2}$. Thus, essentially, we must have exactly two components of size roughly $\frac{n}{2}$ in $G - B$, which means they are very dense. As was proven in~\cite{fomin_et_al:LIPIcs:2020:11724}, in this situation, we can always partition a component into paths that start and end at the given vertices, and span the whole component. We restate their result formally in the next lemma.
\begin{lemma}[Lemma 1 in \cite{fomin_et_al:LIPIcs:2020:11724}]
\label{lemma:many_paths}
Let $G$ be an $n$-vertex graph and $p$ be a positive integer such that $\delta(G) \ge \max\{5p - 3, n - p\}$. Let $\{s_1, t_1\}$, \ldots, $\{s_r, t_r\}$, $r \le p$, be a collection of pairs of vertices of $G$ such that (i) $s_i \notin \{s_j, t_j\}$ for all $i \ne j$, $i, j \in \{1, \ldots, r\}$, and (ii) there is at least one index $i \in \{1, \ldots, r\}$ such that $s_i \ne t_i$.
Then there is a family of pairwise vertex-disjoint paths $\mathcal{P} = \{P_1, \ldots, P_r\}$
in $G$ such that each $P_i$ is an ($s_i$, $t_i$)-path and $\cup_{i = 1}^r V(P_i) = V(G)$, that is, the paths cover all vertices of $G$. \end{lemma} We note that the proof of Lemma~\ref{lemma:many_paths} given in \cite{fomin_et_al:LIPIcs:2020:11724} is actually constructive. That is, there is a polynomial time algorithm that given $G$, $p$, and the respective set of pairs of vertices, returns the family of paths $\mathcal{P}$ from the statement of Lemma~\ref{lemma:many_paths}.
For simplicity, suppose there is a Hamiltonian cycle $C$ in $G$. The cycle induces a certain partition of $B$ into paths. On the other hand, if we are able to find any such path cover $\mathcal{P}$, we can construct the whole Hamiltonian cycle. Namely, on each component $H$ of $G - B$, we invoke Lemma~\ref{lemma:many_paths} with a collection of pairs being a certain matching on ends of $\mathcal{P}$ belonging to $H$. In this way we connect the paths together while also visiting every vertex of $H$. If the pairs are selected in a certain way in both components, the union of all these parts will actually form a Hamiltonian cycle. We find the path cover itself with the help of dynamic programming and the color coding technique of Alon, Yuster, and Zwick~\cite{AlonYZ95}. In what follows, we prove the above in more detail.
\begin{lemma}
\label{lemma:hamiltonian_separator}
Let $G$ be a given $2$-connected graph on $n$ vertices and let $k\ge 0$ be a given integer.
Let $B\subseteq V(G)$ be such that $|B|\le k$, $\delta(G - B) \ge \frac{n}{2} - k$, and the graph $G- B$ is not connected.
There is a $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$ running time algorithm that finds the longest simple cycle in $G$. \end{lemma} \begin{proof}
Assume $n \ge 12k$, otherwise we invoke the general $2^{\mathcal{O}(n)}$ algorithm for the \textsc{Longest Cycle} problem from \Cref{prop:longest_cycle} polynomial number of times to find the longest cycle in $G$.
First, observe that there are exactly two connected components in $G - B$. There must be at least two of them since $G - B$ is not connected. Suppose there are at least three components.
Each of them contains a vertex of degree at least $\frac{n}{2} - k$ in $G - B$, therefore the size of each component is at least $\frac{n}{2} - k + 1$. The total number of vertices is then at least $3 \frac{n}{2} - 3k + 3 = n + (\frac{n}{2} - 3k) + 3 > n$. This is a contradiction.
From now on, let $H_1$ and $H_2$ be the two connected components of $G - B$.
Consider the longest cycle $C$ in $G$. Recall that by Theorem~\ref{thm:relaxed_long_cycle} the length of $C$ is at least $\min\{n - 2k,n-|B|\}\ge n-2k$, thus it necessarily contains vertices from all of $H_1$, $H_2$ and $B$. We say that $C$ induces a path cover $\mathcal{P}$ of $B$, where $\mathcal{P}$ is the set of paths that $C$ forms when restricted to the edges incident to $B$. In other words, remove from $C$ all the edges that are not incident to $B$, and all the vertices that became isolated after that. The resulting collection of vertex-disjoint paths is the path cover $\mathcal{P}$. Note that $\mathcal{P}$ satisfies the following properties.
\begin{enumerate}
\item Every path $P \in \mathcal{P}$ starts and ends in $V(G) \setminus B$.
\item Each path $P \in \mathcal{P}$ has at least one vertex in $B$ and no two consecutive vertices in $V(G) \setminus B$.
\item The paths of $\mathcal{P}$ contain at most $3|B|$ vertices in total.
\item The number of paths in $\mathcal{P}$ that start and end in different components of $G - B$ is even and at least two.
\end{enumerate}
Since for every vertex of $B$, its degree in $C$ is exactly two even when restricted to the edges incident to $B$, the property (1) follows. Each path goes through $B$, and two vertices in $V(G) \setminus B$ cannot be adjacent via an edge incident to $B$, thus (2) follows. Property (3) follows directly from property (2). Finally, (4) holds since $C$ must leave both $H_1$ and $H_2$ an even number of times. Moreover, if there are no paths in $\mathcal{P}$ that start and end in different components, $H_1$ and $H_2$ cannot be connected via $C$, thus $C$ is not a cycle of length at least $n - 2k$.
We call a set of vertex-disjoint paths in $G$ satisfying (1)--(4) \emph{a good path cover}. Now we claim that any good path cover can be used to construct a long cycle in $G$, i.e. we can collect all the vertices of $V(G) \setminus B$ in a cycle by going along the paths in the cover. The proof is essentially by pairing endpoints of the paths carefully and then applying Lemma~\ref{lemma:many_paths} to both $H_1$ and $H_2$. The illustration is shown in Figure~\ref{fig:small_sep} and the proof follows next.
\begin{figure}
\caption{Reconstructing the cycle from the good path cover $\mathcal{P} = \{P_1, P_2, P_3, P_4\}$. The paths $Q_1$ and $Q_2$ are obtained by applying Lemma~\ref{lemma:many_paths} to $H_1' = H_1 - \{i_1\}$, the same for the paths $R_1$, $R_2$ and the graph $H_2' = H_2 - \{i_2\}$. The resulting concatenation of paths is a Hamiltonian cycle in $G$.}
\label{fig:small_sep}
\end{figure}
\begin{claim}
\label{claim:good_cover}
There is a polynomial time algorithm that given a good path cover $\mathcal{P}$ finds a cycle of length $n - t$ in $G$, where $t$ is the number of vertices in $B$ not covered by the paths in $\mathcal{P}$.
\end{claim}
\begin{proof}
Denote $\mathcal{P} = \{P_1, \ldots, P_r\}$, and for each $i \in \{1, \ldots, r\}$, denote the two ends of the path $P_i$ by $s_i$ and $t_i$. We may assume that the paths are ordered in a way that paths $P_1$, \ldots, $P_a$ lead from $H_1$ to $H_2$, paths $P_{a + 1}$, \ldots, $P_b$ start and end in $H_1$, and paths $P_{b + 1}$, \ldots, $P_r$ start and end in $H_2$, for certain integers $a$ and $b$, such that $1 < a \le b \le r$, and $a$ is even by property (5). Additionally, for $i \in \{1, \ldots, a\}$ assume that $s_i \in V(H_1)$, $t_i \in V(H_2)$.
Let $I$ be the set of internal vertices of paths in $\mathcal{P}$,
let $H_1' = H_1 - I$, $H_2' = H_2 - I$. The graphs $H_1'$ and $H_2'$ are targets for applying Lemma~\ref{lemma:many_paths}. By property (2), the size of $I \setminus B$ is at most $k$, thus $\delta(H_1') \ge \delta(G - B) - k = \frac{n}{2} - 2k$, and by the same argument $\delta(H_2') \ge \frac{n}{2} - 2k$.
Consider the following set $T_1$ of $b - \frac{a}{2}$ pairs of vertices in $H_1'$. If $b = a$, the pairs are $\{s_1, s_2\}$, $\{s_3,s_4\}$, \ldots, $\{s_{a - 1}, s_a\}$. If $b > a$, the pairs are $\{s_{2i - 1}, s_{2i}\}$ for $1 \le i < \frac{a}{2}$, $\{s_{a - 1}, s_{a + 1}\}$, $\{t_j, s_{j + 1}\}$ for $a + 1 \le j < b$, and $\{t_b, s_a\}$.
Now, we apply Lemma~\ref{lemma:many_paths} to the graph $H_1'$, the set of pairs $T_1$, and we set the parameter $p$ to be $2k$. Since pairs in $T_1$ are disjoint, and $\max\{5p - 3, n - p\} = \max\{10k - 3, n - 2k\} \le \delta(H_1')$, all conditions of the lemma are satisfied. Thus, there exist vertex-disjoint paths $Q_1$, \ldots, $Q_{b - \frac{a}{2}}$ that have the respective endpoints from $T_1$ and cover all vertices of $H_1'$.
We deal with $H_2'$ similarly. We only need to connect $t_1$, \ldots, $t_a$ in a shifted way compared to $s_1$, \ldots, $s_a$, so that we obtain a cycle at the end.
Consider the following set $T_2$ of $\frac{a}{2} + r - b$ pairs of vertices in $H_2'$. If $b = r$, the pairs are $\{t_2, t_3\}$, $\{t_4, t_5\}$, \ldots, $\{t_{a - 2}, t_{a - 1}\}$, and $\{t_1, t_a\}$.
If $b < r$, the pairs are $\{t_{2i}, t_{2i + 1}\}$ for $1 \le i < \frac{a}{2}$, $\{t_a, s_{b + 1}\}$, $\{t_j, s_{j + 1}\}$ for $b + 1 \le j < r$, and $\{t_1, t_r\}$.
Again, we apply Lemma~\ref{lemma:many_paths} to the graph $H_2'$, the set of pairs $T_2$, and $p = 2k$.
We obtain vertex-disjoint paths $R_1$, \ldots, $R_{r - b + \frac{a}{2}}$ that have the respective endpoints from $T_2$ cover all vertices of $H_2'$.
The resulting cycle $C$ with $V(C)=(V(G)\setminus B )\cup I$ is a cyclic concatenation of paths $P_1$, \ldots, $P_r$, $Q_1$, \ldots, $Q_{b - \frac{a}{2}}$, $R_1$, \ldots, $R_{r - b + \frac{a}{2}}$ in a certain order. Namely,
\[C = P_1Q_1P_2R_1\cdots P_{a - 1}Q_{\frac{a}{2}}P_{a + 1}Q_{\frac{a}{2} + 1}P_{a + 2}\cdots P_b Q_{b - \frac{a}{2}} P_a R_{\frac{a}{2}} P_{b + 1}\cdots P_r R_{r - b + \frac{a}{2}},\]
where we understand the notation $PQ$ for paths $P$ and $Q$ with a common endpoint as their natural concatenation. Clearly, $C$ is a cycle, and it spans all the previously defined paths. By construction, these paths cover all vertices in $I$, $V(H_1')$, and $V(H_2')$, thus they cover all vertices in $V(G)$ except those vertices in $B$ that are not covered by $\mathcal{P}$.
\end{proof}
Now it only remains to find a good path cover that covers the maximum number of vertices in $B$. By Claim~\ref{claim:good_cover}, a good path cover immediately gives us a cycle of the corresponding length, and we have also showed that a long cycle in $G$ induces a good path cover.
To find the desired good path cover, first we observe that the number of vertices covered by the paths in the cover is at most $3|B|$ by property (3) of a good path cover. We proceed with a color-coding scheme using $r = 3|B|$ colors: color each vertex in $B$ in its own color, and each vertex in $V(G) \setminus B$ randomly and independently in one of the remaining $r - |B|$ colors, with equal probability for each color. Denote this coloring by $c : V(G) \to \{1, \ldots, r\}$. Now we look for a colored good path cover, that is, a good path cover that covers at most one vertex of each color.
We find a colored good path cover with the help of dynamic programming. Define a \emph{state} as a tuple $(C, v, i, \ell, p)$ where $C$ is a subset of $\{1, \ldots, r\}$, $v$ is a vertex in $V(G)$, $i \in \{1, 2\}$, $\ell \in \{1, \ldots, r\}$, and $p \in \{0, 1, \ldots, |B|\}$.
We call a state $(C, v, i, \ell, p)$ \emph{feasible} if there exists a set of vertex-disjoint paths $\mathcal{P} = \{P_1, \ldots, P_t\}$ in $G$ such that the following holds.
\begin{enumerate}
\item Every path $P_1$, \ldots, $P_{t - 1}$ starts and ends in $V(G) \setminus B$ and has the length of at least three, $P_t$ starts in $V(H_i)$, ends in $v$, and its length is $\ell$.
\item No path $P \in \mathcal{P}$ has two consecutive vertices in $V(G) \setminus B$.
\item The paths in $\mathcal{P}$ cover exactly one vertex of each color in $C$, and no vertices of other colors.
\item The number of paths in $\{P_1, \ldots, P_{t - 1}\}$ that start and end in different components of $G - B$ is exactly $p$.
\end{enumerate}
Note that $\mathcal{P}$ in the definition of a feasible state is essentially an ``unfinished'' good path cover that agrees with the state $(C, v, i, \ell, p)$. Our goal now is to compute the set of all feasible states $S$. We start by setting
\[S_1 = \big\{\big(\{c(v)\}, v, 1, 1, 0\big) : v \in V(H_1)\big\} \cup \big\{\big(\{c(v)\}, v, 2, 1, 0\big) : v \in V(H_2)\big\}.\]
These are our initial states, corresponding to sets containing one path of length one. Trivially, each such state is feasible, and these are all feasible states that use exactly one color. Next, for each $j$ in $\{1, \ldots, r - 1\}$, we show how to compute the set of feasible states $S_{j + 1}$ of size $j + 1$ from $S_j$, the set of feasible states of size $j$. Here by the size of the state $(C, v, i, \ell, p)$ we mean $|C|$, the number of colors used, which is the same as the total number of vertices covered by any set of paths corresponding to the state.
To compute $S_{j + 1}$ from $S_j$, we iterate over all states in $S_j$ and try to extend each of them by an additional vertex. Intuitively, we either extend the unique unfinished path corresponding to the state, or declare it finished and start a new path. Fix a state $(C, v, i, \ell, p) \in S_j$, there is a set of paths $\mathcal{P} = \{P_1, \ldots, P_t\}$ satisfying the feasibility definition for $(C, v, i, \ell, p)$. Consider each $u \in N_G(v)$ such that $c(u) \notin C$. If both $v$ and $u$ are not in $B$, we do nothing. Otherwise, add to $S_{j + 1}$ the state $(C \cup c(u), u, i, \ell + 1, p)$. Clearly, the size of this state is $j + 1$, and it is easy to verify that the set of paths $\mathcal{P}' = \{P_1, \ldots, P_tu\}$ satisfies the feasibility definition for $(C \cup c(u), u, i, \ell + 1, p)$.
For the ``new path'' kind of extending $(C, v, i, \ell, p)$ with $\ell > 2$, consider each vertex $u \in V(G) \setminus B$ such that $c(u) \notin C$. If $v \in B$ do nothing, otherwise add to $S_{j + 1}$ the state $(C \cup c(u), u, i', 1, p')$, where $i'$ is such that $u \in H_{i'}$ and $p' = p$ if $v \in V(H_i)$, or $p' = p + 1$ if $v \notin V(H_i)$. To see that this state is feasible, consider the set of paths $\mathcal{P}' = \{P_1, \ldots, P_t, u\}$.
Indeed, every path among $P_1$, \ldots, $P_{t - 1}$ starts and ends in $V(G) \setminus B$, and $P_t$ as well, since $v \in V(G) \setminus B$. The length of $P_t$ is $\ell$ so at least three, and for $P_1$, \ldots, $P_{t - 1}$ this holds by feasibility of the original state. The last path $u$ starts in $V(H_{i'})$ by definition of $i'$, ends in $u$, and has the length of one. Properties (2) and (3) are preserved automatically. The value $p'$ reflects exactly how $p$ is changed with respect to the newly finished path $P_t$.
Now we show that $S_{j + 1}$ contains all feasible states of size $j + 1$, provided that $S_j$ contains all feasible states of size $j$. Consider a state $(C', u, i', \ell', p') \in S_{j + 1}$ and a corresponding set of paths $\mathcal{P}' = \{P_1, \ldots, P_t\}$. Recall that $|P_t| = \ell'$, if $\ell' > 1$, consider a state $(C, v, i', \ell' - 1, p')$ where $v$ is the previous vertex to $u$ in $P_t$, $C = C' \setminus \{c(u)\}$. Observe that $(C, v, i', \ell' - 1, p')$ is feasible as witnessed by the set of paths $\mathcal{P} = \{P_1, \ldots, P_t'\}$ where $P_t'$ is $P_t$ without its last vertex $u$. Since $u \in N_G(v)$, $c(u) \notin C$, and $v$ and $u$ are not simultaneously in $V(G) \setminus B$ by property (2) for $\mathcal{P}'$, the state $(C', u, i', \ell', p')$ is added to $S_{j + 1}$ when the algorithm considers extending the state $(C, v, i', \ell'-1, p') \in S_j$ by the vertex $u$. If $\ell' = 1$, consider a state $(C, v, i, \ell, p)$ where $v$ is one of the endpoints of $P_{t - 1}$, $i$ is the index of the component of the other endpoint of $P_{t - 1}$, $\ell = |P_{t - 1}|$, $C = C' \setminus \{c(u)\}$, and $p$ is either $p'$ or $p' - 1$, depending on whether $v$ belongs to $H_i$ or not. The set of paths $\{P_1, \ldots, P_{t - 1}\}$ witnesses the feasibility of $(C, v, i, \ell, p)$, and thus $(C', u, i', \ell', p')$ is added to $S_{j + 1}$ on the corresponding ``new path'' step.
Therefore, we have shown that for each $j$ in $\{1, \ldots, r - 1\}$, we correctly compute the set $S_{j + 1}$ from $S_j$, so in the end we have the sets $S_j$ of feasible states of size $j$, for each $j \in \{1, \ldots, r\}$. Finally, we consider a subset $\mathcal{C}$ of the feasible states $(C, v, i, \ell, p) \in \bigcup_{j = 1}^r S_j$, such that $v \notin B$, $\ell > 2$, and $p'$ is at least 2 and even, where $p' = p$ if $v \in H_i$ and $p'= p + 1$ if $v \notin H_i$. Note that $\mathcal{C}$ is not empty since a long cycle in $G$ guaranteed by Theorem~\ref{thm:relaxed_long_cycle} induces a good path cover, and thus a feasible state of the form above. From $\mathcal{C}$, we pick a state maximizing $|C \cap \{1, \ldots, |B|\}|$. The set of paths $\{P_1, \ldots, P_t\}$ corresponding to this state is a good path cover in $G$ that covers the maximum number of vertices in $B$. Note that the actual good path cover may be found by the usual means of backtracking in dynamic programming.
Together with Claim~\ref{claim:good_cover} this concludes the algorithm, and the proof of its correctness.
\textbf{Running time analysis.} In the dynamic programming part, the number of states is at most $2^r \cdot n \cdot 2 \cdot r \cdot (k + 1)$. While considering a state, we update $\mathcal{O}(n)$ other states, thus the total running time of the dynamic programming subroutine is $2^{\mathcal{O}(k)} n^{\mathcal{O}(1)}$. For a fixed long cycle $C$ in $G$, the probability that we guess the coloring that assigns different colors to all vertices of the induced by $C$ good path cover, is at least $e^{-r}$, since there are at most $r$ vertices in the good path cover. By performing $\lceil e^r \rceil$ iterations of the color coding subroutine, we amplify the success probability to at least $1 - (1 - e^r)^{e^r} \ge 1 - e^{-1}$. Therefore, we obtain a Monte Carlo algorithm with constant success probability and running time $\mathcal{O}(k^2 \cdot e^{3k} \cdot 2^{3k} \cdot n^2) = \mathcal{O}(2^{\mathcal{O}(k)} n^{\mathcal{O}(1)})$. Finally, the algorithm could be derandomized in the standard fashion by using perfect hash families~\cite{NaorSS95}. \end{proof}
\subsection{Main theorem} Now we move on to Theorem~\ref{theorem:hamiltonian}, the main result of this section. We restate the theorem here for convenience of the reader.
\theoremhamiltonian*
\begin{proof}
First, we may assume that $n > 40k$, otherwise the problem can be solved by the classical $2^{\mathcal{O}(n)}$ algorithm for \textsc{Longest Cycle}.
Instead of proving the theorem directly, we show that there exists an algorithm that in time $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$ either
\begin{enumerate}
\item finds the longest cycle in $G$, or
\item finds a vertex cover of $G$ of size at most $\frac{n}{2}+9k$, or
\item finds a set $B'\supseteq B$ of size at most $35k$ such that $G-B'$ is not connected.
\end{enumerate}
We say that (1)--(3) are the \emph{terminal states} of the algorithm.
If state (3) is reached, we simply invoke the algorithm from Lemma~\ref{lemma:hamiltonian_separator} with the respective separating set $B'$ of size at most $35k$. This gives us immediately the longest cycle in $G$.
Similarly, reaching terminal state (2) also suffices to solve the problem, as shown in the next claim.
\begin{claim}
If terminal state (2) is reached, the longest cycle in $G$ can be found in $2^{\mathcal{O}(k)}\cdotn^{\mathcal{O}(1)}$ time.
\end{claim}
\begin{claimproof}
Denote the obtained vertex cover of $G$ of size at most $\frac{n}{2}+9k$ by $S$.
We would like to invoke the algorithm given by \Cref{thmVCad}, but we are not guaranteed that the longest cycle in $G$ has length of the form $2\delta(G-B)+k'$ for $k'\ge 0$.
By \Cref{thm:relaxed_long_cycle}, we have that there is a cycle of length at least $\min\{2\delta(G-B), n-|B|\}$ in $G$, as $G$ is $2$-connected.
We aim to achieve $2\delta(G-B)\le n-|B|$.
Each vertex in $G-B$ has at most $|S|$ neighbours.
Take the vertex in $G-B$ with smallest degree.
It has at least $\frac{n}{2}-k> 19k$ neighbours in $G-B$.
Obtain $B'$ by adding $19k$ neighbours of this vertex in $G-B$ to $B$.
We have that $\delta(G-B')=\delta(G-B)-19k$, and $G-B'$ still contains at least one vertex.
Note that $\delta(G-B')\le |S|-19k\le \frac{n}{2}-10k\le \frac{n}{2}-\frac{|B'|}{2}$, as $|B'| \le 20k$, so $2\delta(G-B')\le n-|B'|$.
Thus, by \Cref{thm:relaxed_long_path}, the length of the longest cycle in $G$ is of form $2\delta(G-B')+k'$ for $k'\ge 0$.
The size of the vertex cover $S$ is at most $\frac{n}{2}+9k\le \delta(G-B)+10k= \delta(G-B')+29k$.
Recall that Theorem~\ref{thmVCad} provides a $2^{\mathcal{O}(p + k')}\cdot n^{\mathcal{O}(1)}$-time algorithm that finds a cycle of length at least $2 \delta(G - B') + k'$ given a vertex cover of $G$ of size $\delta(G - B') + p$, if there is any.
By trying all possible $k'$ from $n - 2\delta(G - B') \le 40k$ to zero, we find the longest cycle in $G$ in time $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$ as $p\le 29k$. By Theorem~\ref{thm:relaxed_long_cycle} there is a cycle of length at least $2\delta(G-B')$ in $G$, thus invoking Theorem~\ref{thmVCad} with $k' = 0$ necessarily provides us with a cycle.
\end{claimproof}
Therefore, in what follows we assume that reaching any of the terminal states solves the problem immediately.
Now consider a cycle $C$ of maximum length in $G$. Identically to the proof of Lemma~\ref{lemma:many_paths}, $C$ induces a path cover of a subset of $B$. Namely, in this proof, we call a set of vertex-disjoint paths $\mathcal{P}$ in $G$ \emph{a good path cover} if $\mathcal{P}$ satisfies the following properties.
\begin{enumerate}
\item Every path $P \in \mathcal{P}$ starts and ends in $V(G) \setminus B$.
\item Each path $P \in \mathcal{P}$ has at least one vertex in $B$ and no two consecutive vertices in $V(G) \setminus B$.
\item The paths of $\mathcal{P}$ contain at most $3|B|$ vertices in total.
\end{enumerate}
Note that this definition is the same as in Lemma~\ref{lemma:many_paths}, except for the property (4) there. Intuitively, we do not need it in this lemma since we may now assume that $G - B$ is connected. Since the current definition is strictly less restrictive, it follows immediately from the proof of Lemma~\ref{lemma:many_paths} that
\begin{itemize}
\item for each $0 \le t \le |B|$, if there is a cycle of length $n - t$ in $G$, there is also a good path cover in $G$ that covers all but $t$ vertices of $B$,
\item in time $2^{\mathcal{O}(k)} n^{\mathcal{O}(1)}$ we can find a good path cover $\mathcal{P}$ that covers the maximum number of vertices in $B$, by the combination of color coding and dynamic programming.
\end{itemize}
Note that the empty set is a good path cover, thus a good path cover always exists.
So for the rest of the proof we deal with the case where we have computed a good path cover $\mathcal{P}$ of $G$, possibly an empty one. Denote by $r$ the number of paths in $\mathcal{P}$, and by $B'$ the set of vertices covered by paths in $\mathcal{P}$ together with the rest of vertices of $B$. By definition, $B \subset B'$, and by property (2) of a good path cover $|B'| \le 3k$. If $G - B'$ is not connected, we have a small separator: the algorithm outputs $B'$ and stops, reaching terminal state (3). If $G - B'$ is not $2$-connected, we add to $B'$ an arbitrary cut vertex of $G - B'$ and return $B'$. Thus, from now on we may assume that $G - B'$ is $2$-connected. The minimum degree of $G - B'$ is at least
\[\delta(G - B') \ge \frac{n}{2} - k - |B' \setminus B| \ge \frac{n - |B' \setminus B|}{2} - 2k > \frac{n - |B'| + 2}{3},\]
since $|B' \setminus B| \le 2k$ and $n > 16k$. By Theorem~\ref{proposition:cycle_or_is}, in time $\mathcal{O}(n^3)$ we find either a Hamiltonian cycle $C_0$ in $G - B'$, or an independent set of size $\delta(G - B') + 1$. If an independent set is found, its complement in $G - B'$ together with $B'$ is a vertex cover of $G$ of size at most $\frac{n}{2} + k + 2|B'| \le \frac{n}{2} + 7k$. In this case we output the vertex cover and stop, reaching terminal state (2).
Otherwise, we have a Hamiltonian cycle $C_0$ in $G - B'$. Now, we iteratively insert the paths of $\mathcal{P} = \{P_1, \ldots, P_r\}$ into the cycle. Namely, for each $i \in \{1, \ldots, r\}$ we prove that given a cycle $C$ that contains exactly the vertices of the cycle $C_0$ and the paths $P_1$, \ldots, $P_{i - 1}$,
we can either modify the cycle $C$ such that it satisfies the same property for $i + 1$, i.e. contains the vertices of the path $P_i$ as well, or reach one of the terminal states. Clearly, applying the above for each $i \in \{1, \ldots, r\}$, starting from the cycle $C_0$, proves the theorem. Thus from now on we focus on this statement.
Consider the path $P_i$ and the obtained cycle $C$ that contains all vertices of $C_0$ and $P_1$, \ldots, $P_{i - 1}$. Denote the endpoints of $P_i$ by $s$ and $t$, observe that both $s$ and $t$ have at least $\frac{n}{2} - 3k$ neighbors on $C$. That holds since $s \notin B$, so $\deg_{G - B} (s) \ge \frac{n}{2} - k$, and at most $2k$ vertices of $G$ belong to $B' \setminus B$ and are neither on $C$ nor in $B$, analogously for $t$.
Denote by $C_s$ the set of neighbors of $s$ on $C$, and by $C_t$ the set of neighbors of $t$ on $C$.
Consider a vertex $c_s \in C_t$ and a vertex $c_t \in C_t$. If $c_s$ and $c_t$ are next to each other on $C$ then we can immediately insert $P_i$ in $C$. If these vertices are not adjacent, but are at distance two on $C$ with a vertex $c' \notin B$ between them, we do the following. Insert $P_i$ in $C$ by going from $c_s$ to $c_t$ through $P_i$ and not through $c'$, denote the resulting cycle by $C'$. The vertex $c'$ is the only vertex that is in $V(C) \cup V(P_i)$, but not on $C'$, thus we are done as long as we insert $c'$ back in $C'$. By the same argument as for $s$ and $t$, $c'$ has at least $\frac{n}{2} - 3k$ neighbors on $C'$. If there are two neighbors of $c'$ on $C'$ that are consecutive on $C'$ again we can immediately insert $c'$ in $C'$, thus we assume this is not the case. Now on $C'$ between every two consecutive neighbors of $c'$ there is a group of at least one and possibly several non-neighbors of $c'$. Since there are at least $\frac{n}{2} - 3k$ neighbors of $c'$ on $C'$, there are also at least $\frac{n}{2} - 3k$ such groups of consecutive non-neighbors. Since there are at most $\frac{n}{2} + 3k$ non-neighbors of $c'$ on $C'$, at most $6k$ of the groups may contain more than one vertex. Thus at least $\frac{n}{2} - 9k$ groups consist of a single vertex, denote the set of all such vertices by $I$. Each vertex of $I$ is not adjacent to $c'$, but both of its neighbors on $C'$ are adjacent to $c'$. We claim that if two vertices in $I$ are adjacent in $G$, there is a cycle that goes through $c'$ and all vertices of $C'$. Denote these vertices by $u$ and $v$, go from $c'$ to a neighbor of $u$, then to $v$ along the arc of $C'$ that does not contain $u$, then take the edge $uv$, and finally collect the rest of $C'$ going from $u$ to a neighbor of $v$ and returning to $c'$.
If no two vertices in $I$ are adjacent in $G$, then $I$ is an independent set of size at least $\frac{n}{2} - 9k$ in $G$. Thus the complement of $I$ is a vertex cover of $G$ of size at most $\frac{n}{2} + 9k$, and we are in the terminal state (2).
\begin{figure}
\caption{Inserting the path $P$ (in red) into the cycle $C$ (in blue) in the presence of an edge between an internal $s$-vertex $c_s$ and an internal $t$-vertex $c_t$. The resulting cycle is in solid.}
\label{fig:insertion}
\end{figure}
Now we deal with the case where for every $c_s \in C_s$ and every $c_t \in C_t$, there is either a vertex of $B$ or at least two other vertices between them on $C$. First, we bound the number of common neighbors of $s$ and $t$ on $C$, denote $C_s \cap C_t$ by $C_{st}$. Fix an ordering on $C$, and consider a vertex $u \in C_{st}$ and the next vertex $v$ along the cycle that belongs to either $C_s$ or $C_t$. Between $u$ and $v$, there must be at least two vertices that belong to neither $C_s$ nor $C_t$, or a vertex of $B$. Thus with each vertex of $C_{st}$ we can uniquely associate either two vertices of $V(C) \setminus C_s \setminus C_t$, or a vertex of $B$. We get that apart from the vertices of $C_{st}$, $C_s \setminus C_{st}$ and $C_t \setminus C_{st}$, there are at least $2(|C_{st}| - |B|)$ other vertices in $C$. Summing the sizes of these four disjoint sets together, we get
\begin{align*}
&|C_{st}| + (|C_s| - |C_{st}|) + (|C_t| - |C_{st}|) + 2(|C_{st}| - |B|) \le n,\\
&|C_{st}| \le n - |C_s| - |C_t| + 2 |B| \le 8k,
\end{align*}
since both $C_s$ and $C_t$ contain at least $\frac{n}{2} - 3k$ vertices, and the size of $B$ is at most $k$. From this bound, we also immediately get that the number of vertices on $C$ that are not adjacent to both $s$ and $t$ is at most $n - |C_s| - |C_t| + |C_{st}| \le 14k$. Thus nearly all vertices of $C$, except for $\mathcal{O}(k)$, are adjacent either to $s$ but not to $t$, or to $t$ but not to $s$. As vertices from $C_s$ cannot be next to vertices from $C_t$ on $C$, they must come in large consecutive chunks along the cycle. To formalize this intuition, let us call a vertex in $C_s$ an \emph{internal $s$-vertex} if both of its neighbors along the cycle are also from $C_s$, and \emph{internal $t$-vertices} are defined analogously. We claim that except for $\mathcal{O}(k)$ vertices, all the vertices of $C$ are either internal $s$-vertices or internal $t$-vertices. Vertices from $C_s$ that are not internal $s$-vertices must have at least one neighbor along $C$ that is not from $C_s$ nor $C_t$, and the same holds for $C_t$. However, there are at most $14k$ vertices in $V(C) \setminus C_s \setminus C_t$, and each of them can ``spoil'' at most two vertices of $C_s$ or $C_t$. Also note that a vertex of $C_{st}$ must have vertices of $V(C) \setminus C_s \setminus C_t$ on both sides, as a vertex from $C_s$ cannot lie next to a vertex of $C_t$ on $C$. Thus the total number of internal $s$-vertices and internal $t$-vertices is at least
\begin{multline*}(|C_s| - |C_{st}|) + (|C_t| - |C_{st}|) - 2 (|V(C) \setminus C_s \setminus C_t| - |C_{st}|) \ge 2(\frac{n}{2} - 3k) - 28k = n - 34k.\end{multline*}
Now assume there is an edge between an internal $s$-vertex and an internal $t$-vertex. If this holds, the path $P_i$ can be inserted in $C$ in the same way as in the case of a single high-degree vertex above, see Figure~\ref{fig:insertion} for an illustration.
On the other hand, if there are no edges between internal $s$-vertices and internal $t$-vertices, then the graph induced on the sets of internal $s$-vertices and $t$-vertices is not connected, as these sets are both non-empty.
Then removing at most $34k$ vertices from $G$ leaves these sets disconnected. Thus we arrive to the terminal state (3) where we have a small separator.
In order to apply \Cref{lemma:hamiltonian_separator}, it should contain $B$ as a subset, so after taking the union with $B$ its size is at most $35k$.
\end{proof}
\section{Dirac decomposition\xspace}\label{sec:bananas}
In this section, we define Dirac decomposition\xspace{s} and show that, given a Dirac decomposition\xspace for a cycle in $G$, we can either find a longer cycle or solve the instance $(G,B,k)$ of \pname{Long Dirac Cycle}\xspace in time single-exponential in $k+|B|$.
\begin{figure}
\caption{A schematic example of a Dirac decomposition\xspace, vertices belonging to $B$ are in light gray. Removing the paths $P_1$ and $P_2$ leaves two \ref{enum:cycle_tunnel_path_bic}-type components that correspond to the long arcs $P'$ and $P''$ of the starting cycle $C$, one \ref{enum:cycle_tunnel_path_cut_left}-type component, and a component consisting only of vertices from $B$, denoted by D0. The four Dirac components are in thick blue.}
\label{fig:bananasoncycle}
\end{figure} \begin{definition}[\textbf{Dirac decomposition\xspace and Dirac component\xspace}]
Let $G$ be a 2-connected graph, let $B$ be a subset of $V(G)$, and let $C$ be a cycle in $G$ of length at least $2\delta(G-B)$.
We say that two disjoint paths $P_1$ and $P_2$ in $G$ induce \emph{a Dirac decomposition\xspace for $C$ and $B$} in $G$ if
\begin{itemize}
\item
The cycle $C$ is of the form $C=P_1 {P'}P_2{P''}$, where each of the paths ${P'}$ and ${P''}$ has at least $\delta(G- B)-2$ edges.
\item
Let $G'$ be the graph obtained from $G$ by applying $B$-refinement to every connected component $H$ of $G- V(P_1 \cup P_2)$, except those components $H$ with $V(H)\subseteq B$. Note that no edges of the paths $P_1$ and $P_2$ are contracted.
Then for every connected component $H'$ of $G'-V(P_1 \cup P_2)$, except those with $V(H')\subseteq B$, holds $|V(H')|\ge 3$ and one of the following.
\begin{enumerate}[label=(D\arabic*)]
\item\label{enum:cycle_tunnel_path_bic} $H'$ is $2$-connected and the maximum size of a matching in $G'$ between $V(H')$ and $V(P_1)$ is one, and between $V(H')$ and $V(P_2)$ is also one;
\item\label{enum:cycle_tunnel_path_cut_left} $H'$ is not 2-connected,
exactly one vertex of $P_1$ has neighbors in $H'$, that is,
$|N_{G'}(V(H'))\cap V(P_1)|=1$, and no inner vertex from a leaf-block of $H'$ has a neighbor in $P_2$;
\item\label{enum:cycle_tunnel_path_cut_right} The same as \ref{enum:cycle_tunnel_path_cut_left}, but with $P_1$ and $P_2$ interchanged. That is,
$H'$ is not 2-connected,
$|N_{G'}(V(H'))\cap V(P_2)|=1$, and no inner vertex from a leaf-block of $H'$ has a neighbor in $P_1$.
\end{enumerate}
\item There is exactly one connected component $H$ in $G-V(P_1\cup P_2)$ with $V(H)\setminus B=V(P')\setminus (B\cup\{s',t'\})$, where $s'$ and $t'$ are the endpoints of $P'$.
Analogously, there is exactly one connected component $H$ in $G-V(P_1\cup P_2)$ with $V(H)\setminus B=V(P'')\setminus (B\cup\{s'',t''\})$.
\end{itemize}
The set of \emph{Dirac component\xspace}s for a Dirac decomposition\xspace
is defined as follows.
First, for each component $H'$ of type \ref{enum:cycle_tunnel_path_bic}, $H'$ is a Dirac component\xspace of the Dirac decomposition\xspace.
Second, for each leaf-block of each $H'$ of type \ref{enum:cycle_tunnel_path_cut_left}, or of type \ref{enum:cycle_tunnel_path_cut_right}, this leaf-block is also a Dirac component\xspace of the Dirac decomposition\xspace.
For an example of a Dirac decomposition, see Figure~\ref{fig:bananasoncycle}. \end{definition}
Note that Lemma~\ref{lemma:st_path_banana_consecutive} holds for an arbitrary cycle $C$ if we replace Erd{\H {o}}s-Gallai component\xspace{s} and Erd{\H {o}}s-Gallai decomposition\xspace{s} with Dirac component\xspace{s} and Dirac decomposition\xspace{s}. We give the analogue of this lemma below without proof, since it is identical to the proof of Lemma~\ref{lemma:st_path_banana_consecutive}.
\begin{lemma}\label{lemma:dirac_cycle_banana_consecutive}
Let $G$ be a $2$-connected graph, $B\subseteq V(G)$, $C$ be a cycle in $G$. Let paths $P_1, P_2$ induce a Dirac decomposition\xspace for $C$ and $B$ in $G$.
Let $M$ be a Dirac component\xspace of the Dirac decomposition\xspace and $P$ be a path in $G$ such that $P$ contains at least one vertex in $V(P_1)\cup V(P_2)$.
If $P$ enters $M$, then all vertices of $M$ hit by $P$ appear consecutively on $P$. \end{lemma}
We now want to prove an analogue of \Cref{lemma:st_path_edge_of_banana} showing that if a long cycle in $G$ exists, then it suffices to look for a long cycle entering a Dirac component\xspace. For that we first require the following weaker result.
\begin{lemma}\label{lemma:dirac_b_leaf_block_separator}
Let $G$ be a $2$-connected graph, $B\subseteq V(G)$, $C$ be a cycle in $G$ of length less than $2\delta(G-B)+k$. Let paths $P_1, P_2$ induce a Dirac decomposition\xspace for $C$ and $B$ in $G$.
If $H'$ is a \ref{enum:cycle_tunnel_path_cut_left}-type or a \ref{enum:cycle_tunnel_path_cut_right}-type component of the Dirac decomposition\xspace and $S$ is a $B$-leaf-block separator of $H'$, then there is a cycle of length at least $\frac{1}{2}(5\delta(G-B)-|S|-(k+5))$ that enters a Dirac component\xspace in $G$. \end{lemma} \begin{proof}
Without loss of generality, let $H'$ be a \ref{enum:cycle_tunnel_path_cut_left}-type component of the Dirac decomposition\xspace.
Take a vertex $v \in V(H')$ that is not an inner vertex of a leaf-block of $H'$ and has a neighbour in $P_2$.
Such vertex always exists by definition of a Dirac decomposition\xspace.
Let $S$ be a $B$-leaf-block separator of $H'$.
By \Cref{lemma:separator_in_non_2c}, there is a $(c,v)$-path of length at least $\frac{1}{2}\delta(H'-B)-\frac{1}{2}|S|$ in $H'$ for some cut-vertex $c$ of a leaf-block of $H'$.
Also $\delta(H'-B)\ge \delta(G-B)-|V(P_1)\cup V(P_2)|\ge \delta(G-B)-(k+5)$, since the total length of $P_1$ and $P_2$ is at most $k+3$.
Denote this leaf-block of $H'$ by $L$.
Note that $\delta(L-(B\cup\{c\}))\ge \delta(G-B)-2$ by properties of \ref{enum:cycle_tunnel_path_cut_left}-type components.
By \Cref{thm:relaxed_st_path}, there is a path of length at least $\delta(G-B)-2$ between $c$ and any other vertex in $L$.
Let $u$ be an inner vertex in $L$ that has a neighbour in $P_1$.
Combine the $(u,c)$-path inside $L$ with the $(c,v)$-path going outside $L$ in $H'$.
The obtained path is a $(u,v)$-path of length at least $(\delta(G-B)-2)+(\frac{1}{2}\delta(H'-B)-\frac{1}{2}|S|)$.
Since $u$ and $v$ have neighbours in $V(P_1)$ and $V(P_2)$ respectively, we obtain a chord of $C$ of length at least $\delta(G-B)+\frac{1}{2}\delta(H'-B)-\frac{1}{2}|S|$.
The chord splits $C$ into two arcs, one of which is of length at least $\delta(G-B)$.
Combine this arc with the chord and obtain a cycle of length at least $2\delta(G-B)+\frac{1}{2}\delta(H'-B)-\frac{1}{2}|S|\ge \frac{5}{2}\delta(G-B)-\frac{1}{2}|S|-\frac{1}{2}(k+5)$. \end{proof}
The following lemma is an analogue of \Cref{lemma:st_path_edge_of_banana} for Dirac component\xspace{s}. In contrast to \Cref{lemma:dirac_cycle_banana_consecutive}, the proof is significantly different from the proof of \Cref{lemma:st_path_edge_of_banana}.
\begin{lemma}\label{lemma:dirac_cycle_edge_of_banana}
Let $G$ be a graph, $B\subseteq V(G)$ be a subset of its vertices and $P_1, P_2$ induce a Dirac decomposition\xspace for a cycle $C$ of length less than $2\delta(G-B)+k$ in $G$.
Let $k$ be an integer such that $6k+4|B|+6 < \delta(G-B)$.
If there exists a cycle of length at least $2\delta(G-B)+k$ in $G$ that contains at least one vertex in $V(P_1)\cup V(P_2)$, then there exists a cycle of length at least $2\delta(G-B)+k$ in $G$ that
enters a Dirac component\xspace. \end{lemma} \begin{proof}
Suppose that there exists a cycle $C'$ of length at least $2\delta(G-B)+k$ in $G$ that contains at least one vertex in $V(P_1)\cup V(P_2)$.
If $C'$ already contains an edge of a Dirac component\xspace, we are done.
We now assume that $C'$ does not contain any edge of any Dirac component\xspace.
We show how to use $C'$ and construct a cycle of length at least $2\delta(G-B)+k$ in $G$ that contains an edge of a Dirac component\xspace of the given Dirac decomposition\xspace.
Let $W$ be the set of all vertices of $G$ that are vertices of non-leaf-blocks of \ref{enum:cycle_tunnel_path_cut_left}-type or \ref{enum:cycle_tunnel_path_cut_right}-type components in the Dirac decomposition\xspace.
We start with the following claim.
\begin{claim}
$|W\cap V(C')|> 5k$.
\end{claim}
\begin{claimproof}
This is a counting argument.
Note that $C'$ cannot contain an edge with both endpoints inside a Dirac component\xspace of $G$.
Since Dirac component\xspace{s} of $G$ are \ref{enum:cycle_tunnel_path_bic}-type components of the Dirac decomposition\xspace and leaf-blocks of \ref{enum:cycle_tunnel_path_cut_left}-type or \ref{enum:cycle_tunnel_path_cut_right}-type connected components, each edge of $C'$ has an endpoint either in $V(P_1)\cup V(P_2)\cup B$, or inside a non-leaf-block of a \ref{enum:cycle_tunnel_path_cut_left}-type or a \ref{enum:cycle_tunnel_path_cut_right}-type connected component.
The union of the vertex sets of the non-leaf-blocks form the set $W$.
Hence, $(W\cap V(C'))\cup V(P_1)\cup V(P_2)\cup B$ is a vertex cover of $C'$.
Note that a vertex cover of any cycle consists of at least half of its vertices.
Then $$2|(W\cap V(C'))\cup V(P_1)\cup V(P_2)\cup B|\ge |V(C')|\ge 2\delta(G-B)+k.$$
Immediately we get that \begin{multline*}2|W\cap V(C')|\ge 2\delta(G-B)+k-2|V(P_1)\cup V(P_2)|-2|B|\ge 2\delta(G-B)+k-2(k-2)-2|B|>10k.\end{multline*}
\end{claimproof}
The following claim is useful for constructing long chords of $C'$ going through the Dirac component\xspace{s} that are leaf-blocks.
\begin{claim}\label{claim:hard_lemma_no_inner_vertices}
Let $H'$ be a \ref{enum:cycle_tunnel_path_cut_left}-type or a \ref{enum:cycle_tunnel_path_cut_right}-type component in the Dirac decomposition\xspace.
$C'$ does not contain any inner vertex of the leaf-blocks of $H'$.
\end{claim}
\begin{claimproof}
Suppose that $C'$ contains some vertex $u \in V(H')$ that is an inner vertex of some leaf-block $L$ of $H'$.
As $L$ is a Dirac component\xspace of $G$, $C'$ cannot contain any edge of $L$, so $C'$ should enter $L$ from $V(P_1) \cup V(P_2)$ through $u$ and leave it immediately.
By definition of Dirac decomposition\xspace{s}, the only option to enter or leave $L$ is to go through the only vertex in $V(P_1)$ (if $H'$ is of type \ref{enum:cycle_tunnel_path_cut_left}) or in $V(P_2)$ (if $H'$ is of type \ref{enum:cycle_tunnel_path_cut_right}).
As $C'$ cannot contain any vertex twice, this is not possible.
\end{claimproof}
We now use the above claims to construct either a family of long chords of $C'$ going through Dirac component\xspace{s}, or a $B$-leaf-block separator of some of the \ref{enum:cycle_tunnel_path_cut_left}-type or \ref{enum:cycle_tunnel_path_cut_right}-type components in the Dirac decomposition\xspace.
To construct the first chord of $C'$, take a vertex $w_1 \in W$.
Since $w_1$ is a vertex of a separable component $H'$, there is a cut vertex $c_1$ of a leaf-block $L_1$ of $H'$ reachable from $w_1$ inside $H'$.
The leaf-block $L_1$ contains also at least one vertex $v_1\neq c_1$ that is connected to $V(P_1)$ (if $H'$ is of type \ref{enum:cycle_tunnel_path_cut_left}) or to $V(P_2)$ (if $H'$ is of type \ref{enum:cycle_tunnel_path_cut_right}) outside $H'$.
We know that $\delta(L_1-(B\cup\{c_1\}))\ge \delta(G-(B\cup \{c_1\}))-1\ge \delta(G-B)-2$, since the only outside neighbour of vertices in $L_1-(B\cup{c_1})$, apart from vertices in $B$, is a single vertex in $V(P_1)$ or $V(P_2)$.
By Corollary~\ref{thm:relaxed_st_path}, there exists an $(c_1,v_1)$-path inside $L_1$ of length at least $\delta(G-B)-2$.
Combine this with $(w_1,c_1)$-path inside $H'$ and obtain a $(w_1,v_1)$-path inside $H'$.
Note that the constructed $(w_1,v_1)$-path can contain vertices from $W$ apart from $w_1$.
Let $w'_1 \in W$ be the vertex on the $(w_1,v_1)$-path farthest from $w_1$.
Note that the $(w'_1,v_1)$-subpath does not contain any vertex from $W$ except $w'_1$, and it still contains the $(c_1,v_1)$-path as a subpath by Claim~\ref{claim:hard_lemma_no_inner_vertices}.
Hence, we obtain a $(w'_1,v_1)$-path of length at least $\delta(G-B)-2$ inside $H'$ that does not contain any vertex in $W\setminus \{w'_1\}$.
To obtain a long chord of $C'$, it is left to reach the vertex in $V(P_1)\cup V(P_2)$ from $v_1$ outside $H'$, and then follow the cycle $C$ until a vertex $v'_1$ of $C'$ is reached. This is always possible since $V(C)\cap V(C')\supseteq (V(P_1)\cup V(P_2))\cap V(C')\neq \emptyset$.
We obtain a chord of length at least $\delta(G-B)-1$ connecting $w'_1$ and $v'_1$.
To construct the second chord, we follow the same process for a vertex $w_2 \in W\setminus \{w'_1\}$.
When constructing the path going from $w_2$ to a cut vertex of a leaf-block, we prohibit this path from going through $w'_1$.
If $w'_1$ separates $w_2$ from all leaf-block cut vertices, then we obtain a small $B$-leaf-block separator of $H'$.
Otherwise, we obtain a $(w'_2,v'_2)$-chord of $C'$ of length at least $\delta(G-B)-1$ that does not contain any vertex in $W\setminus\{w'_1,w'_2\}$.
It is important that during the construction of different chords we always follow $C$ in the same direction.
Repeat this process $3k$ times and obtain either a family of $3k$ $(w'_i,v'_i)$-chords of $C'$ of length at least $\delta(G-B)-1$, or a $B$-leaf-block separator of size at most $3k$.
If it is the latter case, then, by Lemma~\ref{lemma:dirac_b_leaf_block_separator}, there is a cycle of length at least $2\delta(G-B)+\frac{1}{2}(\delta(G-B)-3k-(k+5))> 2\delta(G-B)+k$ that enters a Dirac component\xspace.
We now assume that a family of chords is obtained.
The following claim is useful.
\begin{claim}
If for some $i,j\in [3k]$ we have $v'_i\neq v'_j$, then the chords between $w'_i$ and $v'_i$ and between $w'_j$ and $v'_j$ do not have any common vertex.
\end{claim}
\begin{claimproof}
Note that $v'_i$ or $v'_j$ depend only on the vertex in $V(P_1)$ or in $V(P_2)$ which we start following $C$ from.
If $v'_i\neq v'_j$, then they were found when starting from different vertices.
Then the $i^\text{th}$ and the $j^\text{th}$ chords were constructed from different components of the Dirac decomposition, as for each separable component there is only one vertex in $V(P_1) \cup V(P_2)$ that is adjacent to inner vertices of leaf-blocks of this component.
Therefore, the $(w'_i,v_i)$- and $(w'_j,v_j)$-subpaths of the chords do not have common vertices.
The $(v_i,v'_i)$- and $(v_j,v'_j)$-subpaths also cannot have any common vertex as $v'_i$ and $v'_j$ were found as the first vertices from $V(C')$ on $C$ when following $C$ in the same direction.
\end{claimproof}
\begin{definition}
Consider two chords of $C$ that do not have common endpoints.
Denote the endpoints of one chord by $s$ and $t$ and of the other by $p$ and $q$.
We say that these two chords of $C$ \emph{intersect graphically}, if the vertices $s$, $t$, $p$ and $q$ are located in the order $s, p, t, q$ on $C$ when following $C$ in one or the other direction.
In other words, two chords intersect graphically if each chord cuts $C$ into two arcs each containing exactly one endpoint of the other chord.
\end{definition}
We can now show that if two chords in the constructed family intersect graphically, then we can find a long cycle that contains these two chords.
Assume that there are $i, j \in [3k]$ such that $v'_i\neq v'_j$ and the vertices $w'_i,v'_i,w'_j,v'_j$ are located in the order $w'_i,w'_j,v'_i,v'_j$ on $C'$ when following $C'$ in one of the two directions.
These four vertices split $C'$ into four arcs.
A pair of opposite arcs together with the two chords constitute a cycle in $G$.
Take the pair of arcs with the longest total length.
This total length is at least $\frac{1}{2}|V(C')|\ge \delta(G-B)+\frac{k}{2}$.
Combining these two arcs with the two chords, we obtain a cycle of length at least $3\delta(G-B)+\frac{k}{2}-2>2\delta(G-B)+k$.
This cycle enters two Dirac component\xspace{s}, as each chord enters a Dirac component\xspace.
Hence, in this case the desired cycle exists and the lemma is proved.
\begin{figure}
\caption{The family of long chords of the cycle $C'$ that do not intersect graphically pairwise.}
\label{fig:long_chords}
\end{figure}
We now assume that no two chords of the family intersect graphically.
In this case we can arrange them in an order from left to right.
That is, we can choose a permutation $\pi \in S_{3k}$ and for each $i\in[3k]$ either a pair $a_i=w'_{\pi_i}$ and $b_i=v'_{\pi_i}$ or a pair $a_i=v'_{\pi_i}$ and $b_i=w'_{\pi_i}$ in such a way that following the cycle $C'$ starting from $a_1$ one will read the $6k$ vertices in the order $a_1, a_2, \ldots, a_{3k}, b_{3k}, b_{3k-1}, \ldots, b_1$ (see Figure~\ref{fig:long_chords}).
It is important to note that $a_i={a_{i+1}}$ or $b_i=b_{i+1}$ might hold true for any $i\in [3k-1]$,
but at least one of $a_i\neq a_{i+1}$ and $b_i\neq b_{i+1}$ always holds.
Take an arbitrary $i\in[3k]$.
The chord between $a_i$ and $b_i$ splits $C'$ into two arcs.
We call them \emph{left arc of $(a_i,b_i)$}, that is, the arc that contains vertices $a_1,b_1,a_2,b_2,\ldots,a_i,b_i$ and the \emph{right arc of $(a_i,b_i)$}, that is, the arc that contains vertices $a_i,b_i,a_{i+1},b_{i+1},\ldots,a_{3k},b_{3k}$.
If at least one of these arcs has length at least $\delta(G-B)+k+1$, we call the chord between $a_i$ and $b_i$ a \emph{good} chord.
Then the chord together with the longer arc constitute a cycle of length at least $2\delta(G-B)+k$.
This cycle enters a Dirac component\xspace, so the proof is complete if there is a good chord in the constructed family of chords.
We now show that there exists at least one good chord in the family.
If the chord between $a_1$ and $b_1$ is good, then we are done.
Otherwise, both arcs of $(a_1,b_1)$ are of length at most $\delta(G-B)+k$.
Since the length of $C'$ is at least $2\delta(G-B)+k$, it follows that the length of both these arcs is at least $\delta(G-B)$.
Consider the chord between $a_2$ and $b_2$.
Note that the left arc of $(a_2,b_2)$ is longer than the left arc of $(a_1,b_1)$ because $(a_1,b_1)\neq (a_2,b_2)$.
Hence, the length of the left arc of $(a_2,b_2)$ is at least $\delta(G-B)+1$.
Analogously, we can show that for each $i\in[3k]$ the length of the left arc of $(a_i,b_i)$ is at least $\delta(G-B)+i-1$.
Hence, for any $j \in [2k]$ the chord between $a_{k+j}$ and $b_{k+j}$ is a good chord.
The proof of the lemma is complete. \end{proof}
Finally, we state and prove the main theorem of this section.
\begin{theorem}\label{thm:cyclebanana}
Let $(G,B,k)$ be a given instance of \pname{Long Dirac Cycle}\xspace.
There is an algorithm that, given a cycle $C$ in $G$ and two paths $P_1, P_2$ that induce an Dirac decomposition\xspace for $C$ and $B$ in $G$, in time $2^{\mathcal{O}{(k+|B|)}}\cdotn^{\mathcal{O}(1)}$ either
\begin{itemize}
\item Solves $(G,B,k)$, or
\item Finds a cycle longer than $C$ in $G$.
\end{itemize} \end{theorem} \begin{proof}
The algorithm considers several cases.
If the given cycle $C$ is of length at least $2\delta(G-B)+k$, then the algorithm correctly determines that $(G,B,k)$ is a yes-instance.
Hence, we assume that $|V(C)|< 2\delta(G)+k$.
From now on, we also assume that $6k+4|B|+6 < \delta(G-B)$, otherwise the algorithm solves $(G,B,k)$ using the algorithm from Proposition~\ref{prop:longest_cycle}.
Suppose now that $G$ contains a cycle $C'$ of length at least $2\delta(G-B)+k$.
We show how the algorithm finds \emph{some} cycle of length at least $2\delta(G-B)+k$ in $G$ or enlarges $C$ provided that $C'$ exists.
We are now interested in the set $X=V(C')\cap (V(P_1)\cup V(P_2))$.
Depending on its cardinality, there are several cases.
In most of the cases, it is possible for the algorithm to replace one arc of $C$ with a longer arc that is found using the algorithm for \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace given by Theorem~\ref{thmEG}.
The algorithm is usually applied to a component in $G-V(P_1\cup P_2)$ with the goal of finding a path of length at least $\delta(G-B)+k/2$.
Since $\delta(G-(V(P_1\cup P_2)\cup B))> \delta(G-B)-k-4$, the running time of the algorithm is still bounded by $2^{\mathcal{O}(k+|B|)}\cdotn^{\mathcal{O}(1)}$.
\noindent\textbf{Case 1:} $|X|=0$.
Then $C'$ is completely contained in some connected component $H$ of $G-V(P_1\cup P_2)$.
This component cannot be contained in $B$, since $|B| < \delta(G-B)$.
So after the $B$-refinements, this component is a component $H'$ of type \ref{enum:cycle_tunnel_path_bic}, \ref{enum:cycle_tunnel_path_cut_left} or \ref{enum:cycle_tunnel_path_cut_right}.
Note that only the leaf-blocks that have all inner vertices in $B$ are contracted in $H$ to obtain $H'$.
The cycle $C'$ does not pass through cut vertices.
Moreover, its length is greater than $B$.
Hence, no edge of $C'$ is contracted during the $B$-refinements of $H$, so $C'$ is fully contained in $H'$.
By the last property of Dirac decomposition\xspace{s}, there are exactly two connected components in $G-(V(P_1)\cup V(P_2))$ containing vertices of $C$.
Both of them contain at most $\delta(G-B)+k+|B|$ vertices, as the length of $C$ is less than $2\delta(G-B)+k$.
Hence, $H'$ does not share any vertices with the initial cycle $C$, as $|V(H')|\ge |V(C')|\ge 2\delta(G-B)+k$.
If $H'$ is of type \ref{enum:cycle_tunnel_path_bic}, then it contains a path of length at least $|V(C')|/2$ between any pair of vertices by \Cref{lemma:biconnected_cycle_to_any_path}.
Take any pair $(s,t)$ of neighbours of $H'$ in $V(P_1)$ and $V(P_2)$ respectively.
There is an $(s,t)$-path in $G$ of length at least $|V(C')|/2+2>\delta(G-B)+k/2$ that contains only vertices in $V(H')\cup B$ as internal vertices.
One of the arcs of $C$ between $s$ and $t$ have length less than $\delta(G-B)+k/2$, so it can be replaced with the obtained $(s,t)$-path, making $C$ longer.
If $H'$ is of type \ref{enum:cycle_tunnel_path_cut_left}, then it is not $2$-connected.
Denote the only neighbour of $H'$ in $V(P_1)$ by $s$.
Note that the graph $G'[V(H')\cup \{s\}]$ is $2$-connected and still contains the cycle $C'$.
Hence, it contains a path of length at least $|V(C')|/2$ between any pair of vertices.
Now take any neighbour of $H'$ in $V(P_2)$, say $t$.
It is easy to obtain an $(s,t)$-path in $G$ of length at least $|V(C')|/2+1$ going only through vertices in $V(H')\cup B$.
Again, this path is a replacement for one of the arcs between $s$ and $t$ in $G$.
The case of type \ref{enum:cycle_tunnel_path_cut_right} is symmetrical.
\textbf{Conclusion of Case 1.}
To handle this case, the algorithm unconditionally iterates over all components in $G-V(P_1\cup P_2)$ and tries to find a suitable path of length at least $\delta(G-B)+k/2$ in a $2$-connected subgraph of $G$ using the algorithm of Theorem~\ref{thmEG} for \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace.
Note that a subgraph picked by the algorithm is always a graph $H'$ with $\delta(H'-B')\ge \delta(G-B'-(V(P_1)\cup V(P_2))$, where $|B'|\le |B|+1$.
Hence, $\delta(H'-B')\ge \delta(G-B)-(k+5)$, so the algorithm for \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace always runs in $2^{\mathcal{O}(k+|B|)}\cdotn^{\mathcal{O}(1)}$ running time.
If a suitable path is found, $C$ is made longer by the algorithm, and the algorithm outputs the longer cycle and terminates.
Otherwise, there are no long cycles $C'$ with $|X|=0$ in $G$.
\noindent\textbf{Case 2.}
$|X|=1$.
Denote the only vertex in $X$ by $v$.
Note that $C'$ passes through only one connected component in $G-V(P_1\cup P_2)$, since $C'-v$ is a path having no common vertices with $P_1$ or $P_2$.
Denote the component containing $C'-v$ in $G-V(P_1\cup P_2)$ by $H$.
We know that $H$ consists of at least $2\delta(G-B)+k-1$ vertices, so it is not fully contained in $B$ and does not share any vertex with the initial cycle $C$, just as in the previous case.
Without loss of generality, assume that $v \in V(P_1)$.
Denote by $H'$ the connected component $H$ after the $B$-refinements.
Independently of the type of $H'$ in the Dirac decomposition\xspace, there is a vertex in $H$ with a neighbour in $V(P_2)$.
Hence, there is a path starting in a certain vertex $u \in V(P_2)$ and going to a certain vertex $z \in V(C'-v)$ through $H$ in $G$.
Take the longer arc between $z$ and $v$ on $C'$ and combine it with the path between $u$ and $z$.
The obtained path is of length at least $|V(C')|/2+1$ and is a replacement for the shorter arc between $u$ and $v$ on $C$.
The only obstacle here is that $H$ is not necessarily $2$-connected.
However, the graph $G[V(H)\cup \{v\}]$ still contains the whole cycle $C'$.
If $H'$ is of type \ref{enum:cycle_tunnel_path_bic}, then $G[V(H)\cup \{v\}]$ is necessarily $2$-connected after $B$-refinements.
If $H'$ is of type $\ref{enum:cycle_tunnel_path_cut_left}$, then $G[V(H)\cup \{v\}]$ also becomes $2$-connected after $B$-refinements, as $v$ is the only neighbour in $V(P_1)$ connecting all leaf-blocks of $H'$ together.
In either of the two cases, if $z$ is fixed, the path between $v$ and $z$ can be found using the algorithm for \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace.
Finally, if $H'$ is of type \ref{enum:cycle_tunnel_path_cut_right}, then $u$ is the only neighbour of $H'$ in $V(P_2)$ after the $B$-refinements.
Then the graph $G[V(H)\cup\{u,v\}]$ is necessarily $2$-connected after $B$-refinements and the desired $(u,v)$-path can be found inside it.
\textbf{Conclusion of Case 2.}
The algorithm iterates over all suitable pairs of $v$ and $H$, and iterates over all possible options of $z$ or $u$ when necessary.
When this triple is fixed, it is left to apply the algorithm of Theorem~\ref{thmEG} to the corresponding $B$-refinement as described above.
Again, in time $2^{\mathcal{O}(k+|B|)}\cdotn^{\mathcal{O}(1)}$ our algorithm either makes the initial cycle $C$ longer and stops or correctly determines that no long cycle $C'$ with $|X|=1$ exists.
\noindent\textbf{Case 3:}
$|X|=2$.
Let $X=\{s,t\}$.
Starting from this case, we need to consider Dirac component\xspace{s} that $C'$ enters.
By Lemma~\ref{lemma:dirac_cycle_edge_of_banana}, we can assume that $C'$ enters some Dirac component\xspace $M$.
The cycle $C'$ has two arcs between $s$ and $t$.
At least one of them enters $M$ and, by Lemma~\ref{lemma:dirac_cycle_banana_consecutive}, we know that all vertices of $M$ appear consecutively on this arc.
Suppose that both arcs between $s$ and $t$ enter $M$.
If both $s,t \in V(P_1)$ or $s,t\in V(P_2)$, then we obtain a matching of size two between $V(P_i)$ and a Dirac component\xspace, which is not possible by the definition of Dirac decomposition\xspace.
Hence, we can assume that $s \in V(P_1)$ and $t \in V(P_2)$.
Since both arcs enter $M$, there is a connected component $H$ in $G-V(P_1\cup P_2)$ that contains $M$ and both arcs of $C'$.
Thus $C'$ is contained in $G[V(H)\cup\{s,t\}]$.
After the $B$-refinements, $G[V(H)\cup\{s,t\}]$ also contains the whole cycle $C'$ similarly to the arguments above.
Note that after the $B$-refinements this graph is $2$-connected, since if $H'$ is of type \ref{enum:cycle_tunnel_path_cut_left} or of type \ref{enum:cycle_tunnel_path_cut_right}, the vertex $s$ or the vertex $t$ correspondingly is the vertex connecting all its leaf-blocks together.
Hence, the algorithm can look for an $(s,t)$-path of length at least $\delta(G-B)+k/2$ inside the graph $G[V(H)\cup\{s,t\}]$ with applied $B$-refinements.
The component $H$ contains at least $2\delta(G-B)+k-2$ vertices, so it does not share vertices with $C$.
Thus, this $(s,t)$-path is a suitable replacement for a shorter arc between $s$ and $t$ on $C$.
Suppose now that only one arc of $C'$ between $s$ and $t$ enters $M$.
Then, by Lemma~\ref{lemma:dirac_cycle_banana_consecutive}, we can be sure that \emph{all} vertices of $M$ appear consecutively on $C'$.
That is, there are two vertices $u,v \in V(M)\cap V(C')$ such that one of the arcs of $C'$ between $u$ and $v$ is a $(u,v)$-path inside $M$, and the other arc is a $(u,v)$-path in $G$ that does not contain any vertex of $M$ as internal vertex.
In this case, the algorithm can find these two arcs in the following way.
When $s,t$ and $M$ are fixed, the algorithm iterates over all pairs of distinct vertices $u, v \in V(M)$.
Firstly, the algorithm tries to find a path between $u$ and $v$ outside $M$.
Since there is a path of length at least $\delta(G-B)-2$ between any pair of vertices in $M$, the outer path length $\delta(G-B)+k+2$ is sufficient to construct a cycle of length $2\delta(G-B)+k$ in $G$.
Hence, to find the $(u,v)$-path outside $M$, the algorithm removes all vertices in $V(M)\setminus\{u,v\}$ from $G$, and adds a single edge between $u$ and $v$ in $G$.
Note that if the outer path between $u$ and $v$ exists, then $G$ remains $2$-connected after these operations, since $u$ and $v$ still belong to the same cycle.
If $G$ is not $2$-connected, then the choice of $u$ and $v$ was wrong.
Otherwise, we apply the algorithm of Theorem~\ref{thmEG} to the changed graph $G$ to find a long path between $u$ and $v$.
If a $(u,v)$-path of length at least $\delta(G-B)+k+2$ exists, then $(G,B,k)$ is a yes-instance.
Otherwise, the algorithm finds the longest path between $u$ and $v$.
This is done in $2^{\mathcal{O}(k+|B|)}\cdotn^{\mathcal{O}(1)}$ time.
Note that a $(u,v)$-path of length at least $\delta(G-(V(P_1)\cup V(P_2)\cup B\cup \{u,v\}))\ge \delta(G-B)-k-6$ always exists in the modified graph $G$ by \Cref{thm:relaxed_st_path}.
If the outer $(u,v)$-path is found, it is left for us to find a long path between $u$ and $v$ inside $M$.
If this path is of length at least $\delta(G-B)+2k+6$, then $(G,B,k)$ is a yes-instance of \pname{Long Dirac Cycle}\xspace, since the outer $(u,v)$-path is of length at least $\delta(G-B)-k-6$.
Thus, using the algorithm for \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace, we either find a sufficiently long path between $u$ and $v$ inside $M$, such that the total length of this path and the outer path is at least $2\delta(G-B)+k$, or conclude that none exists and move on to the next choice of $u$ and $v$.
\textbf{Conclusion of Case 3.}
To handle this case, the algorithm iterates over all possible pairs of $s$ and $t$.
To handle the case when $C'$ enters just one connected component of $G-V(P_1\cup P_2)$, the algorithm behaves similarly to previous cases.
Additionally, to handle the case when $C'$ contains a consecutive path inside a Dirac component\xspace, the algorithm iterates over all possible Dirac component\xspace{s} $M$, and pairs $u,v \in V(M)$ and tries to construct a long cycle using two calls to the algorithm for \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace.
\textbf{Case 4.}
$|X|\ge 3$.
Then $X$ contains three distinct vertices $v_1, v_2, v_3$.
These vertices split $C'$ into three arcs $A_1, A_2, A_3$, where $A_1$ is the arc between $v_1$ and $v_2$ that does not contain $v_3$, $A_2$ is the arc between $v_2$ and $v_3$ that does not contain $v_1$, and $A_3$ is the arc between $v_3$ and $v_1$ that does not contain $v_2$.
By Lemma~\ref{lemma:dirac_cycle_edge_of_banana}, we can assume that $C'$ enters a Dirac component\xspace $M$.
Without loss of generality, assume that $A_1$ enters $M$.
By Lemma~\ref{lemma:dirac_cycle_banana_consecutive}, all vertices of $M$ appear consecutively on this arc.
\begin{claim}
$A_2$ and $A_3$ do not contain any vertex of $M$.
\end{claim}
\begin{claimproof}
Take the arc between $v_1$ and $v_3$ that contains $v_2$, i.e.\ the union of $A_1$ and $A_2$.
We now that this arc enters $M$, so by Lemma~\ref{lemma:dirac_cycle_banana_consecutive} all vertices of $M$ appear consecutively on it.
But $A_1$ contains at least two vertices of $M$.
Hence, $A_2$ cannot contain any vertex of $M$, as $v_2 \notin V(M)$ separates $A_1$ and $A_2$ on the arc between $v_1$ and $v_3$.
To show that $A_3$ does not contain any vertex of $M$, take the arc between $v_3$ and $v_2$ that contains $v_1$, i.e.\ the union of $A_3$ and $A_1$.
Again, by Lemma~\ref{lemma:dirac_cycle_banana_consecutive} this arc contains vertices of $M$ consecutively, but $v_1$ divides $A_3$ and $A_1$ on the arc.
Since $A_1$ contains at least two vertices of $M$, $A_3$ cannot contain any of them.
\end{claimproof}
The claim shows that vertices of $M$ induce an arc of $C'$, similarly to the second part of Case 3.
Hence, this case can be handled by the algorithm in exactly the same way as in Case 3.
\textbf{Conclusion of Case 4.}
To cover this case, our algorithm first fixes $v_1, v_2 \in V(P_1\cup P_2)$.
Then it iterates over all Dirac component\xspace{s} of the Dirac decomposition\xspace and tries to combine a long cycle from two paths, one inside the Dirac component\xspace, and one outside.
This is done in exactly the same way as in the second part of Case 3.
The list of cases is exhaustive, so if $C'$ exists, our algorithm enlarges the initial cycle $C$ or finds a cycle of length at least $2\delta(G-B)+k$ in $G$, determining that $(G,B,k)$ is a yes-instance.
If $C'$ does not exist, the algorithm does not find any long arc or long cycle in $G$, and safely decides that $(G,B,k)$ is a no-instance.
This concludes the proof. \end{proof}
\section{Long Dirac Cycle: Putting all together}\label{sec:longDC}
In this section we finalize the proof of \Cref{theorem:main} by combining the main results of previous sections. This relies crucially on the following lemma. The most important part of this lemma is the construction of a Dirac decomposition\xspace.
\begin{lemma}\label{lemma:main_cycle_lemma}
Let $G$ be an $n$-vertex $2$-connected graph, $B\subseteq V(G)$, and $k$ be an integer such that $0< k \le \frac{1}{24}\delta(G-B)$, and
\[2k+2|B|+12\leq \delta(G- B)< \frac{n}{2}-\frac{|B|+k}{2}.
\]
Then there is an algorithm that, given a cycle $C$ of length less than $2\delta(G- B)+k$ in polynomial time finds either
\begin{itemize}
\item Longer cycle in $G$, or
\item Vertex cover of $G-B$ of size at most $\delta(G-B)+2k$, or
\item Two paths $P_1, P_2$ that induce a Dirac decomposition\xspace for $C$ and $B$ in $G$.
\end{itemize} \end{lemma}
Before proceeding with the proof of the lemma, we show how to use it for the proof of \Cref{theorem:main}.
\subsection{Proof of \Cref{theorem:main}}
We combine the main results of Sections \ref{sec:vcalgo}, \ref{sec:HamCycles}, \ref{sec:bananas}, and \Cref{lemma:main_cycle_lemma}. Let $(G, B, k)$ be an instance of \pname{Long Dirac Cycle}\xspace. First we consider the cases that do not fit the conditions of \Cref{lemma:main_cycle_lemma}.
If $\delta(G-B)<12$ or if $24 k > \delta(G-B)$, we can find a cycle of length at least $2\delta(G-B)+k> 48k+24$ in time $2^{\mathcal{O}(k)}\cdotn^{\mathcal{O}(1)}$ by calling the algorithm for \textsc{Longest Cycle} from \Cref{prop:longest_cycle}.
If $2k+2|B|+12> \delta(G- B)$, we have that $5k+4|B|+48> 2\delta(G- B)+k$. By \Cref{prop:longest_cycle}, a cycle of length at least $5k+4|B|+48$ could be found in time $2^{\mathcal{O}(k+|B|)}\cdotn^{\mathcal{O}(1)}$. If
$\delta(G-B)\ge \frac{n}{2}-\frac{|B|+k}{2}$, we apply the algorithmic results of Section~\ref{sec:HamCycles}.
We put $k'=\max\{|B|,\frac{|B|+k}{2}\}$, and obtain that $\delta(G-B)\ge \frac{n}{2}-k'$ for $|B|\le k'$. We apply Theorem~\ref{theorem:hamiltonian} for $G, B$ and $k'$. Then the problem is solvable in time $2^{\mathcal{O}(k')}\cdotn^{\mathcal{O}(1)}=2^{\mathcal{O}(k+|B|)}\cdotn^{\mathcal{O}(1)}$.
From now we assume that $k$ and $B$ satisfy the conditions of \Cref{lemma:main_cycle_lemma}. In particular, now $\min\{2\delta(G-B),n-|B|\}=2\delta(G-B)$, so we are looking for a cycle of length at least $2\delta(G-B)+k$ for $k\ge 0$. By \Cref{lemma:main_cycle_lemma} in polynomial time we either find a longer cycle, a vertex cover, or a Dirac decomposition\xspace of $G$. If a longer cycle is found and the length of this cycle is still less than $2\delta(G-B)+k$, we call
\Cref{lemma:main_cycle_lemma} with the longer cycle.
If a vertex cover of $G-B$ of size at most $\delta(G-B)+2k$ is found, then the vertex cover of $G$ is at most $\delta(G-B)+2k +|B|.$ We
apply Theorem~\ref{thmVCad} to solve the problem in time $2^{\mathcal{O}(k+|B|)}\cdotn^{\mathcal{O}(1)}$. Finally, if a Dirac decomposition\xspace for $C$ and $B$ is found in $G$, we use Theorem~\ref{thm:cyclebanana} to solve $(G,B,k)$ in running time single-exponential in $k+|B|$ or find a longer cycle in $G$ and repeat the application of \Cref{lemma:main_cycle_lemma}.
The proof of Theorem~\ref{theorem:main} (up to the proof of \Cref{lemma:main_cycle_lemma}) is complete. \qed
\subsection{Last piece: proof of \Cref{lemma:main_cycle_lemma}} The remaining part of the section is devoted to the postponed proof of \Cref{lemma:main_cycle_lemma}.
\begin{proof}[Proof of \Cref{lemma:main_cycle_lemma}] The proof is algorithmic. We try
to replace an arc of $C$, that is, a path in $C$, with a path in $G- V(C)$.
This process of \emph{enlarging} $C$ is similar to the process of enlarging a path in Lemma~\ref{lemma:st_path_or_tunnel}.
We consider connected components $H$ in $G-V(C)$ that contain at least one vertex in $V(G)\setminus B$.
Note that at least one such component exists since by the conditions of the lemma, $V(G-V(C)-B)\ge n-(2\delta(G-B)+k-1)-|B|>1$.
To simplify our job, we first apply $B$-refinements to all connected components in $G-B$.
Without loss of generality, we assume that $G$ is a graph with all possible $B$-refinements applied, i.e., $\gbref{B}{H}=G$ for any connected component $H$ in $G-V(C)$ with $V(H)\not\subseteq B$.
Note that this assumption preserves all resulting points of the lemma statement: if a longer cycle, or a vertex cover, or a Dirac decomposition\xspace is found for the graph with applied $B$-refinements, they can be easily restored in the original graph.
Similarly to the proof of Lemma~\ref{lemma:st_path_or_tunnel}, we consider several cases depending on the structure of a connected component $H$ with $V(H)\not\subseteq B$.
The difference is that isolated vertices in $G-V(C)$ now do not lead to an immediate enlargement of $C$.
However, we show that they contribute to a construction of a vertex cover of $G-B$.
In what follows we prove the following. If there is a component $H$ with $G-V(C)$ with exactly two vertices, then cycle $C$ can be always enlarged. If there is a component $H$ with at least $3$ vertices, call it a large component, then either $C$ can be enlarged, or $H$ has a very special structure. The special structure of large components is used twice. First, we show that if there is at least one single-vertex component and at least one large component, then $C$ can be enlarged. Thus if we cannot enlarge $C$, it means that either $G-V(C)$ is an independent set or all components are large. In the first case, we prove that the vertex cover of $G-B$ is at most $\delta(G-B)+2k$. In the second case, the structural properties of large components are used to construct a Dirac decomposition\xspace.
We start with two claims that will be used in several places of the proof. The first claim shows that if there is a pair of distant consecutive neighbors of a vertex $h\not\in V(C)$ in $C$, then $C$ can be enlarged.
\begin{claim}\label{claim:dirac_cycle_isolated_close_neighbors}
Let $h \in V(G)\setminus V(C)$ be a vertex with at least $\delta(G-B)-2$ neighbors in $V(C)$ and such that there is a pair of neighbors $u,v$ of $h$ on $C$ such that one of the $(u,v)$-arcs is of length at least $8k$ containing no other neighbors of $h$. Then $C$ can be enlarged in polynomial time.
\end{claim}
\begin{claimproof}
Suppose that there are two neighbors of $h$, say $u, v \in V(C)$ such that one arc of $C$ between $u$ and $v$ is of length at least $8k$ and does not contain any neighbor of $h$. Hence
the other arc between $u$ and $v$ contains all neighbors of $h$ on $C$. Moreover, since the length of $C$ is at most $2\delta(G-B)+k-1$, the length of this arc is at most $2\delta(G-B)-7k-1$.
There are at least $\delta(G-B)-2$ neighbors of $h$ on $C$.
Since $2(\delta(G-B)-3)>2\delta(G-B)-7k-1$, by the pigeonhole principle, there is a pair of neighbors of $h$ that are adjacent $C$.
Then $h$ can be inserted in $C$ between these neighbors so the length of $C$ increases by one.
\end{claimproof}
The following claim allows to eliminate the existence of large connected components in $G-V(C)$, when there are isolated vertices in $G-V(C)$.
This claim will be useful later in this proof. Recall that a chord of a cycle $C$ is a path connecting two vertices of $C$ and containing no other vertices of $C$.
\begin{claim}\label{claim:dirac_cycle_isolated_and_chord}
If there is a vertex $h \in V(G)\setminus V(C)$ with at least $\delta(G-B)$ neighbors in $V(C)$ and there is a chord of $C$ of length at least $16k$ that does not pass through $h$, then $C$ can be enlarged in polynomial time.
\end{claim}
\begin{claimproof}
By Claim~\ref{claim:dirac_cycle_isolated_close_neighbors}, we can assume that for every pair of neighbors $u,v$ of $h$ on $C$, each of the $(u,v)$-arcs is either of length less than $8k$ or contains other neighbors of $h$.
Let the endpoints of the chord be $c_1, c_2 \in V(C)$.
If the distance between $c_1$ and $c_2$ in $C$ is less than the length of the chord, then $C$ can be made longer by replacing an arc between $c_1$ and $c_2$ with the chord.
Otherwise, both arcs between $c_1$ and $c_2$ are of length at least $16k$.
Each of these two arcs should contain a neighbor of $h$ as an internal vertex. Select one of the two arcs between $c_1$ and $c_2$.
Let $v_1\neq c_1$ be the neighbor of $h$ that is closest to $c_1$ on this arc. Since there are no other neighbors of $h$ between $c_1$ and $v_1$, the
distance in $C$ between $c_1$ and $v_1$ is at most $8k$.
Analogously, take the other arc between $c_1$ and $c_2$ and let $v_2$ be the neighbor of $h$ on this arc that is closest to $c_2$, but is different from it.
Again, the distance between $c_2$ and $v_2$ is at most $8k$.
Now construct the following path between $v_1$ and $v_2$: go from $v_1$ to $c_2$ following the first arc, then go from $c_2$ to $c_1$ following the chord, then go from $c_1$ to $v_2$ following the second arc. See \Cref{fig:chord}.
This path contains all but at most $16k$ edges of the cycle $C$, since $c_i$ and $v_i$ are close to each other on $C$ for each $i \in \{1,2\}$.
Additionally, this path contains at least $16k$ edges of the chord.
Hence, the length of the constructed $(v_1,v_2)$-path is at least the length of the cycle $C$.
This path does not contain $h$, so adding two edges between $v_1$ and $h$ and between $h$ and $v_2$ to it, yields a cycle of length at least $|V(C)|+2$.
\begin{figure}
\caption{Rerouting through a chord.}
\label{fig:chord}
\end{figure}
\end{claimproof}
Depending on the number of vertices in a component $H$ of $G-V(C)$, we consider difference cases. We start with the simplest case.
\noindent\textbf{Case 1:}
\emph{At least one component $H$ consists of two vertices.} In this case we can always enlarge $C$ in polynomial time.
Let $V(H)=\{h_1,h_2\}$ for $h_1\neq h_2$. Then both $h_1$ and $h_2$ have at least $\delta(G-B)-1$ neighbors in $V(C)$ and are connected by an edge.
In this case, $C$ can be made longer in polynomial time.
We formulate this slightly more generally in the following claim.
\begin{claim}\label{claim:dirac_cycle_two_connected_isolated}
If there are two distinct vertices $h_1, h_2 \in V(G)\setminus V(C)$, each having at least $\delta(G-B)-1$ neighbors in $V(C)$, and that are connected by a path in $G-V(C)$, then the length of $C$ can be increased in polynomial time.
\end{claim}
\begin{claimproof}
Let $S$ be the set of neighbors of $h_1$ and $h_2$ in $V(C)$.
Let $a$ be the number of the common neighbors of $h_1$ and $h_2$ in $S$.
Then $|S|\ge 2\delta(G-B)-2-a$ and $S$ splits $C$ into at least $2\max\{a,\delta(G-B)-1\}-a$ arcs. If we have an arc of length $1$, we can always enlarge $C$ by inserting one or both of the $h_i$. Moreover, if one of the endpoints of an arc is a common neighbor of $h_1$ and $h_2$, then the length of this arc should be at least $3$. Indeed, if an arc having a common neighbor of $h_1$ and $h_2$ as its endpoint and is of length less than three, then we can insert a path between $h_1$ and $h_2$ and two boundary edges instead of this arc in $C$; thus $C$ becomes longer.
Therefore, if $C$ cannot be enlarged, its length is at least
$2(|S|-a)+3a$. By the conditions of the lemma, we have that $\delta(G-B)\geq 2k+12$.
If $a\geq \delta(G-B) -1$, then
\[2(|S|-a)+3a\geq 3a \geq 3(\delta(G-B) -1)> 2\delta(G-B)+k.
\] If $a< \delta(G-B) -1$, then
\begin{multline*}2(|S|-a)+3a=2|S|+a \geq 2(2\delta(G-B)-2-a)+a =4\delta(G-B)-a -8 > 2\delta(G-B)+k.\end{multline*}
In both cases, we have that the length of cycle $C$ is more than $ 2\delta(G-B)+k$. This contradicts our assumption that $|V(C)|< 2\delta(G-B)+k$.
\end{claimproof}
The next two cases consider the situation when a component $H$ of $G-V(C)$ contains at least 3 vertices. Then $H$ could be $2$-connected or it contains a cut-vertex.
\noindent\textbf{Case 2:} \emph{$H$ is $2$-connected.} We show that either we can enlarge $H$, or $H$ has very specific properties described in
\Cref{claim:dirac_cycle_matching} and \Cref{claim:dirac_cycle_matching_degree}. These properties will be used in handling isolated components and in constructing Dirac decomposition\xspace.
\begin{claim}\label{claim:dirac_cycle_matching}
Either the maximum size of a matching between $V(H)$ and $V(C)$ in $G$ is two, or $C$ can be enlarged in polynomial time.
\end{claim}
\begin{claimproof} Since $G$ is 2-connected, the maximum matching size between $V(H)$ and $V(C)$ is always at least $2$.
Suppose first that at most one vertex in $V(H-B)$ has neighbors in $V(C)$.
If such vertex exists, let $h \in V(H-B)$ be that vertex, otherwise let $h$ be an arbitrary vertex in $H-B$.
We know that $\delta(H-(B\cup \{h\}))\ge \delta(G-B)-1$, since $H$ is a connected component in $G-V(C)$.
We now claim that if there is a matching of size at least three between $V(H)$ and $V(C)$ in $G$, then $C$ can be made longer by replacing one of its arcs with a path in $H$.
By Theorem~\ref{thm:relaxed_st_path}, there is a path of length at least $\delta(H-(B\cup\{h\}))\ge \delta(G-B)-1$ between an arbitrary pair of vertices in $H$.
The endpoints of the matching in $V(C)$ split $C$ into at least three arcs.
If at least one of these arcs is of length less than $(\delta(G-B)-1)+2$, it can be replaced with a path in $H$ connecting corresponding endpoints of the matching.
Hence, if $C$ cannot be made longer, its length is at least $3\delta(G-B)+3$.
Since $|V(C)|<2\delta(G-B)+k<3\delta(G-B)+3$, we obtain that either $C$ can be made longer or the maximum matching size between $V(H)$ and $V(C)$ in $G$ is two.
Now we assume that at least two vertices in $V(H-B)$ have neighbors in $V(C)$.
Take the vertices $h_1, h_2 \in V(H-B)$ that have the most and the second most number of neighbors in $V(C)$.
Denote $n_i=|N_G(h_i)\cap V(C)|$ for each $i\in\{1,2\}$. Thus $n_1\geq n_2$.
By Theorem~\ref{thm:relaxed_st_path}, there is a path of length at least $\delta(H-(B\cup \{h_1\}))$ between $h_1$ and $h_2$ in $H$.
Let $t=\max\{\delta(H-(B\cup\{h_1\})), 1\}$.
Note that the path between $h_1$ and $h_2$ is of length at least $t$.
Assume that $\delta(H-(B\cup \{h_1\}))<\delta(G-B)-1-n_2$.
Then at least one vertex in $V(H-(B\cup \{h_1\}))$ has at most $\delta(G-B)-2-n_2$ neighbors in $V(H-(B\cup \{h_1\}))$.
Hence, it has at most $\delta(G-B)-1-n_2$ neighbors in $V(H-B)$.
All other neighbors of this vertex in $V(G-B)$ are from $V(C)$, so this vertex should have at least $n_2+1$ neighbors in $V(C)$.
This contradicts the choice of $h_2$ and $n_2$.
Thus, $t\ge \delta(G-B)-1-n_2$, or $n_2\ge \delta(G-B)-t-1$.
Denote by $S$ the set of all neighbors of $h_1$ and $h_2$ in $V(C)$, i.e.\ $S=(N_G(h_1)\cup N_G(h_2))\cap V(C)$.
Let $a$ be the number of common neighbors of $h_1$ and $h_2$ in $S$, i.e.\ $a=|N_G(h_1)\cap N_G(h_2)\cap S|$.
Observe that vertices in $S$ split $C$ into $|S|=n_1+n_2-a$ arcs.
Note that each arc is of length at least two, otherwise we enlarge $C$. Moreover, every arc whose endpoint is a common neighbor of $h_1$ and $h_2$ should have length at least $t+2$, because otherwise $C$ can be made longer.
Hence, $|V(C)|\ge 2|S|+at$.
Since $2\delta(G-B)+k> |V(C)|$, we have that \begin{multline*}
2\delta(G-B)+k> 2(n_1+n_2-a)+at \geq
4n_2-2a+at \geq 4(\delta(G-B)-t-1)+a(t-2). \end{multline*}
Therefore,
\begin{multline*}
k>2\delta(G-B)-4t-4+12-12+a(t-2)>2\delta(G-B)-4(t-2)+a(t-2)-12,
\end{multline*}
and hence
\[
(4-a)(t-2)>2\delta(G-B)-k-12. \]
In particular, $(4-a)(t-2)>0$.
If $t=1$, then $a>2\delta(G-B)-k-8$.
But $|V(C)|\ge 2|S|+at\ge 2a+at=a(t+2)\ge 3a$, so $3a<2\delta(G-B)+k$.
It follows that $3(2\delta(G-B)-k-8)<2\delta(G-B)+k$, or $4\delta(G-B)<4k+24$, which contradicts the assumptions of the lemma.
Thus $t\neq 1$.
Since $t-2\neq 0$, we obtain that $t>2$, and, consequently, $a<4$.
Then $3(t-2)\ge (4-a)(t-2)>2\delta(G-B)-k-12$, or $3t>2\delta(G-B)-k-6$.
It yields that $t\geq\frac{1}{2}\delta(G-B)$.
Assume now that there is a matching in $G$ between $V(H)$ and $V(C)$ of size three.
Let $c_1, c_2, c_3$ be the endpoints of this matching in $V(C)$, and $v_1, v_2, v_3$ be the corresponding endpoints in $V(H)$.
Without loss of generality, we assume that $v_1=h_2$, as if $h_2 \notin \{v_1,v_2,v_3\}$ we can always change the matching to include the vertex $h_2$.
Denote by $T$ the set of all neighbors of $v_1, v_2$ and $v_3$ in $V(C)$, i.e.,\ $T=N_G(\{v_1,v_2,v_3\})\cap V(C)$.
Note that $|T|\ge |N_G(v_1)\cap V(C)|=n_2\ge \delta(G-B)-t-1$.
Unless $C$ can be made longer, the vertices of $T$ split $C$ into $|T|$ arcs of length at least two.
Additionally, at least three arcs (the arcs that are incident to $c_1,c_2,c_3\in T$) should be of length at least $t+2$, as there is a path of length at least $t$ between $v_i$ and $v_j$ in $H$ for any $i\neq j$.
We obtain that $|V(C)|\ge 2|T|+3t\ge 2\delta(G-B)+t-2>2\delta(G-B)+k$ unless $C$ can be made longer.
\end{claimproof}
\begin{claim}\label{claim:dirac_cycle_matching_degree}
Either between any pair of vertices in $H$ there is a path in $H$ of length at least $\delta(G-B)-2$, or $C$ can be made longer in polynomial time.
\end{claim}
\begin{claimproof}
The proof is identical to the proof of \Cref{claim:eg_path_bic_degree}.
\end{claimproof}
\noindent\textbf{Case 3:} \emph{$|V(H)|\ge 3$ and $H$ contains a cut-vertex.}
Since $H$ contains a cut-vertex, it contains at least two leaf-blocks.
Denote the leaf-blocks of $H$ by $L_1, L_2, \ldots, L_p$ and their respective cut-vertices by $c_1, c_2, \ldots, c_p$, where $p\ge 2$.
Since $L_i$ is $2$-connected or $|V(L_i)|=2$, we can proceed similarly to Case~2 with $L_i$ and $B\cup \{c_i\}$ instead of $H$ and $B$, and make $C$ longer or conclude that the maximum matching size between $V(L_i)$ and $V(C)$ in $G$ is at most two.
We now assume that for each $i\in[p]$ the maximum matching size between $V(L_i)$ and $V(C)$ is at most two.
Then for any $i\in[p]$, accordingly to \Cref{claim:dirac_cycle_matching_degree} applied to $L_i$ and $B\cup\{c_i\}$ instead of $H$ and $B$, we obtain that there is a path of length at least $\delta(G-(B\cup\{c_i\})-2\ge\delta(G-B)-3$ between any pair of vertices in $L_i$, if $|V(L_i)|> 2$.
\begin{claim}\label{claim:dirac_cycle_leaf_block_single_neighbor}
$\left|\bigcup_{i=1}^p N_G(V(L_i-\{c_i\}))\right|=1$, or $C$ can be made longer in polynomial time.
\end{claim}
\begin{claimproof}
We first show that if there exists $i \in [p]$ with $|V(L_i)|=2$, then $C$ can be made longer in polynomial time.
Assume that there exists $L_i$ with $|V(L_i)|=2$.
Then $V(L_i)=\{u, c_i\}$ for some vertex $u\neq c_i$.
As $\gbref{B}{H}=G$, it is true that $u\notin B$.
Hence, $u$ has at least $\delta(G-B)-1$ neighbors in $V(C)$.
If $u$ has two consecutive vertices of $C$ as neighbors, then $C$ can be made longer with inserting $u$.
Now take $j \in [p]\setminus\{i\}$ and consider the leaf-block $L_j$.
If $|V(L_j)|=2$, then $V(L_j)=\{u',c_j\}$, where $u'$ has at least $\delta(G-B)-1$ neighbors in $V(C)$.
Note that $u$ and $u'$ are connected by a path in $G-V(C)$.
By \Cref{claim:dirac_cycle_two_connected_isolated}, $C$ can be made longer in polynomial time in this case.
If $|V(L_j)|>2$, then $L_j$ is $2$-connected, so there is a path of length at least $\delta(G-B)-3$ between any pair of vertices in $L_j$.
Hence, each inner vertex of $L_j$ is connected with $u$ by a path of length at least $\delta(G-B)-2$.
Take a vertex $u'\in V(L_j-\{c_j\})$ that has a neighbor $v' \in V(C)$.
By \Cref{claim:dirac_cycle_isolated_close_neighbors}, there is a vertex $v \in V(C)$ that is a neighbor of $u$ and is on a distance at least one and at most $8k$ from $v'$ on $C$.
We obtain a $(v,v')$-chord of $C$ that is of length at least $\delta(G-B)$ but the distance between $v$ and $v'$ on $C$ is at most $8k<\delta(G-B)$.
Hence, $C$ can be made longer in polynomial time.
We now assume that $|V(L_i)|\ge 3$ for each $i \in [p]$.
Then there is a path of length at least $\delta(G-B)-3$ for any pair of vertices in any $L_i$.
Assume that $\bigcup_{i=1}^p N_G(V(L_i-\{c_i\})) \supseteq \{v_1, v_2\}$, where $v_1, v_2 \in V(C)$ and $v_1 \neq v_2$.
Then either there exist $i\neq j$ such that $L_i-\{c_i\}$ contains a neighbor of $v_1$ and $L_j-\{c_j\}$ contain a neighbor of $v_2$, or there only exists $i$ such that $L_i-\{c_i\}$ contains both a neighbor of $v_1$ and a neighbor of $v_2$.
In the latter case, we can pick $j\neq i$ and $v_3\in V(C)$ with $v_3\neq v_1$ or $v_3 \neq v_2$ such that $L_j-\{c_j\}$ contains a neighbor of $v_3$.
Thus, without loss of generality we assume that $L_1-\{c_1\}$ contains a neighbor $u_1$ of $v_1 \in V(C)$ and $L_2-\{c_2\}$ contains a neighbor $u_2$ of $v_2 \in V(C)$ and $v_1 \neq v_2$.
Observe that there exists a $(u_1,u_2)$-path in $H$ of length at least $2\delta(G-B)-6$.
Hence, this path can be prolonged to a $(v_1, v_2)$-chord of $C$ of length at least $2\delta(G-B)-4$.
Note that at least one of $(v_1, v_2)$-arcs of $C$ is of length at most $\delta(G-B)-\frac{k-1}{2}<2\delta(G-B)-4$, so $C$ can be made longer in polynomial time.
\end{claimproof}
The following claim shows that $H$ yields at least one long chord of $C$.
\begin{claim}\label{claim:dirac_cycle_separable_long_path}
Either for any $i \in [p]$ and any $u \in V(L_i - c_i)$, $v \in V(H)\setminus u$, there is a $(u,v)$-path of length at least $\delta(G-B)-2$ in $H$, or $C$ can be made longer in polynomial time.
\end{claim}
\begin{claimproof}
Take $i \in [p]$.
From \Cref{claim:dirac_cycle_leaf_block_single_neighbor} follows that $\delta(L_i-(B\cup\{c_i\}))\ge \delta(G-B)-2$, as each vertex in $V(L_i-c_i)$ has at most one neighbour outside $L_i$.
By \Cref{thm:relaxed_st_path}, there is a path of length at least $\delta(G-B)-2$ between any pair of vertices inside $L_i$.
Take $u \in V(L_i-c_i)$ and $v \in V(H)\setminus u$.
If $v \in V(L_i)$, then we are done.
If $v$ is outside $L_i$, then a path between $u$ and $v$ should go through $c_i$.
Since $u\neq c_i$, there is a $(u,c_i)$-path of length at least $\delta(G-B)-2$ inside $L_i$.
Combine this path with any $(c_i,v)$-path outside $L_i$ in $H$ to obtain the required $(u,v)$-path.
\end{claimproof}
\noindent\textbf{Case 4:} \emph{At least one component $H$ of $G-V(C)$ consists of one vertex.} In this case we show that either we can enlarge $C$ in polynomial time, or construct a vertex cover of $G-B$ of size at most $\delta(G-B)+2k$.
Let $V(H)=\{h\}$ for some vertex $h\in V(G-B)$.
All neighbors of $h$ are from $V(C)$, so $h$ has at least $\delta(G-B)$ neighbors in $V(C)$.
We first claim that if $G-V(C)$ contains both an isolated vertex and some non-isolated connected component, then we can make $C$ longer.
\begin{claim}\label{claim:dirac_cycle_isolated_and_non_isolated}
Let $H_1$ and $H_2$ be two connected components in $G-V(C)$ with $V(H_i)\not\subseteq B$.
If $|V(H_1)|=1$ and $|V(H_{2})|\neq 1$, then $C$ can be made longer in polynomial time.
\end{claim}
\begin{claimproof}
We can assume that $|V(H_2)|\ge 3$, so $V(H_2)$ is either $2$-connected or contains a cut-vertex.
In both of the cases, by \Cref{claim:dirac_cycle_matching_degree} and \Cref{claim:dirac_cycle_separable_long_path}, we can find a chord of $C$ of length at least $\delta(G-B)-2> 16k$ that passes through $H_2$.
By \Cref{claim:dirac_cycle_isolated_and_chord} , the single vertex of $H_1$ and the chord passing through $H_2$ help making $C$ longer in polynomial time.
\end{claimproof}
By \Cref{claim:dirac_cycle_isolated_and_non_isolated}, we can assume that if there is one connected component of $G-V(C)$ which is an isolated vertex, then all other components are also isolated vertices.
Our next step is to to show that if an isolated vertex exists, then we can find a large independent set in $C$ that has no neighbors outside $C$.
For an isolated vertex $h$ in $G-V(C)$, we define the set of its \emph{$101$-neighbors}.
A vertex $v \in V(C)$ is a $101$-neighbor of $h$, if it is not a neighbor of $h$, i.e., $v \notin N_G(h)$, but both neighbors of $v$ in $C$ are also the neighbors of $h$.
In other words, the set of all $101$-neighbors of $h$ is the set of all isolated vertices in $C-N_G(h)$.
We now claim that if $C$ cannot be enlarged, then $101$-neighbors of a vertex $h$ form an independent set in $C$ and do not have neighbors in $V(G)\setminus V(C)$.
\begin{claim}\label{claim:vczeroone}
Let $h\notin B$ be an isolated vertex in $G-V(C)$.
If at least one $101$-neighbor of $h$ on $C$ is not in $B$ and has at least one neighbor in $V(G-V(C)-B)$ or two $101$-neighbors of $h$ are connected by an edge, then $C$ can be made longer in polynomial time.
\end{claim}
\begin{claimproof}
Suppose first that two $101$-neighbors of $h$, say $v_1, v_2 \in V(C)$, are connected by an edge in $G$.
Let the neighbors of $v_i$ on $C$ be $u_i$ and $w_i$ for $i \in \{1,2\}$.
Without loss of generality, we assume that the six vertices appear in the order $u_1, v_1, w_1, u_2, v_2, w_2$ when following $C$, and possibly $w_1=u_2$ or $w_2=u_1$.
Then construct a new cycle as following: $u_1 \rightarrow v_1 \rightarrow v_2 \rightarrow u_2 \rightsquigarrow w_1 \rightarrow h \rightarrow w_2 \rightsquigarrow u_1$, where $\rightarrow$ corresponds to following a single edge in $G$, while $\rightsquigarrow$ corresponds to following an arc of $C$. See \Cref{fig:vc_101}.
Note that the vertex set of the new cycle is $V(C)\cup\{h\}$, so $C$ is enlarged in this case.
\begin{figure}
\caption{Rerouting through adjacent 101-neighbors.}
\label{fig:vc_101}
\end{figure}
Now suppose that a $101$-neighbor of $h$, say $v \in V(C)$, has a neighbor outside $V(C)$, say $h' \in V(G-V(C)-B)$.
By \Cref{claim:dirac_cycle_isolated_and_non_isolated}, we can assume that all vertices in $G-V(C)$ are isolated.
Assume that $h'$ is the only neighbor of $v$ in $V(G-V(C))$.
Then replace $v$ with $h$ in $C$, so $v$ becomes a vertex outside $C$.
Then $v$ and $h'$ form a connected component of size two in $G-V(C)$.
Since $v \notin B$, $v$ has at least $\delta(G-B)-1$ neighbors in $V(C)$.
By \Cref{claim:dirac_cycle_two_connected_isolated}, $C$ can be made longer in polynomial time.
If $h'$ is not the only neighbor of $v$, then after the replacement $v$ connects two vertices with at least $\delta(G-B)-1$ neighbors in $V(C)$.
We can again apply \Cref{claim:dirac_cycle_two_connected_isolated} and make $C$ longer.
\end{claimproof}
\noindent\textbf{Constructing vertex cover.}
The construction of vertex cover of $G-B$ of size at most $\delta(G-B)+2k$ is possible when there is at least one isolated vertex in $G-V(C)$.
Take an isolated vertex $h$ in $G-V(C)-B$.
Denote by $a$ the number of its $101$-neighbors.
The neighbors of $h$ on $C$ split $C$ into arcs.
Since each $101$-neighbor corresponds to an arc of length two, and all other arcs are of length at least three, we obtain that $2a+3(\delta(G-B)-a)\ge |V(C)|$, so $a\ge 3\delta(G-B)-|V(C)|>\delta(G-B)-k$.
Now denote by $S$ the set of all $101$-neighbors of an isolated vertex in $G-V(C)$, so $|S|=a$. By \Cref{claim:vczeroone}, $(V(G)\setminus V(C)) \cup (S\setminus B)$ is an independent set in $G$, so $V(C)\setminus (S \cup B)$ is a vertex cover of $G-B$.
The size of this vertex cover is at most $(2\delta(G-B)+k-1)-(\delta(G-B)-k+1)<\delta(G-B)+2k$. Finally, the
the desired vertex cover of $G-B$ can be trivially found in polynomial time by taking an isolated component $h$ and constructing the set of its $101$-neighbors.
\noindent\textbf{Constructing Dirac decomposition\xspace.}
When no isolated vertex is presented in $G-V(C)-B$, then $G-V(C)$ consists of non-empty connected components, apart from components that are completely contained in $B$.
We show how to construct a Dirac decomposition\xspace in this case.
Before proceeding with claims, it is convenient to define the following notion agreeing with the definition of Dirac decomposition\xspace{s}.
\begin{definition}[Dirac layouts]
We say that a vertex set $X \subseteq V(C)$ is in \emph{Dirac layout} on $C$, if the vertices of $X$ split $C$ into arcs such that two of these arcs are of length at least $\delta(G-B)$.
\end{definition}
In what follows, we show that neighbors of $V(G-V(C))$ on $C$ are in Dirac layout, unless $C$ can be made longer.
We start showing this first for every connected component in $G-V(C)$.
\begin{claim}\label{claim:dirac_cycle_component_dirac_layout}
Let $H$ be a connected component in $G-V(C)$ with $|V(H)|\ge 3$ and $V(H)\not\subseteq B$.
If $N_G(V(H))$ is not in Dirac layout on $C$, then $C$ can be made longer in polynomial time.
\end{claim}
\begin{claimproof}
Denote $S=N_G(V(H))$.
Note that $S\subseteq V(C)$.
We know that $S$ splits $C$ into $|S|$ arcs.
Denote $S=\{v_1,v_2,\ldots, v_t\}$, where $v_1,v_2,\ldots,v_t$ are the vertices of $S$ on $C$ in the order when following $C$ in some direction.
We also assume that $v_{t+1}=v_1$.
Assume first that $H$ is $2$-connected.
Then assign to each vertex $v_i \in S$ a set of its neighbors in $H$.
That is, make an assignment $\sigma: S\to 2^{V(H)}$ with $\sigma(v_i)=N_G(v_i)\cap V(H)$.
As $G$ is $2$-connected, $|\bigcup_{i=1}^t \sigma(v_i)|\ge 2$.
If for at least one $i\in [t]$ holds $|\sigma(v_i)|=2$, then there exist at least two $j \in [t]$ with $\max\{|\sigma(v_j)|,|\sigma(v_{j+1})|\}\ge 2$.
For each such $j$, we can pick $h_j \in \sigma(v_j)$ and $h_{j+1} \in \sigma(v_{j+1})$ with $h_j\neq h_{j+1}$.
Since there is a path of length at least $\delta(G-B)-2$ in $H$, the length of the arc between $s_j$ and $s_{j+1}$ should be at least $\delta(G-B)$.
Otherwise we can make $C$ longer.
Now consider that $|\sigma(v_i)|=1$ for each $i \in [t]$.
But not all values of $\sigma(v_i)$ are equal, since their union is of size at least two.
Then there exist at least two $j \in [t]$ with $\sigma(v_j)\neq \sigma(v_{j+1})$.
Hence, we can again assign distinct $h_j$ and $h_{j+1}$ and obtain that the $(v_j,v_{j+1})$-arc should be of length at least $\delta(G-B)$.
It is left to consider the case when $H$ is not $2$-connected.
We again make an assignment $\sigma$, but now this assignment is slightly different and is denoted $\sigma:S\to 2^{\{0,1\}}$.
If a vertex $v_i$ has a neighbor in $H$ that is an inner vertex of a leaf-block of $H$, then $1 \in \sigma(v_i)$.
If $v_i$ has a neighbor in $H$ that is not an inner vertex of a leaf-block, put $0 \in \sigma(v_i)$.
Thus, $\sigma(v_i)$ denotes the set of types of neighbors that $v_i$ has in $V(H)$.
Note that $\bigcup_{i=1}^t\sigma(v_i)=\{0,1\}$ by \Cref{claim:dirac_cycle_leaf_block_single_neighbor} and $2$-connectivity of $G$.
Analogously to the $2$-connected case, there are two $j \in [t]$ with $0\in\sigma(v_j)$ and $1\in\sigma(v_{j+1})$ or vice versa.
Since there is a path of length at least $\delta(G-B)-2$ between any inner leaf-block vertex and any other vertex, we obtain that the arcs between $v_j$ and $v_{j+1}$ should be of length at least $\delta(G-B)$.
\end{claimproof}
\begin{claim}
Assume that $G-V(C)$ contains no isolated vertex.
Let $X$ be the union of vertex sets of all connected components $H$ in $G-V(C)$ with $V(H)\not\subseteq B$.
If $N_G(X)$ is not in Dirac layout on $C$, then $C$ can be made longer in polynomial time.
\end{claim}
\begin{claimproof}
Take a connected component $H$ in $G-V(C)$ with $V(H)\not\subseteq B$.
By \Cref{claim:dirac_cycle_component_dirac_layout}, we assume that $N_G(V(H))$ is in Dirac layout on $C$.
Hence, the vertices in $N_G(V(H))$ can be covered by two arcs of $C$ of total length at most $|V(C)|-2\delta(G-B)$ and the distance between these arcs on $C$ is at least $\delta(G-B)$.
Let $u_1,u_2$ and $v_2,v_1$ be the endpoints of these arcs.
Among all possible ways to choose the arcs we choose the way when the total length of the $(u_1,u_2)$-arc and $(v_2,v_1)$-arc is the minimum possible.
Hence, $u_1, u_2, v_1, v_2 \in N_G(V(H))$ and the $(u_1,u_2)$-arc and the $(v_2,v_1)$-arc together contain all neighbors of $N_G(V(H))$ on $C$.
Note that these arcs can be of zero length.
For example, if $|N_G(V(H))|=2$, then $u_1=u_2$ and $v_1=v_2$, so $N_G(V(H))=\{u_1,v_1\}$.
We also assume that the order of the vertices on $C$ is $u_1, u_2, v_2, v_1$ when following $C$ in some direction.
Thus, the chords between $u_1$ and $v_1$ and between $u_2$ and $v_2$ do not intersect graphically but can only coincide in one or two endpoints.
From the proof of \Cref{claim:dirac_cycle_component_dirac_layout} follows that $H$ yields a $(u_1,v_1)$-chord or a $(u_2,v_2)$-chord of $C$ of length at least $\delta(G-B)$.
Let $S$ be the union of the sets $\{u_1,u_2,v_1,v_2\}$ among all connected components of $G-V(C)$.
It is easy to see that $S$ is in Dirac layout on $C$ if and only if $S$ is on Dirac layout on $C$.
It is left to show that $S$ is in Dirac layout on $C$ or $C$ can be made longer in polynomial time.
Consider the vertices in $S$ on $C$.
They are connected by chords of length at least $\delta(G-B)$ yielded by their connected components.
If there is a pair of these chords that intersect graphically, then the chords in this pair correspond to distinct connected components of $G-V(C)$.
Hence, if such a pair exists, we can enlarge $C$ as we did in the proof of \Cref{lemma:dirac_cycle_edge_of_banana}.
We can now assume that no two chords of $S$ intersect graphically.
But we also know that no chord splits $C$ into two arcs such that one of them is shorter than $\delta(G-B)$.
Hence, there are two arcs of length at least $\delta(G-B)$ that do not contain any vertex in $S$ as inner vertex.
Then $S$ is in Dirac layout on $C$ by definition.
\end{claimproof}
The claim shows that $N_G(X)$ can be covered by two arcs of $C$ of total length at most $k-1$ at a distance at least $\delta(G-B)$ between them.
Let $P_1$ and $P_2$ be these two arcs chosen in the unique way that minimizes their total length.
It is left for us to show that $P_1$ and $P_2$ induce a Dirac component\xspace for $C$ and $B$ in $G$.
The first property from the defition of Dirac component\xspace is satisfied by the way $P_1$ and $P_2$ are constructed.
It is easy to verify the second property for each connected component in $G-V(C)$: $2$-connected components form \ref{enum:cycle_tunnel_path_bic}-type components and components containing cut vertices form \ref{enum:cycle_tunnel_path_cut_left} and \ref{enum:cycle_tunnel_path_cut_right}-type components of the Dirac decomposition\xspace.
If the matching size conditions are not satisfied for one of these components, then $C$ can be trivially made longer in polynomial time using a long chord yielded by the component.
It is important to verify that the second property holds for all connected components in $G-V(P_1\cup P_2)$.
Note that a connected component $H$ in $G-V(C)$ with $V(H)\not\subseteq B$ is a connected component in $G-V(P_1\cup P_2)$ as well.
Connected components that appear in $G-V(C)$ but do not appear in $G-V(P_1\cup P_2)$ are connected components that contain vertices in $V(C)\setminus V(P_1\cup P_2)$.
Note that there is either one or two such connected components, because the vertex set $V(C)\setminus V(P_1\cup P_2)$ is a union of vertex sets of two arcs of $C$.
If there is just one such connected component $H$, then $V(C)\setminus V(P_1\cup P_2)\subseteq V(H)$.
We claim that if such $H$ exists in $G-V(P_1\cup P_2)$, then $C$ can be made longer in polynomial time (except in some very specific cases).
\begin{figure}
\caption{A schematic picture of an existence of a chord between $P'$ and $P''$ passing through $B$.
A blue chord represents a chord of $C$ passing through a component of $G-V(C)$.}
\label{fig:dirac_cycle_p'_p''}
\end{figure}
Assume that such $H$ exists.
Then two arcs of $C$ of length at least $\delta(G-B)$ (denoted by $P'$ and $P''$ in the definition of Dirac decomposition\xspace and here) are connected by a chord that can pass internally only through vertices in $B$.
Note that the length $\delta(G-B)$ does not match the lower bound in the definition of Dirac decomposition\xspace{s}.
This is intentional.
In one of the cases below, we have to expand the paths $P_1$ and $P_2$ and reduce the length of $P'$ and $P''$ by one or two.
Denote by $s'$ and $t'$ and by $s''$ and $t''$ the endpoints of the arcs $P'$ and $P''$ respectively.
Note that $V(P'-\{s',t'\})\cup V(P''-\{s'',t''\})\subseteq V(H)$, but $s',t',s'',t'' \notin V(H)$.
Since $P_1$ and $P_2$ are an $(s',s'')$-arc and an $(t',t'')$-arc of $C$ respectively.
Hence, the chord connecting $P'$ and $P''$ has endpoints in inner vertices of $P'$ and $P''$.
For clarity of presentation, we formulate the following intermediate claim.
\begin{claim}
If there exists a connected component $H$ in $G-V(P_1\cup P_2)$ with $V(C)\setminus V(P_1\cup P_2)\subseteq V(H)$, then either the only chords connecting $P'-\{s',t'\}$ and $P''-\{s'',t''\}$ are between their respective endpoints or $C$ can be made longer in polynomial time.
\end{claim}
\begin{claimproof}
Let $u \in V(P'-\{s',t'\})$ and $v \in V(P''-\{s'',t''\})$ be the endpoints of this chord.
Denote by $a'$ and $b'$ the length of the paths that $u$ splits $P'$ into.
Analogously, by $a''$ and $b''$ denote the length of the paths that $v$ splits $P''$ into, as shown in \Cref{fig:dirac_cycle_p'_p''}.
If $\max\{a'+b'',a''+b'\}+\delta(G-B)\ge |V(C)|$, then we can find a cycle longer than $C$ in polynomial time using a chord passing though some connected component in $G-V(C)$ and the $(u,v)$-chord of $C$.
Note that if $u$ is the neighbor of $s'$ in $P'$ and $v$ is the neighbor of $t''$ in $P_2$, then $b'\ge \delta(G-B)-1$ and $a''\ge \delta(G-B)-1$ so $a''+b'+\delta(G-B)>|V(C)|$ and $C$ can be made longer.
The situation when $u$ is the neighbor of $t'$ and $v$ is the neighbor of $s''$ is symmetrical.
We now assume that $a'+b''< \delta(G-B)+k$ and $a''+b'<\delta(G-B)+k$ (and, consequently, $a'+b''\ge \delta(G-B)$ and $a''+b'\ge \delta(G-B)$) for each choice of $u$ and $v$.
That is, each such $(u,v)$-chord should split $C$ in a way that the difference between $a'+b''$ and $a''+b'$ is at most $k$.
Consider a fixed $u \in V(P')$.
Without loss of generality, we assume that $u$ is not the neighbor of $s'$ in $P'$.
Note that distinct choices of $v \in V(P'')$ provides distinct values of $a''$ and $b''$ with fixed sum.
Hence, if there are at least $2k+1$ choices of a pair $(u,v)$ for a fixed $u$, there are $2k+1$ different values of $a'+b''$.
Since the sum of $a', b', a'', b''$ is also fixed, in at least one of these choices the difference between $a'+b''$ and $a''+b'$ is at least $k+1$.
It follows that if $u$ has at least $2k+1$ neighbors in $V(P'')$, then $C$ can be made longer in polynomial time.
Note that the same arguments apply to a fixed choice of $v \in V(P'')$.
We have that for each $u \in V(P')$, $|N_G(u)\cap V(P'')|\le 2k$.
As soon as vertices in $P'-\{s',t'\}$ can have neighbors outside only in $V(P'')$, $V(P_1\cup P_2)$ and $B$, we have that, $\delta(G[V(P'-\{s',t'\})]-B)\ge \delta(G-B)-|V(P_1\cup P_2)|-2k$.
Since the total length of $P_1$ and $P_2$ is at most $k-1$ and $|V(P_1\cup P_2)|\le k+2$, we have that $\delta(G[V(P'-\{s',t'\})]-B)\ge \delta(G-B)-3k-1$.
Denote by $H'$ the graph $G[V(P'-\{s',t'\})]-B$.
As the length of $P'$ is less than $\delta(G-B)+k$, we have that $|V(H')|\le \delta(G-B)+k-2$.
Hence, $\delta(H')> |V(H')|-4k$.
On the other hand, $\delta(H')\ge \delta(G-B)-4k \ge 20k$.
We can now apply \Cref{lemma:many_paths} to $H'$ with $p=4k, r=1$ and $\{s_1,t_1\}=\{u',u\}$, where $u'$ is the neighbor of $s'$ in $P'$.
Note that $u' \neq u$ by our assumption.
By \Cref{lemma:many_paths}, there is a Hamiltonian $(u',u)$-path in $H'$.
This path is of length at least $\delta(G-B)-2-|B|$ that is found in polynomial time.
Hence, we obtain a $(s',u)$-path of length at least $\delta(G-B)-1-|B|$ that contains only vertices of $P'-t'$.
The arguments of constructing a $(s',u)$-path for $P'$ are applicable for constructing a $(t'',v)$-path for $P''$, if $v$ is not the neighbor of $t''$ in $P''$.
Then we are able to construct a $(t'',v)$-path of length at least $\delta(G-B)-1-|B|$.
Combine the $(s',u)$-path with $P_1$ and the $(t'',v)$-path and two chords: the $(u,v)$-chord and a $(s'',t'')$-chord of length at least $\delta(G-B)$ (it is depicted in \Cref{fig:dirac_cycle_p'_p''}) to obtain a cycle of length at least $3\delta(G-B)-2|B|-1\ge 2\delta(G-B)+k$.
The last chord always exists by the construction of $P_1$ and $P_2$.
Note that we only required in the above construction that if $v$ is not the neighbor of $t''$ in $P''$.
If $v$ is the neighbor of $t''$, then we can consider constructing a $(t'',u)$-path instead of $(s',u)$-path, but only if $u$ is not the neighbor of $t'$ in $P'$.
The long path between $s''$ and $v$ required for construction is then given by $P''$, and is of length $a''\ge\delta(G-B)-1$.
\end{claimproof}
We are left with the cases when $u$ and $v$ are simultaneously the neighbors of $t'$ and $t''$ in $P'$ and $P''$ respectively, or the neighbors of $s'$ and $s''$ in $P'$ and $P''$ respectively.
That is, the cases when $b'=b''=1$ or $a'=a''=1$.
In these cases, we cannot construct a pair of long paths and combine them with two chords, because we cannot apply \Cref{lemma:many_paths} to both $u$ (e.g.\ to $\{u',u\}$, as $u'=u$) and $v$.
In other cases, we can make $C$ longer in polynomial time.
We now assume that the only two $(u,v)$-chords between $P'$ and $P''$ can be only a chord between the neighbor of $s'$ in $P'$ and the neighbor of $s''$ in $P''$ and a chord between the neighbor of $t'$ in $P'$ and the neighbor of $t''$ in $P''$.
In this case, we need to expand $P_1$ or $P_2$ to contain two more vertices.
If there is a chord between the neighbors of $s'$ and $s''$, expand $P_1$ with two edges so it starts containing these neighbors.
Analogously, expand $P_2$ if there is a chord between the neighbors of $t'$ and $t''$.
Observe that such expansion of $P_1$ or $P_2$ with two edges does not influence the properties for connected components in $G-V(C)$.
We now have that $V(P'-(V(P_1)\cup V(P_2)))$ and $V(P''-(V(P_1)\cup V(P_2)))$ belong to distinct connected components in $G-(V(P_1)\cup V(P_2))$.
The length of $P'$ and $P''$ is now at least $\delta(G-B)-2$ and the total length of $P_1$ and $P_2$ is at most $k+4$.
The first and the last properties of a Dirac decomposition\xspace are satisfied by $P_1$ and $P_2$.
Denote the two connected components of $G-(V(P_1)\cup V(P_2))$ that contain inner vertices of $P'$ and $P''$ by $H'$ and $H''$ respectively.
We know that$|V(H')|,|V(H'')|\le \delta(G-B)+|B|$ while $\delta(H'-B)\ge \delta(G-B)-|V(P_1)\cup V(P_2)|\ge \delta(G-B)-k-6\ge \frac{1}{2}\delta(G-B)+|B|$.
Consider the $B$-refinements of $H'$ and $H''$.
If one of them is not $2$-connected, then it should contain two leaf-blocks each consisting of at least $\frac{1}{2}\delta(G-B)+|B|+1$ vertices.
Then, the total number of vertices in this component would be $2(\frac{1}{2}\delta(G-B)+|B|+1)-1>\delta(G-B)+|B|$, which is not possible.
Hence, the $B$-refinements of $H'$ and $H''$ are $2$-connected.
It is left to prove that they satisfy the properties of \ref{enum:cycle_tunnel_path_bic}-type components of Dirac decomposition\xspace{s}.
That is, we have to prove that the maximum matching size between $V(H')$ or $V(H'')$ and $V(P_1)$ or $V(P_2)$ is exactly one after the $B$-refinements.
Consider that the matching size between $V(H')$ and $V(P_1)$ equals two.
If the path $P_1$ was not expanded, then there is a long chord of $C$ passing though a component in $G-V(C)$ and connecting $s'$ with a vertex in $P_2$.
Hence, we can take a cycle of length at least $2\delta(G-B)-2$ combined of this chord, $P''$, $P_1$ and a part of $P_2$.
Then \Cref{thm:relaxed_st_path} and the maximum matching between $V(H')$ and $V(P_1)$ yields a chord of this cycle with endpoints in $V(P_1)$ of length at least $\delta(H'-B)+2\ge \delta(G-B)-k-4$.
Since the length of $P_1$ is at most $k$, we can enlarge this cycle and obtain a cycle of length at least $(2\delta(G-B)-2)+(\delta(G-B)-k-4)-k\ge 3\delta(G-B)-2k-6> 2\delta(G-B)+k$.
If $P_1$ was expanded, then there is no long chord of $C$ connecting the common endpoint of $P_1$ and $P'$ with a vertex in $P_2$.
However, then there exists a short chord of $C$ connecting the endpoints of $P_1$ and passing only through $B$ without visiting $H'$ or $H''$ or any component in $G-V(C)$.
Also, there is still a chord connecting $s'$ with some vertex in $V(P_2)$, though $s'$ now is not an endpoint of $P_1$ but the neighbor of the common endpoint of $P_1$ and $P'$ in $P_1$.
If the endpoints of the matching in $V(P_1)$ do not include either $s'$ or the endpoint of $P_1$, we can proceed in the same way as when $P_1$ was not expanded.
As $V(H')$, the matching and the edge between $s'$ and the endpoint of $P_1$ produce the required long chord of the new cycle.
The case that requires explanation is when the endpoints of the matching are $s'$ and the endpoint of $P_1$.
Then $V(H')$ only yields an $(s',s')$-chord, which is not appropriate.
In this case, we have to use the chord between the endpoints of $P_1$ instead of the edge between $s'$ and the endpoint of $P_1$.
It is easy to see that $V(H')$ together with the matching and this chord provide a long chord between $s'$ and the other endpoint of $P_1$ (the one closer to $t'$).
Note that this endpoint is different from $s'$, as the length of $P_1$ is at least two.
We have shown that if the matching size between $V(H')$ and $V(P_1)$ is at least two, then we can find a longer cycle in polynomial time.
The other cases are symmetrical.
Hence, $H'$ and $H''$ satisfy the properties of \ref{enum:cycle_tunnel_path_bic}-type components.
This concludes the proof of the lemma. \end{proof}
\todo[backgroundcolor=yellow,inline]{this section describes reduction from cycle to path and probably is not needed} \begin{comment} In this section, we prove Theorem~\ref{theorem:main_path}. The proof crucially relies on the following lemma.
\begin{lemma}\label{lemma:main_lemma_path}
Let $G$ be a $2$-connected graph and $B\subseteq V(G)$ be a subset of its vertices such that $\delta(G-B)\ge 5$.
Let $C$ be a cycle of length at least $2\delta(G-B)$ in $G$ and let $k< \frac{2}{5}\delta(G-B)$ be a non-negative integer.
There is an algorithm that in polynomial time finds either:
\begin{itemize}
\item Longer cycle in $G$, or
\item Path of length at least $2\delta(G-B)+k$ in $G$, or
\item Vertex cover of $G-B$ of size at most $\delta(G-B)+2k$.
\end{itemize} \end{lemma} \begin{proof}
\todo[inline,backgroundcolor=yellow]{write this proof} \end{proof}
\begin{proof}[Proof of Theorem~\ref{theorem:main_path}]
\todo[backgroundcolor=yellow]{this is very sketchy}
We assume that $G$ is connected, otherwise we apply the algorithm to each of its connected components.
If the $B$-refinement of $G$ is not $2$-connected, apply the algorithm for \textsc{\pname{Long Dirac Path}\xspace in Separable Graphs}.
Otherwise, apply algorithm of Lemma~\ref{lemma:main_lemma_path} exhaustively to $(G,B,k)$ while the longer cycle is found.
If the length of the cycle is at least $2\delta(G-B)+k+1$ or a path of length at least $2\delta(G-B)+k$ is found by this algorithm, we can decide that the input instance is a yes-instance.
Otherwise, we obtain a vertex cover of the initial graph $G$ that is of size at least $2k+|B|$ and contains $B$ as a subset.
Finish using the FPT-algorithm for \textsc{\pname{Long Dirac Cycle}\xspace/Vertex Cover Above Degree}. \end{proof} \end{comment}
\section{Conclusion}\label{sec:conclusion}
In this paper, we developed an algorithmic extension of the classical theorem of Dirac. Our main result, Theorem~\ref{theorem:main}, is
that \pname{Long Dirac Cycle}\xspace is solvable in $2^{\mathcal{O} (k+|B|)} \cdot n^{\mathcal{O} (1)}$ time on 2-connected graphs.
An important step in the proof of Theorem~\ref{theorem:main} is Theorem~\ref{thmEG}: \pname{Long Erd{\H{o}}s-Gallai $(s,t)$-Path}\xspace is solvable in $2^{\mathcal{O}(k+|B|)}\cdot n^{\mathcal{O}(1)}$ time on $2$-connected graphs. In this section we provide lower bounds complementing Theorems~\ref{theorem:main} and~\ref{thmEG}, and then conclude with open questions for further research.
\subsection{Tightness of results}
We have already observed that the dependency on $k$ in the running times of Theorems~\ref{theorem:main} and \ref{thmEG} is tight up to ETH. Here we show that the dependency on $|B|$ is similarly tight. Additionally, we show that for any $\varepsilon > 0$, it is $\ensuremath{\operatorClassNP}$-hard to find a cycle of length at least $(1 + \varepsilon)2\delta(G)$, meaning that our starting bound of $2\delta(G)$ is tight. We start with the first hardness result.
\begin{theorem}
Unless ETH fails, there is no algorithm solving \pname{Long Dirac Cycle}\xspace or \pname{Long Dirac Path}\xspace in time $2^{o(|B|)} \cdot |V(G)|^{\mathcal{O}(1)}$, even when $k = 1$.
\label{thm:hard_on_B} \end{theorem} \begin{proof}
\begin{figure}
\caption{An illustration to the hardness reduction in Theorem~\ref{thm:hard_on_B}, from \textsc{Hamiltonian Path} to \pname{Long Dirac Cycle}\xspace. The graph $H$ is the starting \textsc{Hamiltonian Path} instance. The reduction to \pname{Long Dirac Path}\xspace looks similarly, only without the vertex $t$.}
\label{fig:hardnessB}
\end{figure}
First, we show a reduction from \textsc{Hamiltonian Path} to \pname{Long Dirac Cycle}\xspace. Consider an instance $H$ of \textsc{Hamiltonian Path}, let $n = |V(H)|$. Take a disjoint union of $H$ and two disjoint copies of $K_{n - 1}$, the clique on $(n - 1)$ vertices. Add two additional vertices $s$ and $t$ that are adjacent to all previously listed vertices (but not to each other). This finishes the description of the graph $G$ that our reduction constructs from $H$, see Figure~\ref{fig:hardnessB} for the illustration. Finally, set $B$ to $V(H) \subset V(G)$, and $k$ to one. Observe that $2\delta(G - B)$ and $|V(G - B)|$ are both equal to $2n$. Our aim is now to show that $H$ has a Hamiltonian path if and only if $G$ has a cycle of length at least $2n + 1 = \min\{2\delta(G-B), |V(G)|-|B|\}+k$.
In the forward direction, if there is a Hamiltonian path $P$ in $H$, consider a Hamiltonian path $P'$ in one of the $K_{n - 1}$ components. Connect $P$ and $P'$ in a cycle by going through the vertices $s$ and $t$. This results in a cycle of length $|V(G)| + |V(K_{n - 1})| + 2 = 2n + 1$.
In the other direction, let $C$ be a cycle of length at least $2n + 1$ in $G$. Since $|V(G - B)| = 2n$, $C$ necessarily intersects $B$, and since $|B| = n$, $C$ also intersects $V(G - B)$. Since in $G - \{s, t\}$ the set $B$ is disconnected from the rest of the graph, the cycle $C$ necessarily enters $B$ from $s$ and exits via $t$. The two $K_{n - 1}$ copies are also disconnected in $G - \{s, t\}$, thus $C$ intersects exactly one of the cliques. Thus, $C$ has at most $n + 1$ vertices outside of $B$. Since $|C| = 2n + 1$ and $|B| = n$, $C$ must traverse all vertices of $B$. Since $C$ induces a path on $B$, this path is also a Hamiltonian path in $H$. This finishes the proof of correctness of the reduction.
Finally, following the reduction above, a $2^{o(|B|)} \cdot |V(G)|^{\mathcal{O}(1)}$-time algorithm for \pname{Long Dirac Cycle}\xspace would immeidately imply a
$2^{o(n)}$-time algorithm for \textsc{Hamiltonian Path} since $|B| = n$ and $|V(G)| = \mathcal{O}(n)$, and the existence of the latter would contradict ETH.
For \pname{Long Dirac Path}\xspace, the reduction follows a similar idea. From an instance $H$ of \textsc{Hamiltonian Path}, construct a graph $G$ as follows. Take a disjoint union of $H$ and two disjoint copies of $K_{n - 1}$, and add an additional apex vertex $s$. Set $B$ to be $V(H) \subset V(G)$, and set $k$ to one. Clearly, $2 \delta(G - B) = |V(G)| - |B| - 1 = 2n - 2$. If there is a Hamiltonian path in $H$, it extends to a path of length $2n - 1$ in $G$ by continuing through $s$ into one of the cliques, as it is always possible to traverse through all vertices of the clique. On the other hand, if there is a path $P$ of length at least $2n - 1$ in $G$, it necessarily goes from $B$ to $V(G) \setminus B$, since $|B| = n$ and $|V(G) \setminus B| = 2n - 1$. Such a path can only go through $s$ to one of the cliques while completely avoiding the other, since $s$ is an articulation point. Thus, outside of $B$ the path $P$ visits at most $n$ vertices, and since a path of length at least $2n - 1$ has to visit at least $2n$ distinct vertices, $P$ necessarily traverses through all vertices of $V(H)$, yielding a Hamiltonian path in $H$. This finishes the proof for \pname{Long Dirac Path}\xspace. \end{proof}
Next, we show that the bound $2\delta(G)$ cannot be improved unless $\ensuremath{\operatorClassP}=\ensuremath{\operatorClassNP}$ by proving the following theorem.
\begin{theorem}\label{thm:tightness}
For every positive $\varepsilon<1$, it is \ensuremath{\operatorClassNP}-complete to decide whether
\begin{itemize}
\item[(a)] a $2$-connected graph $G$ with two given vertices $s$ and $t$ has an $(s,t)$-path of length at least $(1+\varepsilon)\delta(G)$;
\item[(b)] a $2$-connected graph $G$ has a cycle of length at least $(2+\varepsilon)\delta(G)$.
\end{itemize}
\end{theorem}
\begin{proof}
Both claims are shown by reduction from the classical \textsc{Hamiltonian Path} problem that is well-known to be \ensuremath{\operatorClassNP}-complete~\cite{GareyJ79}. Both reductions exploit the same idea. We first show the claim for an $(s,t)$-path and then explain how to modify the reduction for the second claim.
\begin{figure}
\caption{Construction of $H$ and $H'$.}
\label{fig:hard}
\end{figure}
Let $0<\varepsilon<1$ and let $G$ be an $n$-vertex graph with $n\geq 2$. We select a positive integer $p$ such that $\lceil\varepsilon(p+1)\rceil=n$. Clearly, such an integer exists, because $\varepsilon<1$ and $n\geq 2$. Then we construct the following graph $H$ (see Figure~\ref{fig:hard}). \begin{itemize} \item Construct a copy of $G$. \item Construct a vertex $t$ and make it adjacent to every vertex of $G$. \item Construct a vertex $s$. \item For every vertex $v\in V(G)\cup \{t\}$, construct a clique $Q_v$ with $p$ vertices and make the vertices of $Q_v$ adjacent to $v$ and $s$. \end{itemize} Notice that $H$ is 2-connected and $\delta(H)=p+1$. We claim that $G$ is has a Hamiltonian path if and only if $H$ has an $(s,t)$-path of length at least $(1+\varepsilon)\delta(H)$.
In one direction, let $P$ be a Hamiltonian path in $G$ and denote by $x$ and $y$ its endpoints. Because $Q_x$ is a clique, $H$ has an $(s,x)$-path $R$ with $V(R)=Q_x\cup\{s,x\}$. That is, $R$ is a Hamiltonian path in $H[Q_x\cup\{s,x\}]$. Consider path $P'$ obtained by concatenating $R$, $P$, and $yt$. Then $P'$ is an $(s,t)$-path in $H$. Observe that the length of $P'$ is \begin{multline*} (p+1)+(n-1)+1=p+n+1=p+1+\lceil\varepsilon(p+1)\rceil\geq (1+\varepsilon)(p+1)=(1+\varepsilon)\delta(H) \end{multline*} as required.
For the opposite direction, assume that $P'$ is an $(s,t)$-path in $H$ of length at least $(1+\varepsilon)\delta(H)$. Then the length of $P'$ is at least \begin{equation*} \lceil(1+\varepsilon)\delta(H)\rceil=\delta(H)+\lceil\varepsilon \delta(G)\rceil=(p+1)+\lceil\varepsilon(p+1)\rceil=p+1+n. \end{equation*} By the construction of $H$, $P'$ is the concatenation of paths $R$ and $S$ such that $R$ is an $(s,v)$-path for some $v\in V(G)\cup\{t\}$ where $V(R)\subseteq V(Q_v)\cup\{s,v\}$ and $S$ is a $(v,t)$-path with $V(S)\subseteq V(G)\cup\{t\}$. The length of $R$ is at most $p+1=\delta(H)$. Therefore, the length of $S$ is at least $n$. Consider the path $P$ obtained from $S$ by deleting $t$. We have that $V(P)\subseteq V(G)$ and the length of $P$ is at least $n-1$. We obtain that $P$ is a Hamiltonian path in $G$. This concludes the proof of the first claim.
The proof of (b) is similar. Let $0<\varepsilon<1$ and let $G$ be an $n$-vertex connected graph with $n\geq 3$. Now we select a positive integer $p$ such that $\lceil\varepsilon(p+1)\rceil=n-1$.
We construct graph $H'$ that is, in fact, the graph obtained from $H$ constructed above by deleting $t$ and the vertices of $Q_t$ (see Figure~\ref{fig:hard}). Formally, $H'$ is constructed as follows.
\begin{itemize} \item Construct a copy of $G$. \item Construct a vertex $s$. \item For every vertex $v\in V(G)$, construct a clique $Q_v$ with $p$ vertices and make the vertices of $Q_v$ adjacent to $v$ and $s$. \end{itemize} Because $G$ is a connected graph with at least three vertices, $\delta(H')=p+1$. Because $G$ is connected, $H'$ is 2-connected.
We claim that $G$ has a Hamiltonian path if and only if $H'$ has a cycle of length at least $(2+\varepsilon)\delta(H)$.
Suppose that $P$ is a Hamiltonian path in $G$ and let $x$ and $y$ be its endpoints. Note that since $G$ has at least two vertices, $x\neq y$. Because $Q_x$ and $Q_y$ are cliques, $H$ has an $(s,x)$-path $R_x$ with $V(R_x)=Q_x\cup\{s,x\}$ and a $(y,s)$-path $R_y$ with
$V(R_y)=Q_y\cup\{s,y\}$. Observe that the concatenation of $R_x$, $P$, and $R_y$ is a cycle. Denote this cycle by $C$. The length of $C$ is
\begin{multline*} (p+1)+(n-1)+(p+1)=2(p+1)+n-1=2(p+1)+\lceil\varepsilon(p+1)\rceil\geq (2+\varepsilon)(p+1)=(1+\varepsilon)\delta(H). \end{multline*}
Finally, let $C$ be a cycle of $G$ of length at least $(2+\varepsilon)\delta(H)$. Then the length of $C$ is at least \begin{equation*} \lceil(2+\varepsilon)\delta(H)\rceil=2\delta(H)+\lceil\varepsilon \delta(G)\rceil=2(p+1)+\lceil\varepsilon(p+1)\rceil=2(p+1)+n-1. \end{equation*}
Suppose that $s\notin V(C)$. Then, by the construction of $H'$, either $C$ is a cycle in $H'[Q_v\cup\{v\}]$ for some $v\in V(G)$ or $C$ is a cycle of $G$. In the first case the length of $C$ is at most $|Q_v|+1=p+1$, and in the second case the length of $C$ is at most $n$. In both cases, we have that the length of $C$ is strictly less that $2(p+1)+n-1$. This implies that $s\in V(C)$. If $|V(C)\cap V(G)|\leq 1$, then $V(C)\subseteq Q_v\cup\{s,v\}$ for some $v\in V(G)$. However, $|V(C)|\leq p+2<2(p+1)+n-1$ in this case. Hence, $|V(C)\cap V(G)|\geq 2$. Then the construction of $H'$ implies that $|V(C)\cap V(G)|=2$. Let $\{x,y\}=V(C)\cap V(G)$. It is easy to verify that $C$ can be seen as the concatenation of three paths $R_x$, $P$, and $R_y$, where $R_x$ is an $(s,x)$-path with $V(R_x)\subseteq Q_x\cup\{s,x\}$, $P$ is an $(x,y)$-path in $G$, and $R_y$ is a $(y,s)$-path with $V(R_y)\subseteq Q_y\cup \{s,y\}$. The length of $R_x$ and the length of $R_y$ is at most $p+1$. This means, that the length of $P$ is at least $n-1$. Therefore, $P$ is a Hamiltonian path in $G$. This concludes the proof.
\end{proof}
For simplicity, we proved Theorem~\ref{thm:tightness} for the case when $\varepsilon<1$ but let us remark that the claim also holds for $\varepsilon\geq 1$. Moreover, it can be assumed that $\varepsilon$ not a constant but an appropriate function of $\delta(G)$ like $\varepsilon(\delta)=\delta^c$ for some constant $c>-1$.
\subsection{Open questions} Dirac's theorem is the first fundamental result in Extremal Hamiltonian Graph Theory. The area contains many deep and interesting theorems but it remains largely unexplored from the algorithmic perspective. Here we present several open questions hoping that these questions would trigger further research in this fascinating area.
The first question is from \cite{fomin_et_al:LIPIcs:2019:11168}. Recall that the \emph{average degree} of a graph $G$ is \[\frac{1}{|V(G)|}\sum_{v\in V(G)}\deg_G(v)=2|E(G)|/|V(G)|.\] The following was shown by Erd{\H{o}}s and Gallai~\cite{ErdosG59}.
\begin{proposition}[\cite{ErdosG59}]\label{prop:ad} Every graph $G$ with average degree at least $d\geq 2$ has a cycle of length at least $d$. \end{proposition}
Similarly, it can be shown that a graph $G$ with average degree at least $d$ has a path of length at least $d$. This leads to the following question.
\begin{problem}[\textbf{Path/cycle above average degree}]\label{prob:ad}
Given a $2$-connected (connected, respectively) graph $G$ and a nonnegative integer $k$, how difficult is to decide whether $G$ has a cycle (a path, respectively) of length at least $2|E(G)|/|V(G)|+k$?
\end{problem}
We do not know whether the problem is \ensuremath{\operatorClassFPT}\xspace parameterized by $k$, \classW1-hard, or \ensuremath{\operatorClassParaNP}\xspace. Even the simplest variant of the question: whether a path of length $2|E(G)|/|V(G)|+1$ could be computed in polynomial time, is open.
Our second open question concerns the problem of finding a cycle containing a specified set of vertices.
The study of this problem can be traced back to another fundamental theorem of Dirac from 1960s about the existence of a cycle in $h$-connected graph passing through a given set of $h$ vertices \cite{Dirac1960}. According to Kawarabayashi \cite{Kawarabayashi08} \emph{``...cycles through a vertex set or an edge set are one of central topics in all of graph theory."} Such type of problems have been a popular and important topic in algorithms as well. See, e.g., Bj{\"{o}}rklund, Husfeldt and Taslaman~\cite{BjorklundHT12} and Wahlstr{\"{o}}m~\cite{Wahlstrom13}, and Kawarabayashi \cite{Kawarabayashi08}.
In Extremal Hamiltonian Graph Theory, the following theorem of
Egawa, Glas, and Locke~\cite{Egawa1991} is well-known.
\begin{theorem}[\cite{Egawa1991}] Let $G$ be an $h$-connected graph, $h\geq 2$, with minimum degree $d$, and at least $2d-1$ vertices. Let $X$ be a set of $h$ vertices of $G$. Then $G$ has a cycle $C$ of length at least $2d$ such that every vertex of $X$ is on $C$. \end{theorem}
This brings us to the following algorithmic problem.
\begin{problem}[\textbf{Cycle above Egawa, Glas, and Locke condition}]\label{prob:through} Given an $h$-connected graph $G$, a set of vertices $X\subseteq V(G)$ of size $h$, and a nonnegative integer $k$, how difficult is to decide whether $G$ has a cycle of length at least $2\delta(G)+k$ containing every vertex of $X$?
\end{problem}
As for Open Question~\ref{prob:ad} the question is open even for $k=1$.
Finally, let us mention the area of directed graphs; we refer to the book of Bang-Jensen and Gutin~\cite{BangJensenG09}
and the survey of Bermond and Thomassen~\cite{BermondT81} for extremal theorems for directed graphs. In particular, the classical result of Ghouila-Houri~\cite{GhouilaHouri1960} from 1960, generalizes Theorem~\ref{thm:diracs}. Recall that a digraph $D$ is \emph{strong} if for every two vertices $u$ and $v$, $D$ has directed $(u,v)$ and $(v,u)$-path, and the degree $\deg_D(v)$ of a vertex $v$ is the sum of its \emph{in-degree} $\deg_D^-(v)$ and \emph{out-degree} $\deg_D^+(v)$.
\begin{theorem}[\cite{GhouilaHouri1960}]\label{thm:GH} If for every vertex $v$ of a
strong digraph $D$ with $n$ vertices $\deg_D(v)\geq n$, then $D$ has a Hamiltonian cycle. \end{theorem}
The following question is the variant of the question discussed by Jansen, Kozma and Nederlof in~\cite{DBLP:conf/wg/Jansen0N19} for undirected graphs.
\begin{problem}[\textbf{Cycle above Ghouila-Houri condition}]\label{prob:GH} Given an $n$-vertex strong digraph $D$ and a nonnegative integer $k$ such that at least $n-k$ vertices have degree at least $n$, how difficult is to decide whether $D$ is Hamiltonian? \end{problem} Again, the simplest variant---whether there is a polynomial time algorithm for for $k=1$---is open. We also do not know the complexity of the problem when
every vertex has degree at least $n-k$.
A digraph $D$ with at least two vertices is \emph{$2$-connected} if it is strong and remains strong after deleting an arbitrary vertex.
Thomassen in~\cite{Thomassen81} proved the following analog of Theorem~\ref{thm:circum}.
\begin{theorem}[\cite{Thomassen81}]\label{thm:Thom} Let $D$ be a $2$-connected digraph with at least $2d+1$ vertices such that $\deg_D^{-}(v)\geq d$ and $\deg_D^+(v)\geq d$ for every $v\in V(D)$. Then $D$ contains a cycle of length at least $2d$. \end{theorem}
Whether Thomassen's theorem can be extended algorithmically is our last open question. \begin{problem}[\textbf{Cycle above Thomassen condition}] \label{prob:Thom} What is the (parameterized) complexity of the following problem. Given a $2$-connected digraph $D$ such that $\deg_D^{-}(v)\geq d$ and $\deg_D^+(v)\geq d$ for every $v\in V(D)$, and a nonnegative integer $k$. Decide whether $D$ contains a cycle of length at least $2d+k$. \end{problem}
As in Questions~\ref{prob:ad}--\ref{prob:GH}, even the existence of a polynomial time algorithm for $k=1$ in Question~\ref{prob:Thom} is open.
\end{document} | arXiv |
How many distinct arrangements of the letters in the word "monkey"' are there?
Let's consider building such an arrangement. We can choose the first letter in 6 ways. After we have chosen the first letter, we can choose the second in 5 ways. Similarly, the third letter then has 4 ways of being chosen, the next letter 3, the next 2, and the last only 1. Thus the total number of arrangements is $6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1 = \boxed{720}$. | Math Dataset |
5.5 Averages and Probability
Calculate the mean of a set of numbers
Find the median of a set of numbers
Find the mode of a set of numbers
Apply the basic definition of probability
One application of decimals that arises often is finding the average of a set of numbers. What do you think of when you hear the word average? Is it your grade point average, the average rent for an apartment in your city, the batting average of a player on your favorite baseball team? The average is a typical value in a set of numerical data. Calculating an average sometimes involves working with decimal numbers. In this section, we will look at three different ways to calculate an average.
5.5.1 Calculate the Mean of a Set of Numbers
The mean is often called the arithmetic average. It is computed by dividing the sum of the values by the number of values. Students want to know the mean of their test scores. Climatologists report that the mean temperature has, or has not, changed. City planners are interested in the mean household size.
Suppose Ethan's first three test scores were $85,88$, and $94$. To find the mean score, he would add them and divide by $3$.
$\large \frac{85+88+94}{3}$
$\large \frac{267}{3}$
His mean test score is $89$ points.
THE MEAN
The mean of a set of $n$ numbers is the arithmetic average of the numbers.
$\large \mathrm{mean} = \frac{\mathrm{sum\ of\ values\ in\ data\ set}}{n}$
HOW TO: Calculate the mean of a set of numbers.
Write the formula for the mean
Find the sum of all the values in the set. Write the sum in the numerator.
Count the number, 𝑛,n, of values in the set. Write this number in the denominator.
Check to see that the mean is reasonable. It should be greater than the least number and less than the greatest number in the set.
Find the mean of the numbers $8,12,15,9$, and $6$.
Write the formula for the mean: $\mathrm{mean} = \frac{\mathrm{sum\ of\ values\ in\ data\ set}}{n}$
Write the sum of the numbers in the numerator. $\mathrm{mean} = \frac{8+12+15+9+6}{n}$
Count how many numbers are in the set. There are $5$ numbers in the set, so $n=5$. $\mathrm{mean} = \frac{8+12+15+9+6}{5}$
Add the numbers in the numerator. $\mathrm{mean} = \frac{50}{5}$
Then divide. $\mathrm{mean} = 10$
Check to see that the mean is 'typical': $10$ is neither less than $6$ nor greater than $15$. The mean is $10$.
The ages of the members of a family who got together for a birthday celebration were $16,26,53,56,65,70,93$, and $97$ years. Find the mean age.
Write the sum of the numbers in the numerator. $\mathrm{mean} = \frac{16+26+53+56+65+70+93+97}{n}$
Count how many numbers are in the set. Call this $n$ and write it in the denominator. $\mathrm{mean} = \frac{16+26+53+56+65+70+93+97}{8}$
Simplify the fraction. $\mathrm{mean} = \frac{476}{8}$
$\mathrm{mean} = 59.5$
Is $59.5$ 'typical'? Yes, it is neither less than $16$ nor greater than $97$. The mean age is $59.5$ years.
Did you notice that in the last example, while all the numbers were whole numbers, the mean was $59.5$, a number with one decimal place? It is customary to report the mean to one more decimal place than the original numbers. In the next example, all the numbers represent money, and it will make sense to report the mean in dollars and cents.
For the past four months, Daisy's cell phone bills were $ \$ 42.75, \$ 50.12, \$41.54, \$ 48.15$. Find the mean cost of Daisy's cell phone bills.
Count how many numbers are in the set. Call this $n$ and write it in the denominator. $\mathrm{mean} = \frac{\mathrm{sum\ of\ values\ in\ data\ set}}{4}$
Write the sum of the numbers in the numerator. $\mathrm{mean} = \frac{42.75+50.12+41.54+48.15}{4}$
Simplify the fraction. $\mathrm{mean} = \frac{182.56}{4}$
$\mathrm{mean} = 45.64$
Does $ \$45.64$ seem 'typical' of this set of numbers? Yes, it is neither less than $ \$41.54$ nor greater than $ \$ 50.12$.
The mean cost of her cell phone bill was $ \$ 45.64$.
5.5.2 Find the Median of a Set of Numbers
When Ann, Bianca, Dora, Eve, and Francine sing together on stage, they line up in order of their heights. Their heights, in inches, are shown in the table below.
$59$ $60$ $65$ $68$ $70$
Dora is in the middle of the group. Her height, $65$", is the median of the girls' heights. Half of the heights are less than or equal to Dora's height, and half are greater than or equal. The median is the middle value.
The median of a set of data values is the middle value.
Half the data values are less than or equal to the median.
Half the data values are greater than or equal to the median.
What if Carmen, the pianist, joins the singing group on stage? Carmen is $62$ inches tall, so she fits in the height order between Bianca and Dora. Now the data set looks like this:
$59,60,62,65,68,70$
There is no single middle value. The heights of the six girls can be divided into two equal parts.
Statisticians have agreed that in cases like this the median is the mean of the two values closest to the middle. So the median is the mean of $62$ and $65$, $\frac{62+65}{2}$. The median height is $63.5$ inches.
Notice that when the number of girls was $5$, the median was the third height, but when the number of girls was $6$, the median was the mean of the third and fourth heights. In general, when the number of values is odd, the median will be the one value in the middle, but when the number is even, the median is the mean of the two middle values.
HOW TO: Find the median of a set of numbers.
Step 1. List the numbers from smallest to largest.
Step 2. Count how many numbers are in the set. Call this $n$.
Step 3. Is $n$ odd or even?
If $n$n is an odd number, the median is the middle value.
If $n$ is an even number, the median is the mean of the two middle values.
Find the median of $12,13,19,9,11,15$, and $18$.
List the numbers in order from smallest to largest. $9,11,12,13,15,18,19$
Count how many numbers are in the set. Call this $n$. $n=7$
Is $n$ odd or even? odd
The median is the middle value.
The middle is the number in the $4$th position. So the median of the data is $13$.
Kristen received the following scores on her weekly math quizzes:
$83,79,85,86,92,100,76,90,88$, and $64$. Find her median score.
Find the median of $83,79,85,86,92,100,76,90,88$, and $64$.
List the numbers in order from smallest to largest. $64,76,79,83,85,86,88,90,92,100$
Count the number of data values in the set. Call this $n$. $n=10$
Is $n$ odd or even? even
The median is the mean of the two middle values, the $5$th and the $6$th numbers.
Find the mean of $85$ and $86$. $\mathrm{mean} = \frac{85+86}{2}$
Kirsten's median score is $85.5$.
5.5.3 Identify the Mode of a Set of Numbers
The average is one number in a set of numbers that is somehow typical of the whole set of numbers. The mean and median are both often called the average. Yes, it can be confusing when the word average refers to two different numbers, the mean and the median! In fact, there is a third number that is also an average. This average is the mode. The mode of a set of numbers is the number that occurs the most. The frequency, is the number of times a number occurs. So the mode of a set of numbers is the number with the highest frequency.
The mode of a set of numbers is the number with the highest frequency.
Suppose Jolene kept track of the number of miles she ran since the start of the month, as shown in the figure below.
If we list the numbers in order it is easier to identify the one with the highest frequency.
$2,3,5,8,8,8,15$
Jolene ran $8$ miles three times, and every other distance is listed only once. So the mode of the data is $8$ miles.
HOW TO: Identify the mode of a set of numbers.
List the data values in numerical order.
Count the number of times each value appears.
The mode is the value with the highest frequency.
The ages of students in a college math class are listed below. Identify the mode. $18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, $
$21, 21, 22, 22, 22, 22, 22, 23, 24, 24, 25, 29, 30, 40, 44$.
The ages are already listed in order. We will make a table of frequencies to help identify the age with the highest frequency.
Now look for the highest frequency. The highest frequency is $7$, which corresponds to the age $20$. So the mode of the ages in this class is $20$ years.
The data lists the heights (in inches) of students in a statistics class. Identify the mode.
$56$ $61$ $63$ $64$ $65$ $66$ $67$ $67$
$61$ $63$ $64$ $65$ $66$ $67$ $67$
List each number with its frequency.
Now look for the highest frequency. The highest frequency is $6$, which corresponds to the height $67$ inches. So the mode of this set of heights is $67$ inches.
Some data sets do not have a mode because no value appears more than any other. And some data sets have more than one mode. In a given set, if two or more data values have the same highest frequency, we say they are all modes.
5.5.4 Use the Basic Definition of Probability
The probability of an event tells us how likely that event is to occur. We usually write probabilities as fractions or decimals.
For example, picture a fruit bowl that contains five pieces of fruit – three bananas and two apples.
If you want to choose one piece of fruit to eat for a snack and don't care what it is, there is a $\frac{3}{5}$ probability you will choose a banana, because there are three bananas out of the total of five pieces of fruit. The probability of an event is the number of favorable outcomes divided by the total number of outcomes.
PROBABILTY
The probability of an event is the number of favorable outcomes divided by the total number of outcomes possible.
$\large \mathrm{Probability} = \frac{\mathrm{number\ of\ favorable\ outcomes}}{\mathrm{total\ number\ of\ outcomes}}$
Converting the fraction $\frac{3}{5}$ to a decimal, we would say there is a $0.6$ probability of choosing a banana.
$\large \mathrm{Probability\ of\ choosing\ a\ banana} = \frac{3}{5}$
$\large \mathrm{Probability\ of\ choosing\ a\ banana} = 0.6$
This basic definition of probability assumes that all the outcomes are equally likely to occur. If you study probabilities in a later math class, you'll learn about several other ways to calculate probabilities.
The ski club is holding a raffle to raise money. They sold $100$ tickets. All of the tickets are placed in a jar. One ticket will be pulled out of the jar at random, and the winner will receive a prize. Cherie bought one raffle ticket.
Find the probability she will win the prize.
Convert the fraction to a decimal.
What are you asked to find? The probability Cherie wins the prize.
What is the number of favorable outcomes? $1$, because Cherie has $1$ ticket.
Use the definition of probability. $\mathrm{Probability\ of\ an\ event} = \frac{\mathrm{number\ of\ favorable\ outcomes}}{\mathrm{total\ number\ of\ outcomes}}$
Substitute into the numerator and denominator. $\mathrm{Probability\ Cherie\ wins} = \frac{1}{100}$
Write the probability as a fraction. $\mathrm{Probability} = \frac{1}{100}$
Convert the fraction to a decimal. $\mathrm{Probability} = 0.01$
Three women and five men interviewed for a job. One of the candidates will be offered the job.
Find the probability the job is offered to a women.
What are you asked to find? The probability the job is offered to a woman.
What is the number of favorable outcomes? $3$, because there are three women.
What are the total number of outcomes? $8$, because $8$ people interviewed.
Substitute into the numerator and denominator. $\mathrm{Probability} = \frac{3}{8}$
Write the probability as a fraction. $\mathrm{Probability} = \frac{3}{8}$
Convert the fraction to a decimal. $\mathrm{Probability} = 0.375$
Marecek, L., Anthony-Smith, M., & Mathis, A. H. (2020). Use the Language of Algebra. In Prealgebra 2e. OpenStax. https://openstax.org/books/prealgebra-2e/pages/5-5-averages-and-probability. License: CC BY 4.0. Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction | CommonCrawl |
What is the sum of all positive integers $r$ that satisfy $$\mathop{\text{lcm}}[r,700] = 7000~?$$
Note the prime factorizations $700=2^2\cdot 5^2\cdot 7$ and $7000=2^3\cdot 5^3\cdot 7$.
If $\mathop{\text{lcm}}[r,700]=7000$, then in particular, $r$ is a divisor of $7000$, so we can write $r=2^\alpha\cdot 5^\beta\cdot 7^\gamma$, where $0\le\alpha\le 3$, $0\le\beta\le 3$, and $0\le\gamma\le 1$.
Moreover, we know that $\mathop{\text{lcm}}[r,700]=2^{\max\{\alpha,2\}}\cdot 5^{\max\{\beta,2\}}\cdot 7^{\max\{\gamma,1\}}$, and we know that this is equal to $7000=2^3\cdot 5^3\cdot 7$. This is possible only if $\alpha=3$ and $\beta=3$, but $\gamma$ can be $0$ or $1$, giving us two choices for $r$: $$r = 2^3\cdot 5^3\cdot 7^0 = 1000 \text{~~or~~} r=2^3\cdot 5^3\cdot 7^1 = 7000.$$So the sum of all solutions is $1000+7000=\boxed{8000}$. | Math Dataset |
\begin{document}
\title{ Energetic and entropic effects of bath-induced coherences }
\author{C.L. Latune$^1$, I. Sinayskiy$^{1,2}$, F. Petruccione$^{1,2,3}$} \affiliation{$^1$Quantum Research Group, School of Chemistry and Physics, University of KwaZulu-Natal, Durban, KwaZulu-Natal, 4001, South Africa\\ $^2$National Institute for Theoretical Physics (NITheP), KwaZulu-Natal, 4001, South Africa\\ $^3$School of Electrical Engineering, KAIST, Daejeon, 34141, Republic of Korea}
\date{\today} \begin{abstract} The unavoidable interaction of a quantum system with its surrounding (bath) is not always detrimental for quantum properties. For instance, under some specific conditions (that we identify as {\it indistinguishability}), a many-body system can gain internal coherences thanks to the interaction with its bath. The most famous consequence of this phenomenon is superradiance. Beyond that, the thermodynamic effects on the system of these bath-induced coherences have been mostly unexplored. We show here, for a simple and common system (a pair of two-level systems), that the energetic and entropic impacts can indeed be dramatic and diverse, including amplification of the action of the bath but also its mitigation. Our results can be tested experimentally. They suggest that bath-induced coherences can be harnessed to enhance thermodynamic tasks, opening up interesting perspectives for thermal machines, quantum battery charging, natural or artificial energy harvesting systems, and state preparation and protection. \end{abstract}
\maketitle \section{Introduction} The fundamental importance of quantum coherences in quantum thermodynamics has been shown in a growing number of problems ranging from thermal machines \cite{Scully_2003,Zhang_2007,Dillenschneider_2009,Scully_2011, Rahav_2012, Dorfman_2013, Brandner_2015, Uzdin_2015, Niedenzu_2015, Gelbwaser_2015, Leggio_2015b, Mitchison_2015, Killoran_2015, Korzekwa_2016, Uzdin_2016, Chen_2016, Su_2016,Turkpence_2016, Dag_2016, Niedenzu_2016, Mehta_2017, Dag_2018, Levy_2018, Xu_2018, Holubec_2018,Wertnik_2018}, quantum battery charging \cite{Campaioli_2017,Ferraro_2018,Campaioli_2018}, heat flow \cite{apptemppaper}, energy transport \cite{Caruso_2009}, photovoltaic energy conversion \cite{Scully_2010}, and photosynthesis \cite{Romero_2014}. Still, the role of coherences and the full extend of its impact is far from being fully understood.
Beyond its use, the production, manipulation and conservation of coherences represent serious challenges particularly due to the unavoidable influence of baths which tends to leave any system in thermal states (with no coherences left to use or extract). However, under some specific conditions (to be detailed), the interaction with the bath becomes beneficial for coherences. It is a well-known phenomenon, brilliantly exploited in superradiance \cite{Dicke_1954, Gross_1982} and generation of entanglement \cite{Benatti_2003, Benatti_2010, Passos_2018}. Recently, a study \cite{Cakmak_2017} focused specifically on the amount of coherences that can be generated through bath interaction.
Here, aiming at using bath-induced coherences for thermodynamic applications, we focus on alternative aspects, mostly unexplored so far, namely the energetic and entropic impacts for the system of bath-induced coherences. We show that the effects can indeed be diverse and drastic, and persist for most initial states and any bath temperature. As a special case we study initial thermal states, which are particularly important for thermodynamics and experiments. We find that for a pair of two-level systems, one of the simplest and experimentally accessible system exhibiting bath-induced coherences, phenomena of mitigation of the bath effects can happen, resulting in large reduction or increase of both the steady state energy and entropy of the system. Even more interesting, phenomena of amplification of the bath effects can also happen, implying as well a large reduction or increase of the system's steady state energy.
The underlying phenomenon responsible for such effects is identified as the indistinguishability of the subsystems from the point of view of the bath. Focusing on a pair of two-level systems presents several advantages like allowing for simple experimental realisations and providing the opportunity to better understand the consequences of bath-induced coherences before dealing with more complex systems.
The phenomena of bath mitigation due to bath-induced coherences was already pointed out in \cite{apptemppaper}. Here, we extend the analysis to arbitrary initial states (instead of only the ground state), negative effective bath temperatures (which emerges for instance when two baths interact with the system \cite{Brunner_2012}, or in autonomous thermal machines \cite{autonomous}), and include also entropy considerations. Thus, the scope of the results is greatly increased and the phenomenon of amplification of the bath effects is uncovered. The relation between energy amplification/attenuation, entropy amplification/attenuation, and bath-induced coherences is made explicit. An apparent paradox emerges since coherences do not contribute to the energy so that the common sense tells us that coherences are not able to affect the energy of a system. An intuitive explanation of this paradox is provided in the light of the apparent temperature introduced in \cite{apptemppaper}. Our results have promissing applications in quantum thermodynamics, thermal machines, and quantum battery charging, but also potential applications for natural or artificial energy harvesting devices, and state preparations or protection for computation and quantum error correction.
\section{Model} We consider a pair of two-level systems (two-level atoms or spins $1/2$) of energy transition $\omega$ ($\hbar=1$) and interacting with a bath in a thermal state at inverse temperature $\beta_B$. Importantly, we considered that $\beta_B$ can be positive or negative since the combination of several baths or systems can result in an effective bath having some transitions (pair of levels) with negative temperatures \cite{Brunner_2012}. If such transitions are resonant with the two-level systems, the effective bath behaves formally as a bath at negative temperature. Similarly, the quantum battery of an autonomous thermal machine reaches a steady state with an (apparent) temperature which can take negative values \cite{autonomous}, as if the evolution of the quantum battery was driven by a thermal state in a negative temperature.
Furthermore, we assume that the interaction with the bath gives raise to {\it collective} dissipation. We explain in the following what we mean by collective dissipation and what are the underlying conditions. Taking the seminal example of two-level atoms interacting with the free space electric field (which can be extended straightforwardly to spins), the interaction is of the form (under the dipole approximation) \cite{Gross_1982} \begin{equation}\label{gencoupling} V= -\sum_{i=1,2} \vec D_i.\vec E(\vec r_i),
\end{equation} where $\vec E(\vec r_i)$ is the electric field operator at the position $\vec r_i$ of the atom $i$, and $\vec D_i = d(\sigma_i^{+} + \sigma_i^{-})\vec\epsilon_i$ is the dipole operator of the atom $i$ with $d$ the electric dipole (identical for both atoms) and $\vec\epsilon_i$ the polarisation of the atomic transition between the ground state $|0\rangle_i$ to the excited state $|1\rangle_i$. The operators $\sigma_i^{+}=|1\rangle_i\langle 0|$ and $\sigma_i^{-}=|0\rangle_i\langle 1|$ are the ladder operators of the atom $i$, realising such transition. The above interaction Hamiltonian \eqref{gencoupling} can be rewritten in the form, \begin{equation}\label{gencoupling2} V= -\sum_{i=1,2} (\sigma_i^{+} + \sigma_i^{-}) B_i \end{equation} where $B_i := d \vec \epsilon_i.\vec E(\vec r_i)$ corresponds to the bath operator interacting with the atom $i$. If the two atoms are far apart (typically separated by a distance much larger than the emission wavelength $\lambda_a = c/\omega$), one can show that the expectation value $\langle B_1 B_2 \rangle_{\rho_{\rm bath}} :={\rm Tr} B_1 B_2 \rho_{\rm bath}$ is equal to zero (assuming that the electromagnetic field is in a thermal state, see more detail in \cite{Gross_1982} and Appendix \ref{apppol}). Thus, one can consider that each atom interacts effectively with their own independent bath represented by the operators $B_1$ and $B_2$.
Alternatively, if the atoms have orthogonal polarisations $\vec \epsilon_1.\vec \epsilon_2 =0$ (while confined in a volume much smaller than the emission wavelength $\lambda_a$), one can show (Appendix \ref{apppol}) that the bath operators $B_i$ still satisfy $\langle B_1 B_2 \rangle_{\rho_{\rm bath}} =0$ implying again that each atom interacts effectively with its own independent bath.
Under such conditions and assuming a weak bath coupling so that the Markov and Born approximations \cite{Cohen_Book, Petruccione_Book} are valid, the dissipative dynamics of the system is given by a master equation of the form \cite{Gross_1982} \begin{eqnarray}\label{medis} \dot{\rho}_S^I &=& -i\Omega_L [\sigma_1^{+}\sigma_1^{-}+\sigma_2^{+}\sigma_2^{-},\rho_S^I] \nonumber\\ &&+ g [n(\omega) +1]\sum_{i=1}^2 (2\sigma_i^-\rho_S^I \sigma_i^+ -\sigma_i^+\sigma_i^-\rho_S^I - \rho_S^I\sigma_i^+\sigma_i^-)\nonumber\\ &&+ g n(\omega) \sum_{i=1}^2(2\sigma_i^+\rho_S^I \sigma_i^- -\sigma_i^-\sigma_i^+\rho_S^I - \rho_S^I\sigma_i^-\sigma_i^+),\nonumber\\ \end{eqnarray} where $n(\omega)$ is the bath mean excitation number at the frequency $\omega$, $\Omega_L$ is the Lamb shift, and $g= \frac{d^2\omega^3}{6\pi c^3\hbar \epsilon_0}$ is the effective coupling strength with $c$ the vacuum light velocity and $\epsilon_0$ is the vacuum permeability. The above master equation corresponds to two atoms interacting with their own independent bath of same characteristics (temperature and coupling strength). \\
The opposite situation is when the atoms are at the same position (or at least confined in a volume much smaller than the emission wavelength $\lambda_a = c/\omega$) and with parallel polarisation, $\vec \epsilon_1=\vec \epsilon_2$, so that the two bath operator $B_1$ and $B_2$ are equal: the two atoms are effectively {\it indistinguishable} to the bath.
In such a situation the dissipation is given by the following dynamics \cite{Gross_1982,Cakmak_2017} (still assuming a weak coupling with the bath),
\begin{eqnarray}\label{meind} \dot{\rho}_S^I &=& -i\Omega_L [\sigma_1^{+}\sigma_1^{-} + \sigma_2^{+}\sigma_2^{-},\rho_S^I]-i\Omega_{I}[\sigma_1^{+}\sigma_{2}^{-}+\sigma_1^{-}\sigma_{2}^{+},\rho_S^I] \nonumber\\ &&+ g [n(\omega) +1] (2S^-\rho_S^I S^+ -S^+S^-\rho_S^I - \rho_S^IS^+S^-)\nonumber\\ &&+ g n(\omega) (2S^+\rho_S^I S^- -S^-S^+\rho_S^I - \rho_S^IS^-S^+),\nonumber\\ \end{eqnarray} where $S^{\pm} = \sigma_1^{\pm} +\sigma_2^{\pm}$ are the {\it collective} ladder operators, and $\Omega_I$ characterises the interaction strength between the two atoms (which appears due to the spatial confinement of the pair, see \cite{Gross_1982} and Appendix \ref{tlatoms}). Note that the work in \cite{Cakmak_2017} studies the generation of coherences induced by the bath in a pair of two-level atoms considering also the above dynamics \eqref{medis}. However, they focus mainly on the situation where the pair is initialised in the ground state (and on the temporary impact of the atom interaction $\Omega_I$ for some specific initial states). Moreover, all the considerations on the energetic and entropic impact on the steady state of the pair, which constitute the main contributions of our paper, are absent of \cite{Cakmak_2017}. \\
In the following, we refer to the dynamics described by \eqref{medis} as {\it independent dissipation}. As shown above, it results from the different position or polarisation of the two atoms. In other words, the two atoms bear different individual characteristics which makes them {\it distinguishable} from the point of view of the bath. This concept can be extended to any system. We therefore call {\it distinguishable} any pair (or larger ensemble) of subsystems bearing different individual characteristics resulting in an independent interaction of each subsystem with its own effective bath (as described by Eq. \eqref{medis}). By contrast, we refer to the dynamics \eqref{meind} as {\it collective dissipation}, which was shown to be a consequence of the same individual characteristic of the two atoms, making them {\it indistinguishable} to the bath. Therefore, we call {\it indistinguishable} any pair (or larger ensemble) of subsystems bearing the same characteristics so that each subsystem interacts with exactly the same bath (as described by Eq. \eqref{meind}). Note that indistinguishability can also be enforced through bath engineering (as for instance by inserting an optical cavity \cite{Woods_2014}).
The above example of a pair of two-level atoms interacting with the free space electromagnetic field is merely an illustration of these notions of distinguishability and indistinguishability.
In the remainder of the paper we focus on their consequences for the steady states and the associated thermodynamic properties.
\section{Steady states} The steady state of the independent dynamics Eq. \eqref{medis} is well-known since it corresponds to the independent dissipation of each subsystems. The steady state is therefore the product of the steady state of each subsystems, namely, thermal states at the bath temperature \cite{Petruccione_Book}, \begin{eqnarray} \rho^{\rm th}(\beta_B) &=& Z^{-1}(\beta_B) e^{-H_0\beta_B} \nonumber\\
&=&Z^{-1}(\beta_B) \Big[e^{-2\omega\beta_B}|1\rangle|1\rangle\langle 1|\bra1|\\
+&e^{-\omega\beta_B}&(|1\rangle|0\rangle\langle 1|\bra0|+|0\rangle|1\rangle\bra0|\bra1|) + |0\rangle|0\rangle\bra0|\bra0|\Big]\nonumber
\end{eqnarray} (the tensor product order is taken to be the same for ``bras'' and ``kets''), where $Z(\beta_B):=1+2e^{-\omega\beta_B} + e^{-2\omega\beta_B}$, and \begin{equation}\label{freeH} H_0:= \omega (\sigma_1^{+}\sigma_1^{-} + \sigma_2^{+}\sigma_2^{-}), \end{equation}
is the free Hamiltonian of the pair of two-level systems.\\
The steady state of the collective dissipation, denoted by $\rho^{\rm ss}(\beta_B,r)$, can be easily found by projecting Eq. \eqref{meind} onto the basis of symmetric and anti-symmetric states $|\psi_{\pm}\rangle :=(|0\rangle|1\rangle\pm|1\rangle|0\rangle)/\sqrt{2}$, $|\psi_{0}\rangle := |0\rangle|0\rangle$, and $|\psi_1\rangle:=|1\rangle|1\rangle$. We obtain (see Appendix \ref{dynamics}), \begin{eqnarray}\label{genss}
&& \rho^{\rm ss}(\beta_B,r) = (1-r)|\psi_{-}\rangle\langle\psi_{-}| + rZ_{+}^{-1}(\beta_B) \\
&&\hspace{0.7cm}\times\Big(e^{-2\omega\beta_B}|\psi_1\rangle\langle \psi_1|+e^{-\omega\beta_B}|\psi_+\rangle\langle\psi_+| + |\psi_0\rangle\langle\psi_0|\Big),\nonumber
\end{eqnarray} where $r:= \langle \psi_0|\rho(0)|\psi_0\rangle + \langle \psi_1|\rho(0)|\psi_1\rangle + \langle \psi_{+}|\rho(0)|\psi_{+}\rangle$ and $Z_{+}(\beta_B):= 1+e^{-\omega\beta_B} +e^{-2\omega\beta_B}$. The steady state of the collective dynamics contrasts largely with the thermal state $\rho^{\rm th}(\beta_B)$, which brings several observations. First, $\rho^{\rm ss}(\beta_B,r)$ is not unique and depends on the initial state through $r$, which is a striking difference from the steady state $\rho^{\rm th}(\beta_B)$. Secondly, the steady state $\rho^{\rm ss}(\beta_B,r)$ can contain non-vanishing coherences as opposed to $\rho^{\rm th}(\beta_B)$. This is the object of the following Section. \\
\section{Steady state coherences}\label{sectionsscoh}
In this Section we show explicitly that the steady state of the collective dissipation, $\rho^{\rm ss}(\beta_B,r)$, can contain (global) coherences in the form of coherent superposition of the states $|0\rangle|1\rangle$ and $|1\rangle|0\rangle$ (corresponding to non-diagonal terms in the basis $\{|0\rangle|0\rangle,|1\rangle|0\rangle,|0\rangle|1\rangle,|1\rangle|1\rangle\}$). Such coherences are induced (or maintained, if initially present) by the bath. Roughly speaking, since the bath does not distinguish between the two-level systems, each time an excitation is absorbed from the bath or emitted to the bath, both two-level systems gain or lose simultaneously an excitation (if in the ground or excited state), generating correlations between them.
More precisely, during these processes of absorption and emission the transitions $|1\rangle|1\rangle \leftrightarrow |\psi_{+}\rangle$ and $|0\rangle|0\rangle \leftrightarrow|\psi_{+}\rangle$ take place, which involves coherent superpositions of $|0\rangle|1\rangle$ and $|1\rangle|0\rangle$ ($|\psi_{+}\rangle= (|1\rangle|0\rangle +|0\rangle|1\rangle)/\sqrt{2}$), generating (or maintaining) coherences in the system. In Appendix \ref{appindistin} we detail more this idea around the role of indistinguishability and draw a parallel with entanglement generation in quantum optics.
One should note that the coherences between $|1\rangle|0\rangle$ and $|0\rangle|1\rangle$ corresponds also to correlations between the two-level systems. This can be seen be observing that the reduced steady state of each two-level system is always diagonal, so that the tensor product of the local states is different from $\rho^{\rm ss}(\beta_B,r)$, indicating the presence of correlations. Therefore, the coherences between $|1\rangle|0\rangle$ and $|0\rangle|1\rangle$ can be seen alternatively as correlations between the two two-level systems.
The expression of the steady state coherences is obtained directly from the expression of the steady state \eqref{genss},
\begin{eqnarray}
\langle 0|\bra1|\rho^{\rm ss}(\beta_B,r)|1\rangle|0\rangle &=&\langle 1|\bra0|\rho^{\rm ss}(\beta_B,r)|0\rangle|1\rangle \nonumber\\
&=& \frac{1}{2}\left(\frac{r}{z(\beta_B)} -1\right),
\end{eqnarray}
with $z(\beta_B):= \frac{Z_+(\beta_B)}{Z(\beta_B)}$. It is convenient for the remainder of the paper to define $c:= \langle 0|\bra1|\rho^{\rm ss}(\beta_B,r)|1\rangle|0\rangle + \langle 1|\bra0|\rho^{\rm ss}(\beta_B,r)|0\rangle|1\rangle=\left(r/z(\beta_B)-1\right)$ the sum of the steady state coherences, which can take any value within the interval $[-1;\frac{1}{3}]$. One should note that a proper measure of coherence as defined in \cite{Baumgratz_2014} is for instance the $l_1$ norm of coherence (the sum of the absolute value of each coherence), which would give here ${\cal C}_{l_1}[\rho^{\rm ss}(\beta_B,r)]= | \langle 0|\bra1|\rho^{\rm ss}(\beta_B,r)|1\rangle|0\rangle| + | \langle 1|\bra0|\rho^{\rm ss}(\beta_B,r)|0\rangle|1\rangle|= |c|$, taking value in $[0;1]$. Note that one recovers the result of \cite{Cakmak_2017} when the pair is initialised in the ground state by taking $r=1$, giving an amount of steady state coherence equal to ${\cal C}_{l_1}[\rho^{\rm ss}(\beta_B,1)]=1/z(\beta_B)-1$. Nevertheless, we will see in the following that the sign of the coherences is essential, therefore the important quantity in this problem is $c$.
Crucially, the coherences (and therefore $c$) are strictly positive as soon as $r > z(\beta_B)$, strictly negative when $r < z(\beta_B)$, and null when $r = z(\beta_B)$. Anticipating the remainder of the paper, we stress that this observation is fundamental since, as shown in \cite{apptemppaper}, the sign of $c$ determines whether the apparent temperature \cite{apptemppaper} is increased or decreased and consequently the steady state energy (see more detail in the following).
Up to now our considerations are valid for any initial state. It is however particularly interesting to look at the restricted class of initial states made of thermal states, particularly important for thermodynamics and the most common and accessible experimentally. Moreover, thermal states contain no coherence so that any coherence in the steady state is induced by the bath. Considering an initial state $\rho^{\rm th}(\beta_0)$ at the inverse temperature $\beta_0$ (allowed to be negative), the steady state is given by the general expression \eqref{genss} with $r$ equal to
\begin{eqnarray} r&=& \langle \psi_0|\rho^{\rm th}(\beta_0)|\psi_0\rangle + \langle \psi_1|\rho^{\rm th}(\beta_0)|\psi_1\rangle \nonumber\\
&&\hspace{2.4cm}+ \langle \psi_{+}|\rho^{\rm th}(\beta_0)|\psi_{+}\rangle\nonumber\\ &=& z(\beta_0). \end{eqnarray}
$z(\beta_0)$ is a strictly monotonic increasing function of $\beta_0$ on $]0;+\infty]$ (from $3/4$ to 1) and decreasing on $]-\infty;0[$ (from 1 to $3/4$). In the remainder of the paper we denote by $\rho^{\rm ss}(\beta_B,\beta_0):=\rho^{\rm ss}[\beta_B,r=z(\beta_0)]$ the steady state reached by the pair when initialised in $\rho^{\rm th}(\beta_0)$.
Coming back to the considerations on the steady state coherences, we reach an enlightening conclusion. The bath-induced coherences
take the simple form $c=z(\beta_0)/z(\beta_B) -1$, which are strictly positive (negative) if and only if $|\beta_0|>|\beta_B|$ ($|\beta_0|<|\beta_B|$). Moreover, the only situation with no bath-induced coherences is when $|\beta_0| = |\beta_B|$. It coincides with the fact that $\rho^{\rm th}(\beta_B)$ is naturally a steady state of the collective dissipation, meaning that for $\beta_0=\beta_B$ we have $\rho^{\rm ss}(\beta_B,\beta_B) = \rho^{\rm th}(\beta_B)$ which does not contain coherences. The crucial role of the steady state coherences in determining the major thermodynamic properties (energy and entropy) of the pair of two-level systems is shown in the following.\\
\begin{figure}
\caption{Graph of the steady state energy $E^{\rm ss}(\beta_B,\beta_0)$ as a function of the initial inverse temperature $\beta_0$, for (a) $\omega\beta_B =2$, and (b) $\omega\beta_B=-2$. In each graph the value of the thermal energy $E^{\rm th}(\beta_B)$ is indicated as reference by the Black line. }
\label{ssenergy}
\end{figure}
\section{Steady state energy}\label{sectionenergy}
The steady state energy of the collective dissipation is given by
\begin{eqnarray}\label{ssen1}
E^{\rm ss}(\beta_B,r) &: =& {\rm Tr} H_0 \rho^{\rm ss}(\beta_B,r) \nonumber\\
&=& \omega \frac{r}{Z_{+}(\beta_B)} (2e^{-2\omega\beta_B} + e^{-\omega \beta_B}) +\omega(1-r),\nonumber\\
\end{eqnarray}
to be compared with the thermal energy
\begin{equation}
E^{\rm th}(\beta_B) := {\rm Tr} H_0 \rho^{\rm th}(\beta_B) = 2\omega (e^{\omega\beta_B}+1)^{-1},
\end{equation}
reached by the independent dissipation. One can already see from \eqref{ssen1} that on top of a strong dependence on $r$ and consequently on the initial state, the steady state energy can be smaller or larger than the thermal energy $E^{\rm th}(\beta_B)$ (as detailed in the following).
Moreover, $E^{\rm ss}(\beta_B,r)$ can be simply expressed in term of the thermal energy and bath-induced coherences,
\begin{eqnarray}\label{energycoh}
E^{\rm ss}(\beta_B,r) &=& E^{\rm th}(\beta_B) - \omega \frac{1-e^{-\omega\beta_B}}{1+e^{-\omega\beta_B}}c.
\end{eqnarray}
This is a rather surprising result as the coherences do not carry energy.
How do the steady state coherences end up contributing to the steady state energy? We answer to this question in the following. First, one can notice that for positive bath temperature, $E^{\rm ss}(\beta_B,r)$ is increased with respect to the thermal energy precisely when the bath-induced coherences $c$ are negative, and conversely, $E^{\rm ss}(\beta_B,r)$ is reduced when $c$ is positive. For negative bath temperature, the above conclusions are inverted. Thus, the steady state energy $E^{\rm ss}(\beta_B,r)$ can be amplified or attenuated in a controlled way (determined by the sign of $c$ which is itself related to the initial conditions). This may have useful applications, in particular in thermodynamics and quantum thermal machines.
\subsection{Mitigation and amplification of the bath effects}
In a perspective of thermodynamic applications, we consider an initial thermal state at inverse temperature $\beta_0$ as in the previous Section. Similarly with the steady state, we denote by $E^{\rm ss}(\beta_B,\beta_0):=E^{\rm ss}[\beta_B,r=z(\beta_0)]$ the steady state energy when the pair is initialised in the state $\rho^{\rm th}(\beta_0)$. We obtain the following insight. The steady state energy $E^{\rm ss}(\beta_B,\beta_0)$ is strictly larger (smaller) than $E^{\rm th}(\beta_B)$ if and only if $|\beta_0|<\beta_B$ ($|\beta_0|>\beta_B$) for positive bath temperatures, as shown in Fig. \ref{ssenergy} (a). Conversely, the steady state energy is strictly larger (smaller) than the thermal energy if and only if $|\beta_0|>\beta_B$ ($|\beta_0|<\beta_B$) for negative bath temperatures, see Fig. \ref{ssenergy} (b). Interestingly, for $\beta_0/\beta_B>-1$ (which corresponds to the regimes of parameters denoted by ``Mitigation'' in Fig. \ref{ssenergy}), the collective interaction mitigates the bath dissipation since the pair of two-level systems starts and ends with an energy larger (or smaller) than the thermal energy.
This might be useful for preparing or maintaining a pair in a low (or high) energy state.
By contrast, when $\beta_0/\beta_B<-1$ (which corresponds to the regimes of parameters denoted by ``Amplification'' in Fig. \ref{ssenergy}), there is an amplification of the bath effects since the steady state energy goes beyond the thermal energy, meaning that the system starts with an energy larger (smaller) than $E^{\rm th}(\beta_B)$ and ends up with a steady state energy $E^{\rm ss}(\beta_B,\beta_0)$ smaller (larger) than $E^{\rm th}(\beta_B)$. Such effects can be applied to amplify a cooling process. Let's consider a pair of two-level systems undergoing a cooling process either by direct thermal contact with a cold bath at inverse temperature $\beta_B$, either with a thermal machine yielding a steady state inverse temperature $\beta_B$ \cite{autonomous} (which can also be identified as the virtual temperature of the baths acting on the thermal machine \cite{Brunner_2012}). Then, if the pair of two-level systems is initially in an excited state such that $\beta_0<-\beta_B<0$), the pair of systems reaches a steady state energy $E^{\rm ss}(\beta_B,\beta_0)$ strictly lower than the thermal energy $E^{\rm th}(\beta_B)$: the cooling process is amplified.
Similarly, considering the reverse process of loading energy into the pair of two-level systems, one can consider either a direct charging by thermal contact with a bath at inverse temperature $\beta_B$, either a charging through a thermal machine yielding a steady state inverse temperature $\beta_B$. Then, when the effective inverse temperature $\beta_B$ is negative \cite{Brunner_2012, autonomous}, if the pair is initialised in a low temperature state satisfying $\beta_0>-\beta_B>0$ it reaches a steady state energy $E^{\rm ss}(\beta_B,\beta_0)$ strictly larger than the thermal energy $E^{\rm th}(\beta_B)$: the energy charging is increased. This is of great interest for quantum battery charging.
Remarkably, such extra performances (mitigation, super cooling and super energy charging) rely only on indistinguishability (collective interaction with the bath).
\begin{figure}
\caption{(a) Graph of the steady state energy $E^{\rm ss}(\beta_B,\beta_0)$ (Orange curve) and thermal energy $E^{\rm th}(\beta_B)$ (Balck curve) as a function of the bath temperature $\omega \beta_B \in [0;4]$ for $\omega|\beta_0| \gg1$. (b) Corresponding ratio $E^{\rm ss}(\beta_B,\beta_0)/E^{\rm th}(\beta_B)$ as a function of $\omega \beta_B$.}
\label{enposlargeb}
\end{figure}
\begin{figure}
\caption{(a) Graph of the steady state energy $E^{\rm ss}(\beta_B,\beta_0)$ (Orange curve) and thermal energy $E^{\rm th}(\beta_B)$ (Balck curve) as a function of the bath temperature $\omega\beta_B \in [0;4]$ for $\omega|\beta_0| \ll1$. (b) Corresponding ratio $E^{\rm ss}(\beta_B,\beta_0)/E^{\rm th}(\beta_B)$ as a function of $\omega\beta_B$.}
\label{enpossmallb}
\end{figure}
\begin{figure}
\caption{(a) Graph of the steady state energy $E^{\rm ss}(\beta_B,\beta_0)$ (Orange curve) and thermal energy $E^{\rm th}(\beta_B)$ (Balck curve) as a function of the bath temperature $\omega \beta_B \in [-4;0]$ for $\omega|\beta_0| \gg1$. (b) Corresponding ratio $E^{\rm ss}(\beta_B,\beta_0)/E^{\rm th}(\beta_B)$ as a function of $\omega \beta_B$.}
\label{enneglargeb}
\end{figure}
\begin{figure}
\caption{(a) Graph of the steady state energy $E^{\rm ss}(\beta_B,\beta_0)$ (Orange curve) and thermal energy $E^{\rm th}(\beta_B)$ (Balck curve) as a function of the bath temperature $\omega \beta_B \in [-4;0]$ for $\omega|\beta_0| \ll1$. (b) Corresponding ratio $E^{\rm ss}(\beta_B,\beta_0)/E^{\rm th}(\beta_B)$ as a function of $\omega \beta_B$.}
\label{ennegsmallb}
\end{figure}
How important can be these extra performances?
When $\beta_B>0$, the minimal value of the steady state energy $E^{\rm ss}(\beta_B,\beta_0)$ is attained for initial states such that $\omega |\beta_0| \gg 1$ and is equal to
\begin{equation}\label{enbeta0gg1}
E^{\rm ss}(\beta_B,\beta_0) \rightarrow_{\omega|\beta_0|\gg1} E^{\rm th}(\beta_B) - \omega \frac{1-e^{-\omega\beta_B}}{1+e^{-\omega\beta_B}}\frac{e^{-\omega\beta_B}}{Z_{+}(\beta_B)},
\end{equation} which tends to be $50\%$ smaller than $E^{\rm th}(\beta_B)$ when $\omega\beta_B\gg1$. Graphs of $E^{\rm ss}(\beta_B,\beta_0)$, $E^{\rm th}(\beta_B)$, and $E^{\rm ss}(\beta_B,\beta_0)/E^{\rm th}(\beta_B)$ as functions of $\beta_B$ are represented in Fig. \ref{enposlargeb} for $\omega|\beta_0|\gg1$. Consequently, extra cooling of up to $50\%$ (steady state energy up to 50$\%$ smaller) can be achieved due to bath-induced coherences (indistinguishability).
Conversely, the maximal steady state energy, still in presence of a positive bath temperature, is attained for $\omega|\beta_0|\ll1$, and is equal to
\begin{eqnarray}\label{enbeta0ll1}
&&E^{\rm ss}(\beta_B,\beta_0) \rightarrow_{\omega|\beta_0|\ll1} E^{\rm th}(\beta_B) \\
&&\hspace{3.5cm}-\omega \frac{1-e^{-\omega\beta_B}}{1+e^{-\omega\beta_B}}\left(\frac{3}{4}\frac{Z(\beta_B)}{Z_{+}(\beta_B)}-1\right),\nonumber
\end{eqnarray}
which tends to the value $\omega/4$ for $\omega \beta_B\gg1$ whereas $E^{\rm th}(\beta_B)$ tends to 0 for such bath temperatures: strong mitigation effect. Graphs of $E^{\rm ss}(\beta_B,\beta_0)$, $E^{\rm th}(\beta_B)$, and $E^{\rm ss}(\beta_B,\beta_0)/E^{\rm th}(\beta_B)$ as functions of $\beta_B$ are represented in Fig. \ref{enpossmallb} for $\omega|\beta_0|\ll1$.
The above observations are inverted when the bath temperature is negative. Namely, the minimal steady state energy is attained for $\omega|\beta_0|\ll1$, and the maximal steady state energy for $\omega|\beta_0|\gg1$, with the same respective expression \eqref{enbeta0ll1} and \eqref{enbeta0gg1}. The corresponding graphs of $E^{\rm ss}(\beta_B,\beta_0)$, $E^{\rm th}(\beta_B)$, and $E^{\rm ss}(\beta_B,\beta_0)/E^{\rm th}(\beta_B)$ as functions of $\beta_B$ are represented in Fig. \ref{enneglargeb} and Fig. \ref{ennegsmallb} for $\omega|\beta_0|\gg1$ and $\omega|\beta_0| \ll1$, respectively.
Then, in the amplification regime, corresponding to an initial thermal state such that $\beta_0 > |\beta_B|$, the bath-induced coherences can yield an extra load of energy of up to $8\%$ (Fig. \ref{enneglargeb}). \\
\subsection{Local temperature} One more aspect strengthening the above considerations is that locally, each two-level systems ends up in a {\it thermal state} at an inverse temperature $\beta_{\rm Loc}$ {\it different} from the bath temperature. This can be seen directly by tracing out one of the two-level systems in $\rho^{\rm ss}(\beta_B,\beta_0)$. We do not provide here a detailed description of the properties of $\beta_{\rm Loc}$ since it is intimately related to the ones of $E^{\rm ss}(\beta_B,\beta_0)$. We only mention briefly that the mitigation and amplification effects appear also strongly in the steady state local temperature (reaching high levels up to $33\%$ of relative increase or reduction with respect to $\beta_B$). More details can be found in Appendix \ref{localtemp}.
\subsection{The fundamental role of bath-induced coherences} We now come back to the question raised above: how coherences, which do not carry energy, can end up contributing to the steady state energy?
Firstly, as a preliminary observation, the heat flow between the pair of the two-level systems in a state $\rho$
and the bath is characterised by the apparent temperature of the pair defined by \cite{apptemppaper} \begin{equation}\label{defapptemp} {\cal T} :=\omega \left(\log\frac{{\rm Tr}S^{-}S^{+}\rho}{{\rm Tr}S^{+}S^{-}\rho}\right)^{-1}.
\end{equation} Note that if the pair is in the dark state $|\psi_-\rangle\langle \psi_-|$ it does not interact with the bath and therefore there is no heat flow and no apparent temperature can be defined. The apparent temperature ${\cal T}$ determines the direction of the heat flow in the same way as usual temperature does \cite{apptemppaper}: if ${\cal T} > 1/\beta_B$, the heat goes from the pair to the bath. Conversely, if ${\cal T} < 1/\beta_B$, the heat flows from the bath to the pair. We can conclude that a necessary condition for a steady state is to have an apparent temperature equal to the reservoir temperature $1/\beta_B$ (otherwise the heat flow is not null). Indeed, one can verify (Appendix \ref{smapptempss}) that all states of the form \eqref{genss} have an apparent temperature equal to the bath temperature $1/\beta_B$.
Secondly, coherences (correlations) are built up (or maintained) due to indistinguishability, as explained above in Section \ref{sectionsscoh}. Such coherences affect dramatically the apparent temperature \cite{apptemppaper}. More precisely, for positive bath temperature, positive steady state coherences (when $r>z(\beta_B)$) increase the apparent temperature of the pair of two-level systems. Thus, the steady state of the pair cannot have the same excited state populations as the thermal state $\rho^{\rm th}(\beta_B)$ otherwise its apparent temperature would be strictly larger than the bath temperature $1/\beta_B$. Consequently, the steady state must have lower excited state populations than the thermal state $\rho^{\rm th}(\beta_B)$, which implies lower energy.
Conversely, negative coherences (corresponding to $r<z(\beta_B)$) reduce the apparent temperature so that the steady state of the pair must have higher excited state populations than the thermal state $\rho^{\rm th}(\beta_B)$ in order to reach an apparent temperature equal to the bath temperature $1/\beta_B$. The above results are inverted for negative bath temperature, namely positive (negative) coherences decrease (increase) the apparent temperature. Again, if the steady state populations are affected, the steady state energy too.
Therefore, for these reasons, the amount of steady state coherences affect indirectly the steady state energy, as observed in Eq. \eqref{energycoh}.
\subsection{Dark states} Alternatively, the above effects can be understood in terms of dark states.
Considering that the pair of two-level systems is initially in a thermal state $\rho^{\rm th}(\beta_0)$ at inverse temperature $\beta_0$, it can be decomposed as a balanced combination of bright and dark states, $|\psi_+\rangle$ and $|\psi_-\rangle$, respectively (plus the contributions of $|\psi_0\rangle$ and $|\psi_1\rangle$). The dark state component is $e^{-\omega\beta_0}/Z(\beta_0)$, which is a monotonic decreasing function of $\beta_0 \in [0;\infty]$. Then, when $S$ thermalises with the bath at $\beta_B$, only the components $|\psi_0\rangle$, $|\psi_+\rangle$, and $|\psi_1\rangle$ ``thermalise''. If $\beta_B$ is smaller than $\beta_0$, the pair ends up with a deficit of dark state component in comparison to a thermal state at inverse temperature $\beta_B$. Therefore, remembering that the dark states bears an energy $\omega$, the pair sees its energy reduced in relation to the thermal energy $E^{\rm th}(\beta_B)$. Conversely, if $\beta_B$ is larger than $\beta_0$, the pair ends up with a surplus of dark state component in comparison to a thermal state at inverse temperature $\beta_B$. Consequently, the energy of the pair $E^{\rm ss}(\beta_B,\beta_0)$ is found to be higher than the thermal energy $E^{\rm th}(\beta_B)$. This is mitigation of the bath effects.
The above rough reasoning misses some subtleties related to the normalisation factors, but this does not change the central fact that a deficit (surplus) of dark state reduces (increases) the energy of the pair. Similar considerations can be extended to negative temperatures, remembering that $e^{-\omega\beta_0}/Z(\beta_0)$ is monotonic increasing on $[-\infty;0]$, and that a deficit (surplus) of dark state component increases (reduces) the energy of the pair in comparison to the thermal energy $E^{\rm th}(\beta_B)$ for $\beta_B<0$.
\section{Steady state entropy}
We look now at the von Neumann entropy of the steady state. Computing the von Neumann entropy of the steady state \eqref{genss} we have, \begin{eqnarray} &&S^{\rm ss}(\beta_B,r) := S[\rho^{\rm ss}(\beta_B,r)] = -{\rm Tr} \rho^{\rm ss} (\beta_B)\log \rho^{\rm ss} (\beta_B) \nonumber\\ &&\hspace{1.4cm}= -r \log r -(1-r)\log(1-r)\nonumber\\ && \hspace{0.4cm}+ r \log Z_{+}(\beta_B) + \omega\beta_BrZ_{+}^{-1}(\beta_B)(e^{-\omega \beta_B}+2e^{-2\omega \beta_B}) \nonumber\\ &&\hspace{1.4cm}= -r \log r -(1-r)\log(1-r)\nonumber\\ &&\hspace{0.4cm} + r \log Z_{+}(\beta_B) + \beta_B E^{\rm ss}(\beta_B) - \omega\beta_B(1-r), \end{eqnarray} to be compared with the thermal state entropy, \begin{eqnarray} S^{\rm th}(\beta_B) &:=& S[\rho^{\rm th}(\beta_B)] \nonumber\\ &=&\log Z(\beta_B) + \beta_B E^{\rm th}(\beta_B). \end{eqnarray} The steady state entropy $S^{\rm ss}(\beta_B,r)$ can be re-written in terms of $S^{\rm th}(\beta_B) $ and $c$, \begin{eqnarray}\label{entropyc} S^{\rm ss}(\beta_B,r) &=& S^{\rm th}(\beta_B) -(1-r)\log[1-e^{\omega\beta_B}Z_{+}(\beta_B)c]\nonumber\\ && -r\log(1+c)-\omega\beta_B\frac{1-e^{-\omega\beta_B}}{1+e^{-\omega\beta_B}} c. \end{eqnarray} The above relation shows the impact of the steady state coherences on the steady state entropy. As a preliminary observation, one recovers that when the steady state coherences are null ($c=0$), $S^{\rm ss}(\beta_B,r)=S^{\rm th}(\beta_B)$. However, for $c\ne0$, $S^{\rm ss}(\beta_B,r)$ can be amplified or attenuated depending on the steady state coherences and initial conditions, which again can have interesting applications for state preparation, quantum thermal machines and in thermodynamics. More precisely, one can show (see Appendix \ref{appentropy}) that when $c>0$, $S^{\rm ss}(\beta_B,r)< S^{\rm th}(\beta_B)$. By contrast, for $c<0$, there is a critical value $c^{*}<0$ such that $S^{\rm ss}(\beta_B,r)> S^{\rm th}(\beta_B)$ for $c \in ]c^{*};0[$, and $S^{\rm ss}(\beta_B,r)< S^{\rm th}(\beta_B)$ for $c \in [-1;c^{*}[$ (see Appendix \ref{appentropy}).
As previously, for thermodynamic and experimental purposes we consider the special situation where the pair of two-level systems is initially in a thermal state at the inverse temperature $\beta_0$, and we denote by $S^{\rm ss}(\beta_B,\beta_0):=S^{\rm ss}[\beta_B,r=z(\beta_0)]$ the corresponding steady state entropy. It comes the following elegantly simple result. The initial condition $|\beta_0|>|\beta_B|$ (equivalent to $S^{\rm th}(\beta_0)< S^{\rm th}(\beta_B)$) implies $S^{\rm ss}(\beta_B,\beta_0)< S^{\rm th}(\beta_B)$, and conversely, $|\beta_0|<|\beta_B|$ (equivalent to $S^{\rm th}(\beta_0)> S^{\rm th}(\beta_B)$) implies $S^{\rm ss}(\beta_B,\beta_0)> S^{\rm th}(\beta_B)$, see Fig. \ref{entropbb}. Then, differently from the energy, the collective dissipation has always a mitigating effect for the entropy.
We now briefly show how strong the mitigation effect can be. The smallest values of $S^{\rm ss}(\beta_B,\beta_0)$ are obtained for $\omega|\beta_0| \gg 1$, reducing to the following expression,
\begin{eqnarray} S^{\rm ss}(\beta_B,\beta_0) &\rightarrow_{\omega|\beta_0| \gg 1}& S^{\rm th}(\beta_B) + \log r(\beta_B) \\ &&-\omega\beta_B \frac{1-e^{-\omega\beta_B}}{1+e^{-\omega\beta_B}} (r^{-1}(\beta_B)-1),\nonumber
\end{eqnarray} which tends to $\frac{1}{2}S^{\rm th}(\beta_B)$ for $\omega |\beta_B| \gg1$ (see Fig \ref{entropb0} a). This is precisely the regime of low temperatures crucial in so many experiments and computational tasks. Then, for instance, a pair of two-level systems can be maintained in a state of energy and entropy up to twice smaller than the thermal energy $E^{\rm th}(\beta_B)$ and entropy $S^{\rm th}(\beta_B)$ only thanks to indistinguishability. This is also the regime of amplification of bath effects ($\beta_0<-\beta_B<0$ or $\beta_0>-\beta_B>0$) discussed in the previous Section \ref{sectionenergy}. Consequently, not only collective dissipation provides super cooling and super energy charging, but this is achieved with lower steady state entropy, which is always a highly desired in battery charging or refrigeration.
For sake of completeness we mention the regime $\omega|\beta_0| \ll 1$ where $S^{\rm ss}(\beta_B,\beta_0)$ takes its maximum values, given by the following expression, \begin{eqnarray}
&&S^{\rm ss}(\beta_B,\beta_0) \rightarrow_{\omega|\beta_0| \ll 1} S^{\rm th}(\beta_B) -\frac{3}{4}\log \frac{3}{4z(\beta_B)}\nonumber\\ &&\hspace{1.9cm} -\frac{1}{4} \log \left[ 1-e^{\omega\beta_B}Z_{+}(\beta_B)\left(\frac{3}{4z(\beta_B)} -1\right)\right] \nonumber\\ &&\hspace{1.9cm}-\omega\beta_B \frac{1-e^{-\omega\beta_B}}{1+e^{-\omega\beta_B}} \left(\frac{3}{4z(\beta_B)}-1\right),
\end{eqnarray} which tends to $-\frac{1}{4}\log\frac{1}{4}-\frac{3}{4}\log\frac{3}{4}$ while $S^{\rm th}(\beta_B)$ tends to zero when $\omega|\beta_B| \gg1$.
As illustrations, the graph of $S^{\rm ss}(\beta_B,\beta_0)$ as a function of $\omega\beta_0$ for $\omega \beta_B =2$ (or equivalently, for $\omega \beta_B=-2$) is shown in Fig. \ref{entropbb}, with the value of $S^{\rm th}(\beta_B)$ indicated for comparison. Graphs of $S^{\rm ss}(\beta_B,\beta_0)$, $S^{\rm th}(\beta_B)$, and $S^{\rm ss}(\beta_B,\beta_0)/S^{\rm th}(\beta_B)$ as functions of $\beta_B$ are shown in Fig. \ref{entropb0} for different value of $\omega\beta_0$. \\
\begin{figure}
\caption{Graphs of $S^{\rm ss}(\beta_B,\beta_0)$ (Orange curve) as a function of $\beta_0$ for $\omega\beta_B=2$ (or equivalently $\omega \beta_B=-2$). The corresponding value of $S^{\rm th}(\beta_B)$ is indicated by the Black line for comparison.}
\label{entropbb}
\end{figure}
\begin{figure}
\caption{Graphs of $S^{\rm ss}(\beta_B,\beta_0)$ (Orange curve), $S^{\rm th}(\beta_B)$ (Black curve), and $S^{\rm ss}(\beta_B,\beta_0)/S^{\rm th}(\beta_B)$ (Green curve, right panel) as a function of $\beta_B$ for (a) $\omega|\beta_0|\gg1$, (b) $\omega\beta_0=0$, and (c) $\omega|\beta_0|=3$.}
\label{entropb0}
\end{figure}
\section{Conclusion} Systems interacting with a bath can become effectively indistinguishable to that bath when they share the same characteristics with respect to the degrees of freedom that the bath is sensitive to. This was explicitly illustrated with the example of a pair of two-level atoms interacting with the free space electromagnetic field. One should also keep in mind that bath engineering can be helpful to reduce the bath sensitivity to some degrees of freedom and therefore increase the indistinguishability (as for instance adding a cavity field around a system to reduce the relevant electromagnetic modes \cite{Woods_2014}). The consequence of this indistinguishability is a collective dissipation (whose dynamics has been studied in superradiance problems).
The study of the thermodynamic properties of the steady states generated by collective dissipation reveal the crucial role of the bath-induced coherences (fruit of indistinguishability).
They allow the system to reach an apparent temperature equal to the bath temperature while having lower (or higher) populations of excited levels than the thermal state $\rho^{\rm th}(\beta_B)$. In other words, without the bath-induced coherences the system could not have a steady state energy different from the thermal energy $E^{\rm th}(\beta_B)$. The result is either a mitigation or an amplification of the bath action, which manifests itself in both the energy and entropy of the system and can reach dramatic levels (up to $50\%$ of the thermal energy and entropy). In addition, the local steady states of the two-level systems are thermal states at temperatures $\beta_{\rm Loc}$ which can reach values much lower or higher than $\beta_B$ (up to $33\%$), providing one more dimension of the mitigation and amplification of the bath's action. Such effects can be alternatively understood in terms of dark states.
The mitigation of the bath action can be useful for state preparation or protection for computational purposes or quantum error correction.
The amplification of the bath action suggests promising applications in charging of quantum batteries, thermal machines, and potentially also natural energy harvesting systems like photosynthesis.
In particular, the framework of autonomous thermal machines pioneered in \cite{Boukobza_2013,Gelbwaser_2014} and extended in \cite{autonomous} seems well fitted to apply the above results to charge quantum batteries, but also to cool them (loosing their role of proper battery to become systems to be refrigerated). This is particularly interesting since collective effects for cooling target systems have received little attention.
Furthermore, the phenomena described in this paper might be at the origin of the power increase witnessed in several papers on many-body quantum batteries \cite{Campaioli_2017,Ferraro_2018,Campaioli_2018} and on thermal machines with many-body working medium \cite{Wang_2009, Altintas_2014,Jaramillo_2016, Barrios_2017, Hardal_2017,Niedenzu_2018}, which would help pinpointing the resource at the origin of the power increase (still highly debated). This certainly deserves further studies.
We emphasise that the interesting and promising effects mentioned above emerge only thanks to the indistinguishability (of the two subsystems from the bath point of view).
Throughout this paper we consider two idealised dynamics, corresponding either to totally distinguishable or totally indistinguishable subsystems. One can wonder for instance what happen when the two atoms are detuned. We expect, in the measure that the detuning is smaller than the inverse of the relaxation time (related to the strength of the bath coupling), that the above effects still hold but on a limited time interval given by the inverse of the detuning.
Future studies should also investigate other relevant intermediary dynamics between totally distinguishable and totally indistinguishable subsystems.
Still, numerous experiments already observed superradiance \cite{Skribanowitz_1973,Gross_1976,Rohlsberger_2010}, confirming that collective dissipation can be achieved in real system. Moreover, the present example with a pair of two-level systems reduces considerably the experimental challenges (in particular the interaction between subsystems does not break the indistinguishability) so that our results could be verified experimentally relatively easily.
Finally, as already mentioned, this phenomenon is not limited to the present system. As long as the subsystems are indistinguishable to the bath, similar phenomena as the ones pointed out in this study should emerge. A future work will investigate the generalisation to larger systems, in particular ensembles of arbitrary number $n$ of arbitrary spin $s$, making the above results even more interesting and useful. \\
\acknowledgements This work is based upon research supported by the South African Research Chair Initiative of the Department of Science and Technology and National Research Foundation.
\appendix \numberwithin{equation}{section}
\section{Distinguishable atoms}\label{apppol} In this Section we evaluate the expectation value $ \langle B_i B_j \rangle_{\rho_{\rm bath}}$ in some specific situations, namely when the two atoms are far apart (having parallel polarisations) and when they have orthogonal polarisations while being at the same position. The free electromagnetic field at the point $\vec r$ decomposed in the plane wave basis can be written as \cite{Gross_1982} $\vec E(\vec r)= \vec E^{+}(\vec r) + \vec E^{-}(\vec r)$ with \begin{equation} \vec E^{+}(\vec r)=-i\sum_{\vec k,\vec \epsilon} \vec{\cal E}_{\vec k,\vec \epsilon} a_{\vec k,\vec \epsilon} e^{i\vec k.\vec r}, \end{equation} where $a_{\vec k, \vec \epsilon}$ is the annihilation operator associated to photons populating the planar mode of wave-vector $\vec k$ and polarisation $\vec \epsilon$. The vector $\vec {\cal E}_{\vec k, \vec \epsilon}=\sqrt{\frac{\hbar c k}{2\epsilon_0 V}} \vec \epsilon $ represents the electric field per photon ($V$ is an arbitrary volume of quantisation, much larger than the system). Therefore $\langle B_i B_j\rangle_{\rho_{\rm bath}}$ takes the following form, \begin{eqnarray} \langle B_iB_j\rangle_{\rho_{\rm bath}} &=& d^2 \sum_{\vec k,\vec \epsilon} \vec e_i .\vec {\cal E}_{\vec k,\vec \epsilon} \vec e_j.\vec {\cal E}_{\vec k,\vec \epsilon} e^{i\vec k.\vec r_{ij}} \langle a_{\vec k,\vec \epsilon} a^{\dag}_{\vec k,\vec \epsilon} \rangle_{\rho_{\rm bath}}\nonumber\\ &+& d^2 \sum_{\vec k,\vec \epsilon} \vec e_i .\vec {\cal E}_{\vec k,\vec \epsilon} \vec e_j.\vec {\cal E}_{\vec k,\vec \epsilon} e^{-i\vec k.\vec r_{ij}} \langle a^{\dag}_{\vec k,\vec \epsilon} a_{\vec k,\vec \epsilon} \rangle_{\rho_{\rm bath}}\nonumber\\ \end{eqnarray} where $\vec r_{ij} := \vec r_i -\vec r_j$, and assuming that the electromagnetic field is in a thermal state so that $\langle a^{\dag}_{\vec k,\vec \epsilon} a_{\vec k',\vec \epsilon'} \rangle_{\rho_{\rm bath}} =\langle a_{\vec k,\vec \epsilon} a^{\dag}_{\vec k',\vec \epsilon'} \rangle_{\rho_{\rm bath}} = 0$ if $\vec k \ne \vec k'$ or $\vec \epsilon \ne \vec\epsilon'$. Transforming the discrete sum over the electromagnetic modes into a continuous one (volume integral) \begin{equation} \frac{1}{V} \sum_{\vec k,\vec \epsilon} \rightarrow \int \frac{d^3\vec k}{(2\pi)^3}\sum_{\vec\epsilon} = \frac{1}{(2\pi)^3}\int k^2dk \int d\Omega \sum_{\vec\epsilon}, \end{equation} where the integral over $\Omega$ denotes the integral over directions, one obtains \begin{eqnarray} \langle B_iB_j\rangle_{\rho_{\rm bath}} &=& \frac{d^2}{(2\pi)^3}\int k^2dk \langle a_{k} a^{\dag}_{k} \rangle_{\rho_{\rm bath}}\int d\Omega e^{i\vec k.\vec r_{ij}} \nonumber\\ &&\hspace{1cm} \times \sum_{\vec \epsilon} \vec e_i .\vec {\cal E}_{\vec k,\vec \epsilon} \vec e_j.\vec {\cal E}_{\vec k,\vec \epsilon} \nonumber\\ &+& \frac{d^2}{(2\pi)^3}\int k^2dk \langle a^{\dag}_{k} a_{k} \rangle_{\rho_{\rm bath}}\int d\Omega e^{-i\vec k.\vec r_{ij}} \nonumber\\ &&\hspace{1cm} \times \sum_{\vec \epsilon} \vec e_i .\vec {\cal E}_{\vec k,\vec \epsilon} \vec e_j.\vec {\cal E}_{\vec k,\vec \epsilon}, \nonumber\\ \end{eqnarray} where we assumed that the occupation number $ \langle a^{\dag}_{\vec k,\vec \epsilon} a_{\vec k,\vec \epsilon} \rangle_{\rho_{\rm bath}}$ of each electromagnetic modes does not depend on the polarisation $\vec \epsilon$ neither on the direction of $\vec k$ but only on the norm of the wave-vector $k$. \\
{\bf Parallel polarisation}. When $\vec e_i =\vec e_j$, \begin{eqnarray} \int d\Omega e^{i\vec k.\vec r_{ij}} \sum_{\vec \epsilon} \vec e_i .\vec {\cal E}_{\vec k,\vec \epsilon} \vec e_j.\vec {\cal E}_{\vec k,\vec \epsilon} &=& \int d\Omega e^{i\vec k.\vec r_{ij}} \sum_{\vec \epsilon} (\vec e_i .\vec {\cal E}_{\vec k,\vec \epsilon} )^2 \nonumber\\ &=& F_{ij}(kr_{ij}) \end{eqnarray} where $F_{ij}(kr_{ij})$ is the function defined in \cite{Gross_1982} which depends on $k, r_{ij}$ and $\vec e_i$. The detail of the expression of $F_{ij}(kr_{ij})$ is not important in our present analysis. However, what is important is that $F_{ij}(kr_{ij})$ tends to zero when $kr_{ij}$ is much larger than 1. Taking into account that only the resonant electromagnetic modes ($ k c = \omega$) interact predominantly with the atoms, we can consider that $\langle B_iB_j\rangle_{\rho_{\rm bath}} = 0$ when $r_{ij}$ is much larger than $\lambda_a = c/\omega$, which corresponds to the emission wavelength, as announced in the main text. \\
{\bf Orthogonal polarisations}. We now consider that the two atoms are confined in a volume much smaller than the emission wavelength $c/\omega$. Therefore, for the modes interacting predominantly with the atoms we can consider that $kr_{ij} \simeq 0$ so that the expectation value is reduced to \begin{eqnarray} \langle B_iB_j\rangle_{\rho_{\rm bath}} &=& \frac{d^2}{(2\pi)^3}\int k^2dk \langle a_{k} a^{\dag}_{k} \rangle_{\rho_{\rm bath}} \nonumber\\ &&\hspace{1cm} \times \int d\Omega \sum_{\vec \epsilon} \vec e_i .\vec {\cal E}_{\vec k,\vec \epsilon} \vec e_j.\vec {\cal E}_{\vec k,\vec \epsilon} \nonumber\\ &+& \frac{d^2}{(2\pi)^3}\int k^2dk \langle a^{\dag}_{k} a_{k} \rangle_{\rho_{\rm bath}} \nonumber\\ &&\hspace{1cm} \times \int d\Omega\sum_{\vec \epsilon} \vec e_i .\vec {\cal E}_{\vec k,\vec \epsilon} \vec e_j.\vec {\cal E}_{\vec k,\vec \epsilon} ~. \nonumber\\ \end{eqnarray} Remembering that for each wave-vector $\vec k$ the sum of $\vec \epsilon$ runs over any two orthonormal vectors belonging to the plan perpendicular to $\vec k$, we have that $\vec e_{i}.\vec e_{j} = e_{i,\vec k}e_{j,\vec k} +\sum_{\vec \epsilon} \vec e_i .\vec {\cal E}_{\vec k,\vec \epsilon} \vec e_j.\vec {\cal E}_{\vec k,\vec \epsilon}$, where $e_{i,\vec k} : = \vec e_i.\vec k/k$. Assuming that the polarisations of the two atoms are orthogonal we have $\vec e_{i}.\vec e_{j}=0$ so that $\sum_{\vec \epsilon} \vec e_i .\vec {\cal E}_{\vec k,\vec \epsilon} \vec e_j.\vec {\cal E}_{\vec k,\vec \epsilon}=-e_{i,\vec k}e_{j,\vec k} $.
Therefore, it follows \begin{eqnarray} \int d\Omega\sum_{\vec \epsilon} \vec e_i .\vec {\cal E}_{\vec k,\vec \epsilon} \vec e_j.\vec {\cal E}_{\vec k,\vec \epsilon} &=& \int d\Omega e_{i,\vec k} e_{j,\vec k}\nonumber\\ &=& \int_{S} dS xy \end{eqnarray} where $S$ is the unit sphere (sphere centered at $k=0$ with radius 1), and $x$ and $y$ stand for the Cartesian coordinates (defined along the axis $\vec e_i$ and $\vec e_j$, so that $x:=e_{i,\vec k}$, and $y:=e_{j,\vec k}$) of a point on the sphere $S$. Such integral is equal to zero implying that the expectation value $\langle B_iB_j\rangle_{\rho_{\rm bath}} $ is null.
\section{Interaction between the two confined atoms}\label{tlatoms} The expression of the coupling constant related to the interaction between the two confined atoms is given by \cite{Gross_1982} $\Omega_{I}=\frac{d^2}{4\pi\epsilon_0r_{12}^3}\left[1-\frac{3(\vec{\epsilon}_i.\vec r_{12})^2}{r_{12}^2}\right]$. The interaction corresponds to the Van der Waals interaction between the two atoms at position $\vec r_1$ and $\vec r_2$, with $\vec r_{12}:=\vec r_1 -\vec r_2$, and $\vec \epsilon_i$ is the polarisation of the electric dipole (assumed to be the same for both atoms). In particular, if the atoms are far apart their interaction vanishes.
\section{Steady state of a pair of indistinguishable two-level systems}\label{dynamics}
In this Section we derive the steady state of the collective dissipation described by a generalisation of the master equation Eq. \eqref{meind} of the main text, \begin{eqnarray}\label{indsm} \dot{\rho}_S^I &=& -i\Omega_L \sum_{i=1}^2 [\sigma_i^{+}\sigma_i^{-},\rho_S^I]-i\Omega_{I}[\sigma_1^{+}\sigma_{2}^{-}+\sigma_1^{-}\sigma_{2}^{+},\rho_S^I] \nonumber\\ &&+ G(\omega) (2S^-\rho_S^I S^+ -S^+S^-\rho_S^I - \rho_S^IS^+S^-)\nonumber\\ &&+ G(-\omega) (2S^+\rho_S^I S^- -S^-S^+\rho_S^I - \rho_S^IS^-S^+),\nonumber\\ \end{eqnarray} where the dissipation rate $G(\omega)$ and the pumping rate $G(-\omega)$ depend on the characteristics of the bath or effective bath (which can be the result of the interaction of several baths or collisional model). This amounts to replace $g[n(\omega)+1]$ by $G(\omega)$ and $gn(\omega)$ by $G(-\omega)$. One should note that the temperature (or apparent temperature \cite{apptemppaper}) of the effective bath is given by $e^{\omega \beta_B} = G(\omega)/G(-\omega)$ which can be larger or smaller than 1, corresponding to $\beta_B$ positive or negative.
The dynamics can be easily solved by considering the basis $\{|\psi_0\rangle,|\psi_+\rangle,|\psi_-\rangle,|\psi_1\rangle\}$ with $|\psi_{\pm}\rangle =(|0\rangle|1\rangle\pm|1\rangle|0\rangle)/\sqrt{2}$, $|\psi_{0}\rangle = |0\rangle|0\rangle$, and $|\psi_1\rangle=|1\rangle|1\rangle$. In such basis the collective ladder operators can be expressed as $S^{+}=\sqrt{2}|\psi_+\rangle \langle\psi_0|+\sqrt{2}|\psi_1\rangle\langle\psi_+|$ and $S^{-}=\sqrt{2}|\psi_0\rangle \langle\psi_{+}|+\sqrt{2}|\psi_{+}\rangle\langle\psi_1|$. From \eqref{indsm} one obtains the following dynamics for the populations $p_i:=\langle \psi_i|\rho_S|\psi_i\rangle$, $i=0,1,+,-$,
\begin{eqnarray}\label{sys}
&&\dot{p}_1 = 4 G(-\omega)p_+ - 4G(\omega)p_1\nonumber\\
&&\dot{p}_0=4G(\omega)p_+ -4G(-\omega)p_0\nonumber\\
&&\dot{p}_+ = 4G(\omega)(p_1-p_+)+4G(-\omega)(p_0-p_+)\nonumber\\
&&\dot{p}_- = 0.
\end{eqnarray}
The steady state populations can be obtained by canceling the time derivatives in the above system of equations. Alternatively, one can also solve the above system. This is simplified by noting that $\dot{p}_1+\dot{p}_0+\dot{p}_+ = 0$, which implies that $r:=p_1+p_0+p_+$ is a constant determined by the initial conditions. The system can therefore be reduced to a system of two linearly independent equations (substituting for instance $p_1$ by $r-p_0-p_+$), \begin{eqnarray}
&&\dot{p}_0=4G(\omega)p_+ -4G(-\omega)p_0\nonumber\\
&&\dot{p}_+ = -4[G(\omega)-G(-\omega)]p_0-4[2G(\omega)+G(-\omega)]p_+ \nonumber\\
&&\hspace{0.9cm}+ 4G(\omega)r.
\end{eqnarray} The reduced system is diagonalised by the quantities $q^{\pm} := p_+ + (1\pm\sqrt{G(-\omega)/G(\omega)})p_0$, with the associated eigenvalues $a^{\pm}:= 4[ \pm \sqrt{G(\omega)G(-\omega)}-G(\omega)-G(-\omega)]$, so that \begin{equation} \dot q^{\pm} = a^{\pm} q^{\pm} + 4G(\omega)r, \end{equation} and \begin{equation} q^{\pm}(t)= e^{a^{\pm} t} q^{\pm}(0) +4G(\omega)r\frac{e^{a^{\pm}t}-1}{a^{\pm}}. \end{equation} From the time evolution of $q^{\pm}(t)$ one obtains straightforwardly the expression for the time evolution of the populations $p_0$, $p_+$, and $p_1$.
Using any of the above methods, the steady state populations are found to be \begin{eqnarray} &&p_0^{\rm ss}=rZ_{+}^ {-1}(\beta_B),\nonumber\\ && p_+^{\rm ss}=rZ_{+}^{-1}(\beta_B)e^{-\omega \beta_B},\nonumber\\ && p_1^{\rm ss}= rZ_{+}^{-1}(\beta_B)e^{-2\omega \beta_B},\nonumber\\ && p_{-}^{\rm ss} = p_{-}(t=0)=1-r,
\end{eqnarray}
with $Z_{+}(\beta_B) := 1 +e^{-\omega\beta_B}+e^{-2\omega\beta_B}$. \\%Alternatively, the steady state populations can be obtained directly from \eqref{sys} by canceling the time derivatives. \\
For the coherences, defined as $\rho_{ij}:=\langle \psi_i|\rho_S^I|\psi_j\rangle$, $i,j \in \{0,1,+,-\}$, one obtains (including the Lamb shift in the interaction picture), \begin{eqnarray} &&\dot{\rho}_{+,-} = -2\big[G(\omega)+G(-\omega) + i\Omega_{I} \big] \rho_{+,-}\nonumber\\ &&\dot{\rho}_{1,-} = -\big[2G(\omega)+i\Omega_{I}\big]\rho_{1,-} \nonumber\\ &&\dot{\rho}_{0,-} = -\big[2G(-\omega)+i\Omega_{I}\big]\rho_{0,-} \nonumber\\ &&\dot{\rho}_{1,0} = -2[G(\omega)+G(-\omega)]\rho_{1,0} \end{eqnarray} which straightforwardly gives $0$ as steady state solution. The dynamics of the two remaining coherences is coupled, \begin{eqnarray} &&\dot{\rho}_{1,+} = -\big[2(2G(\omega)+G(-\omega))-i\Omega_{I}\big]\rho_{1,+} + 4G(-\omega)\rho_{+,0}\nonumber\\ &&\dot{\rho}_{+,0} = -\big[2g(G(\omega)+2G(-\omega))+i\Omega_{I}\big]\rho_{+,0} + 4G(\omega)\rho_{1,+},\nonumber\\ \end{eqnarray} and also leads to $0$ as steady state solution. Finally, one can write the steady state in the form, \begin{eqnarray}
&& \rho^{\rm ss}(\beta_B,r) := (1-r)|\psi_{-}\rangle\langle\psi_{-}| + rZ_{+}^{-1}(\beta_B) \nonumber\\
&&\hspace{0.5cm}\times \Big(e^{-2\omega\beta_B}|\psi_1\rangle\langle \psi_1|+e^{-\omega\beta_B}|\psi_+\rangle\langle\psi_+| + |\psi_0\rangle\langle\psi_0|\Big),\nonumber\\ \end{eqnarray} as announced in the main text.\\
\section{The role of the indistinguishability from the bath}\label{appindistin} In this Section we come back on the role of the indistinguishability from the bath. The underlying mechanism can be generalise in the following way.
Let's consider a system initially in a pure state $|a\rangle$. This system undergoes a process (enters a black box) with two different outputs $|b\rangle$ and $|c\rangle$ (orthogonal states) of same probability $1/2$. We assume that the system's bath, or more generally the surrounding of the system, does not distinguish whether the system is in the state $|b\rangle$ or $|c\rangle$ (we give some example of such situations in the following). This can be alternatively formulated in the following way: no information about the actual system's state is leaked to the bath or surrounding environment. If such conditions are fulfilled, the system is left after the process (black box) in a coherent superposition of $|b\rangle$ and $|c\rangle$.
This phenomenon is well-known in quantum optics for the design of interferometers and in the double slit experiment.
In both experiments, the incident photon goes through a process (the double slit or the beam splitter). The two outcomes are two different paths/modes. If no information about the path used by the photon is leaked to the surrounding environment, the photon is in a coherent superposition of paths which interfere. Conversely, the more information is available about the path used by the photon the less coherent is the superposition and the weaker are the interferences. Thus, the indistinguishability of the two paths from the point of view of the environment enables the coherent superposition of the two paths.
The absorption of one bath excitation by the pair of atoms follows the same scenario. To draw a simple comparison, let's assume both atoms are initially in the ground state. Then, they go through the process which is the absorption of one bath excitation. There are two orthogonal outputs, $|b\rangle \equiv$ {\it atom 1 in the excited state and atom 2 in the ground state} and $|c\rangle \equiv$ {\it atom 1 in the ground state and atom 2 in the excited state}. The indistinguishability of the two atoms from the bath's point of view means that the bath cannot distinguish which atom absorbs the excitation. Therefore, the output states $|b\rangle$ and $|c\rangle$ are indistinguishable for the bath, so that the process generates a coherent superposition of $|b\rangle$ and $|c\rangle$.
Of course this toy model can be extended to situations where there are more than two output states, where the output probabilities are unbalanced, and where partial information is leaked to the bath (reducing the coherence of the superposition). Importantly, we stress that the object/system performing the process does not need to contain any quantum features. The double slit and the beam splitter are classical objects. Similar ideas based on indistinguishability have been exploited in \cite{Lofranco_2016,Lofranco_2018,Kysela_2019} for entanglement generation.
\section{Local Temperature}\label{localtemp} In this Section we provide graphs of the steady state local temperature in function of the bath temperature. The first graph Fig. \ref{loctemp} shows how much the cooling can be amplified when the pair of two-level systems is initialised with highly inverted population, reminiscent of the Mpemba effect \cite{Mpemba_1969,Lasanta_2017,Lu_2017} (expect that here hotter initial states reach colder temperatures). Similar graphs can be obtained showing the mitigation effects.
\begin{figure}
\caption{(a) Graph of the steady state local inverse temperature $\beta_{\rm Loc}$ (Orange curve) and as a function of the bath inverse temperature $\omega \beta_B \in [0;4]$ for $\omega|\beta_0| \gg1$. As a comparison the bath inverse temperature is indicated by the Black curve. (b) Graph of the ratio $\beta_{\rm Loc}/\beta_B$ as a function of $\omega \beta_B$.}
\label{loctemp}
\end{figure}
\section{Apparent temperature of the steady states}\label{smapptempss} Inserting the expression of the steady state \eqref{genss} into the definition of the apparent temperature \eqref{defapptemp} one obtains \begin{eqnarray}
{\cal T} &=& \omega \left(\log\frac{\langle \psi_0|\rho^{\rm ss}(\beta_B,r)|\psi_0\rangle +\langle \psi_+|\rho^{\rm ss}(\beta_B,r)|\psi_+\rangle}{\langle \psi_1|\rho^{\rm ss}(\beta_B,r)|\psi_1\rangle +\langle \psi_+|\rho^{\rm ss}(\beta_B,r)|\psi_+\rangle} \right)^{-1}\nonumber\\
\end{eqnarray} with $\langle\psi_0|\rho^{\rm ss}(\beta_B,r)|\psi_0\rangle = rZ_{+}^{-1}(\beta_B)$, $\langle\psi_+|\rho^{\rm ss}(\beta_B,r)|\psi_+\rangle = rZ_{+}^{-1}(\beta_B)e^{-\omega\beta_B}$, and $\langle\psi_1|\rho^{\rm ss}(\beta_B,r)|\psi_1\rangle = rZ_{+}^{-1}(\beta_B)e^{-2\omega\beta_B}$. As a result, \begin{equation} {\cal T} = 1/\beta_B. \end{equation}
\section{Behaviour of the steady state entropy in term of the coherence}\label{appentropy} In this Section we study the behaviour of the steady state entropy in term of the coherence. Note that for a fixed bath temperature, the steady state coherence is entirely determined by $r$. Using Eq. \eqref{entropyc} we compute the derivative of the steady state entropy with respect to $r$ and obtain, \begin{eqnarray} \frac{\partial S^{\rm ss}(\beta_B,r)}{\partial r}&=& \log \left(\frac{1}{r} -1\right) + \log Z_{+}(\beta_B)\nonumber\\ &&+\omega\beta_B\frac{(e^{-\omega\beta_B}+2e^{-2\omega\beta_B})}{Z_{+}(\beta_B)}. \end{eqnarray} The derivative $\frac{\partial S^{\rm ss}(\beta_B,r)}{\partial r}$ is positive for $r$ in $[0;r_{cr}(\beta_B)]$ and negative on $[r_{cr}(\beta_B),1]$, with \begin{equation} r_{cr}(\beta_B) := \frac{z(\beta_B)}{1 + \frac{e^{-\omega^{*}\beta_B}-e^{-\omega\beta_B}}{Z(\beta_B)}}, \end{equation} and $\omega^{*}:=\omega \frac{e^{-\omega\beta_B}+2e^{-2\omega\beta_B}}{Z_{+}(\beta_B)}$. The graph of $S^{\rm ss}(\beta_B,r)$ as a function of $r$ is shown in Fig. \ref{entropr} (for $\omega\beta_B=2$). Note that $r_{cr}(\beta_B) \leq z(\beta_B) $ for any $\beta_B\ne 0$ (equality only for $\beta_B=0$). Since $S^{\rm ss}(\beta_B,\beta_B) = S^{\rm th}(\beta_B)$ when $r=z(\beta_B)$ (equivalent to $c=0$), we have that $S^{\rm ss}(\beta_B,r) <S^{\rm th}(\beta_B)$ for $c>0$. Furthermore, given that $\frac{\partial S^{\rm ss}(\beta_B,r)}{\partial r}$ is positive for $r\leq r_{cr}(\beta_B)$, and $S^{\rm ss}(\beta_B,r=0)=0$ while $S^{\rm ss}(\beta_B,r)\geq S^{\rm th}(\beta_B)$ for $r=r_{cr}(\beta_B)$, there is a value $r$ belonging to the interval $]0;r_{cr}(\beta_B)]$ such that $S^{\rm ss}(\beta_B,r)=S^{\rm th}(\beta_B)$. We denote by $r^{*}(\beta_B)$ such value, and by $c^{*}(\beta_B)$ the corresponding value of the coherence ($c^{*}(\beta_B) := r^{*}(\beta_B)/r(\beta_B) -1 \leq r_{cr}(\beta_B)/r(\beta_B) -1)$. Therefore, $S^{\rm ss}(\beta_B,r)>S^{\rm th}(\beta_B)$ for $c \in ]c^{*}(\beta_B);0[$, and $S^{\rm ss}(\beta_B,r)<S^{\rm th}(\beta_B)$ for $c \in [-1;c^{*}(\beta_B)[$ (as announced in the main text).
Considering only initial thermal states, the parameter $r$ becomes equal to $z(\beta_B)$ which takes value in the interval $[\frac{3}{4};1]$. On the other hand, one can verify that $r_{\rm cr}(\beta_B)\leq 3/4$ for any $\beta_B$. This guarantees that $S^{\rm ss}(\beta_B,\beta_0)$ stays larger than $S^{\rm th}(\beta_B)$ for any initial inverse temperature $\beta_0$ such that $|\beta_0|<|\beta_B|$ (equivalent to $z(\beta_0)<z(\beta_B)$).
We finally reach the statement made in the main text: $S^{\rm ss}(\beta_B,\beta_0)$ is strictly larger (smaller) than $S^{\rm th}(\beta_B)$ if and only if $|\beta_0| <|\beta_B|$ ($|\beta_0|>|\beta_B|$).
\begin{figure}
\caption{Graphs of $S^{\rm ss}(\beta_B,r)$ (Orange curve) as a function of $r$ for $\beta_B=2$. The value of $S^{\rm th}(\beta_B)$ is indicated by the Black line. Indicated by the dashed lines, the maximum of $S^{\rm ss}(\beta_B,r)$ at the point $r=r_{\rm cr}(\beta_B)$, and the two points such that $S^{\rm ss}(\beta_B,r)=S^{\rm th}(\beta_B)$ at $r=z(\beta_B)$ and $r=r^{*}(\beta_B)$. }
\label{entropr}
\end{figure}
\end{document} | arXiv |
While the mechanism is largely unknown, one commonly mechanism possibility is that light of the relevant wavelengths is preferentially absorbed by the protein cytochrome c oxidase, which is a key protein in mitochondrial metabolism and production of ATP, substantially increasing output, and this extra output presumably can be useful for cellular activities like healing or higher performance.
Popular among computer programmers, oxiracetam, another racetam, has been shown to be effective in recovery from neurological trauma and improvement to long-term memory. It is believed to effective in improving attention span, memory, learning capacity, focus, sensory perception, and logical thinking. It also acts as a stimulant, increasing mental energy, alertness, and motivation.
With all these studies pointing to the nootropic benefits of some essential oils, it can logically be concluded then that some essential oils can be considered "smart drugs." However, since essential oils have so much variety and only a small fraction of this wide range has been studied, it cannot be definitively concluded that absolutely all essential oils have brain-boosting benefits. The connection between the two is strong, however.
A synthetic derivative of Piracetam, aniracetam is believed to be the second most widely used nootropic in the Racetam family, popular for its stimulatory effects because it enters the bloodstream quickly. Initially developed for memory and learning, many anecdotal reports also claim that it increases creativity. However, clinical studies show no effect on the cognitive functioning of healthy adult mice.
REPUTATION: We were blown away by the top-notch reputation that Thrive Naturals has in the industry. From the consumers we interviewed, we found that this company has a legion of loyal brand advocates. Their customers frequently told us that they found Thrive Naturals easy to communicate with, and quick to process and deliver their orders. The company has an amazing track record of customer service and prides itself on its Risk-Free No Questions Asked 1-Year Money Back Guarantee. As an online advocate for consumer rights, we were happy to see that they have no hidden fees nor ongoing monthly billing programs that many others try to trap consumers into.
Evidence in support of the neuroprotective effects of flavonoids has increased significantly in recent years, although to date much of this evidence has emerged from animal rather than human studies. Nonetheless, with a view to making recommendations for future good practice, we review 15 existing human dietary intervention studies that have examined the effects of particular types of flavonoid on cognitive performance. The studies employed a total of 55 different cognitive tests covering a broad range of cognitive domains. Most studies incorporated at least one measure of executive function/working memory, with nine reporting significant improvements in performance as a function of flavonoid supplementation compared to a control group. However, some domains were overlooked completely (e.g. implicit memory, prospective memory), and for the most part there was little consistency in terms of the particular cognitive tests used making across study comparisons difficult. Furthermore, there was some confusion concerning what aspects of cognitive function particular tests were actually measuring. Overall, while initial results are encouraging, future studies need to pay careful attention when selecting cognitive measures, especially in terms of ensuring that tasks are actually sensitive enough to detect treatment effects.
Herbal supplements have been used for centuries to treat a wide range of medical conditions. Studies have shown that certain herbs may improve memory and cognition, and they can be used to help fight the effects of dementia and Alzheimer's disease. These herbs are considered safe when taken in normal doses, but care should be taken as they may interfere with other medications.
The absence of a suitable home for this needed research on the current research funding landscape exemplifies a more general problem emerging now, as applications of neuroscience begin to reach out of the clinical setting and into classrooms, offices, courtrooms, nurseries, marketplaces, and battlefields (Farah, 2011). Most of the longstanding sources of public support for neuroscience research are dedicated to basic research or medical applications. As neuroscience is increasingly applied to solving problems outside the medical realm, it loses access to public funding. The result is products and systems reaching the public with less than adequate information about effectiveness and/or safety. Examples include cognitive enhancement with prescription stimulants, event-related potential and fMRI-based lie detection, neuroscience-based educational software, and anti-brain-aging computer programs. Research and development in nonmedical neuroscience are now primarily the responsibility of private corporations, which have an interest in promoting their products. Greater public support of nonmedical neuroscience research, including methods of cognitive enhancement, will encourage greater knowledge and transparency concerning the efficacy and safety of these products and will encourage the development of products based on social value rather than profit value.
One of the most obscure -racetams around, coluracetam (Smarter Nootropics, Ceretropic, Isochroma) acts in a different way from piracetam - piracetam apparently attacks the breakdown of acetylcholine while coluracetam instead increases how much choline can be turned into useful acetylcholine. This apparently is a unique mechanism. A crazy Longecity user, ScienceGuy ponied up $16,000 (!) for a custom synthesis of 500g; he was experimenting with 10-80mg sublingual doses (the ranges in the original anti-depressive trials) and reported a laundry list of effects (as does Isochroma): primarily that it was anxiolytic and increased work stamina. Unfortunately for my stack, he claims it combines poorly with piracetam. He offered free 2g samples for regulars to test his claims. I asked & received some.
Talk to your doctor, too, before diving in "to ensure that they do not conflict with current meds or cause a detrimental effect," Hohler says. You also want to consider what you already know about your health and body – if you have anxiety or are already sensitive to caffeine, for example, you may find that some of the supplements work a little too well and just enhance anxiety or make it difficult to sleep, Barbour says. Finances matter, too, of course: The retail price for Qualia Mind is $139 for 22 seven-capsule "servings"; the suggestion is to take one serving a day, five days a week. The retail price for Alpha Brain is $79.95 for 90 capsules; adults are advised to take two a day.
Caveats aside, if you do want to try a nootropic, consider starting with something simple and pretty much risk-free, like aromatherapy with lemon essential oil or frankincense, which can help activate your brain, Barbour says. You could also sip on "golden milk," a sweet and anti-inflammatory beverage made with turmeric, or rosemary-infused water, she adds.
It is a known fact that cognitive decline is often linked to aging. It may not be as visible as skin aging, but the brain does in fact age. Often, cognitive decline is not noticeable because it could be as mild as forgetting names of people. However, research has shown that even in healthy adults, cognitive decline can start as early as in the late twenties or early thirties.
Second, users are concerned with the possibility of withdrawal if they stop taking the nootropics. They worry that if they stop taking nootropics they won't be as smart as when they were taking nootropics, and will need to continue taking them to function. Some users report feeling a slight brain fog when discontinuing nootropics, but that isn't a sign of regression.
Many over the counter and prescription smart drugs fall under the category of stimulants. These substances contribute to an overall feeling of enhanced alertness and attention, which can improve concentration, focus, and learning. While these substances are often considered safe in moderation, taking too much can cause side effects such as decreased cognition, irregular heartbeat, and cardiovascular problems.
Enhanced learning was also observed in two studies that involved multiple repeated encoding opportunities. Camp-Bruno and Herting (1994) found MPH enhanced summed recall in the Buschke Selective Reminding Test (Buschke, 1973; Buschke & Fuld, 1974) when 1-hr and 2-hr delays were combined, although individually only the 2-hr delay approached significance. Likewise, de Wit, Enggasser, and Richards (2002) found no effect of d-AMP on the Hopkins Verbal Learning Test (Brandt, 1991) after a 25-min delay. Willett (1962) tested rote learning of nonsense syllables with repeated presentations, and his results indicate that d-AMP decreased the number of trials needed to reach criterion.
A fancier method of imputation would be multiple imputation using, for example, the R library mice (Multivariate Imputation by Chained Equations) (guide), which will try to impute all missing values in a way which mimicks the internal structure of the data and provide several possible datasets to give us an idea of what the underlying data might have looked like, so we can see how our estimates improve with no missingness & how much of the estimate is now due to the imputation:
A number of so-called 'smart drugs' or cognitive enhancers have captured attention recently, from stimulants such as modafinil, to amphetamines (often prescribed under the name Adderall) and methylphenidate (also known by its brand name Ritalin). According to widespread news reports, students have begun using these drugs to enhance their performance in school and college, and are continuing to do so in their professional lives.
The placebos can be the usual pills filled with olive oil. The Nature's Answer fish oil is lemon-flavored; it may be worth mixing in some lemon juice. In Kiecolt-Glaser et al 2011, anxiety was measured via the Beck Anxiety scale; the placebo mean was 1.2 on a standard deviation of 0.075, and the experimental mean was 0.93 on a standard deviation of 0.076. (These are all log-transformed covariates or something; I don't know what that means, but if I naively plug those numbers into Cohen's d, I get a very large effect: \frac{1.2 - 0.93}{0.076}=3.55.)
However, normally when you hear the term nootropic kicked around, people really mean a "cognitive enhancer" — something that does benefit thinking in some way (improved memory, faster speed-of-processing, increased concentration, or a combination of these, etc.), but might not meet the more rigorous definition above. "Smart drugs" is another largely-interchangeable term.
Cocoa flavanols (CF) positively influence physiological processes in ways which suggest that their consumption may improve aspects of cognitive function. This study investigated the acute cognitive and subjective effects of CF consumption during sustained mental demand. In this randomized, controlled, double-blinded, balanced, three period crossover trial 30 healthy adults consumed drinks containing 520 mg, 994 mg CF and a matched control, with a 3-day washout between drinks. Assessments included the state anxiety inventory and repeated 10-min cycles of a Cognitive Demand Battery comprising of two serial subtraction tasks (Serial Threes and Serial Sevens), a Rapid Visual Information Processing (RVIP) task and a mental fatigue scale, over the course of 1 h. Consumption of both 520 mg and 994 mg CF significantly improved Serial Threes performance. The 994 mg CF beverage significantly speeded RVIP responses but also resulted in more errors during Serial Sevens. Increases in self-reported mental fatigue were significantly attenuated by the consumption of the 520 mg CF beverage only. This is the first report of acute cognitive improvements following CF consumption in healthy adults. While the mechanisms underlying the effects are unknown they may be related to known effects of CF on endothelial function and blood flow.
This calculation - reaping only \frac{7}{9} of the naive expectation - gives one pause. How serious is the sleep rebound? In another article, I point to a mice study that sleep deficits can take 28 days to repay. What if the gain from modafinil is entirely wiped out by repayment and all it did was defer sleep? Would that render modafinil a waste of money? Perhaps. Thinking on it, I believe deferring sleep is of some value, but I cannot decide whether it is a net profit.
…It is without activity in man! Certainly not for the lack of trying, as some of the dosage trials that are tucked away in the literature (as abstracted in the Qualitative Comments given above) are pretty heavy duty. Actually, I truly doubt that all of the experimenters used exactly that phrase, No effects, but it is patently obvious that no effects were found. It happened to be the phrase I had used in my own notes.
Because executive functions tend to work in concert with one another, these three categories are somewhat overlapping. For example, tasks that require working memory also require a degree of cognitive control to prevent current stimuli from interfering with the contents of working memory, and tasks that require planning, fluency, and reasoning require working memory to hold the task goals in mind. The assignment of studies to sections was based on best fit, according to the aspects of executive function most heavily taxed by the task, rather than exclusive category membership. Within each section, studies are further grouped according to the type of task and specific type of learning, working memory, cognitive control, or other executive function being assessed.
Dr. Larry Cleary's Lucidal – the critically acclaimed secret formula that has been created, revised, and optimized to the point that it's Dr. Cleary-approved. As a product of Dr. Cleary's extensive years and expertise in the industry, it is his brainchild. Heavily marketed as the pill for reversing memory loss, whilst aiding focus, it's seen some popularity in the last few years. In light of all the hubbub and controversy, we put their claims to the test, to see whether or not Lucidal is able to come forth with flying colors, just as all its acclamation has it to be… Learn More...
A total of 14 studies surveyed reasons for using prescription stimulants nonmedically, all but one study confined to student respondents. The most common reasons were related to cognitive enhancement. Different studies worded the multiple-choice alternatives differently, but all of the following appeared among the top reasons for using the drugs: "concentration" or "attention" (Boyd et al., 2006; DeSantis et al., 2008, 2009; Rabiner et al., 2009; Teter et al., 2003, 2006; Teter, McCabe, Cranford, Boyd, & Guthrie, 2005; White et al., 2006); "help memorize," "study," "study habits," or "academic assignments" (Arria et al., 2008; Barrett et al., 2005; Boyd et al., 2006; DeSantis et al., 2008, 2009; DuPont et al., 2008; Low & Gendaszek, 2002; Rabiner et al., 2009; Teter et al., 2005, 2006; White et al., 2006); "grades" or "intellectual performance" (Low & Gendaszek, 2002; White et al., 2006); "before tests" or "finals week" (Hall et al., 2005); "alertness" (Boyd et al., 2006; Hall et al., 2005; Teter et al., 2003, 2005, 2006); or "performance" (Novak et al., 2007). However, every survey found other motives mentioned as well. The pills were also taken to "stay awake," "get high," "be able to drink and party longer without feeling drunk," "lose weight," "experiment," and for "recreational purposes."
Kennedy et al. (1990) administered what they termed a grammatical reasoning task to subjects, in which a sentence describing the order of two letters, A and B, is presented along with the letter pair, and subjects must determine whether or not the sentence correctly describes the letter pair. They found no effect of d-AMP on performance of this task.
In general, I feel a little bit less alert, but still close to normal. By 6PM, I have a mild headache, but I try out 30 rounds of gbrainy (haven't played it in months) and am surprised to find that I reach an all-time high; no idea whether this is due to DNB or not, since Gbrainy is very heavily crystallized (half the challenge disappears as you learn how the problems work), but it does indicate I'm not deluding myself about mental ability. (To give a figure: my last score well before I did any DNB was 64, and I was doing well that day; on modafinil, I had a 77.) I figure the headache might be food related, eat, and by 7:30 the headache is pretty much gone and I'm fine up to midnight.
In a broad sense, this is enhancement; in a stricter one, it's optimisation. "I think people think about smart drugs the way they think about steroids in athletics," Arnsten says, "but it's not a proper analogy, because with steroids you're creating more muscle. With smart drugs, all you're doing is taking the brain that you have and putting it in its optimal chemical state. You're not taking Homer Simpson and making him into Einstein."
"Cavin has done an amazing job in all aspects of his life. Overcoming the horrific life threatening accident, and then going on to do whatever he can to help others with his contagious wonderful attitude. This book is an easy to understand fact filled manual for anyone, but especially those who are or are caregivers for a loved one with tbi. I also highly recommend his podcast series."
In sum, the evidence concerning stimulant effects of working memory is mixed, with some findings of enhancement and some null results, although no findings of overall performance impairment. A few studies showed greater enhancement for less able participants, including two studies reporting overall null results. When significant effects have been found, their sizes vary from small to large, as shown in Table 4. Taken together, these results suggest that stimulants probably do enhance working memory, at least for some individuals in some task contexts, although the effects are not so large or reliable as to be observable in all or even most working memory studies.
Schroeder, Mann-Koepke, Gualtieri, Eckerman, and Breese (1987) assessed the performance of subjects on placebo and MPH in a game that allowed subjects to switch between two different sectors seeking targets to shoot. They did not observe an effect of the drug on overall level of performance, but they did find fewer switches between sectors among subjects who took MPH, and perhaps because of this, these subjects did not develop a preference for the more fruitful sector.
Interesting. On days ranked 2 (below-average mood/productivity), nicotine seems to have boosted scores; on days ranked 3, nicotine hurts scores; there aren't enough 4's to tell, but even '5 days seem to see a boost from nicotine, which is not predicted by the theory. But I don't think much of a conclusion can be drawn: not enough data to make out any simple relationship. Some modeling suggests no relationship in this data either (although also no difference in standard deviations, leading me to wonder if I screwed up the data recording - not all of the DNB scores seem to match the input data in the previous analysis). So although the 2 days in the graph are striking, the theory may not be right.
"I am nearly four years out from my traumatic brain injury and I have been through 100's of hours of rehabilitation therapy. I have been surprised by how little attention is given to adequate nutrition for recovering from TBI. I'm always looking for further opportunities to recover and so this book fell into the right hands. Cavin outlines the science and reasoning behind the diet he suggests, but the real power in this book comes when he writes, "WE." WE can give our brains proper nutrition. Now I'm excited to drink smoothies and eat breakfasts that look like dinners! I will recommend this book to my friends.
This formula presents a relatively high price and one bottle of 60 tables, at the recommended dosage of two tablets per day with a meal, a bottle provides a month's supply. The secure online purchase is available on the manufacturer's site as well as at several online retailers. Although no free trials or money back guarantees are available at this time, the manufacturer provides free shipping if the desired order exceeds a certain amount. With time different online retailers could offer some advantages depending on the amount purchased, so an online research is advised before purchase, as to assess the market and find the best solution.
Another interpretation of the mixed results in the literature is that, in some cases at least, individual differences in response to stimulants have led to null results when some participants in the sample are in fact enhanced and others are not. This possibility is not inconsistent with the previously mentioned ones; both could be at work. Evidence has already been reviewed that ability level, personality, and COMT genotype modulate the effect of stimulants, although most studies in the literature have not broken their samples down along these dimensions. There may well be other as-yet-unexamined individual characteristics that determine drug response. The equivocal nature of the current literature may reflect a mixture of substantial cognitive-enhancement effects for some individuals, diluted by null effects or even counteracted by impairment in others.
Nootropics are a responsible way of using smart drugs to enhance productivity. As defined by Giurgea in the 1960's, nootropics should have little to no side-effects. With nootropics, there should be no dependency. And maybe the effects of nootropics are smaller than for instance Adderall, you still improve your productivity without risking your life. This is what separates nootropics from other drugs.
The Trail Making Test is a paper-and-pencil neuropsychological test with two parts, one of which requires shifting between stimulus categories. Part A simply requires the subject to connect circled numbers in ascending order. Part B requires the subject to connect circled numbers and letters in an interleaved ascending order (1, A, 2, B, 3, C….), a task that places heavier demands on cognitive control. Silber et al. (2006) analyzed the effect of d-AMP on Trails A and B and failed to find an effect. | CommonCrawl |
\begin{document}
\title[Bubble concentration on spheres for supercritical elliptic problems]{Bubble concentration on spheres for supercritical elliptic problems} \author{Filomena Pacella} \address[Filomena Pacella] {Dipartimento di Matematica "G.Castelnuovo", Universit\`{a} di Roma ``La Sapienza", P.le Aldo Moro 5, 00185 Roma, Italy} \email{[email protected]}
\author{Angela Pistoia} \address[Angela Pistoia] {Dipartimento SBAI, Universit\`{a} di Roma ``La Sapienza", via Antonio Scarpa 16, 00161 Roma, Italy} \email{[email protected]}
\begin{abstract} We consider the supercritical Lane-Emden problem $$(P_\varepsilon)\qquad
-\Delta v= |v|^{p_\varepsilon-1} v \ \hbox{in}\ \mathcal{A} ,\quad u=0\ \hbox{on}\ \partial\mathcal{A} $$
where $\mathcal A$ is an annulus in $ \mathbb{R}^{2m},$ $m\ge2$ and $p_\varepsilon={(m+1)+2\over(m+1)-2}-\varepsilon$, $\varepsilon>0.$
We prove the existence of positive and sign changing solutions of $(P_\varepsilon)$ concentrating and blowing-up,
as $\varepsilon\to0$, on $(m-1)-$dimensional spheres. Using a reduction method (\cite{RS,PS})
we transform problem $(P_\varepsilon)$ into a nonhomogeneous problem in an annulus $\mathcal D\subset \mathbb{R}^{m+1}$ which can be solved by a Ljapunov-Schmidt finite
dimensional reduction.
\end{abstract}
\subjclass[2010]{35J61, 35B25, 35B40}
\date{\today}
\keywords{supercritical problem, concentration on manifolds} \maketitle
\maketitle \section{Introduction}
In this paper we address the question of finding solutions concentrated on manifolds of positive dimension of supercritical elliptic problems of the type
\begin{equation}
\label{prob1}
-\Delta v= |v|^{p-1} v \ \hbox{in}\ \mathcal{A} ,\quad u=0\ \hbox{on}\ \partial\mathcal{A} ,
\end{equation}
where $\mathcal{A} :=\{y\in \mathbb{R}^{d}\ :\ a< |y|< b\} ,$ $a>0,$ is an annulus in $ \mathbb{R}^{d} ,$ $d>2$
and $p>{d+2\over d-2}$ is a supercritical exponent.
We remark that the critical and supercritical Lane-Emden problems are very delicate due to topological and geometrical obstruction
enlightened by the Pohozaev's identity (\cite{po}). We also point out that in the supercritical case a result of Bahri-Coron type (\cite{BC}) cannot hold in general
nontrivially topological domains as shown by a nonexistence result of Passaseo (\cite{pa}), obtained exploiting critical exponents in lower dimensions.
Using similar ideas, some results for exponents $p$ which are subcritical in dimension $n<d$ and instead supercritical in dimension $d$ have been obtained in different kind of domains in \cite{ACP,BCGP,CFP,DMP,GG,KP1,KP2,PPS}.
Here we consider annuli in even dimension $d=2m,$ $m\ge2$ and obtain results about the existence of solutions, both positive and sign changing, of
different type, concentrated on $(m-1)-$dimensional spheres. More precisely, we have
\begin{theorem}
\label{1.1} [Case of positive solutions] Let $\mathcal A\subset \mathbb{R}^{2m},$ $m\ge2$ and define $(\partial \mathcal A)_a:=\left\{y\in\partial\mathcal A\ :\ |y|=a\right\}.$ There exists $\epsilon _{0}>0$ such that for any $\epsilon \in (0,\epsilon _{0}),$ the following supercritical problem
\begin{equation}
\label{prob2}
-\Delta v= |v|^{p_\varepsilon-1} v \ \hbox{in}\ \mathcal{A} ,\quad u=0\ \hbox{on}\ \partial\mathcal{A} ,
\end{equation}
with $p_\varepsilon={(m+1)+2\over(m+1)-2}-\varepsilon$ has:
\begin{itemize} \item [{i)}] a positive solution $v_\varepsilon$ which concentrates and blows-up on a $(m-1)-$dimensional sphere $\Gamma\subset(\partial \mathcal A)_a$ as $\varepsilon\to0,$ \item [{ii)}] a positive solution $v_\varepsilon$ which concentrates and blows-up on two $(m-1)-$dimensional orthogonal spheres $\Gamma_1\subset(\partial \mathcal A)_a$ and $\Gamma_2\subset(\partial \mathcal A)_a$ as $\varepsilon\to0,$
\end{itemize} \end{theorem}
\begin{theorem}
\label{1.2} [Case of sign changing solutions] Let $\mathcal A\subset \mathbb{R}^{2m},$ $m\ge2$ and define $(\partial \mathcal A)_a:=\left\{y\in\partial\mathcal A\ :\ |y|=a\right\}.$ There exists $\epsilon _{0}>0$ such that for any $\epsilon \in (0,\epsilon _{0}),$ the supercritical problem \eqref{prob2} with
$p_\varepsilon={(m+1)+2\over(m+1)-2}-\varepsilon$ has:
\begin{itemize} \item [{i)}] a sign changing solution $v_\varepsilon$ such that $v_\varepsilon^+$ and $ v_\varepsilon^-$ concentrate and blow-up on two $(m-1)-$dimensional orthogonal spheres $\Gamma_+\subset(\partial \mathcal A)_a$ and $\Gamma_-\subset(\partial \mathcal A)_a,$ respectively, as $\varepsilon\to0,$ \item [{ii)}] a sign changing solution $v_\varepsilon$ such that $v_\varepsilon^+$ and $ v_\varepsilon^-$ concentrate and blow-up on the same $(m-1)-$dimensional sphere $\Gamma \subset(\partial \mathcal A)_a,$ as $\varepsilon\to0,$
\item [{iii)}] two sign changing solutions $v^1_\varepsilon$ and $v^2_\varepsilon$ each one is such that $(v_\varepsilon^i)^+$ and $ (v_\varepsilon^i)^-$ concentrate and blow-up on two $(m-1)-$dimensional orthogonal spheres $(\Gamma_i)_+\subset(\partial \mathcal A)_a$ and $(\Gamma_i)_-\subset(\partial \mathcal A)_a,$ respectively, as $\varepsilon\to0,$ $i=1,2.$
\end{itemize} \end{theorem}
We remark that the exponent ${(m+1)+2\over(m+1)-2}-\varepsilon$ which is almost critical in dimension $(m+1)$ is obviously supercritical for problem \eqref{prob2}.
To prove our results we use the reduction method introduced in
\cite{PS} which allows to transform symmetric solutions to \eqref{prob2} into symmetric solutions of a similar nonhomogeneous problem in an annulus $\mathcal D\subset \mathbb{R}^{m+1}.$ This method was inspired by the paper \cite{RS} where a reduction approach was used to pass from a singularly perturbed problem in an annulus in $ \mathbb{R}^4$ to a singularly perturbed problem in an annulus in $ \mathbb{R}^3.$
More precisely let us consider the annulus $\mathcal D\subset \mathbb{R}^{m+1} $ $\mathcal{D} :=\{x\in \mathbb{R}^{m+1}\ :\ {a^2/2}< |y|< {b^2/2}\} ,$
and, write a point $y\in \mathbb{R}^{2m}$ as
$y=(y_1,y_2)$, $y_i\in \mathbb{R}^m,$ $i=1,2.$ Then we consider functions $v$ in $\mathcal A\subset \mathbb{R}^{2m}$ which are radially symmetric in $y_1$ and $y_2$, i.e. $v(y)=w(|y_1|,|y_2|)$ and functions $u$ in $\mathcal D\subset \mathbb{R}^{m+1}$ which are radially symmetric about the $x_{m+1}-$axis, i.e. $u(x)=h(|x|,\varphi)$
with $\varphi=\arccos \({x\over|x|,\underline e_{m+1}}\)$ where $\underline e_{m+1}=(0,\dots,0,1).$ We also set
$$X=\left\{v\in C^{2,\alpha}(\overline A)\ :\ \hbox{$v$ is radially symmetric}\right\}$$
$$Y=\left\{u\in C^{2,\alpha}(\overline D)\ :\ \hbox{$u$ is axially symmetric}\right\}.$$
Then, as corollary of Theorem 1.1 of \cite{PS} we have
\begin{proposition}\label{prop1.1}
There is a bijective correspondence $h$ between solutions $v$ of \eqref{prob2} in $X$ and solutions $u=h(v)$ in $Y$ of the following reduced problem
\begin{equation}
\label{prob3}
-\Delta u={1\over 2|x|}|u|^{p_\varepsilon-1 } u \quad \hbox{in}\ \mathcal D\subset \mathbb{R}^{m+1},\qquad u=0\quad \hbox{on}\ \partial\mathcal D.
\end{equation}
\end{proposition} As a consequence of this result we can obtain solutions of problem \eqref{prob2} by constructing axially symmetric solutions of the lower-dimensional problem \eqref{prob3}. This has the immediate advantage of transforming the supercritical problem \eqref{prob2} into the subcritical problem \eqref{prob3} if the exponent $p_\varepsilon$ is taken as $ {(m+1)+2\over(m+1)-2}-\varepsilon.$ Indeed we will prove Theorem \ref{1.1} and Theorem \ref{1.2} by constructing axially symmetric solutions of \eqref{pro3}, positive or sign changing, which blow-up and concentrate in points of the annulus $\mathcal D\subset \mathbb{R}^{m+1}.$ These solutions will give rise to solutions of \eqref{prob2} concentrating on $(m-1)-$dimensional spheres, because, as a consequence of the proof of Theorem 1.1 of \cite{PS} and of Remark 3.1 of \cite{PS} it holds \begin{proposition}\label{prop1.2} If $u_\varepsilon$ is an axially symmetric solution of \eqref{prob2} concentrating, as $\varepsilon\to0$, on a point $\xi$ which belongs to the $x_{(m+1)}-$axis, i.e. $\xi=(0,\dots,0,t)$ for $t\in \mathbb{R}\setminus\{0\},$ then the corresponding solution $v_\varepsilon=h^{-1}(u_\varepsilon)$ concentrates on a $(m-1)-$dimensional sphere in $ \mathbb{R}^{2m}.$ \end{proposition}
This is because, by symmetry considerations and by the change of variable performed in \cite{PS} to prove Theorem 1.1 any point $\xi$ on the $x_{(m+1)}-$axis in $\mathcal D\subset \mathbb{R}^{m+1}$ is mapped into a $(m-1)-$dimensional sphere in $\mathcal A\subset \mathbb{R}^{2m}.$ We refer to \cite{PS} for all details.
Thus let $\Omega:=\{x\in \mathbb{R}^n\ :\ 1< |x|< r\} $ be an annulus in $ \mathbb{R}^n,$ $n \ge3,$ and consider the problem
\begin{equation}
\label{p1}
-\Delta u={1\over 2|x|}|u|^{p-1-\epsilon} u \quad \hbox{in}\ \Omega,\qquad u=0\quad \hbox{on}\ \partial\Omega,
\end{equation}
where $p={n+2\over n-2} $ and $\varepsilon$ is a small positive parameter.
Let
$U_{\delta,\xi}(x):=\alpha_n{\delta^{n-2\over2}\over(\delta^2+|x-\xi|^2){n-2\over2}}$ with $ \delta>0$ and $ x,\xi\in \mathbb{R}^n,$
be the solutions to the critical problem $-\Delta u=u^p$ in $ \mathbb{R}^n.$ Here $ \alpha_n:=[n(n-2)]^{n-2\over4}.$ We have \begin{theorem} \label{main} There exists $\epsilon _{0}>0$ such that, for each $\epsilon \in (0,\epsilon _{0}),$ problem \eqref{p1} has \begin{itemize} \item[(i)] an axially symmetric positive solution $u_{\epsilon }$ with one simple positive blow-up point which converge to $\xi_0$ as $\varepsilon$ goes to zero, i.e. \begin{equation*} u_{\epsilon }(x)=U_{{\delta }_{\epsilon },{\xi }_{\epsilon }}(x) +o(1)\ \quad \hbox{in}\ H^1_0(\Omega), \end{equation*} with \begin{equation*} \epsilon ^{-{\frac{n-1}{n-2}}}{\delta }_{\epsilon }\rightarrow d >0,\quad {\xi }_{\epsilon }\rightarrow \xi_0 ; \end{equation*} \item[(ii)] an axially symmetric positive solution $u_{\epsilon }$ with two simple positive blow-up points which converge to $\xi_0$ and $-\xi_0$ as $\varepsilon$ goes to zero, i.e. \begin{equation*} u_{\epsilon }(x)=U_{ \delta _ \epsilon , \xi _ \epsilon }(x)+U_{ \delta _ \epsilon , -\xi _ \epsilon }(x)+o(1), \end{equation*} with \begin{equation*} \epsilon ^{-{\frac{n-1}{n-2}}}{\delta }_{\epsilon }\rightarrow d >0,\quad {\xi }_{\epsilon }\rightarrow \xi_0 ; \end{equation*} \item[(iii)] an axially symmetric sign-changing solutions solution $u_{\epsilon }$ with one simple positive and one simple negative blow-up points which converge to $\xi_0$ and $-\xi_0$ as $\varepsilon$ goes to zero, i.e. \begin{equation*} u_{\epsilon }(x)=U_{ \delta _ \epsilon , \xi _ \epsilon }(x)-U_{ \delta _ \epsilon , -\xi _ \epsilon }(x)+o(1), \end{equation*} with \begin{equation*} \epsilon ^{-{\frac{n-1}{n-2}}}{\delta }_{\epsilon }\rightarrow d >0,\quad {\xi }_{\epsilon }\rightarrow \xi_0 ; \end{equation*} \item[(iv)] an axially symmetric sign-changing solutions solution $u_{\epsilon }$ with one double nodal blow-up point which converge to $\xi_0$ as $\varepsilon$ goes to zero, i.e. \begin{equation*} u_{\epsilon }(x)=U_{{\delta _{1}}_{\epsilon },{\xi _{1}}_{\epsilon }}(x)-U_{{ \delta _{2}}_{\epsilon },{\xi _{2}}_{\epsilon }}(x)+o(1), \end{equation*} with \begin{equation*} \epsilon ^{-{\frac{n-1}{n-2}}}{\delta _{i}}_{\epsilon }\rightarrow d_{i}>0,\quad {\xi _{i}}_{\epsilon }\rightarrow \xi_0 \end{equation*} for $i=1,2.$ \item[(v)] two axially symmetric sign-changing solutions solution $u_{\epsilon }$ with two double nodal blow-up points which converge to $\xi_0$ and $-\xi_0$ as $\varepsilon$ goes to zero, i.e. \begin{equation*} u_{\epsilon }(x)=\[U_{{\delta _{1}}_{\epsilon },{\xi _{1}}_{\epsilon }}(x)-U_{{ \delta _{2}}_{\epsilon },{\xi _{2}}_{\epsilon }}(x)\right]+\[U_{{-\delta _{1}}_{\epsilon },{-\xi _{1}}_{\epsilon }}(x)-U_{{ -\delta _{2}}_{\epsilon },{-\xi _{2}}_{\epsilon }}(x)\right]+o(1) \end{equation*} and \begin{equation*} u_{\epsilon }(x)=\[U_{{\delta _{1}}_{\epsilon },{\xi _{1}}_{\epsilon }}(x)-U_{{ \delta _{2}}_{\epsilon },{\xi _{2}}_{\epsilon }}(x)\right]-\[U_{{-\delta _{1}}_{\epsilon },{-\xi _{1}}_{\epsilon }}(x)-U_{{ -\delta _{2}}_{\epsilon },{-\xi _{2}}_{\epsilon }}(x)\right]+o(1) \end{equation*} with \begin{equation*} \epsilon ^{-{\frac{n-1}{n-2}}}{\delta _{i}}_{\epsilon }\rightarrow d_{i}>0,\quad {\xi _{i}}_{\epsilon }\rightarrow \xi_0 \end{equation*} for $i=1,2.$ \end{itemize} \end{theorem}
Obviously Theorem \ref{1.1} and Theorem \ref{1.2} derive from Theorem \ref{main} for $n=m+1$ using Proposition \ref{prop1.1} and Proposition \ref{prop1.2}.
The proof of Theorem \ref{main} relies on a very well known Ljapunov-Schmidt finite dimensional reduction. We omit many details on the finite dimensional reduction because they can be found, up to some minor modifications, in the literature. In Section \ref{uno} we write the approximate solution, we sketch the proof of the Ljapunov-Schmidt procedure and we prove Theorem \ref{main}. In Section \ref{due} we only compute the expansion of the reduced energy, which is crucial in this framework. In the Appendix we recall some well known facts.
\section{The Ljapunov-Schmidt procedure}\label{uno}
We equip ${\rm H}^1_0(\Omega)$ with the inner product $(u,v)=\int\limits_\Omega \nabla u\nabla vdx$ and the corresponding norm $\|u\|^2= \int\limits_\Omega |\nabla u|^2dx .$ For $r\in[1,\infty)$ and $u\in{\rm L}^{r}(\Omega)$ we set $\|u\|_r^r= \int\limits_\Omega | u|^rdx .$
Let us rewrite problem \eqref{p1} in a different way. Let $i^*:{\rm L}^{2n\over n-2}(\Omega)\to{\rm H}^1_0(\Omega)$ be the adjoint operator of the embedding
$i:{\rm H}^1_0(\Omega) \hookrightarrow{\rm L}^{2n\over n-2}(\Omega),$ i.e.
$$i^*(u)=v\quad \Leftrightarrow\quad \(v,\varphi\)=\int\limits_\Omega u(x)\varphi(x)dx\ \forall\ \varphi\in{\rm H}^1_0(\Omega).$$
It is clear that there exists a positive constant $c$ such that
$$\|i^*(u)\|\le c\|u\|^{2n\over n+2}\qquad \forall\ u\in{\rm L}^{2n\over n+2}(\Omega).$$
Setting $f_\varepsilon(s):=|s|^{p-1-\varepsilon}s$ and using the operator $i^*$, problem \eqref{p1} turns out to be equivalent to
\begin{equation}
\label{p2}
u=i^*\[{1\over 2|x|}f_\varepsilon(u)\right],\quad u\in {\rm H}^1_0(\Omega).
\end{equation}
Let $U_{\delta,\xi}:=\alpha_n{\delta^{n-2\over2}\over(\delta^2+|x-\xi|^2)^{n-2\over2}},$ with $\alpha_n:=\[n(n-2)\right]^{n-2\over4} $
be the positive solutions to the limit problem
$$ -\Delta u=u^p\ \hbox{in}\ \mathbb{R}^n.$$ Set
$$\psi^0_{\delta,\xi}(x):={\partial U_{\delta ,\xi }\over \partial\delta }=\alpha_n{n-2\over2}\delta^{n-4\over2}{|x-\xi|^2-\delta^2\over(\delta^2+|x-\xi|^2)^{n/2}}$$
and for any $j= 1,\dots,n$
$$\psi^j_{\delta,\xi}(x):={\partial U_{\delta ,\xi }\over \partial\xi_j }=\alpha_n(n-2)\delta^{n-2\over2}{ x_j-\xi_j\over(\delta^2+|x-\xi|^2)^{n/2}}.$$ It is well known that the space spanned by the $(n+1)$ functions $\psi^j_{\delta,\xi}$ is the set of the solution to the linearized problem $$ -\Delta \psi=pU^{p-1}_{\delta,\xi}\psi\ \hbox{in}\ \mathbb{R}^n.$$
We also denote by $PW$ the projection onto ${\rm H}^1_0(\Omega)$ of a function $W\in D^{1,2}( \mathbb{R}^n),$ i.e.
$$\Delta PW=\Delta W\ \hbox{in}\ \Omega,\quad PW=0\ \hbox{on}\ \partial\Omega.$$
Set $\xi_0:=(0,\dots,0,1).$
We look for two different types of solutions to problem \eqref{p2}.
The solutions of the type (i), (ii) and (iii) of Theorem \ref{main} will be of the form
\begin{equation}
\label{ans1}
u_\varepsilon= PU_{\delta ,\xi }+\lambda PU_{\mu ,\eta }+\phi
\end{equation}
where $\lambda \in\{-1,0,+1\}$ ($\lambda=0$ in case (i), $\lambda=+1$ in case (ii) and $\lambda=-1$ in case (iii)) and the concentration parameters are
\begin{equation}
\label{ans2}
\mu =\delta \quad\hbox{and}\quad \delta :=\varepsilon^{n-1\over n-2}d \ \hbox{for some}\ d >0
\end{equation} while the concentration points satisfy
\begin{equation}
\label{ans3}
\eta =-\xi \quad\hbox{and}\quad \xi =(1+\tau )\xi_0, \ \hbox{with}\ \tau := \varepsilon t ,\ t >0.
\end{equation}
On the other hand, the solutions of the type (iv) and (v) of Theorem \ref{main} will be of the form
\begin{equation}
\label{ans11}
u_\varepsilon=PU_{\delta_1,\xi_1}-PU_{\delta_2,\xi_2}+\lambda\(PU_{\mu_1,\eta_1}-PU_{\mu_2,\eta_2}\)+\phi,
\end{equation} where $\lambda \in\{-1,0,+1\}$ ($\lambda=0$ in case (iv), $\lambda=+1$ in the first case (v) and $\lambda=-1$ in the second case (v)) and the concentration parameters are
\begin{equation}\label{ans21}
\mu_i=\delta_i\quad\hbox{and}\quad \delta_i:=\varepsilon^{n-1\over n-2}d_i \quad\hbox{with}\quad d_i >0
\end{equation} while the concentration points are aligned along the $x_n-$axes and satisfy
\begin{equation}
\label{ans31}
\eta_i=-\xi_i\quad\hbox{and}\quad \xi_i=(1+\tau_i)\xi_0 \ \hbox{with} \ \tau_i:= \varepsilon t_i,\ t_i>0.
\end{equation}
Next, we introduce the configuration space $\Lambda$ where the concentration parameters and the concentration points lie. For solutions of type \eqref{ans1} we
set $ \mathbf{d}=d\in(0,+\infty)$ and $ \mathbf{t}=t\in(0,+\infty)$
and so
$$\Lambda:=\left\{( \mathbf{d}, \mathbf{t})\in (0,+\infty)\times(0,+\infty)\right\},$$
while for solutions of type \eqref{ans11} we
set $ \mathbf{d}=(d_1,d_2)\in(0,+\infty)^2$ and $ \mathbf{t}=(t_1,t_2)\in(0,+\infty)^2$
and so
$$\Lambda:=\{( \mathbf{d}, \mathbf{t})\in(0,+\infty)^4\ :\ t_1<t_2\} .$$
In each of these cases we write
$$V_{ \mathbf{d}, \mathbf{t}}:=PU_{\delta ,\xi }+\lambda PU_{\mu ,\eta }\quad\hbox{or}\quad V_{ \mathbf{d}, \mathbf{t}}:= PU_{\delta_1,\xi_1}-PU_{\delta_2,\xi_2}+\lambda\(PU_{\mu_1,\eta_1}-PU_{\mu_2,\eta_2}\).$$
The remainder term $\phi$ in both cases \eqref{ans1} and \eqref{ans11} belongs to a suitable space which we now define.
We introduce the spaces
$$K_{ \mathbf{d}, \mathbf{t}}:={\rm span}\{P\psi^j_{\delta_i,\xi_i},\ P\psi^\ell_{\mu_\kappa,\xi_\kappa} \ :\ i,\kappa=1,2,\ j,\ell=0,1,\dots,n\} $$ (we agree that if $\lambda=0$ then $K_{ \mathbf{d}, \mathbf{t}}$ is only generated by the $P\psi^j_{\delta_i,\xi_i}$'s) and
$$K_{ \mathbf{d}, \mathbf{t}}^\perp:=\left\{\phi\in \mathcal H_\lambda\ :\ (\phi, \psi )=0\quad \forall\ \psi\in K_{ \mathbf{d}, \mathbf{t}}\right\},$$
where the space $\mathcal H_\lambda$ depends on $\lambda \in\{-1,0,+1\}$ and is defined by
$$\mathcal H_0:=\{\phi\in {\rm H}^1_0(\Omega)\ :\ \phi\ \hbox{is axially symmetric with respect to the $x_n$-axes }\},$$
$$\mathcal H_{+1}:=\{\phi\in \mathcal H_0\ :\ \phi(x_1,\dots,x_n)=\phi(x_1,\dots,-x_n\},$$
$$\mathcal H_{-1}:=\{\phi\in \mathcal H_0\ :\ \phi(x_1,\dots,x_n)=-\phi(x_1,\dots,-x_n\}.$$
Then we introduce the orthogonal projection operators
$\Pi_{ \mathbf{d}, \mathbf{t}}$ and $ \Pi_{ \mathbf{d}, \mathbf{t}}^\perp $ in $H^1_0(\Omega),$ respectively.
As usual for this reduction method, the approach to solve problem \eqref{p1} or \eqref{p2} will be to find a pair $( \mathbf{d}, \mathbf{t})$ and a function $\phi\in K_{ \mathbf{d}, \mathbf{t}}^\perp$ such that \begin{equation}\label{es1}
\Pi_{ \mathbf{d}, \mathbf{t}}^\perp\left\{V_{ \mathbf{d}, \mathbf{t}}+\phi-i^*\[{1\over 2|x|}f_\varepsilon\(V_{ \mathbf{d}, \mathbf{t}}+\phi\)\right]\right\}=0 \end{equation} and \begin{equation}\label{es2}
\Pi_{ \mathbf{d}, \mathbf{t}} \left\{V_{ \mathbf{d}, \mathbf{t}}+\phi-i^*\[{1\over 2|x|}f_\varepsilon\(V_{ \mathbf{d}, \mathbf{t}}+\phi\)\right]\right\}=0 \end{equation}
First, we shall find for any $( \mathbf{d}, \mathbf{t})$ and for small $\varepsilon,$ a function $\phi\in K_{ \mathbf{d}, \mathbf{t}}^\perp$ such that \eqref{es1} holds. To this aim we define a linear operator $L_{ \mathbf{d}, \mathbf{t}}:K_{ \mathbf{d}, \mathbf{t}}^\perp\to K_{ \mathbf{d}, \mathbf{t}}^\perp$ by $$L_{ \mathbf{d}, \mathbf{t}}\phi:=\phi-\Pi_{ \mathbf{d}, \mathbf{t}}^\perp i^*\[ f'_0\(V_{ \mathbf{d}, \mathbf{t}}\)\phi\right].$$
\begin{proposition}\label{pro1} For any compact sets $\mathbf{C}$ in $\Lambda$ there exists $\varepsilon_0,c>0$ such that for any $\varepsilon\in(0,\varepsilon_0)$ and for any $( \mathbf{d}, \mathbf{t})\in \mathbf{C}$ the operator $L_{ \mathbf{d}, \mathbf{t}}$ is invertible and
$$\|L_{ \mathbf{d}, \mathbf{t}}\phi\|\ge c\|\phi\|\ \quad\ \forall\ \phi\in K_{ \mathbf{d}, \mathbf{t}}^\perp.$$ \end{proposition} \begin{proof}
We argue as in Lemma 1.7 of \cite{MP}.
\end{proof}
Now, we are in position to solve equation \eqref{es1}.
\begin{proposition}\label{pro2} For any compact sets $\mathbf{C}$ in $\Lambda$ there exists $\varepsilon_0, c,\sigma>0$ such that for any $\varepsilon\in(0,\varepsilon_0)$ and for any $( \mathbf{d}, \mathbf{t})\in \mathbf{C}$ there exists a unique $\phi ^\varepsilon_{ \mathbf{d}, \mathbf{t}} \in K_{ \mathbf{d}, \mathbf{t}}^\perp $ such that
$$\Pi_{ \mathbf{d}, \mathbf{t}}^\perp\left\{V_{ \mathbf{d}, \mathbf{t}}+\phi^\varepsilon_{ \mathbf{d}, \mathbf{t}}-i^*\[{1\over 2|x|}f_\varepsilon\(V_{ \mathbf{d}, \mathbf{t}}+\phi^\varepsilon_{ \mathbf{d}, \mathbf{t}}\)\right]\right\}=0. $$ Moreover $$
\left\|\phi^\varepsilon_{ \mathbf{d}, \mathbf{t}}\right\|\le c\varepsilon^{{1\over2}+\sigma}. $$ \end{proposition} \begin{proof} First, we estimate the rate of the error term $$
R_{ \mathbf{d}, \mathbf{t}}:=\Pi_{ \mathbf{d}, \mathbf{t}}^\perp\left\{V_{ \mathbf{d}, \mathbf{t}} -i^*\[{1\over |x|}f_\varepsilon\(V_{ \mathbf{d}, \mathbf{t}} \)\right]\right\} $$ as
$$\left\|R_{ \mathbf{d}, \mathbf{t}}\right\|_{2n\over n+2}=O\(\varepsilon^{{1\over2}+\sigma}\)$$ for some $\sigma>0.$ We argue as in Appendix B of \cite{ACP} using estimates of Section \ref{due}. Then we argue exactly as in Proposition 2.3 of \cite{BMP}.\end{proof}
Now, we introduce the energy functional $J_\varepsilon: {\rm H}^1_0(\Omega)\to \mathbb{R}$ defined by
$$J_\varepsilon(u):={1\over2}\int\limits_{\Omega}|\nabla u |^2dx-{1\over p+1-\varepsilon}\int\limits_{\Omega}{1\over 2|x|}| u |^{p+1-\varepsilon}dx,$$ whose critical points are the solutions to problem \eqref{p1}. Let us define the reduced energy $\widetilde J_\varepsilon:\Lambda\to \mathbb{R}$ by $$\widetilde J_\varepsilon( \mathbf{d}, \mathbf{t})=J_\varepsilon\(V_{ \mathbf{d}, \mathbf{t}}+\phi^\varepsilon_{ \mathbf{d}, \mathbf{t}}\).$$ Next, we prove that the critical points of $\widetilde J_\varepsilon$ are the solution to problem \eqref{es2}.
\begin{proposition}\label{pro3}
The function $V_{ \mathbf{d}, \mathbf{t}}+\phi^\varepsilon_{ \mathbf{d}, \mathbf{t}}$ is a critical point of the functional $J_\varepsilon$ if and only if the point $( \mathbf{d}, \mathbf{t})$ is a critical point of the function $\widetilde J_\varepsilon.$ \end{proposition} \begin{proof}
We argue as in Proposition 1 of \cite{BLR}.
\end{proof}
The problem is thus reduced to the search for critical points of $\widetilde J_\varepsilon,$ so it is necessary to compute the asymptotic expansion of $\widetilde J_\varepsilon$.
\begin{proposition}\label{pro4}
It holds true that
$$ \widetilde J_\varepsilon( \mathbf{d}, \mathbf{t})= c_1+ c_2\varepsilon+c_3\varepsilon\log\varepsilon+\varepsilon(1+|\lambda|)\Phi( \mathbf{d}, \mathbf{t})+o(\varepsilon), $$
$C^0-$uniformly on compact sets of $\Lambda,$ where
\begin{itemize}
\item[(i)] in case \eqref{ans1}
$$ \Phi( \mathbf{d}, \mathbf{t}):= c_4 \({d \over 2t }\)^{n-2} +c_5t -c_6 \ln d $$
\item[(ii)] in case \eqref{ans11}
\begin{align*}
\Phi( \mathbf{d}, \mathbf{t}):=&c_4\[\({d_1\over 2t_1}\)^{n-2}+\({d_2\over 2t_2}\)^{n-2}+2\(d_1d_2\)^{n-2\over2}\({1\over |t_1-t_2|^{n-2}}-{1\over |t_1+t_2|^{n-2}}\)\right]
\nonumber\\ &+c_5\(t_1+t_2\)-c_6\(\ln d_1+\ln d_2\).\end{align*}
\end{itemize} Here $c_i$'s are constants and $c_4,$ $c_5$ and $c_6$ are positive. \end{proposition} \begin{proof}
The proof is postponed to Section \ref{due}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{main}] It is easy to verify that the function $\Phi$ of Proposition \ref{pro4} in both cases has a minimum point which is stable under uniform perturbations. Therefore, if $\varepsilon$ is small enough there exists a critical point $( \mathbf{d}_\varepsilon, \mathbf{t}_\varepsilon)$ of the reduced energy $ \widetilde J_\varepsilon.$ Finally, the claim follows by Proposition \ref{pro3}.
\end{proof}
\section{Expansion of the reduced energy}\label{due}
It is standard to prove that $$\widetilde J_\varepsilon( \mathbf{d}, \mathbf{t})=J_\varepsilon\(V_{ \mathbf{d}, \mathbf{t}}\)+ o(\varepsilon)$$ (see for example \cite{BLR,BMP}). So the problem reduces to estimating the leading term $J_\varepsilon\(V_{ \mathbf{d}, \mathbf{t}}\).$
We will estimate it in case \eqref{ans11} with $|\lambda|=1$, because in the other cases its expansion is easier and can be deduced from that. Proposition \ref{pro4} will follow from Lemma \ref{lex1}, Lemma \ref{ley1} and Lemma \ref{lem33}.
For future reference we define the constants \begin{align}\label{g1}
&\gamma_1=\alpha_n^{p+1}\int\limits_{ \mathbb{R}^n}{1\over (1+|y|^2)^n}dy,\\ \label{g2}
&\gamma_2=\alpha_n^{p +1}\int\limits_{ \mathbb{R}^n}{1\over (1+|y|^2) ^{n+2\over2}}dy,\\ \label{g3}
&\gamma_3=\alpha_n^{p+1 }\int\limits_{ \mathbb{R}^n}{1\over (1+|y|^2) ^{n }}\log {1\over (1+|y|^2 )^{n-2\over2}}dy. \end{align}
For sake of simplicity, we set $U_i:=U_{\delta_i,\xi_i}$ and $V_i:=V_{\mu_i,\eta_i}.$
\begin{lemma}\label{lex1} It holds true that
\begin{align*}&{1\over2}\int\limits_\Omega|\nabla V_{ \mathbf{d}, \mathbf{t}}|^2dx=2 \gamma_1 \\&- \gamma_2\varepsilon\[\({d _1\over2t_1}\)^{n-2}+\({d_2\over2t_2}\)^{n-2}
+\(d_1d_2\)^{n-2\over2}\({1\over|t_1-t_2|^{n-2}}-{1\over|t_1+t_2|^{n-2}}\)\right] +o(\varepsilon). \end{align*} \end{lemma} \begin{proof} We have \begin{align}\label{lex11}
\int\limits_\Omega|\nabla V_{ \mathbf{d}, \mathbf{t}}|^2dx=&\int\limits_\Omega|\nabla PU_1|^2dx+\int\limits_\Omega|\nabla PU_2|^2dx-2\int\limits_\Omega \nabla PU_1\nabla PU_2dx\nonumber\\
&+\int\limits_\Omega|\nabla PV_1|^2dx+\int\limits_\Omega|\nabla PV_2|^2dx-2\int\limits_\Omega \nabla PV_1\nabla PV_2dx\nonumber\\ &+2\sum\limits_{i,j=1}^2\lambda\int\limits_\Omega \nabla PU_i \nabla PV_jdx\nonumber\\
=&2\(\int\limits_\Omega|\nabla PU_1|^2dx+\int\limits_\Omega|\nabla PU_2|^2dx-2\int\limits_\Omega \nabla PU_1\nabla PU_2dx\)+o(\varepsilon),\nonumber\\ \end{align} because of the symmetry (see \eqref{ans21} and \eqref{ans31}) and the fact that $$\int\limits_\Omega \nabla PU_i \nabla PV_jdx=O\(\delta_i^{n-2\over2}\mu_j^{n-2\over2}\)=o(\varepsilon).$$ Let us estimate the first term in \eqref{lex11}. The estimate of the second term is similar. We set
\begin{equation}\label{tau}\tau:=\min\left\{{\rm d}(\xi_1,\partial\Omega),{\rm d}(\xi_2,\partial\Omega),{|\xi_1-\xi_2|\over2}\right\}=\min\left\{\tau_1,\tau_2,{|\tau_1-\tau_2|\over2}\right\}.\end{equation} We get
$$ \int\limits_\Omega|\nabla PU_1|^2dx=\int\limits_\Omega U_1^pPU_1dx=\int\limits_{B(\xi_1,\tau )}U_1^pPU_1dx+\int\limits_{\Omega\setminus B(\xi_1,\tau )}U_1^pPU_1dx. $$
By Lemma \ref{lem2} we deduce $$
\int\limits_{\Omega\setminus B(\xi_1,\tau )} U_1^pPU_1dx=O\(\({\delta_1\over\tau }\)^n\)=o(\varepsilon) $$
\begin{align}\label{lex14}
&\int\limits_{B(\xi_1,\tau )} U_1^pPU_1dx= \int\limits_{B(\xi_1,\tau )} U_1^{p+1}dx +\int\limits_{B(\xi_1,\tau )} U_1^p\(PU_1-U_1\)dx, \end{align} with $$
\int\limits_{B(\xi_1,\tau)} U_1^{p+1} =\gamma_1+O\(\({\delta_1\over\tau_1}\)^n\)=\gamma_1+o(\varepsilon). $$
The second term in \eqref{lex14} is estimated in (i) of Lemma \ref{lez1}.
It remains only to estimate the third term in \eqref{lex11}.
\begin{align}\label{lex18} \int\limits_\Omega \nabla PU_1\nabla PU_2dx=\int\limits_\Omega U_1^p PU_2dx=\int\limits_{B(\xi_1,\tau )} U_1^p PU_2dx+ \int\limits_{\Omega\setminus B(\xi_1,\tau )} U_1^p PU_2dx. \end{align}
We have \begin{align*}
&\int\limits_{\Omega\setminus B(\xi_1,\tau )} U_1^p PU_2dx=O\({\delta_1^{n+2\over2}\delta_2^{n-2\over2} } \int\limits_{\Omega\setminus B(\xi_1,\tau )}{1\over|x-\xi_1|^{n+2}}{1\over|x-\xi_2|^{n-2}}dx\)\nonumber\\ &
=O\({\delta_1^{n+2\over2}\delta_2^{n-2\over2}\over\tau^n } \int\limits_{ \mathbb{R}^n\setminus B(0,1 )}{1\over|y|^{n+2}}{1\over|y+{\xi_1-\xi_2\over\tau}|^{n-2}}dy\)=O\({\delta_1^{n+2\over2}\delta_2^{n-2\over2}\over\tau^n }\)=o(\varepsilon). \end{align*}
The first term in \eqref{lex18} is estimated in (ii) of Lemma \ref{lez1}.
The claim then follows collecting all the previous estimates and taking into account the choice of $\delta_i'$s and $\tau_i'$s made in \eqref{ans1} and \eqref{ans2}.
\end{proof}
\begin{lemma}\label{ley1} It holds true that \begin{align*}
&{1\over p+1}\int\limits_\Omega{1\over |x|}| V_{ \mathbf{d}, \mathbf{t}}|^{p+1}dx=2\[{2\over p+1} \gamma_1-{1\over p+1} \gamma_1\varepsilon\(t_1+t_2\)\right]\\&-2\gamma_2\varepsilon\[\({d_1\over2t_1}\)^{n-2}+\({d_2\over2t_2}\)^{n-2}
+2\(d_1d_2\)^{n-2\over2}\({1\over|t_1-t_2|^{n-2}}-{1\over|t_1+t_2|^{n-2}}\)\right] +o(\varepsilon). \end{align*} \end{lemma} \begin{proof} We have \begin{align}\label{ley11}
& \int\limits_\Omega{1\over |x|}| V_{ \mathbf{d}, \mathbf{t}}|^{p+1}dx=\int\limits_\Omega{1\over |x|}| PU_1-PU_2+\lambda\(PV_1-PV_2\)|^{p+1}dx\nonumber\\ &
=\int\limits_\Omega{1\over |x|}\(| PU_1-PU_2+\lambda\(PV_1-PV_2\)|^{p+1}-| U_1 |^{p+1}-| U_2|^{p+1}-| V_1 |^{p+1}-| V_2|^{p+1}\)dx\nonumber\\ &
+\int\limits_\Omega{1\over |x|}\( | U_1 |^{p+1} +| U_2|^{p+1}+| V_1 |^{p+1}+| V_2|^{p+1}\)dx \nonumber\\ &
=\int\limits_\Omega{1\over |x|}\(| PU_1-PU_2+\lambda\(PV_1-PV_2\)|^{p+1}-| U_1 |^{p+1}-| U_2|^{p+1}-| V_1 |^{p+1}-| V_2|^{p+1}\)dx\nonumber\\ &
+2\int\limits_\Omega{1\over |x|}\( | U_1 |^{p+1} +| U_2|^{p+1} \)dx, \end{align} because of the symmetry (see \eqref{ans21} and \eqref{ans31}).
The last two terms in \eqref{ley11} are estimated in (v) of Lemma \ref{lez1}. Let
$\tau$ as in \eqref{tau}.
We split the first integral as \begin{align}\label{ley12}
& \int\limits_\Omega{1\over |x|}\(| PU_1-PU_2+\lambda\(PV_1-PV_2\)|^{p+1}-| U_1 |^{p+1}-| U_2|^{p+1}-| V_1 |^{p+1}-| V_2|^{p+1}\)dx\nonumber\\ & =\int\limits_{B(\xi_1,\tau)}\dots+\int\limits_{B(\xi_2,\tau)}\dots+\int\limits_{B(-\xi_1,\tau)}\dots+\int\limits_{B(-\xi_2,\tau)}\dots\nonumber\\ &+\int\limits_{\Omega\setminus\(B(\xi_1,\tau)\cup B(\xi_2,\tau)\cup B(-\xi_1,\tau)\cup B(-\xi_2,\tau)\)}\dots \end{align}
By Lemma \ref{lem2} we deduce
\begin{align*} & \int\limits_{\Omega\setminus\(B(\xi_1,\tau)\cup B(\xi_2,\tau)\cup B(-\xi_1,\tau)\cup B(-\xi_2,\tau)\)}\dots\nonumber\\ & =O\( \int\limits_{\Omega\setminus\(B(\xi_1,\tau)\cup B(\xi_2,\tau)\cup B(-\xi_1,\tau)\cup B(-\xi_2,\tau)\)} \( U_1 ^{p+1}+ U_2 ^{p+1}+V_1 ^{p+1}+ V_2 ^{p+1}\)dx\) \nonumber\\ &=O\({\delta_1^n\over\tau^n}+{\delta_2^n\over\tau^n}\)=o(\varepsilon). \end{align*}
We now estimate the integral over $B(\xi_1,\tau)$ in \eqref{ley12}. \begin{align}\label{ley14}
& \int\limits_{ B(\xi_1,\tau) }{1\over |x|}\(| PU_1-PU_2+\lambda\(PV_1-PV_2\)|^{p+1}-| U_1 |^{p+1}-| U_2|^{p+1}-| V_1 |^{p+1}-| V_2|^{p+1}\)dx\nonumber\\ &=
(p+1) \int\limits_{ B(\xi_1,\tau) }{1\over |x|} U_1 ^ p\( PU_1-U_1 - PU_2+\lambda\(PV_1-PV_2\)\) dx\nonumber\\ &+
{p(p+1)\over2} \int\limits_{ B(\xi_1,\tau) }{1\over |x|} | U_1+\theta \rho |^{p-1} \rho^2dx
-\int\limits_{ B(\xi_1,\tau) }{1\over |x|}\( | U_2|^{p+1} +| V_1 |^{p+1}-| V_2|^{p+1}\)dx\nonumber\\ &=
(p+1) \int\limits_{ B(\xi_1,\tau) }{1\over |x|} U_1 ^ p\( PU_1-U_1 \)dx-(p+1) \int\limits_{ B(\xi_1,\tau) }{1\over |x|} U_1 ^ p PU_2 dx+o(\varepsilon), \end{align} where $\rho:= PU_1-U_1-PU_2+\lambda\(PV_1-PV_2\)$. Indeed, by Lemma \ref{lem2} one can easily deduce that
$$\int\limits_{ B(\xi_1,\tau) }{1\over |x|} U_1 ^ p \(PV_1-PV_2 \) dx, \int\limits_{ B(\xi_1,\tau) }{1\over |x|} | U_2|^{p+1} dx,\int\limits_{ B(\xi_1,\tau) }{1\over |x|}
| V_i |^{p+1} dx=o(\varepsilon) $$ and also \begin{align*}
& {p(p+1)\over2} \int\limits_{ B(\xi_1,\tau) }{1\over |x|} | U_1+\theta\rho|^{p-1} \rho^2dx\le c\int\limits_{ B(\xi_1,\tau) } | U_1|^{p-1} \rho^2dx+\int\limits_{ B(\xi_1,\tau) } |\rho|^{p+1} dx \\ & \nonumber\\ &\le c\int\limits_{ B(\xi_1,\tau) } U_1 ^{p-1} \( PU_1-U_1 \)^2dx+c\int\limits_{ B(\xi_1,\tau) } U_1 ^{p-1}\(PU_2\)^2dx+c \int\limits_{ B(\xi_1,\tau) } U_1 ^{p-1} \(PV_1-PV_2 \)^2dx
\nonumber\\ &+ c\int\limits_{ B(\xi_1,\tau) }|PU_1-U_1 |^{p+1}+c\int\limits_{ B(\xi_1,\tau) }| U_2 |^{p+1}+c\int\limits_{ B(\xi_1,\tau) }\(| V_1 |^{p+1}+| V_2 |^{p+1}\)dx\nonumber\\ & =o(\varepsilon). \end{align*}
The first term and the second term in \eqref{ley14} are estimated in (iii) and (iv) of Lemma \ref{lez1}, respectively.
Therefore, the claim follows. \end{proof}
\begin{lemma}\label{lem33} It holds true that \begin{align*}
& {1 \over p + 1 -\varepsilon} \int\limits_\Omega {1\over |x|}|V_{ \mathbf{d}, \mathbf{t}}|^{p+1-\varepsilon}={1 \over p + 1} \int\limits_\Omega {1\over |x|}|V_{ \mathbf{d}, \mathbf{t}}|^{p+1} \\
&+ \(1+|\lambda|\)\[{\gamma_1\over (p+1)^2}-\alpha_n{\gamma_1\over (p+1) }-{\gamma_3\over (p+1) }\varepsilon +{n-2\over2(p+1)}\(\ln\delta_1+\ln\delta_2\)\right]+ o(\varepsilon). \end{align*} \end{lemma} \begin{proof} We argue exactly as in Lemma 3.2 of \cite{DFM}. \end{proof}
\begin{lemma}\label{lez1} Let $\tau$ as in \eqref{tau}. It holds true that \begin{itemize} \item[(i)] $$\int\limits_{ B(\xi_1,\tau) } U_1 ^ p\( PU_1-U_1\)dx=-\gamma_2\({\delta_1\over2\tau_1}\)^{n-2}+o(\varepsilon) $$
\item[(ii)] $$\int\limits_{ B(\xi_1,\tau) } U_1 ^ pPU_2dx= \gamma_2\({\delta_1\delta_2 }\)^{n-2\over2}\({1\over|\tau_1-\tau_2|^{n-2}}-{1\over|\tau_1+\tau_2|^{n-2}}\)+o(\varepsilon)$$
\item[(iii)] $$\int\limits_{ B(\xi_1,\tau) }{1\over |x|} U_1 ^ p\( PU_1-U_1\)dx=-\gamma_2\({\delta_1\over2\tau_1}\)^{n-2}+o(\varepsilon) $$
\item[(iv)] $$\int\limits_{ B(\xi_1,\tau) }{1\over |x|} U_1 ^ pPU_2dx=-\gamma_2\({\delta_1\over2\tau_1}\)^{n-2}+o(\varepsilon)$$
\item[(v)] $$\int\limits_\Omega{1\over |x|}U_1 ^{p+1}dx=\gamma_1-\gamma_1\tau_1+o(\varepsilon).$$
\end{itemize} \end{lemma} \begin{proof}
{\em Proof of (i)} By Lemma \ref{lem2} we get \begin{align*}
&\int\limits_{B(\xi_1,\tau )} U_1^p \( PU_1-U_1\)dx= \int\limits_{B(\xi_1,\tau )} U_1^p \( -\alpha_n\delta_1^{n-2\over 2}H(x,\xi_1)+R_{\delta_1,\xi_1}\)dx\nonumber \\
=& -\alpha_n\delta_1^{n-2\over 2}\int\limits_{B(\xi_1,\tau )} U_1^pH(x,\xi_1)dx+\int\limits_{B(\xi_1,\tau )} U_1^{p }R_{\delta_1,\xi_1} dx, \end{align*} with $$ \int\limits_{B(\xi_1,\tau )} U_1^{p }R_{\delta_1,\xi_1} dx= O\(\({\delta_1\over\tau_1}\)^n\). $$ By Lemma \ref{lem3} we get \begin{align*}
&\alpha_n\delta_1^{n-2\over 2}\int\limits_{B(\xi_1,\tau )} U_1^pH(x,\xi_1)dx =\alpha_n^{p+1}\delta_1^{n-2}\int\limits_{B(0,\tau /\delta_1)} H(\delta_1y+\xi_1,\xi_1){1\over (1+|y|^2)^{n+2\over2}}dy\nonumber\\ & =
\alpha_n^{p+1}\({\delta_1\over\tau_1}\)^{n-2} \int\limits_{B(0,\tau /\delta_1)}\tau_1^{n-2}H( \delta_1y+\xi_1,\xi_1){1\over (1+|y|^2)^{n+2\over2}}dy\nonumber\\&=\alpha_n^{p+1}\({\delta_1\over\tau_1}\)^{n-2} \[{1\over 2^{n-2}}\int\limits_{ \mathbb{R}^n)} {1\over(1+|y|^2)^{n+2\over2}}dy+o(1)\right]. \end{align*}
{\em Proof of (ii)} By Lemma \ref{lem2} and Lemma \ref{lem3} we get \begin{align*}
&\int\limits_{B(\xi_1,\tau )} U_1^p PU_2dx=\int\limits_{B(\xi_1,\tau )} U_1^p \(U_2-\alpha_n\delta_2^{n-2\over2}H(x,\xi_2)+R_{\delta_2,\xi_2}\)dx\nonumber \\&
=\alpha_n^{p+1} (\delta_1\delta_2)^{n-2\over2}\int\limits_{B(0,\tau /\delta_1)}{1\over(1+|y|^2)^{n+2\over2}}
{1\over(\delta_2^2+|\delta_1y+\xi_1-\xi_2|^2)^{n-2\over2}}dy\nonumber\\
& -\alpha_n^{p+1} (\delta_1\delta_2)^{n-2\over2}\int\limits_{B(0,\tau /\delta_1)}{1\over(1+|y|^2)^{n+2\over2}} H(\delta_1y+\xi_1,\xi_2) dy\nonumber \\ &+\alpha_n^{p+1} (\delta_1\delta_2)^{n-2\over2}\int\limits_{B(0,\tau /\delta_1)}{1\over(1+|y|^2)^{n+2\over2}}
R_{\delta_2,\xi_2}(\delta_1y+\xi_1)dy=\nonumber \\&
=\alpha_n^{p+1} {(\delta_1\delta_2)^{n-2\over2}\over |\tau_1-\tau_2|^{n-2}}\int\limits_{B(0,\tau /\delta_1)}{ 1\over(1+|y|^2)^{n+2\over2}}
{|\tau_1-\tau_2|^{n-2}\over(\delta_2^2+|\delta_1y+\xi_1-\xi_2|^2)^{n-2\over2}}dy\nonumber\\
& -\alpha_n^{p+1} {(\delta_1\delta_2)^{n-2\over2}\over |\tau_1+\tau_2|^{n-2}}\int\limits_{B(0,\tau /\delta_1)}{|\tau_1+\tau_2|^{n-2}\over(1+|y|^2)^{n+2\over2}} H(\delta_1y+\xi_1,\xi_2) dy\nonumber \\ &+O\( (\delta_1\delta_2)^{n-2\over2}{\delta_2^{n+2\over2}\over\tau_2^{n}}\)=\nonumber \\&
=\alpha_n^{p+1} {(\delta_1\delta_2)^{n-2\over2}\over |\tau_1-\tau_2|^{n-2}}\[\int\limits_{ \mathbb{R}^n}{ 1\over(1+|y|^2)^{n+2\over2}}dy+o(1)\right]
\nonumber\\
& -\alpha_n^{p+1} {(\delta_1\delta_2)^{n-2\over2}\over |\tau_1+\tau_2|^{n-2}}\[\int\limits_{ \mathbb{R}^n}{1\over(1+|y|^2)^{n+2\over2}} dy +o(1)\right]\nonumber \\ &+o\( {(\delta_1\delta_2)^{n-2\over2} \over\tau_2^{n-2}}\).
\end{align*}
{\em Proof of (iii) and (iv)} We argue as in the proof of (i) and (ii) using estimates \eqref{lew15} and \eqref{lew16}.
{\em Proof of (v)} We have \begin{align}\label{lew11}
\int\limits_\Omega{1\over |x|}U_1 ^{p+1}dx=\int\limits_{B(\xi_1,\tau)}U_1 ^{p+1}dx+ \int\limits_{\Omega\setminus B(\xi_1,\tau)}U_1 ^{p+1}dx,\end{align} with $$
\int\limits_{\Omega\setminus B(\xi_1,\tau)} {1\over |x|}U_1 ^{p+1}dx= O\({\delta_1^n\over\tau^n}\),$$ So, we only have to estimate the first term in \eqref{lew11}. We split it as
$$ \int\limits_{B(\xi_1,\tau)} {1\over |x|} U_1 ^{p+1}dx= \int\limits_{B(\xi_1,\tau)} U_1 ^{p+1}dx+\int\limits_{B(\xi_1,\tau)}\({1\over |x|}-1\)U_1 ^{p+1}dx.$$
We have $$ \int\limits_{B(\xi_1,\tau)} U_1 ^{p+1}dx=\gamma_1+O\({\delta_1^n\over\tau^n}\).$$
Since $\xi_1=\xi_0(1+\tau_1)$ and $|\xi_0|=1$, by the mean value theorem we get \begin{align}\label{lew15}
{1\over|\delta_1 y+\tau_1\xi_0+\xi_0|}-1=-\tau_1-\delta_1 \<y, \xi_0\right\rangle+R(y),\end{align} where $R$ satisfies the uniform estimate \begin{align}\label{lew16}
|R(y )|\le c\(\delta_1^2|y|^2+\delta_1\tau_1|y|+\tau_1^2\)\ \hbox{for any}\ y\in B(0,\tau/\delta_1).\end{align} Therefore we conclude \begin{align*}
& \int\limits_{B(\xi_1,\tau)}\({1\over |x|}-1\)U_1 ^{p+1}dx= \alpha_n^{p+1}\int\limits_{B(0,\tau/\delta_1)}\({1\over |\delta_1 y+\tau_1\xi_0+\xi_0|}-1\){1\over(1+|y|^2)^{n}}dy\nonumber\\ &=\alpha_n^{p+1}\int\limits_{B(0,\tau/\delta_1)}\(-\tau_1-\delta_1\tau_1\<y, \xi_0\right\rangle+R(y)\){1\over(1+|y|^2)^{n}}dy =-\gamma_1\tau_1+o(\tau).\end{align*} Collecting all the previous estimates we get the claim. \end{proof}
\begin{lemma}\label{lem3} Let $\tau$ as in \eqref{tau}. It holds true that
\begin{itemize}\item[(i)]$$\int\limits_{B(0,\tau /\delta_1)}\tau_1^{n-2}H( \delta_1y+\xi_1,\xi_1){1\over (1+|y|^2)^{n+2\over2}}dy={1\over 2^{n-2}}\int\limits_{ \mathbb{R}^n} {1\over(1+|y|^2)^{n+2\over2}}dy+o(1) ,$$
\item[(ii)]$$\int\limits_{B(0,\tau /\delta_1)}{|\tau_1+\tau_2|^{n-2}\over(1+|y|^2)^{n+2\over2}} H(\delta_1y+\xi_1,\xi_2) dy= \int\limits_{ \mathbb{R}^n} {1\over(1+|y|^2)^{n+2\over2}}dy+o(1) ,$$
\item[(iii)]$$\int\limits_{B(0,\tau /\delta_1)}{ 1\over(1+|y|^2)^{n+2\over2}}
{|\tau_1-\tau_2|^{n-2}\over(\delta_2^2+|\delta_1y+\xi_1-\xi_2|^2)^{n-2\over2}}dy=\int\limits_{ \mathbb{R}^n} {1\over(1+|y|^2)^{n+2\over2}}dy+o(1) .$$
\end{itemize} \end{lemma} \begin{proof} We are going to use Lebesgue's dominated convergence Theorem together with Lemma \ref{lemacca} . First of all, taking into account that $ \xi_1=(1+\tau_1)\xi_0$ and $ \bar \xi_1=(1-\tau_1)\xi_0$ we deduce that
$$\tau_1^{n-2}H( \delta_1y+\xi_1,\xi_1){1\over (1+|y|^2)^{n+2\over2}}\to {1\over 2^{n-2}}{1\over (1+|y|^2)^{n+2\over2}}\ \hbox{a.e. in}\ \mathbb{R}^n $$ and also that
$$ H( \delta_1y+\xi_1,\xi_1) \le C_2 {1\over|\delta_1y+\xi_1-\bar \xi_1|^{n-2}}=C_2{1\over|\delta_1y+2\tau_1\xi_0|^{n-2}}\le C_2{1\over\tau_1^{n-2}},$$ since
$$|\delta_1y+2\tau\xi_0|\ge 2\tau_1-|\delta_1y|\ge \tau_1\ \hbox{for any} \ y\in B(0,\tau /\delta_1).$$
That proves (i).
In a similar way, taking into account that $ \xi_1=(1+\tau_1)\xi_0$ and $ \bar \xi_2=(1-\tau_2)\xi_0$
we get
$$(\tau_2+\tau_1)^{n-2}H( \delta_1y+\xi_1,\xi_2){1\over (1+|y|^2)^{n+2\over2}}\to {1\over (1+|y|^2)^{n+2\over2}}\ \hbox{a.e. in}\ \mathbb{R}^n $$ and also that
$$ H( \delta_1y+\xi_1,\xi_2) \le C_2 {1\over|\delta_1y+\xi_1-\bar \xi_2|^{n-2}}=C_2{1\over|\delta_1y+(\tau_1+\tau_2)\xi_0|^{n-2}}\le C_2{1\over\tau_2^{n-2}},$$ since
$$|\delta_1y+(\tau_1+\tau_2)\xi_0|\ge \tau_1+\tau_2 -|\delta_1y|\ge \tau_2\ \hbox{for any} \ y\in B(0,\tau /\delta_1).$$
That proves (ii).
Finally, we have
$${ 1\over(1+|y|^2)^{n+2\over2}}
{|\tau_1-\tau_2|^{n-2}\over(\delta_2^2+|\delta_1y+\xi_1-\xi_2|^2)^{n-2\over2}}\to {1\over (1+|y|^2)^{n+2\over2}}\ \hbox{a.e. in}\ \mathbb{R}^n $$ and also that $$
{1\over(\delta_2^2+|\delta_1y+\xi_1-\xi_2|^2)^{n-2\over2}}\le {1\over |\delta_1y+\xi_1-\xi_2| ^{n-2 }}\le {2^{n-2}\over|\tau_1-\tau_2|^{n-2}}$$ since
$$|\delta_1y+\xi_1-\xi_2| \ge |\xi_1-\xi_2| -|\delta_1y|\ge {|\xi_1-\xi_2|\over2} \ \hbox{for any} \ y\in B(0,\tau /\delta_1).$$
That proves (iii).
\end{proof}
\appendix \section{}
Here we recall some well known facts which are useful to get estimates in Section \ref{due}.
We denote by $G(x,y)$ the Green's function associated to $-\Delta$ with Dirichlet boundary condition and $H(x,y)$ its regular part, i.e. $$-\Delta_x G(x,y) = \delta_y (x) \quad \text{for } x \in \Omega,\quad G(x,y) = 0 \quad \text{for } x \in \partial\Omega, $$ and
$$G(x,y) = \gamma_n \( \frac{1}{|x-y|^{n-2}} - H(x,y) \) \quad \text{ where} \quad \gamma_n = \frac{1}{(n-2)|S^{n-1}|}$$
($|S^{n-1}| = (2 \pi^{n/2})/\ \Gamma(n/2)$ denotes the Lebesgue measure of the $(n-1)$-dimensional unit sphere).
The following lemma was proved in \cite{R}.
\begin{lemma}\label{lem2} It holds true that $$PU_{\delta,\xi}(x)=U_{\delta,\xi}(x)-\alpha_n\delta^{n-2\over2} H(x,\xi)+O\({\delta^{n+2\over2}\over {\rm dist}(\xi,\partial\Omega)^n}\)$$ for any $x\in\Omega.$\end{lemma}
Since $\Omega$ is smooth, we can choose small $\epsilon > 0$ such that, for every $x \in \Omega$ with
${\rm dist}(x, \partial\Omega) \le \epsilon$, there is a unique point $x_{\nu} \in \partial\Omega$ satisfying ${\rm dist}(x, \partial\Omega) = |x - x_{\nu}|$. For such $x \in \Omega$, we define $x^* = 2x_{\nu} - x$ the reflection point of $x$ with respect to $ \partial\Omega $.
The following two lemmas are proved in \cite{ACP}. \begin{lemma}\label{lemacca} It holds true that
$$\left| H(x,y)-{1\over |\bar x-y|^{n-2}}\right|=O\({{\rm dist}(x,\partial\Omega)^n \over |\bar x-y|^{n-2}}\)$$ and
$$\left|\nabla_x\( H(x,y)-{1\over |\bar x-y|^{n-2}}\)\right|=O\({1 \over |\bar x-y|^{n-2}}\)$$ for any $x \in \Omega$ with ${\rm dist}(x, \partial\Omega) \le \epsilon$.\end{lemma}
\end{document} | arXiv |
Orbital stability analysis and photometric characterization of the second Earth Trojan asteroid 2020 XL5
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T. Santana-Ros ORCID: orcid.org/0000-0002-0143-94401,2,
M. Micheli ORCID: orcid.org/0000-0001-7895-82093,
L. Faggioli ORCID: orcid.org/0000-0002-5447-432X3,
R. Cennamo ORCID: orcid.org/0000-0003-1121-93933,
M. Devogèle ORCID: orcid.org/0000-0002-6509-63604,
A. Alvarez-Candal ORCID: orcid.org/0000-0002-5045-96755,6,7,
D. Oszkiewicz ORCID: orcid.org/0000-0002-5356-64338,
O. Ramírez9,
P.-Y. Liu6,
P. G. Benavidez ORCID: orcid.org/0000-0001-6569-02231,6,
A. Campo Bagatin ORCID: orcid.org/0000-0001-9840-22161,6,
E. J. Christensen10,
R. J. Wainscoat ORCID: orcid.org/0000-0002-1341-095211,
R. Weryk ORCID: orcid.org/0000-0002-0439-934112,
L. Fraga ORCID: orcid.org/0000-0003-0680-197913,
C. Briceño ORCID: orcid.org/0000-0001-7124-409414 &
L. Conversi ORCID: orcid.org/0000-0002-6710-84763,15
Nature Communications volume 13, Article number: 447 (2022) Cite this article
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Asteroids, comets and Kuiper belt
Inner planets
Trojan asteroids are small bodies orbiting around the L4 or L5 Lagrangian points of a Sun-planet system. Due to their peculiar orbits, they provide key constraints to the Solar System evolution models. Despite numerous dedicated observational efforts in the last decade, asteroid 2010 TK7 has been the only known Earth Trojan thus far. Here we confirm that the recently discovered 2020 XL5 is the second transient Earth Trojan known. To study its orbit, we used archival data from 2012 to 2019 and observed the object in 2021 from three ground-based observatories. Our study of its orbital stability shows that 2020 XL5 will remain in L4 for at least 4 000 years. With a photometric analysis we estimate its absolute magnitude to be \({H}_{r}=18.5{8}_{-0.15}^{+0.16}\), and color indices suggestive of a C-complex taxonomy. Assuming an albedo of 0.06 ± 0.03, we obtain a diameter of 1.18 ± 0.08 km, larger than the first known Earth Trojan asteroid.
The classical work of J. L. Lagrange published in 1772 on the three-body problem had to wait until 1906 to find an empirical verification with the discovery of asteroid (588) Achilles. This asteroid is orbiting around a theoretical point located 60° ahead of Jupiter along its orbit. After discovering (588) Achilles, many other objects were found orbiting around nearly the same point or its mirroring position 60° behind Jupiter. Both points are the so-called triangular Lagrange points and are commonly known as L4 (for the former) and L5 (for the latter)1. Asteroids orbiting around these points of a planet-Sun system are known as Trojan asteroids.
Although Trojan asteroids have been known for decades in other Solar System planets2 such as Venus3, Mars4, Jupiter5, Uranus6, and Neptune7, it wasn't until 2011 that asteroid 2010 TK7 was found to be the first (and hitherto unique) Earth Trojan (ET) asteroid8. ET asteroids have been broadly debated and proved to be feasible from the point of view of celestial mechanics. In particular, their possible existence has been shown using theoretical studies by means of numerical simulations9,10. Observational strategies have been defined trying to detect new ETs but all the dedicated surveys performed so far have failed to discover any new member of this population11,12,13,14, including an in situ survey15 performed by the OSIRIS-REx spacecraft within the L4 region and observations16 of L5 by the Hayabusa2 spacecraft on its way to asteroid (162173) Ryugu. Despite failing in the detection of new ET asteroids, some of these surveys provided population constraints regarding their number and their size12,13,14,15.
The reason behind this low discovery success rate is related to the unfavorable viewing geometry of an object orbiting Earth-Sun's L4 or L5 points as seen from our planet17. In short, these objects are often observable very close to the Sun (i.e., at low Solar elongations) and under large phase angles (the Sun-object-observer angle), meaning that a significant fraction of the object is shadowed as seen from the Earth, which in turn implies the object being faint. Under such geometries, observations must be acquired at high airmass, where seeing is typically worse, which, together with higher background from zodiacal light, further increases the difficulty of these searches. For both ETs known to date, opportunities for better observing geometries at larger solar elongations might exist thanks to their higher eccentricity values.
Asteroid 2020 XL5 was discovered by the Pan-STARRS1 survey on 2020 December 12. Shortly after the discovery, follow-up observations were gathered from different stations, allowing for an initial orbit determination. The orbit behavior suggested that 2020 XL5 could have been a candidate to become the second known ET, but the orbit uncertainty due to the short arc covered with observations was still too large to confirm a current Trojan engagement with Earth18.
Here we present the results of our study based on our new observations of 2020 XL5 which confirms that 2020 XL5 will be an ET for at least 4000 years.
Orbit stability
The nominal orbit of 2020 XL5 has been computed at the epoch MJD = 58444.1 using the European Space Agency's (ESA) AstOD orbit determination software19,20, based on the methods described in the literature21, taking as input the full observations dataset described in the observation sections of Methods. The resulting Keplerian orbital elements and their uncertainties are presented in Table 1. The dynamical model used in the orbit determination and propagation takes into account the N-body gravitational attraction of the Sun, eight planets, the Earth's Moon, and the parameterized post-Newtonian relativistic contribution and the oblateness of Sun and Earth. The software makes use of the JPL Planetary and Lunar Ephemerides DE43122.
Table 1 Orbital elements.
In order to investigate the stability of the object in the L4 Lagrange point of the Earth, we performed a set of numerical simulations by integrating the nominal orbit, together with 800 clone orbits sampling its uncertainty, over a time span of ~29,000 years. The integration of the orbits has been performed using the ESA AstOD orbit determination software. The 800 clone orbits have been computed by sampling the orbit covariance matrix. The Trojan-like behavior of an object is seen in a reference frame co-rotating with the Earth's orbital motion23. The key parameter to quantify the state is the relative mean longitude λr is defined as the difference between the mean longitude of the asteroid λa and the mean longitude of the Earth λE: when the resonant angle λr librates around 60°, i.e., 0° < λr < 180° the asteroid is called an L4 Trojan, when the resonant angle λr librates around 300°, i.e., 180° < λr < 360° the asteroid is called an L5 Trojan, when λr circulates then the asteroid leaves the Trojan-like orbit24. Nevertheless, Trojans can have a displacement of a maximum of \(\omega =2{5}^{\circ }11^{\prime}\) from the typical equilateral location for eccentric orbits25. We found that in the case of 2020 XL5 the mean longitude (λr) evolution of the nominal orbit in Table 1 shows a transient ET behavior for this object (see Fig. 1). We define t0 = 0 as the mean epoch of the observations arc (corresponding to November 2018). The plot shows that before the time t1 ≃ −500 years, λr circulates, and therefore the asteroid could not be considered to be in a Trojan-like orbit. Starting from t1, 2020 XL5 is an L4 ET librating around λr ~ 75°, being stably located at the Lagrangian point for a time interval of about 4500 years, until the time t2 ≃ 4000.
Fig. 1: Mean longitude evolution analysis.
a The relative mean longitude (λr) evolution of the 2020 XL5 nominal orbit as in Table 1 and clones orbit over 29,000 years, where each clone orbit is represented by a different color, while the green line represents the nominal orbit. b Evolution of λr for the nominal orbit of 2020 XL5 over 29,000 years. c Stack plot representing the behavior of the nominal orbit and the 800 clone orbits. In the plot the time t0 = 0 is the mean epoch of the orbits. Source data are provided as a Source Data file.
The observed stability time interval of the clone orbits appears to be consistent with that of the nominal orbit, as shown in Fig. 1, where a different color for each clone orbit has been used, with the green line representing the nominal orbit. Some clone orbits escape from stability before the nominal one, as shown in the zoomed Fig. 2.
Fig. 2: Beginning of the non-deterministic regime.
The relative mean longitude (λr) evolution of the 2020 XL5 nominal orbit as in Table 1 and clones orbit from year ~3300 until year ~4300, where each clone orbit is represented by a different color. Some clone orbits escape from the stability before the nominal one. Source data are provided as a Source Data file.
To better visualize the results we generated a stack plot classifying the behavior of the nominal and clone orbits in three different states: librating around L4, librating around L5, and circulating (Fig. 1). The stack plot covers the entire time span of the simulations, with the time t0 = 0 set as the mean epoch of the observation arch. The plot shows that some orbits begin to show instability after time t ~ 3500, and by time t = 5000 less than 40% of the orbits are still in L4. Integrating backward, all the orbits become unstable by time t ~ −500, and less than 5% of orbits are located in L4 by t ~ −2000 and earlier. Moreover, the plot shows that less than 10% of the orbits are librating around L5 both before t ~ −1000 and during the interval between t ~ 6000 and t ~ 8000.
These transitions between libration points are a well-known behavior for objects in co-orbital motion, especially for those in orbits with large enough eccentricity and/or high enough inclination26.
Physical characterization
In addition to the orbital analysis discussed above, we also investigated the physical characteristics of 2020 XL5, by photometrically analyzing the new imaging data gathered during the 2020–2021 apparition and the precovery datasets. The SNR of the target was not sufficient to perform time-series photometry, since the object was only marginally detectable on single images, and could only be measured after stacking multiple frames. Moreover, since the observations were obtained close to dawn and with the telescope pointing very low over the horizon due to the object's proximity to the Sun, the background signal increased very fast from frame to frame, and only the first 4 images for each band were useful for photometric purposes. Nevertheless, we obtained V-band photometry on 2021 February 22, Sloan g′, r′, and i′ photometry on 2021 March 9, and Sloan r′ photometry on 2021 March 13 and 16. Using these measurements, we investigated the taxonomy of the object; it is compatible with the C-complex, as shown in Fig. 3, although the low SNR of the measurements results in a large uncertainty for all the determined colors. The computed a* color is a* = −0.9 ± 0.6 for the first night and a* = −0.4 ± 1.0 for the second night, which are compatible with C-complex objects. The S-complex objects typically have a* > 0 and on average a* ~ 0.15 and the C-complex objects a* < 0 with average value of a* ~ −0.127. In addition, deviations due to lightcurve amplitude effects could not be corrected, since the spin state of the object is still unknown. In light of the significant uncertainty of the photometric measurements, and the impossibility to compensate for rotational effects, this taxonomic classification is to be seen as provisional, and needs to be confirmed by further observations, ideally when the object presents a more favorable viewing geometry. We further investigated the photometric behavior of the object by analyzing the dependence of its brightness on the phase angle at the different nights of observations, to obtain the object's phase curve (see Methods subsection Phase curve). After calibrating the data we obtained \({H}_{r}=18.5{8}_{-0.15}^{+0.16}\) for a C-type asteroid28 with \({G}_{{1}_{r}}^{* }=0.83\) and \({G}_{{2}_{r}}^{* }=0.02\). Thus, assuming a C-complex class, and therefore an albedo of 0.06 ± 0.0329, we estimate the diameter of 2020 XL5 as \(1.1{8}_{-0.08}^{+0.08}\,{{\mbox{km}}}\,\). The uncertainty interval corresponds to the 16th and the 84th percentile, encompassing the 68% of the underling distribution of possible values. The inferred diameter for 2020 XL5 is larger than the value known for the first ET asteroid, 2010 TK7, which was estimated8 to have a diameter of ~0.3 km.
Fig. 3: Color indices.
The g-r and r-i colors indices of 2020 XL5 from the two observing nights, the color scale is according to the SDSS a* parameter. Gray lines represent one sigma color uncertainty. The 2043 color points represent 1308 asteroids from all taxonomic types present in the SDSS MOC DR3 catalog. The C-complex objects typically have a* < 0 with an average value of −0.1 and the S-complex object a* > 0 with an average value of 0.1527. 2020 XL5 is located in the C-complex area. Source data are provided as a Source Data file.
Delta-v budget
ETs, among the Earth co-orbital objects, are considered to be potential candidates for rendezvous and even sample return missions, due to the low-energy requirements expected, as shown in previous works30 with a theoretical population of ETs. Therefore, we decided to investigate the required delta-v budget for both a rendezvous and a fly-by mission to the two ETs known, 2020 XL5 and 2010 TK7, in order to decide if they would be good candidates for a mission. The pykep tool31 used for the analysis is described in the Methods subsection Delta-v budget.
Figure 4 shows, for both objects, the minimum total delta-v required for each departure date and each of the considered scenarios, i.e., a launch from Low Earth orbit (LEO) or Geostationary transfer orbit (GTO), and a space mission to a rendezvous or a fly-by with the asteroid.
Fig. 4: Delta-V of trajectories.
Minimum delta-v trajectories to 2020 XL5 (solid line) and 2010 TK7 (dotted line) for the next 5 years. Four different scenarios are considered: departure from LEO and/or GTO orbit, and a rendezvous and/or fly-by mission to the asteroid. Source data are provided as a Source Data file.
First, for a rendezvous mission to 2020 XL5, the absolute minimum total delta-v is estimated to be between 7.9 and 10.3 km/s, depending on the launch conditions. Launching from LEO directly to escape is very expensive, and thus, the resulting required delta-v budget is not feasible. Getting to the GTO orbit via a shared launch significantly reduces the delta-v down to a value lower than 8 km/s, but it is still too high to be considered an ideal target for a rendezvous mission.
When comparing the results for 2020 XL5 to the other known ET, we can see that the latter presents slightly lower values for the absolute minimum total delta-v, which is estimated to be between 6 and 8.5 km/s, also far from the expected very low-energy requirements for rendezvous to theoretical ETs30. The main reason for such large velocities is the relatively high inclination of both 2020 XL5 and 2010 TK7, resulting in an additional plane-change maneuver that is extremely costly. In a previous large-scale statistical survey of delta-v to Earth co-orbital asteroids, similar conclusions were derived regarding the importance of inclination when considering a low delta-v budget to rendezvous asteroids with Earth-like orbits30. It might be preferable to consider another Earth co-orbital object closer to Earth's orbital plane, or instead of a rendezvous mission, consider a fly-by mission to the asteroid. Therefore, we conclude that neither of the known ETs are good candidates for a space mission.
Figure 4 also shows the total delta-v required to perform a fly-by, which is significantly lower than for a rendezvous with the asteroid, since there is no need to match the asteroid's orbit and hence no additional maneuver is performed. In addition, similar absolute minimum total delta-v values are obtained for both ETs, between 0.9 and 3.3 km/s depending on the launch conditions, making both of them potential fly-by targets. It is however important to highlight that 2020 XL5 presents a flatter minimum total delta-v with respect to the departure date. Therefore, 2020 XL5 might be a better candidate for a fly-by mission to an ET since it provides more flexibility to choose a suitable launch date.
Several observational surveys have been devoted to discover ETs near the L4 and L5 points12,13,14,15,16. Despite these efforts, only two objects have been discovered so far: 2010 TK7 and 2020 XL5. However, both asteroids are transient ETs, meaning that their stability around L4 has been shown to be in the scale of thousands of years, far from the stability time-scale of a theoretical primordial ET population, which are thought to be remnants from the Earth's formation period32. Although no primordial ETs has been found yet, some constraints have been provided on their population. The most recent and restrictive values on their magnitude limit for L414 are NET < 1 for H = 13.93, NET < 10 for H = 16, and NET < 938 for H = 22, while for L513 are NET < 1 for H < 15.5, NET = 60 − 85 for H < 19.7, and NET = 97 for H = 20.4.
The discovery of a second ET asteroid may enhance our knowledge of the dynamics of this elusive population. By comparing the orbital nature of the two ETs known so far, we can better understand the mechanisms that allow for their transient stability. For instance, the librating point of both asteroids is displaced from the expected 60° due to their inclined and eccentric orbits. This might suggest that captured ET asteroids may likely be found in orbits displaced from the libration points.
Regarding the physical characterization of 2020 XL5, the improvement of its orbit and therefore its ephemeris to the arc-second level provided by this work is opening new interesting observational possibilities. In particular, it is now possible to plan observations with instruments having small fields of view. For instance, gathering new photometric data of the object during its yearly favorable observing window from November to December, will help to reduce the uncertainty on the color indices and therefore enhance its taxonomic classification. More specific studies resulting from infrared or spectroscopic observations will also enhance our knowledge of this dynamically exotic object. The latter will provide better constraints on its size estimation and composition. If follow-up observations confirm its C-complex nature, one reasonable explanation of this body's orbit would be a transient capture from the main belt, after being ejected from the main belt by any of the 2:1 and 5:2 resonance complex with Jupiter33. Nonetheless, it has been shown34 that the Hungaria region could be a possible source of co-orbital bodies in the inner Solar System, with libration periods around ~10 kyr, in agreement with our results for 2020 XL5. On top of that, it has been found35 that the Hungaria region is mostly dominated by C and S-types, especially at small sizes, which is also consistent with our taxonomic classification of 2020 XL5.
Future surveys of the L4 and L5 regions will allow to derive tighter constraints on the ET populations and, maybe, discover their primordial bodies. Their study could help enhance the Solar System evolution models and therefore can be very valuable to understand its formation. Discovering ETs having lower orbital inclination and eccentricity might have another important implication: unlike 2020 XL5, objects librating near to the Lagrangian points with low inclinations could be reached from the Earth with a very low delta-v budget30. Therefore these objects may become ideal targets for space missions and, in the more distant future, to settle human bases or install scientific hardware that would benefit from their peculiar location.
New observations
The astrometric dataset of 2020 XL5, including information about the telescopes and the observing conditions, is presented in Table 2. In the following section, we describe the observations performed in our follow-up observation campaign for each of the telescopes used.
Table 2 Astrometry.
In order to extend the observed arc of 2020 XL5 into 2021, we gathered optical images of 2020 XL5 with two 4 m class telescopes (the Southern Astrophysical Research telescope and the Lowell Discovery Telescope) and a 1.0 m telescope (ESA's Optical Ground Station) from 2021 February 9 to 2021 March 16, covering an additional orbital arc of 35 days.
The target's viewing geometry and observational circumstances were extremely challenging during the time of all these observations: the target was very close to the Sun (low elongation), and also overlapping with the galactic plane earlier in the observation period.
We designed our observing strategy for the purely astrometric datasets prioritizing the goal of achieving the minimum signal-to-noise ratio (SNR) necessary in order to make the object measurable from an astrometric point of view. We, therefore, used broad-band filters in all our observations and pointed to the object as soon as it reached the minimum elevation allowed by the telescopes' specifications. All in all, we had a few minutes per night to observe the object until the background quickly started saturating due to twilight.
Additionally, the length of the exposure times was limited by the apparent motion of 2020 XL5. Therefore to achieve the best possible astrometric accuracy, while still keeping the ability to track non-linearly on the motion of the object, and reject bright overlapping stars, we decided to limit our single exposures to at most 20s. Astrometric measurements were then extracted from stacked images for each night, whereas a quality test for the background levels was performed to select the frames qualifying for the photometric analysis.
Optical ground station (OGS)
ESA's Optical Ground Station (OGS) 1.0 m telescope (MPC code J04) in Tenerife, Canary Island, Spain, is regularly used by ESA's Planetary Defence Office to obtain astrometric observations of NEOs. In the context of such routine monitoring and follow-up activities, our team obtained and reported observations of the target on 2020 December 13, providing observational confirmation of the object's existence just 14 h after the initial discovery. Other stations reported additional data over the following nights, but the observational coverage available at the Minor Planet Center ended in early 2021 January. By the end of the month, the object's trajectory brought it towards the skyplane location of the galactic center, and observations became consequently more challenging. We nevertheless decided to attempt further observations in early February with the OGS telescope. The field containing the object was exposed on three consecutive mornings, from 2021 February 9 to 11. On the first night, we could detect a possible faint candidate on a small subset of frames not significantly affected by background stars. On the subsequent nights, the stellar confusion of the field was too significant to achieve any detection. All the OGS observations were obtained with the ESA Space Debris Camera 2 (SDC2), equipped with a 4k imager used in 2 × 2 binning mode and sidereal tracking. This instrument configuration results in a \(47.5^{\prime}\) FoV, with 1.39″ binned pixels, optimally sampling the ~2.5″ to 3″ FWHM of the system's PSF for astrometric purposes.
Lowell discovery telescope (LDT)
Observations at the 4.3 m Lowell Discovery Telescope (G37) were obtained on 2021 February 22 during astronomical twilight when 2020 XL5 was at an airmass between 4.2 and 3.8. We used the Large Monolithic Imager (LMI)36,37. LMI is a e2v CCD231-SN-10382-14-0 of 6144 × 6160 15 μm pixels. On the LDT, LMI provides a field of view of \(12.3^{\prime} \times 12.3^{\prime}\) with a pixel scale of 0.12″/pixel when operated unbinned. For the 2020 XL5 observation we used a binning mode of 5 × 5 providing a pixel scale of 0.60″/pixel. We obtained 13 individual exposures of 20 s each. The telescope was tracking at a non-sidereal rate matching the motion of the asteroid of 3.44″/pixel providing star trails of 1.15″ that are much smaller than the ~3″ seeing at such high airmass. The observations were performed in the VR filter (bandpass from 0.480 ± 0.005 to 0.721 ± 0.005 μm). The resulting stacked image is shown in Fig. 5.
Fig. 5: Example of detections.
a A mosaic showing the Pan-STARRS pre-discovery observations of 2020 XL5. The orange circles highlight the position of the object. b Overall stack of the 13 frames obtained with the Lowell Discovery Telescope on 2021 February 22. The orange circle highlights the position of the object.
Southern astrophysical research (SOAR)
We used the 4.1 m Southern Astrophysical Research Telescope (I33) on the nights of 2021 March 9 (NOIRLab program 2018B-927, P.I., S. Zepf, Michigan State, University), 13 (NOIRLab Astronomical Event Observatory Network (AEON) 2021A queue), and 16 (Brazil DDT night, PI: L. Fraga). We used the Goodman optical imager which provides a field of view of \(7.2^{\prime}\) and a pixel scale of 0.15″/pix. Observations were very challenging due to the extremely low solar elongation of the object at that time (between 32° and 34° away from the Sun). As a result, the object was observable only during a few minutes before dawn and very close to the horizon (15° elevation), with an airmass between 3.7 and 3.3. We used the Goodman HTS imaging camera equipped with SDSS filters and 3 × 3 binning mode in order to reduce the readout time. We tracked non-sidereally and used short exposures (20 s) for each of the g′, r′, and i′ filters. The background was quickly saturating and only the first four frames of each band were useful for photometric analyses.
Additional observational attempts
We also attempted observations of 2020 XL5 using the 0.8 m Joan Oró (TJO) telescope (MPC code C65) and the 2.2 m Calar Alto (CAHA) telescope (MPC code 493) on 2021 February 11 and 2021 March 2 respectively. However, on both attempts, we could not confidently detect the object, even after stacking all the frames, and therefore we are not including these data in the analyses presented in this work.
Precovery observations
The new observational data gathered by our team and discussed above, combined with the astrometry available at the Minor Planet Center, cover the entire observable arc from the time of the earliest submitted precovery observation (2020 November 26, by code G96), to the disappearance of the object into solar conjunction (elongation <30°, at the end of 2021 March), which marked the end of the discovery apparition. Overall, the arc covers almost 111 days, sufficient for a rough analysis of the stability of the object, but not ideal for a long-term study of the behavior of the object18.
In order to increase the observed arc without the need to wait for an additional future apparition, we performed a thorough search for precovery detections in archival data. We began with the archive of the Catalina Sky Survey: the search revealed a promising field exposed with the 1.5 m Mt. Lemmon telescope (G96) on 2019 October 27, at a time when the object had an ephemeris uncertainty of just ±5″ (1σ confidence) based on the data from the 2020–2021 apparition. A careful analysis of the area of the images corresponding to the prediction revealed a possible faint candidate with a signal-to-noise ratio (SNR) of ~3.
Despite not being a convincing detection, we temporarily assumed its correctness and used it to extend our search further back in time. Another promising field was located, exposed by the same telescope on 2017 October 24. The uncertainty of the ephemeris at that time, assuming the correctness of the 2019 detection, was only ±2.3″. No source was visible in the single images, but another very faint source with SNR ~3 appeared in the stack of all four frames of that night. These two tentative detections, individually not sufficiently strong to claim a certain identification, when taken together provide reasonable evidence that they could indeed be real. Their orbit improvement potential was so great that they now made it possible to determine the object's ephemeris to better than ±5″ as far back as 2012, allowing for many additional precovery opportunities at multiple apparitions.
We, therefore, extended our search to two additional archives: the online repository of all images obtained by the wide-field DECam instrument with the 4.1m Víctor M. Blanco Telescope on Cerro Tololo, Chile, and the archive of the 1.8m Pan-STARRS survey. The DECam archive provided a solid detection compatible with the tentative G96 precoveries on 2014 November 4, while Pan-STARRS thoroughly confirmed the chain of precovery observations providing eleven additional measurements covering a time span from 2012 to 2015 (see Fig. 5).
Orbit determination
Clone orbits used in the numerical simulations were generated applying the Cholesky method for multivariate normal distributions38. The clone orbits are generated using Python 3.639, starting from the nominal orbit and its covariance matrix computed with the AstOD software. Alternatively, freely available software such as OrbFit40 can be used for the analysis.
The Cholesky method consists of the factorization of a Hermitian, positive-definite matrix, as the product of a lower triangular matrix and its transpose conjugate. In our case, the covariance matrix C is thus decomposed as C = LLT, where L is a lower triangular matrix. The Keplerian orbital elements ei of the i-th clone orbit are then defined as follow:
$${{{{{{{{\bf{e}}}}}}}}}_{i}={{{{{{{{\bf{e}}}}}}}}}_{{{{{{{{\bf{0}}}}}}}}}+L{{{{{{{{\bf{r}}}}}}}}}_{{{{{{{{\bf{i}}}}}}}}},$$
where e0 is the vector of the orbital elements of the nominal orbit, and ri is a 6-dim random vector, whose components are generated following a normal distribution with mean 0 and variance 1 (\({{{{{{{\mathcal{N}}}}}}}}(0,1)\)).
Since the uncertainties in the nominal orbital elements are very small differences between the initial conditions of the clone orbits and those of the nominal orbit are small as well.
Starting from these initial conditions, we have integrated the 800 clone orbits along a time span of ~29,000 years. Considering as initial time t0 = 0 the mean epoch of the observations, the forward propagation has been executed along 15,000 years, where this limit in time is defined by the JPL Planetary and Lunar Ephemerides DE43122 whilst the backward propagation has been executed for 14,000 years. As a second step, we studied the evolution of the relative mean longitude λr of 2020 XL5 with respect to the Earth. The relative mean longitude λr is defined as the difference between the mean longitude of the asteroid λa and the mean longitude of the Earth λE: when the resonant angle λr librates around 60°, i.e., 0° < λr < 180° the asteroid is called an L4 Trojan, when the resonant angle λr librates around 300°, i.e., 180° < λr < 360° the asteroid is called an L5 Trojan, when λr circulates then the asteroid leaves the Trojan-like orbit24.
We have reproduced the same calculations using the public software OrbFit40, obtaining consistent results for the deterministic part of the orbital evolution.
Validation of the orbital study method
In order to validate the methods adopted during the analysis of the orbital stability of 2020 XL5, we applied the same approach also to 2010 TK7, the other known ET. In Fig. 6 we report the evolution of its λ over the integration interval. In this case, the t0 = 0 is set as the mean epoch of the observations arc of 2010 TK7, which corresponds to 2012 August. The results obtained on the stability in L4 are fully consistent with the ones reported in the literature8,24, confirming the validity of our computational methods and approach.
Fig. 6: Mean longitude evolution analysis of 2010 TK7.
Evolution of λr for the nominal orbit of 2010 TK7 along 29,000 years. Source data are provided as a Source Data file.
In order to perform our photometric measurements, we optimized the object's SNR by creating different stacked images on the motion direction of 2020 XL5. In order to create the stacked images and to make the photometric measurements, we used the publicly available Tycho Tracker software41. For each night, we started selecting all the frames gathered for each filter to create a stacked image and measure the object's SNR. We then repeated the process after removing the last frame ordered chronologically and we iterated the process until we were left with only four images to stack. We then compared the SNR values obtained for the different stacks and selected the image with the target having the highest SNR. With this process, we managed to obtain the best possible photometry, despite the nearly saturated background due to the twilight proximity. We then used 3.5 pixel apertures to measure the flux of 2020 XL5. As comparison stars, we selected all the solar analog stars in the field and we used the values of the ATLAS catalog42 to compute the absolute photometry for the different bands used in our observations.
In order to select the stacked images suitable for photometry, we applied a threshold filter of SNR >5. From the follow-up observations gathered with SOAR, we could obtain three \(r^{\prime}\) measurements and two with \(i^{\prime}\) and \(g^{\prime}\) filters. Observations with LDT allowed us to extract a measurement in the V filter. We also checked the recovery data and managed to extract two \(r^{\prime}\) measurements from the Pan-STARRS data. Unfortunately, DECam's data had an SNR <3, which was not enough to extract any photometric measurement. The photometric measurements for the different bands and nights are presented in Table 3.
Table 3 Photometry.
Color indices
We estimated the reflectance values at the wavelengths of standard Sloan Digital Sky Survey (SDSS) filters following a procedure from the literature43. We first computed the solar-corrected44 colors and albedos, normalized at the r band, and their corresponding uncertainties.
Furthermore, we computed the a* parameter45 a* = 0.9285(g − r) + 0.3712(r − i) − 0.66204, which turns out to be a* = −0.9 ± 0.6 for 2020 XL5 on the first observing night, and is more consistent with C-complex objects44; on the other hand, we obtained a* = −0.4 ± 1.0 for the second night, compatible with all possible complexes44. Additionally, we plotted the g − r and r − i colors indices of 2020 XL5 together with objects from the SDSS MOV catalog release 3 (see Fig. 3). From the SDSS catalogs, we selected only those objects which fulfill the quality criteria as found in the literature46. The estimated colors lie outside the space occupied by most other asteroids from the SSDS but are offset towards the C-complex objects. The uncertainties of colors estimated from the second night are compatible with all possible taxonomic complexes.
Phase curve
We used the magnitudes in the r filter to construct a phase curve for 2020 XL5 (see Fig. 7). We first normalized the magnitudes to unit distances to the Earth and the Sun according to \(r(\alpha )=r(R,{{\Delta }},\alpha )-5{{{{{{\mathrm{log}}}}}}}\,(R{{\Delta }})\), where α is the phase angle, R is the heliocentric distance, and Δ the geocentric distance (both in au). We then applied the HG12* model47 using the online version of the tool48. Because of the large phase angles and the low coverage of the phase curve, the best fit we obtained assumes a single free parameter, the absolute magnitude, while G12* is fixed, according to different taxonomical types. In our case, the best fit was obtained for the P and C types. The error in the magnitudes is estimated using Monte Carlo methods and the lower (upper) uncertainty corresponds to the 16th (84th) percentile. Note that the last point in the phase curve, at about α = 80°, was obtained from the measured V magnitude using the transformations from Sloan's ugriz to UBVRI49.
Fig. 7: Photometric phase curve.
Phase curve in the r filter of 2020 XL5 using the six observing nights. The dashed line indicates the adopted solution, while the dot-dashed line indicates the uncertainty interval of the absolute magnitude. Source data are provided as a Source Data file.
Delta-v budget: patched-conics approach
Despite the proximity of the ETs orbits, they are still deep-space targets, and thus, any practical mission to this kind of orbits is likely to make use of gravity-assists and/or low-thrust solar electric propulsion (SEP). However, for the purpose of this paper, we limited our search to sub-optimal delta-v trajectories using a patched-conics approach with ballistic trajectory.
For the heliocentric segment of the transfer trajectory, we solved Lambert's boundary value problem (BVP) in order to determine the Keplerian orbit that connects the spacecraft and the target asteroid in space in a given elapsed time. Lambert's BVP solver is very computationally fast, which allows us to do a quick survey of ballistic transfers from Earth to 2020 XL5 for different departure dates and time-of-flights (TOF). For this paper, we used the multi-revolutions Lambert's problem solver50 implemented within the pykep software31 by the mission analysis team at ESA/ESOC.
For the geocentric portion of the trajectory, the escape orbit is calculated starting from both a low Earth orbit (LEO) at approximately 300 km, and a geostationary transfer orbit (GTO) with a perigee altitude of also 300 km and the apogee at GEO altitude. In general, similar research works only deal with a simple launch from LEO; however, it is very expensive to go from LEO directly to escape with only chemical propulsion. For this reason, we also considered the GTO orbit, which is more common in actual space missions (especially if we assume a shared launch) and reduces significantly the cost of getting out of the Earth's sphere of influence (SOI).
Estimation of the total delta-v
The total delta-v for this mission is estimated as the sum of the delta-v required to go from launch conditions to Earth's escape orbit, dv1, and the delta-v required to do the asteroid-rendezvous, dv2, which is zero for a fly-by mission. For a given transfer trajectory by Lambert's problem, we can compute dv1 as the excess velocity required at the perigee of the starting orbit in order to enter an Earth's escape trajectory with a velocity at infinity (vinf) equal to the relative velocity difference between the Earth and the calculated starting velocity of the transfer trajectory at the launch date. Similarly, dv2 can be computed as the velocity difference between the target asteroid and the calculated ending velocity of the transfer trajectory at the arrival date.
For each combination of departure date and TOF, we can compute a single set of dv1 and dv2. Therefore, in order to find the optimal transfer trajectory, we have performed a scan in TOF, between 180 days and 3 years and a time step of 1 day, and we selected the minimum total delta-v (dv1 + dv2) as the optimal solution for the departure date considered. We have repeated this process for the next departure date, which is about 0.5 days later. The launch window considered goes from 2021 January 1 until 2026 January 1. This time window is enough for the purposes of this study, since the results are completely periodic, and thus, similar results can be expected for other departure dates. Additionally, we have validated our results using the JPL Small-Body Mission-Design online Tool51, which provided very similar results.
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request. DECam data are available in the NOIRLab Archive Astro Data Archive (https://astroarchive.noirlab.edu). Pan-STARRS data are available upon request to R. Wainscoat ([email protected]). Catalina Sky Survey data are available upon request to E. Christensen ([email protected]). Lowell Discovery Telescope data and SOAR data are available upon request to the corresponding author. Astrometric measurements are available on the MPC site (http://www.minorplanetcenter.net/db_search/show_object?object_id=2020+XL5). Source data are provided with this paper.
The ESA NEO Coordination Centre's orbit determination and impact monitoring AstOD software used for the study of the orbit stability is proprietary (https://neo.ssa.esa.int/about-neocc). The Tycho Tracker software was used for data processing and photometric analysis (http://www.tycho-tracker.com). The pykep scientific library was used for the delta-v budget calculations (http://esa.github.io/pykep).
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We thank Federica Spoto for her assistance on the independent check of the orbit calculations done with the OrbFit software. The work of TS-R was carried out through grant APOSTD/2019/046 by Generalitat Valenciana (Spain). This work was (partially) funded by the Spanish MICIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe" by the "European Union" through grant RTI2018-095076-B-C21, and the Institute of Cosmos Sciences University of Barcelona (ICCUB, Unidad de Excelencia 'María de Maeztu') through grant CEX2019-000918-M. P-YL acknowledges NEO-MAPP project under H2020-SPACE-2019 GA 870377. PGB, and ACB acknowledge funding from the Spanish MICINN project RTI2018-099464-B-I00. AA-C acknowledges support from the State Agency for Research of the Spanish MCIU through the "Center of Excellence Severo Ochoa" award to the Instituto de Astrofísica de Andalucía (SEV-2017-0709). DO was supported by National Science Centre, Poland, grants numbers 2017/26/D/ST9/00240 and 2017/25/B/ST9/00740. Based on observations obtained at the Southern Astrophysical Research (SOAR) telescope, which is a joint project of the Ministério da Ciência, Tecnologia e Inovações (MCTI/LNA) do Brasil, the US National Science Foundation's NOIRLab, the University of North Carolina at Chapel Hill (UNC), and Michigan State University (MSU). The Catalina Sky Survey is funded by NASA's Planetary Defense Coordination Office, under grant 80NSSC18K1130. The Joan Oró Telescope (TJO) of the Montsec Observatory (OdM) is owned by the Catalan Government and operated by the Institute for Space Studies of Catalonia (IEEC). MD acknowledges funding from NASA NEOO grant 80NSSC21K0045 in support of the Second Lunation NEO Follow-up. These results made use of the Lowell Discovery Telescope (LDT) at Lowell Observatory. Lowell is a private, non-profit institution dedicated to astrophysical research and public appreciation of astronomy and operates the LDT in partnership with Boston University, the University of Maryland, the University of Toledo, Northern Arizona University, and Yale University. The Large Monolithic Imager (LMI) was built by Lowell Observatory using funds provided by the National Science Foundation (AST-1005313). Operation of the Pan-STARRS telescope is supported by the National Aeronautics and Space Administration under Grant No. 80NSSC18K0971 issued through the SSO Near-Earth Object Observations Program. This project used public archival data from the Dark Energy Survey (DES) as distributed by the Astro Data Archive at NSF's NOIRLab. Funding for the DES Projects has been provided by the DOE and NSF (USA), MISE (Spain), STFC (UK), HEFCE (UK), NCSA (UIUC), KICP (U. Chicago), CCAPP (Ohio State), MIFPA (Texas A&M), CNPQ, FAPERJ, FINEP (Brazil), MINECO (Spain), DFG (Germany), and the collaborating institutions in the Dark Energy Survey, which are Argonne Lab, UC Santa Cruz, University of Cambridge, CIEMAT-Madrid, University of Chicago, University College London, DESBrazil Consortium, University of Edinburgh, ETH Zürich, Fermilab, University of Illinois, ICE (IEECCSIC), IFAE Barcelona, Lawrence Berkeley Lab, LMU München and the associated Excellence Cluster Universe, University of Michigan, NOIRLab, University of Nottingham, Ohio State University, OzDES Membership Consortium, University of Pennsylvania, University of Portsmouth, SLAC National Lab, Stanford University, University of Sussex, and Texas A&M University. Based on observations at Cerro Tololo Inter-American Observatory, a program of NOIRLab (NOIRLab Prop. ID 2012B-0001; PI: J. Frieman), which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. This research has made use of data and/or services provided by the International Astronomical Union's Minor Planet Center.
Departamento de Fisica, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Carr. de San Vicente del Raspeig, s/n, 03690 San Vicente del Raspeig, Alicante, Spain
T. Santana-Ros, P. G. Benavidez & A. Campo Bagatin
Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Carrer de Martí i Franquès, 1, 08028, Barcelona, Spain
T. Santana-Ros
ESA NEO Coordination Centre, Largo Galileo Galilei, 1, 00044, Frascati, Italy
M. Micheli, L. Faggioli, R. Cennamo & L. Conversi
Arecibo Observatory, University of Central Florida, HC3 Box 53995, Arecibo, PR, 00612, USA
M. Devogèle
Instituto de Astrofísica de Andalucía, CSIC, Apartado 3004, 18080, Granada, Spain
A. Alvarez-Candal
Instituto de Física Aplicada a las Ciencias y las Tecnologías, Universidad de Alicante, San Vicente del Raspeig, 03080, Alicante, Spain
A. Alvarez-Candal, P.-Y. Liu, P. G. Benavidez & A. Campo Bagatin
Observatório Nacional / MCTIC, R. Gen. José Cristino, 77, Rio de Janeiro, 20921-400, Brazil
Astronomical Observatory Institute, Faculty of Physics, A. Mickiewicz University, Słoneczna 36, 60-286, Poznań, Poland
D. Oszkiewicz
Solenix Deutschland GmbH, Spreestraße 3, 64295, Darmstadt, Germany
O. Ramírez
The University of Arizona, Lunar and Planetary Laboratory, 1629 E University Blvd, Tucson, AZ, 85721, USA
E. J. Christensen
Institute for Astronomy, University of Hawaii, 2680 Woodlawn Dr, Honolulu, HI, 96822, USA
R. J. Wainscoat
Department of Physics and Astronomy, The University of Western Ontario, 1151 Richmond St, London, ON, N6A 3K7, Canada
R. Weryk
Laboratório Nacional de Astrofísica LNA/MCTIC, R. dos Estados Unidos, 154, Itajubá, 37504-364, Brazil
L. Fraga
Cerro Tololo Inter-American Observatory/NSF's NOIRLab, Casilla 603, La Serena, Chile
C. Briceño
ESA ESRIN, Largo Galileo Galilei, 1, 00044, Frascati, Italy
L. Conversi
M. Micheli
L. Faggioli
R. Cennamo
P.-Y. Liu
P. G. Benavidez
A. Campo Bagatin
T.S.-R. coordinated the observation campaign, contributed to the writing of the proposals, performed the photometric measurements, and participated in all aspects of the discussion and to the paper writing; M.M. contributed to the writing of the proposals, performed the astrometric measurements, and participated in all aspects of the discussion and to the paper writing; L.Fa. and R.C. performed the orbit stability analysis, participated to the discussion of the paper results, and contributed to the paper writing; M.D. contributed to the writing of the proposals, performed observations, and contributed to the paper writing; A.A.-C. contributed to the writing of the proposals, performed the phase curve analysis, and contributed to the paper writing; D.O. perform the color indices analysis and contributed to the paper writing; O.R. and P.-Y.L. performed the delta-v budget analysis and contributed to the paper writing; P.G.B. and A.C.B. participated to the discussion. C.B. and L.Fr. carried out observations at SOAR. R.W. measured the PS data and participated in the discussion. E.J.C., R.J.W., and L.C. provided access to data archives and participated in the discussion. All authors reviewed the manuscript.
Correspondence to T. Santana-Ros.
Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work.
Santana-Ros, T., Micheli, M., Faggioli, L. et al. Orbital stability analysis and photometric characterization of the second Earth Trojan asteroid 2020 XL5. Nat Commun 13, 447 (2022). https://doi.org/10.1038/s41467-022-27988-4
Astronomy and planetary science
A new entry for the elusive Earth Trojans
Morgan Hollis
Nature Astronomy Research Highlight 11 Feb 2022 | CommonCrawl |
Germ (mathematics)
In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind that captures their shared local properties. In particular, the objects in question are mostly functions (or maps) and subsets. In specific implementations of this idea, the functions or subsets in question will have some property, such as being analytic or smooth, but in general this is not needed (the functions in question need not even be continuous); it is however necessary that the space on/in which the object is defined is a topological space, in order that the word local has some meaning.
Name
The name is derived from cereal germ in a continuation of the sheaf metaphor, as a germ is (locally) the "heart" of a function, as it is for a grain.
Formal definition
Basic definition
Given a point x of a topological space X, and two maps $f,g:X\to Y$ (where Y is any set), then $f$ and $g$ define the same germ at x if there is a neighbourhood U of x such that restricted to U, f and g are equal; meaning that $f(u)=g(u)$ for all u in U.
Similarly, if S and T are any two subsets of X, then they define the same germ at x if there is again a neighbourhood U of x such that
$S\cap U=T\cap U.$
It is straightforward to see that defining the same germ at x is an equivalence relation (be it on maps or sets), and the equivalence classes are called germs (map-germs, or set-germs accordingly). The equivalence relation is usually written
$f\sim _{x}g\quad {\text{or}}\quad S\sim _{x}T.$
Given a map f on X, then its germ at x is usually denoted [f ]x. Similarly, the germ at x of a set S is written [S]x. Thus,
$[f]_{x}=\{g:X\to Y\mid g\sim _{x}f\}.$
A map germ at x in X that maps the point x in X to the point y in Y is denoted as
$f:(X,x)\to (Y,y).$
When using this notation, f is then intended as an entire equivalence class of maps, using the same letter f for any representative map.
Notice that two sets are germ-equivalent at x if and only if their characteristic functions are germ-equivalent at x:
$S\sim _{x}T\Longleftrightarrow \mathbf {1} _{S}\sim _{x}\mathbf {1} _{T}.$
More generally
Maps need not be defined on all of X, and in particular they don't need to have the same domain. However, if f has domain S and g has domain T, both subsets of X, then f and g are germ equivalent at x in X if first S and T are germ equivalent at x, say $S\cap U=T\cap U\neq \emptyset ,$ and then moreover $f|_{S\cap V}=g|_{T\cap V}$, for some smaller neighbourhood V with $x\in V\subseteq U$. This is particularly relevant in two settings:
1. f is defined on a subvariety V of X, and
2. f has a pole of some sort at x, so is not even defined at x, as for example a rational function, which would be defined off a subvariety.
Basic properties
If f and g are germ equivalent at x, then they share all local properties, such as continuity, differentiability etc., so it makes sense to talk about a differentiable or analytic germ, etc. Similarly for subsets: if one representative of a germ is an analytic set then so are all representatives, at least on some neighbourhood of x.
Algebraic structures on the target Y are inherited by the set of germs with values in Y. For instance, if the target Y is a group, then it makes sense to multiply germs: to define [f]x[g]x, first take representatives f and g, defined on neighbourhoods U and V respectively, and define [f]x[g]x to be the germ at x of the pointwise product map fg (which is defined on $U\cap V$). In the same way, if Y is an abelian group, vector space, or ring, then so is the set of germs.
The set of germs at x of maps from X to Y does not have a useful topology, except for the discrete one. It therefore makes little or no sense to talk of a convergent sequence of germs. However, if X and Y are manifolds, then the spaces of jets $J_{x}^{k}(X,Y)$ (finite order Taylor series at x of map(-germs)) do have topologies as they can be identified with finite-dimensional vector spaces.
Relation with sheaves
The idea of germs is behind the definition of sheaves and presheaves. A presheaf ${\mathcal {F}}$ of abelian groups on a topological space X assigns an abelian group ${\mathcal {F}}(U)$ to each open set U in X. Typical examples of abelian groups here are: real valued functions on U, differential forms on U, vector fields on U, holomorphic functions on U (when X is a complex space), constant functions on U and differential operators on U.
If $V\subseteq U$ then there is a restriction map $\mathrm {res} _{VU}:{\mathcal {F}}(U)\to {\mathcal {F}}(V),$ satisfying certain compatibility conditions. For a fixed x, one says that elements $f\in {\mathcal {F}}(U)$ and $g\in {\mathcal {F}}(V)$ are equivalent at x if there is a neighbourhood $W\subseteq U\cap V$ of x with resWU(f) = resWV(g) (both elements of ${\mathcal {F}}(W)$). The equivalence classes form the stalk ${\mathcal {F}}_{x}$ at x of the presheaf ${\mathcal {F}}$. This equivalence relation is an abstraction of the germ equivalence described above.
Interpreting germs through sheaves also gives a general explanation for the presence of algebraic structures on sets of germs. The reason is that formation of stalks preserves finite limits. This implies that if T is a Lawvere theory and a sheaf F is a T-algebra, then any stalk Fx is also a T-algebra.
Examples
If $X$ and $Y$ have additional structure, it is possible to define subsets of the set of all maps from X to Y or more generally sub-presheaves of a given presheaf ${\mathcal {F}}$ and corresponding germs: some notable examples follow.
• If $X,Y$ are both topological spaces, the subset
$C^{0}(X,Y)\subseteq {\mbox{Hom}}(X,Y)$
of continuous functions defines germs of continuous functions.
• If both $X$ and $Y$ admit a differentiable structure, the subset
$C^{k}(X,Y)\subseteq {\mbox{Hom}}(X,Y)$
of $k$-times continuously differentiable functions, the subset
$C^{\infty }(X,Y)=\bigcap \nolimits _{k}C^{k}(X,Y)\subseteq {\mbox{Hom}}(X,Y)$
of smooth functions and the subset
$C^{\omega }(X,Y)\subseteq {\mbox{Hom}}(X,Y)$
of analytic functions can be defined ($\omega $ here is the ordinal for infinity; this is an abuse of notation, by analogy with $C^{k}$ and $C^{\infty }$), and then spaces of germs of (finitely) differentiable, smooth, analytic functions can be constructed.
• If $X,Y$ have a complex structure (for instance, are subsets of complex vector spaces), holomorphic functions between them can be defined, and therefore spaces of germs of holomorphic functions can be constructed.
• If $X,Y$ have an algebraic structure, then regular (and rational) functions between them can be defined, and germs of regular functions (and likewise rational) can be defined.
• The germ of f : ℝ → Y at positive infinity (or simply the germ of f) is $\{g:\exists x\forall y>x\,f(y)=g(y)\}$. These germs are used in asymptotic analysis and Hardy fields.
Notation
The stalk of a sheaf ${\mathcal {F}}$ on a topological space $X$ at a point $x$ of $X$ is commonly denoted by ${\mathcal {F}}_{x}.$ As a consequence, germs, constituting stalks of sheaves of various kind of functions, borrow this scheme of notation:
• ${\mathcal {C}}_{x}^{0}$ is the space of germs of continuous functions at $x$.
• ${\mathcal {C}}_{x}^{k}$ for each natural number $k$ is the space of germs of $k$-times-differentiable functions at $x$.
• ${\mathcal {C}}_{x}^{\infty }$ is the space of germs of infinitely differentiable ("smooth") functions at $x$.
• ${\mathcal {C}}_{x}^{\omega }$ is the space of germs of analytic functions at $x$.
• ${\mathcal {O}}_{x}$ is the space of germs of holomorphic functions (in complex geometry), or space of germs of regular functions (in algebraic geometry) at $x$.
For germs of sets and varieties, the notation is not so well established: some notations found in literature include:
• ${\mathfrak {V}}_{x}$ is the space of germs of analytic varieties at $x$. When the point $x$ is fixed and known (e.g. when $X$ is a topological vector space and $x=0$), it can be dropped in each of the above symbols: also, when $\dim X=n$, a subscript before the symbol can be added. As example
• ${_{n}{\mathcal {C}}^{0}},{_{n}{\mathcal {C}}^{k}},{_{n}{\mathcal {C}}^{\infty }},{_{n}{\mathcal {C}}^{\omega }},{_{n}{\mathcal {O}}},{_{n}{\mathfrak {V}}}$ are the spaces of germs shown above when $X$ is a $n$-dimensional vector space and $x=0$.
Applications
The key word in the applications of germs is locality: all local properties of a function at a point can be studied by analyzing its germ. They are a generalization of Taylor series, and indeed the Taylor series of a germ (of a differentiable function) is defined: you only need local information to compute derivatives.
Germs are useful in determining the properties of dynamical systems near chosen points of their phase space: they are one of the main tools in singularity theory and catastrophe theory.
When the topological spaces considered are Riemann surfaces or more generally complex-analytic varieties, germs of holomorphic functions on them can be viewed as power series, and thus the set of germs can be considered to be the analytic continuation of an analytic function.
Germs can also be used in the definition of tangent vectors in differential geometry. A tangent vector can be viewed as a point-derivation on the algebra of germs at that point.[1]
Algebraic properties
As noted earlier, sets of germs may have algebraic structures such as being rings. In many situations, rings of germs are not arbitrary rings but instead have quite specific properties.
Suppose that X is a space of some sort. It is often the case that, at each x ∈ X, the ring of germs of functions at x is a local ring. This is the case, for example, for continuous functions on a topological space; for k-times differentiable, smooth, or analytic functions on a real manifold (when such functions are defined); for holomorphic functions on a complex manifold; and for regular functions on an algebraic variety. The property that rings of germs are local rings is axiomatized by the theory of locally ringed spaces.
The types of local rings that arise, however, depend closely on the theory under consideration. The Weierstrass preparation theorem implies that rings of germs of holomorphic functions are Noetherian rings. It can also be shown that these are regular rings. On the other hand, let ${\mathcal {C}}_{0}^{\infty }(\mathbf {R} )$ be the ring of germs at the origin of smooth functions on R. This ring is local but not Noetherian. To see why, observe that the maximal ideal m of this ring consists of all germs that vanish at the origin, and the power mk consists of those germs whose first k − 1 derivatives vanish. If this ring were Noetherian, then the Krull intersection theorem would imply that a smooth function whose Taylor series vanished would be the zero function. But this is false, as can be seen by considering
$f(x)={\begin{cases}e^{-1/x^{2}},&x\neq 0,\\0,&x=0.\end{cases}}$
This ring is also not a unique factorization domain. This is because all UFDs satisfy the ascending chain condition on principal ideals, but there is an infinite ascending chain of principal ideals
$\cdots \subsetneq (x^{-j+1}f(x))\subsetneq (x^{-j}f(x))\subsetneq (x^{-j-1}f(x))\subsetneq \cdots .$
The inclusions are strict because x is in the maximal ideal m.
The ring ${\mathcal {C}}_{0}^{0}(\mathbf {R} )$ of germs at the origin of continuous functions on R even has the property that its maximal ideal m satisfies m2 = m. Any germ f ∈ m can be written as
$f=|f|^{1/2}\cdot {\big (}\operatorname {sgn} (f)|f|^{1/2}{\big )},$
where sgn is the sign function. Since |f| vanishes at the origin, this expresses f as the product of two functions in m, whence the conclusion. This is related to the setup of almost ring theory.
See also
• Analytic variety
• Catastrophe theory
• Gluing axiom
• Riemann surface
• Sheaf
• Stalk
References
1. Tu, L. W. (2007). An introduction to manifolds. New York: Springer. p. 11.
• Nicolas Bourbaki (1989). General Topology. Chapters 1-4 (paperback ed.). Springer-Verlag. ISBN 3-540-64241-2., chapter I, paragraph 6, subparagraph 10 "Germs at a point".
• Raghavan Narasimhan (1973). Analysis on Real and Complex Manifolds (2nd ed.). North-Holland Elsevier. ISBN 0-7204-2501-8., chapter 2, paragraph 2.1, "Basic Definitions".
• Robert C. Gunning and Hugo Rossi (1965). Analytic Functions of Several Complex Variables. Prentice-Hall., chapter 2 "Local Rings of Holomorphic Functions", especially paragraph A "The Elementary Properties of the Local Rings" and paragraph E "Germs of Varieties".
• Ian R. Porteous (2001) Geometric Differentiation, page 71, Cambridge University Press ISBN 0-521-00264-8 .
• Giuseppe Tallini (1973). Varietà differenziabili e coomologia di De Rham (Differentiable manifolds and De Rham cohomology). Edizioni Cremonese. ISBN 88-7083-413-1., paragraph 31, "Germi di funzioni differenziabili in un punto $P$ di $V_{n}$ (Germs of differentiable functions at a point $P$ of $V_{n}$)" (in Italian).
External links
• Chirka, Evgeniǐ Mikhaǐlovich (2001) [1994], "Germ", Encyclopedia of Mathematics, EMS Press
• Germ of smooth functions at PlanetMath.
• Mozyrska, Dorota; Bartosiewicz, Zbigniew (2006). "Systems of germs and theorems of zeros in infinite-dimensional spaces". arXiv:math/0612355. Bibcode:2006math.....12355M. {{cite journal}}: Cite journal requires |journal= (help) A research preprint dealing with germs of analytic varieties in an infinite dimensional setting.
| Wikipedia |
\begin{document}
\title[Central Extension]{Yet another construction of the central extension of the loop group. } \author{Michael K. Murray} \address[Michael K. Murray] {Department of Pure Mathematics\\ University of Adelaide\\ Adelaide, SA 5005 \\ Australia} \email[Michael K. Murray]{[email protected]} \thanks{The first author acknowledges the support of the Australian Research Council.} \author{Daniel Stevenson} \email[Daniel Stevenson]{[email protected]} \thanks{The second author acknowledges the support of the Australian Research Council.}
\subjclass{}
\begin{abstract} We give a characterisation of central extensions of a Lie group $G$ by $\C^\times$ in terms of a differential two-form on $G$ and a differential one-form on $G \times G$. This is applied to the case of the central extension of the loop group. \end{abstract} \maketitle \section{Introduction} Let $G$ and $A$ be groups. A {\em central extension} of $G$ by $A$ is another group $\hat G$ and a homomorphism $\pi\colon\hat G \to G$ whose kernel is isomorphic to $A$ and in the center of $\hat G$. Note that because $A$ is in the center of $\hat G$ it is necessarily abelian. We will be interested ultimately in the case that $G = \Omega(K)$ the {\em loop group} of all smooth maps from the circle $S^1$ to a Lie group $K$ with pointwise multiplication but the theory developed applies to any Lie group $G$.
\section{Central extension of groups} Consider first the case when $G$ is just a group and ignore questions of continuity or differentiability. In this case we can choose a {\em section} of the map $\pi$. That is a map $s \colon G \to \hat G$ such that $\pi(s(g)) = g $ for every $g \in G$. From this section we can construct a bijection $$ \phi \colon A \times G \to \hat G $$ by $\phi(g, a) = \iota(a)s(g)$ where $\iota \colon A \to \hat G$ is the identification of $A$ with the kernel of $\pi$. So we know that as a set $\hat G$ is just the product $A \times G$. However as a group $\hat G$ is not generally a product. To see what it is note that $\pi(s(g)s(h)) =\pi(s(g))\pi(s(h)) = gh = \pi(s(gh))$ so that $s(g)s(h) = c(g, h) s(gh)$ where $c \colon G \times G \to A$. The bijection $\phi \colon A \times G \to \hat G$ induces a product on $A \times G$ for which $\phi$ is a homomorphism. To calculate this product we note that
\begin{align*}
\phi(a, g) \phi(b, h) &= \iota(a)s(g)\iota(b)s(h) \\
&= \iota(ab)s(g)s(h)\\
&= \iota(ab)c(gh)s(gh).
\end{align*} Hence the product on $A \times G$ is given by $(a, g) \star (b, h) = (abc(g,h) gh)$ and the map $\phi$ is a group isomorphism between $\hat G$ and $A \times G$ with this product.
Notice that if we choose a different section $\tilde s$ then $\tilde s = sh$ were $h \colon G \to A$.
It is straightforward to check that if we pick any $c \colon G \times G \to A$ and define a product on $A \times G$ by $(a, g) \star (b, h) = (abc(g,h) gh)$ then this is an associative product if and only if $c$ satisfies the {\em cocycle condition} $$ c(g,h)c(gh,k) = c(g, hk)c(h,k) $$ for all $g$, $h$ and $k$ in $G$.
If we choose a different section $\tilde s$ then we must have $\tilde s = s d$ for some $d \colon G \to A$. If $\tilde c$ is the cocycle determined by $\tilde s$ then a calculation shows that \begin{equation}
\label{eq:equiv_ext}
c(g, h) = \tilde c(g, h) d(gh) d(g)^{-1}d(h)^{-1}. \end{equation}
We have now essentially shown that all central extensions of $G$ by $A$ are determined by cocycles $c$ modulo identifying two that satisfy the condition \eqref{eq:equiv_ext}. Let us recast this result in a form that we will see again in this talk.
Define maps $d_i \colon G^{p+1} \to G^{p}$ by \begin{equation}
\label{eq:barcomplex} d_{i}(g_{1},\ldots,g_{p+1}) = \begin{cases}
(g_{2},\ldots,g_{p+1}), & i = 0, \\
(g_{1},\ldots,g_{i-1}g_{i},g_{i+1},
\ldots,g_{p+1}), & 1\leq i\leq p-1, \\
(g_{1},\ldots,g_{p}), & i = p.
\end{cases} \end{equation} If $M^p(G;A) = \text{Map}(G^p, A)$ then we define $\delta \colon M^p(G; A)\to M^{p-1}(G; A) $ by $\delta(c) = (c\circ d_1) (c \circ d_2)^{-1} (c\circ d_3) \dots$. This satisfies $\delta^2 = 1$ and defines a complex $$ M^0(G;A) \stackrel{\delta}{\to} M^1(G;A)\stackrel{\delta}{\to} M^2(G;A) \stackrel{\delta}{\to} \dots $$ The cocycle condition can be rewritten as $\delta(c) = 1$ and the condition that two cocycles give rise to the same central extension is that $c =\tilde c \delta(d)$. If we define $$ H^p(G; A) =\frac{\text{kernel\ } \delta \colon M^p(G;A) \to M^{p-1}(G;A)} {\text{image\ } \delta \colon M^{p+1}(G;A) \to M^{p}(G;A)} $$ then we have shown that central extensions of $G$ by $A$ are classified by $H^2(G; A)$.
\section{Central extensions of Lie groups} In the case that $G$ is a topological or Lie group it is well-known that there are interesting central extensions for which no continuous or differentiable section exists. For example consider the central extension $$ \mathbb Z_2 \to SU(2) = \text{Spin}(3) \to SO(3) $$ of the three dimensional orthogonal group $SO(3)$ by its double cover $\text{Spin}(3)$. Here $SU(2)$ is known to be the three sphere but if a section existed then we would have $SU(2)$ homeomorphic to $\mathbb Z_2 \times SO(3)$ and hence disconnected.
From now on we will concentrate on the case when $A = \mathbb C^\times$. Then $\hat G \to G$ can be thought of as a $\mathbb C^\times $ principal bundle and a section will only exist if this bundle is trivial. The structure of the central extension as a $\mathbb C^\times$ bundle is important in what follows so we digress to discuss them in more detail.
\subsection{$\C^\times$ bundles}
Let $P \to X$ be a $\C^\times$ bundle over a manifold $X$. We denote the fibre of $P$ over $x \in X$ by $P_x$. Recall \cite{Bry} that if $P$ is a
$\C^\times$ bundle over a manifold $X$ we can define the dual bundle $P^*$ as the same space $P$ but with the action $p^* g = (pg^{-1})^*$ and, that if $Q$ is another such bundle, we can define the product bundle $P\otimes Q$ by $(P\otimes Q)_x = (P_x \times Q_x)/\C^\times $ where $\C^\times$ acts by $(p,q)w = (pw, qw^{-1})$. We denote an element of $P\otimes Q$ by $p \otimes q$ with the understanding that $(pw) \otimes q = p \otimes (qw) = (p\otimes q)w$ for $w \in \C^\times$. It is straightforward to check that $P\otimes P^*$ is canonically trivialised by the section $x \mapsto p \otimes p^*$ where $p$ is any point in $P_x.$
If $P$ and $Q$ are $\C^\times$ bundles on $X$ with connections $\mu_P$ and $\mu_Q$ then $P \otimes Q$ has an induced connection we denote by $\mu_P \otimes \mu_Q$. The curvature of this connection is $R_P + R_Q$ where $R_P$ and $R_Q$ are the curvatures of $\mu_P$ and $\mu_Q$ respectively. The bundle $P^*$ has an induced connection whose curvature
is $-R_P$.
Recall the maps $d_i \colon G^p \to G^{p-1}$ defined by \eqref{eq:barcomplex}. If $P \to G^p$ is a $\C^\times$ bundle then we can define a $\C^\times$ bundle over $G^{p+1}$ denoted $\delta(P)$ by $$ \delta(P) = \pi_1^{-1}(P) \otimes \pi_2^{-1}(P)^* \otimes \pi_3^{-1}(P) \otimes \dots. $$ If $s$ is a section of $P$ then it defines $\delta(s)$ a section of $\delta(P)$ and if $\mu$ is a connection on $P$ with curvature $R$ it defines a connection on $\delta(P)$ which we denote by $\delta(\mu)$. To define the curvature of $\delta(\mu)$ let us denote by $\Omega^q(G^p)$ the space of all differentiable $q$ forms on $G^p$. Then we define a map \begin{equation}
\label{delta_forms} \delta \colon \Omega^q(G^p) \to \Omega^q(G^{p+1}) \end{equation} by $\delta = \sum_{i=0}^p d_i^*$, the alternating sum of pull-backs by the various maps $d_i \colon G^{p+1} \to G^p$. Then the curvature of $\delta(\mu)$ is $\delta(R)$. If we consider $\delta(\delta(P))$ it is a product of factors and every factor occurs with its dual so $\delta(\delta(P))$ is canonically trivial. If $s $ is a section of $P$ then under this identification $\delta\delta(s) = 1$ and moreover if $\mu$ is a connection on $P$ then $\delta\delta(\mu) $ is the flat connection on $\delta\delta(P)$ with respect to $\delta(\delta(s))$.
\section{Central extensions} Let $ G$ be a Lie group and consider a central extension $$ \C^\times\to \hat{ G}\stackrel{\pi}{\to} G. $$ Following Brylinski and McLaughlin \cite{BryMac} we think of this as a $\C^\times$ bundle $\hat G \to G$ with a product $M \colon \hat G \times \hat G \to \hat G$ covering the product $m = d_1 \colon G \times G \to G$.
Because this is a central extension we must have that $M(pz, qw) = M(p,q)zw$ for any $p, q \in P$ and $z, w \in \C^\times$. This means we have a section $s$ of $\delta(P)$ given by $$ s(g, h) = p \otimes M(p, q) \otimes q $$ for any $p \in P_g$ and $q \in P_h$. This is well-defined as $pw \otimes M(pw, qz) \otimes qz = pw \otimes M(p, q)(wz)^{-1} \otimes qz = p \otimes M(p, q) \otimes q$. Conversely any such section gives rise to an $M$.
Of course we need an associative product and it can be shown that $M$ being associative is equivalent to $\delta(s) = 1$. To actually make $\hat G$ into a group we need more than multiplication we need an identity $\hat e \in \hat G$ and an inverse map. It is straightforward to check that if $e \in G$ is the identity then, because $M \colon \hat G_e \times \hat G_e \to \hat G_e$, there is a unique $\hat e \in \hat G_e$ such that $M(\hat e, \hat e) = \hat e$. It is also straightforward to deduce the existence of a unique inverse.
Hence we have the result from \cite{BryMac} that a central extension of $ G$ is a $\C^\times$ bundle $P \to G$ together with a section $s $ of $\delta(P) \to G\times G$ such that $\delta(s) = 1$. In \cite{BryMac} this is phrased in terms of simplicial line bundles. Note that this is a kind of cohomology result analogous to that in the first section. We have an object (in this case a $\C^\times$ bundle) and $\delta$ of the object is `zero' i.e. trivial as a $\C^\times$ bundle.
For our purposes we need to phrase this result in terms of differential forms. We call a connection for $\hat G \to G$, thought of as a $\C^\times$ bundle, a connection for the central extension. An isomorphism of central extensions with connection is an isomorphism of bundles with connection which is a group isomorphism on the total space $\hat G$. Denote by $C( G)$ the set of all isomorphism classes of central extensions of $ G$ with connection.
Let $\mu \in \Omega^1(\hat G)$ be a connection on the bundle $\hat{ G} \to G$ and consider the tensor product connection $\delta(\mu)$. Let
$\alpha = s^*(\delta(\mu))$. We then have that \begin{align*}
\delta(\alpha) &= (\delta(s)^*) (\delta(\mu)) \\ &= (1)^*(\delta^2(\mu))\\ &= 0 \end{align*} as $\delta^2(\mu) $ is the flat connection on $\delta^2(P)$. Also $d\alpha = s^*(d\delta(\mu)) = \delta(R)$.
Let $\Gamma( G)$ denote the set of all pairs $(\alpha, R)$ where $R$ is a closed, $2\pi i $ integral, two form on $ G$ and $\alpha$ is a one-form on $ G \times G$ with $\delta(R) = d\alpha$ and $\delta(\alpha) = 0$.
We have constructed a map $C( G) \to \Gamma( G)$. In the next section we construct an inverse to this map by showing how to define a central extension from a pair $(\alpha, R)$. For now notice that isomorphic central extensions with connection clearly give rise to the same $(\alpha, R)$ and that if we vary the connection, which is only possible by adding on the pull-back of a one-form $\eta$ from $ G$, then we change $(\alpha, R)$ to $(\alpha + \delta(\eta), R + d\eta)$.
\subsection{Constructing the central extension} Recall that given $R$ we can find a principal $\C^\times$ bundle $P \to G$ with connection $\mu$ and curvature $R$ which is unique up to isomorphism. It is a standard result in the theory of bundles that if $P \to X$ is a bundle with connection $\mu$ which is flat and $\pi_1(X) = 0$ then $P$ has a section $s \colon X \to P$ such that $s^*(\mu) = 0$. Such a section is not unique of course it can be multiplied by a (constant) element of $\C^\times$. As our interest is in the loop group $G$ which satisfies $\pi_1(G)= 0$ we shall assume, from now on, that $\pi_1(G) = 0$. Consider now our pair $(R, \alpha)$ and the bundle $P$. As $\delta(R) = d\alpha$ we have that the connection $\delta(w) - \pi^*(\alpha)$ on $\delta(P) \to G\times G$ is flat and hence (as $\pi_1(G \times G) = 0$) we can find a section $s$ such that $s^*(\delta(w)) = \alpha$.
The section $s$ defines a multiplication by $$ s(p, q) = p \otimes M(p, q)^* \otimes q. $$ Consider now $\delta(s)$ this satisfies $\delta(s)^*(\delta(\delta(w))) = \delta(s^*(\delta(w)) = \delta(\alpha) = 0$.
On the other hand the canonical section $1$ of $\delta(\delta(P))$ also satisfies this so they differ by a constant element of the group. This means that there is a $w \in \C^\times$ such that for any $p$, $q$ and $r$ we must have $$ M(M(p, q), r) = w M(p, M(q, r)). $$ Choose $p \in \hat G_e$ where $e$ is the identity in $ G$. Then $M(p, p ) \in \hat G_e$ and hence $M(p, p) = pz$ for some $z \in \C^\times$. Now let $p=q=r$ and it is clear that we must have $w=1$.
So from $(\alpha, R)$ we have constructed $P$ and a section $s$ of $\delta(P)$ with $\delta(s) = 1$. However $s$ is not unique but this is not a problem. If we change $s$ to $s' = sz$ for some constant $z \in \C^\times$ then we have changed $M$ to $M' = M z$. As $\C^\times$ is central multiplying by $z$ is an isomorphism of central extensions with connection. So the ambiguity in $s$ does not change the isomorphism class of the central extension with connection. Hence we have constructed a map $$ \Gamma( G) \to C( G) $$ as required. That it is the inverse of the earlier map follows from the definition of $\alpha$ as $s^*(\delta(\mu))$ and the fact that the connection on $P$ is chosen so its curvature is $R$.
\section{Conclusion: Loop groups}
In the case where $ G = L (K)$ there is a well known expression for the curvature $R$ of a left invariant connection on $\hat{L(K)}$ --- see \cite{PreSeg}. We can also write down a 1-form $\alpha$ on $L(K)\times L(K)$ such that $\delta(R) = d\alpha$ and $\delta(\alpha) = 0$. We have: \begin{align*}
R(g)(gX,gY) &= \frac{1}{4\pi^{2}}\int_{S^1} \langle X , \partial_\theta Y \rangle d\theta \\
\alpha(g_{1},g_{2})(g_{1}X_{1},g_{2}X_{2}) &= \frac{1}{4\pi^{2}} \int_{S^{1}} \langle X_{1} ,(\partial_\theta g_{2}){\partial_\theta} g_{2}^{-1} \rangle d\theta. \end{align*} Here $\langle\ ,\ \rangle$ is the Killing form on $\mathfrak{k}$ normalised so the longest root has length squared equal to $2$ and $\partial_\theta$ denotes differentiation with respect to $\theta \in S^1$. Note that $R$ is left invariant and that $\alpha$ is left invariant in the first factor of $ G\times G$. It can be shown that these are the $R$ and $\alpha$ arising in \cite{Mur}.
In \cite{MurSte} we apply the methods of this talk to give an explicit construction of the `string class' of a loop group bundle and relate it to earlier work of Murray on calorons.
\end{document} | arXiv |
\begin{definition}[Definition:Composition Functor on Slice Categories]
Let $\mathbf C$ be a metacategory.
Let $C$ and $D$ be objects of $\mathbf C$.
Let $\mathbf C / C$ and $\mathbf C / D$ be the associated slice categories.
Let $g: C \to D$ be a morphism of $\mathbf C$.
Then $g$ defines a '''composition functor''' $g_* : \mathbf C / C \to \mathbf C / D$:
{{begin-axiom}}
{{axiom|lc= Object functor:
|m = g_* f := g \circ f
|rc= The composition $\circ$ is taken in $\mathbf C$
}}
{{axiom|lc= Morphism functor:
|m = g_* a := a
}}
{{end-axiom}}
That it is in fact a functor is shown on Composition Functor on Slice Categories is Functor.
The effect of $g_*$ is captured in the following commutative diagram:
::$\begin{xy}
<-3em,0em>*+{X} = "X",
<3em,0em>*+{X'} = "X2",
<0em,-4em>*+{C} = "C",
<0em,-8em>*+{D} = "D",
"X";"X2" **@{-} ?>*@{>} ?*!/_1em/{a},
"X";"C" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{f},
"X2";"C" **@{-} ?>*@{>} ?<>(.4)*!/_.6em/{f'},
"C";"D" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{g},
"X";"D" **\crv{<-5em,-4em>} ?>*@{>} ?*!/^1.6em/{g_* f = \\ g \circ f},
"X2";"D" **\crv{<5em,-4em>} ?>*@{>} ?*!/_1.6em/{g_* f' = \\ g \circ f'},
\end{xy}$
\end{definition} | ProofWiki |
Journal of Fluid Mechanics (15)
Laser and Particle Beams (1)
Ryan Test (15)
Richtmyer–Meshkov instability on a quasi-single-mode interface
Yu Liang, Zhigang Zhai, Juchun Ding, Xisheng Luo
Journal: Journal of Fluid Mechanics / Volume 872 / 10 August 2019
Experiments on Richtmyer–Meshkov instability of quasi-single-mode interfaces are performed. Four quasi-single-mode air/ $\text{SF}_{6}$ interfaces with different deviations from the single-mode one are generated by the soap film technique to evaluate the effects of high-order modes on amplitude growth in the linear and weakly nonlinear stages. For each case, two different initial amplitudes are considered to highlight the high-amplitude effect. For the single-mode and saw-tooth interfaces with high initial amplitude, a cavity is observed at the spike head, providing experimental evidence for the previous numerical results for the first time. For the quasi-single-mode interfaces, the fundamental mode is the dominant one such that it determines the amplitude linear growth, and subsequently the impulsive theory gives a reasonable prediction of the experiments by introducing a reduction factor. The discrepancy in linear growth rates between the experiment and the prediction is amplified as the quasi-single-mode interface deviates more severely from the single-mode one. In the weakly nonlinear stage, the nonlinear model valid for a single-mode interface with small amplitude loses efficacy, which indicates that the effects of high-order modes on amplitude growth must be considered. For the saw-tooth interface with small amplitude, the amplitudes of the first three harmonics are extracted from the experiment and compared with the previous theory. The comparison proves that each initial mode develops independently in the linear and weakly nonlinear stages. A nonlinear model proposed by Zhang & Guo (J. Fluid Mech., vol. 786, 2016, pp. 47–61) is then modified by considering the effects of high-order modes. The modified model is proved to be valid in the weakly nonlinear stage even for the cases with high initial amplitude. More high-order modes are needed to match the experiment for the interfaces with a more severe deviation from the single-mode one.
Effects of non-periodic portions of interface on Richtmyer–Meshkov instability
Xisheng Luo, Yu Liang, Ting Si, Zhigang Zhai
Journal: Journal of Fluid Mechanics / Volume 861 / 25 February 2019
The development of a non-periodic $\text{air}\text{/}\text{SF}_{6}$ gaseous interface subjected to a planar shock wave is investigated experimentally and theoretically to evaluate the effects of the non-periodic portions of the interface on the Richtmyer–Meshkov instability. Experimentally, five kinds of discontinuous chevron-shaped interfaces with or without non-periodic portions are created by the extended soap film technique. The post-shock flows and the interface morphologies are captured by schlieren photography combined with a high-speed video camera. A periodic chevron-shaped interface, which is multi-modal (81 % fundamental mode and 19 % high-order modes), is first considered to evaluate the impulsive linear model and several typical nonlinear models. Then, the non-periodic chevron-shaped interfaces are investigated and the results show that the existence of non-periodic portions significantly changes the balanced position of the initial interface, and subsequently disables the nonlinear model which is applicable to the periodic chevron-shaped interface. A modified nonlinear model is proposed to consider the effects of the non-periodic portions. It turns out that the new model can predict the growth of the shocked non-periodic interface well. Finally, a method is established using spectrum analysis on the initial shape of the interface to separate its bubble structure and spike structure such that the new model can apply to any random perturbed interface. These findings can facilitate the understanding of the evolution of non-periodic interfaces which are more common in reality.
Mach stem deformation in pseudo-steady shock wave reflections
Xiaofeng Shi, Yujian Zhu, Jiming Yang, Xisheng Luo
The deformation of the Mach stem in pseudo-steady shock wave reflections is investigated numerically and theoretically. The numerical simulation provides the typical flow patterns of Mach stem deformation and reveals the differences caused by high-temperature gas effects. The results also show that the wall jet, which causes Mach stem deformation, can be regarded as a branch of the mainstream from the first reflected shock. A new theoretical model for predicting the Mach stem deformation is developed by considering volume conservation. The theoretical predictions agree well with the numerical results in a wide range of test conditions. With this model, the wall-jet velocity and the inflow velocity from the Mach stem are identified as the two dominating factors that convey the influence of high-temperature thermodynamics. The mechanism of high-temperature gas effects on the Mach stem deformation phenomenon are then discussed.
An elaborate experiment on the single-mode Richtmyer–Meshkov instability
Lili Liu, Yu Liang, Juchun Ding, Naian Liu, Xisheng Luo
Journal: Journal of Fluid Mechanics / Volume 853 / 25 October 2018
Published online by Cambridge University Press: 23 August 2018, R2
Print publication: 25 October 2018
High-fidelity experiments of Richtmyer–Meshkov instability on a single-mode air/ $\text{SF}_{6}$ interface are carried out at weak shock conditions. The soap-film technique is extended to create single-mode gaseous interfaces which are free of small-wavelength perturbations, diffusion layers and three-dimensionality. The interfacial morphologies captured show that the instability evolution evidently involves the smallest experimental uncertainty among all existing results. The performances of the impulsive model and other nonlinear models are thoroughly examined through temporal variations of the perturbation amplitude. The individual growth of bubbles or spikes demonstrates that all nonlinear models can provide a reliable forecast of bubble development, but only the model of Zhang & Guo (J. Fluid Mech., vol. 786, 2016, pp. 47–61) can reasonably predict spike development. The distinct images of the interface morphology obtained also provide a rare opportunity to extract interface contours such that a spectral analysis of the interfacial contours can be performed, which realizes the first direct validation of the high-order nonlinear models of Zhang & Sohn (Phys. Fluids, vol. 9, 1997, pp. 1106–1124) and Vandenboomgaerde et al. (Phys. Fluids, vol. 14 (3), 2002, pp. 1111–1122) in terms of the fundamental mode and high-order harmonics. It is found that both models show a very good and almost identical accuracy in predicting the first two modes. However, the model of Zhang & Sohn (1997) becomes much more accurate in modelling the third-order harmonics due to the fewer simplifications used.
Long-term effect of Rayleigh–Taylor stabilization on converging Richtmyer–Meshkov instability
Xisheng Luo, Fu Zhang, Juchun Ding, Ting Si, Jiming Yang, Zhigang Zhai, Chih-yung Wen
The Richtmyer–Meshkov instability on a three-dimensional single-mode light/heavy interface is experimentally studied in a converging shock tube. The converging shock tube has a slender test section so that the non-uniform feature of the shocked flow is amply exhibited in a long testing time. A deceleration phenomenon is evident in the unperturbed interface subjected to a converging shock. The single-mode interface presents three-dimensional characteristics because of its minimum surface feature, which leads to the stratified evolution of the shocked interface. For the symmetry interface, it is quantitatively found that the perturbation amplitude experiences a rapid growth to a maximum value after shock compression and finally drops quickly before the reshock. This quick reduction of the interface amplitude is ascribed to a significant Rayleigh–Taylor stabilization effect caused by the deceleration of the light/heavy interface. The long-term effect of the Rayleigh–Taylor stabilization even leads to a phase inversion on the interface before the reshock when the initial interface has sufficiently small perturbations. It is also found that the amplitude growth is strongly suppressed by the three-dimensional effect, which facilitates the occurrence of the phase inversion.
On the interaction of a planar shock with a three-dimensional light gas cylinder
Juchun Ding, Ting Si, Mojun Chen, Zhigang Zhai, Xiyun Lu, Xisheng Luo
Experimental and numerical investigations on the interaction of a planar shock wave with two-dimensional (2-D) and three-dimensional (3-D) light gas cylinders are performed. The effects of initial interface curvature on flow morphology, wave pattern, vorticity distribution and interface movement are emphasized. In experiments, a wire-restriction method based on the soap film technique is employed to generate N $_{2}$ cylinders surrounded by SF $_{6}$ with well-characterized shapes, including a convex cylinder, a concave cylinder with a minimum-surface feature and a 2-D cylinder. The high-speed schlieren pictures demonstrate that fewer disturbance waves exist in the flow field and the evolving interfaces develop in a more symmetrical way relative to previous studies. By combining the high-order weighted essentially non-oscillatory construction with the double-flux scheme, numerical simulation is conducted to explore the detailed 3-D flow structures. It is indicated that the shape and the size of 3-D gas cylinders in different planes along the vertical direction change gradually due to the existence of both horizontal and vertical velocities of the flow. At very early stages, pressure oscillations in the vicinity of evolving interfaces induced by complex waves contribute much to the deformation of the 3-D gas cylinders. As time proceeds, the development of the shocked volume would be dominated by the baroclinic vorticity deposited on the interface. In comparison with the 2-D case, the oppositely (or identically) signed principal curvatures of the concave (or convex) SF $_{6}$ /N $_{2}$ boundary cause complex high pressure zones and additional vorticity deposition, and the upstream interface from the symmetric slice of the concave (or convex) N $_{2}$ cylinder moves with an inhibition (or a promotion). Finally, a generalized 3-D theoretical model is proposed for predicting the upstream interface movements of different gas cylinders and the present experimental and numerical findings are well predicted.
Experimental study on a sinusoidal air/SF $_{6}$ interface accelerated by a cylindrically converging shock
Fan Lei, Juchun Ding, Ting Si, Zhigang Zhai, Xisheng Luo
Journal: Journal of Fluid Mechanics / Volume 826 / 10 September 2017
Print publication: 10 September 2017
Ritchmyer–Meshkov instability on an air/SF $_{6}$ interface is experimentally studied in a coaxial converging shock tube by a high-speed laser sheet imaging technique. An unperturbed case is first examined to obtain the characteristics of the converging shock and the shocked interface. For sinusoidal interfaces, the wave pattern and the interface morphology of the whole process are clearly observed. It is quantitatively found that the perturbation amplitude first decreases due to the shock compression, then experiences a rapid growth to a maximum value and finally drops quickly before the reshock. The reduction of growth rate is ascribed to the Rayleigh–Taylor stabilization caused by the interface deceleration motion that is present in the converging circumstance. It is noted that the influence of the wavenumber on the amplitude growth is very little before the reshock, but becomes significant after the reshock.
The Richtmyer–Meshkov instability of a 'V' shaped air/ $\text{SF}_{6}$ interface
Xisheng Luo, Ping Dong, Ting Si, Zhigang Zhai
The Richtmyer–Meshkov instability on a 'V' shaped air/SF $_{6}$ gaseous interface is experimentally studied in a shock tube. By the soap film technique, a discontinuous interface without supporting mesh is formed so that the initial conditions of the interface can be accurately controlled. Five 'V' shaped air/ $\text{SF}_{6}$ interfaces with different vertex angles ( $60^{\circ }$ , $90^{\circ }$ , $120^{\circ }$ , $140^{\circ }$ and $160^{\circ }$ ) are created where the ratio of the initial interface amplitude to the wavelength varies to highlight the effects of initial condition on the flow characteristics. The wave patterns and interface morphologies are clearly identified in the high-speed schlieren sequences, which show that the interface deforms in a less pronounced manner with less vortices generated as the vertex angle increases. A regime change is observed in the interface width growth rate near a vertex angle of $160^{\circ }$ , which provides an experimental evidence for the numerical results obtained by McFarland et al. (Phys. Scr. vol. T155, 2013, 014014). The growth rate of interface width in the linear phase is compared with the theoretical predictions from the classical impulsive model and a modified linear model, and the latter is proven to be effective for a moderate to large initial amplitude. It is found that the initial growth rate of the interface width is a non-monotone function of the initial vertex angle (amplitude–wavelength ratio), i.e. the interface width growth rate in the linear stage experiences an increase and then a decrease as the vertex angle increases. A similar conclusion was also reached by Dell et al. (Phys. Plasmas, vol. 22, 2015, 092711) numerically for a sinusoidal interface. Finally, the general behaviour of the interface width growth in the nonlinear stage can be well captured by the nonlinear model proposed by Dimonte & Ramaprabhu (Phys. Fluids, vol. 22, 2010, 014104).
Experimental investigation of cylindrical converging shock waves interacting with a polygonal heavy gas cylinder
Ting Si, Tong Long, Zhigang Zhai, Xisheng Luo
Journal: Journal of Fluid Mechanics / Volume 784 / 10 December 2015
Print publication: 10 December 2015
The interaction of cylindrical converging shock waves with a polygonal heavy gas cylinder is studied experimentally in a vertical annular diaphragmless shock tube. The reliability of the shock tube facility is verified in advance by capturing the cylindrical shock movements during the convergence and reflection processes using high-speed schlieren photography. Three types of air/SF6 polygonal interfaces with cross-sections of an octagon, a square and an equilateral triangle are formed by the soap film technique. A high-speed laser sheet imaging method is employed to monitor the evolution of the three polygonal interfaces subjected to the converging shock waves. In the experiments, the Mach number of the incident cylindrical shock at its first contact with each interface is maintained to be 1.35 for all three cases. The results show that the evolution of the polygonal interfaces is heavily dependent on the initial conditions, such as the interface shapes and the shock features. A theoretical model for circulation initially deposited along the air/SF6 polygonal interface is developed based on the theory of Samtaney & Zabusky (J. Fluid Mech., vol. 269, 1994, pp. 45–78). The circulation depositions along the initial interface result in the differences in flow features among the three polygonal interfaces, including the interface velocities and the perturbation growth rates. In comparison with planar shock cases, there are distinct phenomena caused by the convergence effects, including the variation of shock strength during imploding and exploding (geometric convergence), consecutive reshocks on the interface (compressibility), and special behaviours of the movement of the interface structures (phase inversion).
On the interaction of a planar shock with an $\text{SF}_{6}$ polygon
Xisheng Luo, Minghu Wang, Ting Si, Zhigang Zhai
Journal: Journal of Fluid Mechanics / Volume 773 / 25 June 2015
The interaction of a planar shock wave ( $M\approx 1.2$ ) with an $\text{SF}_{6}$ polygonal inhomogeneity surrounded by air is experimentally investigated. Six polygons including a square, two types of rectangle, two types of triangle, and a diamond are generated by the soap film technique developed in our previous work, in which thin pins are used as angular vertexes to avoid the pressure singularities caused by the surface tension. The evolutions of the shock-accelerated $\text{SF}_{6}$ polygons are captured by a high-speed schlieren system from which wave systems and the interface characteristics can be clearly identified. Both regular and irregular refraction phenomena are observed outside the volume, and more complex wave patterns, including transmitted shock, refracted shock, Mach stem and the interactions between them, are found inside the volume. Two typical irregular refraction phenomena (free precursor refraction, FPR, and free precursor von Neumann refraction, FNR) are observed and analysed, and the transition from FPR to FNR is found, providing the experimental evidence for the transition between different wave patterns numerically found in the literature. Combined with our previous work (Zhai et al., J. Fluid Mech., vol. 757, 2014, pp. 800–816), the reciprocal transitions between FPR and FNR are experimentally confirmed. The velocities and trajectories of the triple points are further measured and it is found that the motions of the triple points are self-similar or pseudo-stationary. Besides the shock dynamics phenomena, the evolutions of these shocked heavy polygonal volumes, which are quite different from the light ones, are captured and found to be closely related to their initial shapes. Specifically, for square and rectangular geometries, the different width–height ratios result in different behaviours of shock–shock interaction inside the volume, and subsequently different features for the outward jet and the interface. Quantitatively, the time-variations of the interface scales, such as the width and the normalized displacements of the edges, are obtained and compared with those from previous work. The comparison illustrates the superiority of the interface formation method and the significant effect of the initial interface shape on the interface features. Furthermore, the characteristics of the vortex core, including the velocity and vortex spacing, are experimentally measured, and the vortex velocity is compared with those from some circulation models to check the validity of the models. The results in the present work enrich understanding of the shock refraction phenomenon and the database of research into Richtmyer–Meshkov instability (RMI).
On the interaction of a planar shock with a light polygonal interface
Zhigang Zhai, Minghu Wang, Ting Si, Xisheng Luo
The interaction of a planar shock wave with a polygonal $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\mathrm{N}}_2$ volume surrounded by ${\mathrm{SF}}_6$ is investigated experimentally and numerically. Three polygonal interfaces (square, triangle and diamond) are formed by the soap film technique developed in our previous work, in which thin pins are introduced as angular vertexes to connect adjacent sides of polygonal soap films. The evolutions of the shock-accelerated polygonal interfaces are then visualized by a high-speed schlieren system. Wave systems and interface structures can be clearly identified in experimental schlieren images, and agree well with the numerical ones. Quantitatively, the movement of the distorted interface, and the length and height of the interface structures are further compared and good agreements are achieved between experimental and numerical results. It is found that the evolution of these polygonal interfaces is closely related to their initial shapes. In the square interface, two vortices are generated shortly after the shock impact around the left corner and dominate the flow field at late stages. In the triangular and diamond cases, the most remarkable feature is the small ' ${\mathrm{SF}}_6$ jet' which grows constantly with time and penetrates the downstream boundary of the interface, forming two independent vortices. These distinct morphologies of the three polygonal interfaces also lead to the different behaviours of the interface features including the length and height. It is also found that the velocities of the vortex pair predicted from the theory of Rudinger and Somers (J. Fluid Mech., vol. 7, 1960, pp. 161–176) agree with the experimental ones, especially for the square case. Typical free precursor irregular refraction phenomena and the transitions among them are observed and analysed, which gives direct experimental evidence for wave patterns and their transitions at a slow/fast interface. The velocities of triple points and shocks are experimentally measured. It is found that the transmitted shock near the interface boundary has weakened into an evanescent wave.
Experimental study of Richtmyer-Meshkov instability in a cylindrical converging shock tube
Ting Si, Zhigang Zhai, Xisheng Luo
Journal: Laser and Particle Beams / Volume 32 / Issue 3 / September 2014
The interaction of a cylindrical converging shock wave with an initially perturbed gaseous interface is studied experimentally. The cylindrical converging shock is generated in an ordinary shock tube but with a specially designed test section, in which the incident planar shock wave is directly converted into a cylindrical one. Two kinds of typical initial interfaces involving gas bubble and gas cylinder are employed. A high-speed video camera combined with schlieren or planar Mie scattering photography is utilized to capture the evolution process of flow structures. The distribution of baroclinic vorticity on the interface induced by the cylindrical shock and the reflected shock from the center of convergence results in distinct phenomena. In the gas bubble case, the shock focusing and the jet formation are observed and the turbulent mixing of two fluids is promoted because of the gradually changed shock strength and complex shock structures in the converging part. In the gas cylinder case, a counter-rotating vortex pair is formed after the impact of the converging shock and its rotating direction may be changed when interacting with the reflected shock for a relatively long reflection distance. The variations of the interface displacements and structural dimensions with time are further measured. It is found that these quantities are different from those in the planar counterpart because of the shock curvature, the Mach number effect and the complex shock reflection within the converging shock tube test section. Therefore, the experiments reported here exhibit the great potential of this experimental method in study of the Richtmyer-Meshkov instability induced by converging shock waves.
The Richtmyer–Meshkov instability of a three-dimensional air/SF6 interface with a minimum-surface feature
Xisheng Luo, Xiansheng Wang, Ting Si
Journal: Journal of Fluid Mechanics / Volume 722 / 10 May 2013
Published online by Cambridge University Press: 04 April 2013, R2
Print publication: 10 May 2013
A novel method to create a discontinuous gaseous interface with a minimum-surface feature by the soap film technique is developed for three-dimensional (3D) Richtmyer–Meshkov instability (RMI) studies. The interface formed is free of supporting mesh and the initial condition can be well controlled. Five air/SF6 interfaces with different amplitude are realized in shock-tube experiments. Time-resolved schlieren and planar Mie-scattering photography are employed to capture the motion of the shocked interface. It is found that the instability at the linear stage in the symmetry plane grows much slower than the predictions of previous two-dimensional (2D) impulsive models, which is ascribed to the opposite principal curvatures of the minimum surface. The 2D impulsive model is extended to describe the general 3D RMI. A quantitative analysis reveals a good agreement between experiments and the extended linear model for all the configurations including both the 2D and 3D RMIs at their early stages. An empirical model that combines the early linear growth with the late-time nonlinear growth is also proposed for the whole evolution process of the present configuration.
On condensation-induced waves
WAN CHENG, XISHENG LUO, M. E. H. van DONGEN
Complex wave patterns caused by unsteady heat release due to cloud formation in confined compressible flows are discussed. Two detailed numerical studies of condensation-induced waves are carried out. First, the response of a flow of nitrogen in a slender Laval nozzle to a sudden addition of water vapour at the nozzle entrance is considered. Condensation occurs just downstream of the nozzle throat, which initially leads to upstream- and downstream-moving shocks and an expansion fan downstream of the condensation front. Then, the flow becomes oscillatory and the expansion fan disappears, while upstream and much weaker downstream shocks are repeatedly generated. For a lower initial humidity, only a downstream starting shock is formed and a steady flow is established. Second, homogeneous condensation in an unsteady expansion fan in humid nitrogen is considered. In the initial phase, two condensation-induced shocks are found, moving upstream and downstream. The upstream-moving shock changes the shape of the expansion fan and has a strong influence on the condensation process itself. It is even quenching the nucleation process locally, which leads to a renewed condensation process more downstream. This process is repeated with asymptotically decreasing strength. The repeated interaction of the condensation-induced shocks with the main expansion wave leads to a distortion of the expansion wave towards its shape that can be expected on the basis of phase equilibrium, i.e. a self-similar wave structure consisting of dry part, a plateau of constant state and a wet part. The strengths of the condensation-induced waves, as well for the Laval nozzle flow as for the expansion fan, appear to be in qualitative agreement with the results from the analytical Rayleigh–Bartlmä model.
Effects of homogeneous condensation in compressible flows: Ludwieg-tube experiments and simulations
XISHENG LUO, GRAZIA LAMANNA, A. P. C. HOLTEN, M. E. H. VAN DONGEN
Journal: Journal of Fluid Mechanics / Volume 572 / February 2007
Effects of homogeneous nucleation and subsequent droplet growth in compressible flows in humid nitrogen are investigated numerically and experimentally. A Ludwieg tube is employed to produce expansion flows. Corresponding to different configurations, three types of experiment are carried out in such a tube. First, the phase transition in a strong unsteady expansion wave is investigated to demonstrate the mutual interaction between the unsteady flow and the condensation process and also the formation of condensation-induced shock waves. The role of condensation-induced shocks in the gradual transition from a frozen initial structure to an equilibrium structure is explained. Second, the condensing flow in a slender supersonic nozzle G2 is considered. Particular attention is given to condensation-induced oscillations and to the transition from symmetrical mode-1 oscillations to asymmetrical mode-2 oscillations in a starting nozzle flow, as first observed by Adam & Schnerr. The transition is also found numerically, but the amplitude, frequency and transition time are not yet well predicted. Third, a sharp-edged obstacle is placed in the tube to generate a starting vortex. Condensation in the vortex is found. Owing to the release of latent heat of condensation, an increase in the pressure and temperature in the vortex core is observed. Condensation-induced shock waves are found, for a sufficiently high initial saturation ratio, which interact with the starting vortex, resulting in a very complex flow. As time proceeds, a subsonic or transonic free jet is formed downstream of the sharp-edged obstacle, which becomes oscillatory for a relatively high main-flow velocity and for a sufficiently high humidity.
On phase transition in compressible flows: modelling and validation
XISHENG LUO, BART PRAST, M. E. H. van DONGEN, H. W. M. HOEIJMAKERS, JIMING YANG
A physical model for compressible flows with phase transition is described, in which all the processes of phase transition, i.e. nucleation, droplet growth, droplet evaporation and de-nucleation, are incorporated. The model is focused on dilute mixtures of vapour and droplets in a carrier gas with typical maximum liquid mass fraction smaller than 0.02. The new model is based on a reinterpretation of Hill's method of moments of the droplet size distribution function. Starting from the general dynamic equation, it is emphasized that nucleation or de-nucleation correspond to the rates at which droplets enter or leave droplet size space, respectively. Nucleation and de-nucleation have to be treated differently in agreement with their differences in physical nature. Attention is given to the droplet growth model that takes into account Knudsen effects and temperature differences between droplets and gas. The new phase transition model is then combined with the Euler equations and results in a new numerical method: ASCE2D. The numerical method is first applied to the problem of shock/expansion wave formation in a closed shock tube with humid nitrogen as a driver gas. Nucleation and droplet growth are induced by the expansion wave, and in turn affect the structure of the expansion wave. When the main shock, reflected from the end wall of the low-pressure section, passes the condensation zone, evaporation and de-nucleation occur. As a second example, the problem of the flow of humid nitrogen in a pulse-expansion wave tube, designed to study nucleation and droplet growth in monodisperse clouds, is investigated experimentally and numerically. | CommonCrawl |
A competitive analysis of EU ports by fixing spatial and economic dimensions
Claudio Quintano1,
Paolo Mazzocchi ORCID: orcid.org/0000-0002-6632-314X1 &
Antonella Rocca1
The purpose of this paper is to evaluate the efficiencies of ten of the leading European ports. The motivation of the research refers to the relevant topic of selection of indicators that can be involved in the comparative analysis. Concerning the theoretical model, the authors' efforts are especially directed towards the usage of the stochastic frontier analysis (SFA) and of the data envelopment analysis (DEA). These techniques have been widely adopted for benchmarking and performance evaluation by involving indicators based on data from National Accounts. If one of these indicators, such as labour force consistency, is not available at a specific level of aggregation, detailed assumptions are needed to address this complication. The present study proposes an additive model in order to provide an estimation of ports' economic activities by fixing the port activity boundaries and the spatial perimeter of the firms investigated. Several NUTS (Nomenclature of Territorial Units for Statistics) levels and NACE (EU Statistical Classification of Economic Activities) codes are fixed to offer a useful comparable labour indicator. Empirical results reveal that each port area presents a combination of the NACE categories which significantly impact the efficiency that can reach very high performance values through both the SFA and DEA techniques. Since the managers can choose which sectors to improve, which particular improvement strategies to support, which specific service to add, their decisions impact this performance evaluation, and their performance can be verified through the approaches proposed.
Port authorities and port operators manage the new context of supply and logistics chains. Increasing globalization has improved the strategic relevance of ports, and the attention to port efficiency has consequently grown. The traditionally strong competition among the ports affects port performance at intra-port and inter-port levels (Castelein et al. 2019). This competitiveness has encouraged management to address performance evaluation methods and benchmarking models (Figueiredo De Oliveira and Cariou 2015; European Commission 2016; Wiegmans and Witte 2017; Ferreira et al. 2018; Ha et al. 2019). The performance evaluation approaches also dedicate increasing attention to sustainability criteria (IAPH - International Association of Ports and Harbours 2007; Baynes et al. 2011; Chang and Wang 2012; Lam and Notteboom 2014; Laxe et al. 2016; Roos and Kliemann-Neto 2017; Chang et al. 2018).
Despite the existing remarkable literature on port performance, the subject is still quite debated. One main problem is the complexity of the port structure since various characteristics determine maritime performance, such as the number of activities linked to it, the development of intermodal transportation and undesirable outputs (OECD 2016; Madeira Jr et al. 2012; Bulut and Durur 2018; Munim and Schramm 2018; Shobayo and Van Hassel 2019). An additional issue is that there is no reliable database of collective information of international port dimensions (Cheon et al. 2010).
Concerning the selection of the measurements that can be used in the competitive analysis, several authors referred to the empirical criterion that considers the availability of inputs and outputs, while other authors suggested considering the measurements commonly adopted in previous studies (Cullinane et al. 2006). In the current work special prominence has been dedicated to the collection of data of a specific indicator, the labour consistency. In fact, the availability of labour data sources—in addition to capital and port land—represents a relevant topic in port benchmarking models (Dowd and Leschine 1990). In order to enhance this dimension, since these data are very difficult to collect, two perspectives exist in past literature. On the one hand, Tongzon (2001), Estache et al. (2002), Barros (2003), Min and Park (2005), Cullinane et al. (2006) and Turnbull (2012) proposed solutions targeted to include a proxy of the number of port' employees. On the other hand, Demirel et al. (2012) suggested the involvement of input indicators strictly connected to labour force consistency. Since both perspectives share the effort aimed at avoiding the exclusion of the labour indicator, the present paper contributes to the debate attempting to address the availability issue by means of the usage of spatial and economic patterns. The authors argue (1) that the geographical concentration of the maritime firms and (2) that an inventory of the NACE (European Statistical Classification of Economic Activities) classes related to the maritime sector can be assumed to fix homogenous and comparable indicators connected to ports. In the authors' opinion, the involvement of firms which operate in well-defined territorial districts and in specific port activities could be a good way to analyse the port performances in future research. Specifically, this paper aims to analyse the efficiency of ten of the leading European container ports focusing on the labour force estimation, and considering as a case study the port of Antwerp compared to the port of Rotterdam. The model results can be considered as implications for policymakers.
As for the theoretical model, both parametric stochastic frontier analysis (SFA) and non-parametric data envelopment analysis (DEA) were undertaken to achieve the performance investigation. Liu (1995) was among the first researchers to utilize SFA in the port sector and Barros (2005) and Cullinane et al. (2006) significantly contributed to this approach. DEA has also been widely adopted for the benchmarking and environmental performance in transportation (Roll and Hayuth 1993; Cullinane et al. 2004; Barros 2006). Among others, Ensslin et al. (2018) provided an overview of the most common port efficiency techniques.
As for the remaining content of this article, the following section discusses the model assumptions. Section three briefly reviews the theoretical background and data. Section four combines the results and discussion. Section five refers to the case study. Section six considers the concluding remarks.
Model assumptions
In general, a performance quantitative method requires some specifications: the sample size must be appropriate, several conditions must be preserved and the results must be validated. Concerning the availability of one or more indicators, it does not represent a problem since a specific database contains the corresponding figures at a specific level of aggregation. In different conditions, the estimation of one indicator for comparative analysis requires additional assumptions. As aforementioned, in the port sector a large number of factors—such as the port features connected to the structural dimension and/or company attributes, manpower, advanced technology and port institutional reforms—need to be considered (Cheon et al. 2010; Van Den Bos and Wiegmans 2018). Nevertheless, in the current research the efficiency measurements were calculated using a limited number of variables, one input and one output in addition to the labour dimension. These indicators—discussed more in depth later—have been obtained from the following databases: Eurostat, Bureau van Dijk, World Port Source and Harbours Review. The present paper focuses on 2016, and it was selected because it has the most comprehensive data availability.
As argued in the introduction, authors assume that the issue of the availability of the labour force consistency can be addressed considering an additive model that fixes (1) the port activity boundaries (economic activities) strictly depending on maritime activities and (2) the spatial perimeter (territorial districts) of the firms investigated.
Ports' behaviour of providing services to several economic sectors has been discussed in recent literature by, among others, Van Der Lugt and De Langen (2005), De Langen and Haezendonck (2012) and Alijohani and Thompson (2016). The NACE codes refer to the System of National Accounts, and the present research proposes the usage of four-digit NACE codes (classes) of port sector quoted in Table 1.
Table 1 Four-digit NACE (rev 2) classes and descriptions of the economic activities considered for each EU port
The firms were selected by fixing 'active' companies throughout a Boolean search strategy via the Bureau van Dijk database. Nevertheless, this selection entails several weaknesses. First of all, the Bureau van Dijk database contains key establishment information, including firm name, type of activity (NACE code), number of employees and address. This classification is based on the activity declared by the establishment upon creation. Therefore, the assigned code could not exactly reflect the economic activity and/or there could be changes in the NACE classification over several years. Furthermore, the number of firms could be underestimated since some firms involved in maritime activities could have a main (primary or secondary) activity that is different from the classifications considered in the present research. Another difficulty is that some firms provide auxiliary services for maritime transportation and a distinction may be necessary. As a consequence, in addition to the NACE codes, Surís-Regueiro et al. (2013) suggested analysing the contribution of maritime activities to GDP (Gross Domestic Product) by using specific weights, which referred to economic activities that are fully or partially involved in the maritime economy. Bruno et al. (1999) proposed the entropic average as a useful indicator to investigate highly asymmetrical distributions. Interesting findings also derived from Oum and Park (2004), Fernández-Macho et al. (2016) and Heitz et al. (2018). Using a different standpoint, Baynes et al. (2011) referred to the input-output (I/O) approach as proposed by Leontief (1936).
Territorial districts
If one assumes that a firm's location near a port increases its probability of depending upon the port to exist, then comparison of the spatial dimension appears to be a sustainable perspective. The approach prioritizes the proximity as a key element in defining an appropriate cluster of activities and it has been analysed by, among others, Rivera et al. (2014). These authors defined the clusters in logistics and transportation by considering the geographic concentration of firms providing logistics services. NUTS2 (Nomenclature of Territorial Units for Statistics) classifications represent territorial districts allowing for harmonized and comparable socio-economic analyses. Therefore, the usage of the NUTS2 level appears to be suitable to ensure a high degree of homogeneity of the geographical division. Eurostat (2009, 2017) referred to the NUTS2 codes to analyse different maritime policies and several tourism flows across the EU. According to this perspective, in the current paper labour force consistency has been estimated by fixing the firms located in the NUTS2 regions mentioned in Table 2, and those involved in the NACE codes quoted in the above-mentioned Table 1. Data from a sample of 11,849 active firms has been considered. Table 2 also shows the number of firms involved in each NUTS2 level.
Table 2 NUTS2 levels considered for the ports analysed in the present research, and number of firms involved
Theoretical background and data
The literature differentiates between two fundamental methodologies for measuring efficiency through the functional form: the non-parametric linear programming technique—DEA—and the parametric model—SFA. As discussed in the introduction, both the SFA and DEA approaches have been commonly considered in the port performance literature. Barros et al. (2011, b), Odeck and Bråthen (2012) and Lampe and Hilgers (2015) presented an extensive description, assumptions and differences between the SFA and DEA perspectives.
Selected studies connected to the usage of DEA and SFA in previous port literature can be found in Table 3. This table also summarises the approaches proposed by each research paper and the indicators used in each work.
Table 3 Input and output variables used in previous port studies
In this paper the variables were selected after reviewing the existing literature quoted in Table 3; the first input dimension is the number of employees (Roll and Hayuth 1993; Coto-Millan et al. 2000; Notteboom et al. 2000; Estache et al. 2002; Barros 2003, 2006, 2012; Min and Park 2005; Rios and Maçada 2006; Barros and Peypoch 2007; Panayides et al. 2011; Gong et al. 2019). De Langen and Pallis (2006), Turnbull and Wass (2007) and Murphy et al. (2016) highlighted that, even though the capital-intensive paradigm is increasing in the port sector, the labour remains an important dimension in port competition. The economic efficiency of the labour market significantly influences the productivity, thus inefficient work procedures can cause inefficiencies in port operations. Notteboom (2010, 2012) emphasized that several features impact the port labour cost and competitiveness, for instance direct and indirect (or hidden) costs (such as strikes, absenteeism, inactivity for accidents/sickness), technological innovations, introduction of new cargo handling equipment, etc. Port labour environment also changes as a consequence of port reform, new port security regulation, labour port schemes, etc.
In the present work authors assume that the managers' efforts are addressed to maximize the goods handled in each port involved in the analysis (the correlation matrix of the input and output variables presents a positive relationship among the indicators). This perspective represents one of two different schools of thought on labour assumptions. In fact, on the one hand, if one assumes this positive correlation, port policy measures can be targeted to increase the port throughput to expand the labour component (Ferrari et al. 2010; Bottasso et al. 2013). On the other hand, different authors, such as Grobar (2008) and Deng et al. (2013), noted that recent advancements in transportation technology have modified the role of ports in local economic development. For instance, in the container sector, the transportation activity has made the process of goods movement much more capital intensive, thus decreasing the local employment benefits of having a port. If this standpoint prevails, the implications on the port throughput are less clear.
The second input measurement refers to the terminal quay length of each port. This dimension has strategic importance in terms of time waiting as a performance indicator (Notteboom et al. 2000; Cullinane et al. 2006; Rios and Maçada 2006; Almawsheki and Shah 2015; Barros 2003, 2012; Panayides et al. 2011; Demirel et al. 2012; Nguyen et al. 2016; Suárez-Alemán et al. 2016). This dimension appears to be as a more neutral measurement than the container quay length, since ports can have a different division between diverse output products in their trade (for instance, Rotterdam is traditionally focused on bulk).
In regard to the output, current research considers the total gross weight of goods handled in each port (bulk and containers) which is expressed in thousands of tons. According to Table 3, also this dimension represents a widely accepted indicator of port output. Eurostat (2020) highlighted that Rotterdam was the largest European port for all types of cargo in 2019, with almost 110 million tons for each quarter. The second port in the same year was Antwerp which handled close to half of the tonnage recorded by Rotterdam, while the third port was Hamburg. Considering only the container cargo segment, the ranking is similar and Rotterdam, Antwerp and Hamburg remained the three main European ports in 2019, followed by the two Spanish ports of Algeciras and Valencia. In contrast, slight differences in ranking appear observing diverse types of bulk.
Table 4 provides the descriptive statistics of the input and output measurements included in the model.
Table 4 Descriptive statistics of the indicators involved in the DEA and SFA approaches
DEA represents a widely utilised method to obtain a multi-variate frontier estimation and to measure the efficiency of multiple homogeneous DMUs (decisions making units) with the same set of inputs and outputs. The original idea behind the DEA model can be traced back to Farrell (1957), while the model was significantly advanced by Charnes et al. (1978) and Banker et al. (1984). This technique does not require a specific functional relationship among inputs and outputs. Both the input and output orientations can be used, and several technologies are available: constant returns to scale (CRS, or CCR), variable returns to scale (VRS, or BCC), and non-increasing returns to scale (NIRS). Following the definition proposed by Cook and Zhu (2005), eq. (1) summarizes the two-stage input DEA approach.
$$ {\displaystyle \begin{array}{l}\min {\theta}_0-\varepsilon \left(\sum \limits_{i=1}^m{s}_i^{-}+\sum \limits_{r=1}^s{s}_r^{+}\right)\\ {} subject\ to\\ {}\sum \limits_{j=1}^n{\lambda}_j{x}_{ij}+{s}_i^{-}={\theta}_0{x}_{i0}\kern1.75em \left(i=1,\dots, m\right)\\ {}\sum \limits_{j=1}^n{\lambda}_j{y}_{jh}-{s}_r^{+}={y}_{r0}\kern2em \left(r=1,.\dots, s\right)\\ {}{\lambda}_j,{s}_r^{+},{s}_i^{-}\ge 0\kern5em 1\end{array}} $$
In this equation θ denotes the efficiency score for each DMU; \( {s}_i^{-} \) and \( {s}_r^{+} \) represent input and output slacks; the non-Archimedean ε allows the minimization involving the slacks; xi is the i-th input of m inputs; yr indicates the r-th output of s outputs; λj is a non-negative scalar. In addition to the radial approach, the non-radial efficiency measurements have also been considered in the DEA models. Zhou et al. (2007) highlighted that the non-radial DEA seems to be more effective in measuring the environmental performance, since this approach has a high discriminating power in evaluating the DMU's efficiencies. Cook and Seiford (2009) presented a detailed review of the DEA techniques, while Sahoo et al. (2016) and Liu et al. (2016) proposed innovative DEA approaches.
The SFA approach was developed by Aigner et al. (1977) and Meeusen and Van den Broeck (1977). Battese and Coelli (1992, 1995) significantly expanded the basic model. In contrast to DEA, SFA requires the specification of a parametric function. The most popular parametrization of the model refers to the Cobb-Douglas (log) function, which can be exhibited in form of multiplicative specifications as shown in eq. (2).
$$ y=f\left(x;\beta \right)\mathit{\exp}\left(v-u\right) $$
In this equation, y is a scalar output, while x is a vector of the inputs. β is a vector of the technology parameters. The composed error refers to the decomposition of the error term ε into the two components represented by ε= v- \( u.v\sim N\left(0,{\sigma}_v^2\right) \) is the first error component, that concerns the effects of the statistical noise, and it is unrestricted in sign. u is the second error component, and it considers the effects of technical inefficiency (u ≥ 0). u is considered to have a distribution such as the exponential or half normal \( u\sim {N}_{+}\left(0,{\sigma}_u^2\right) \), to ensure that it produces only non-negative values. The model assumes that the corresponding log-likelihood function needs to be maximized, by using the maximum likelihood method (Kumbhakar and Lovell 2003). The stochastic version of output-oriented technical efficiency proposed by Coelli et al. (2005) is shown in eq. (3):
$$ TE=\frac{y}{f(x){e}^v}=\frac{f(x){e}^{-u}{e}^v}{f(x){e}^v}=\mathit{\exp}\left(-u\right) $$
where TE indicates the technical efficiency of production obtained as a ratio between the observed output (y) and the corresponding stochastic frontier output; e−u denotes the inefficiency; ev represents the noise. Different functional forms can be used, instead of using the traditional Cobb-Douglas (log) function. In this article the performance evaluation derived from the translog SFA as proposed by Christensen et al. (1973). Translog SFA represents a less restrictive approach compared to the standard Cobb-Douglas function.
In the prevalent literature, the SFA technique is less frequently used than the DEA. One main reason is that multiple input and output measurements limit the usage of the SFA technique in the standard version. In fact, when the SFA model needs to consider multiple outputs instead of a single one, the standard econometric approach requires that input and output prices are available. An extensive consideration of DEA and SFA techniques is beyond the scope of this paper while the actual purpose of this paper is to include the radial DEA and SFA techniques to (1) verify if the model provides consistent results and (2) evaluate a specific labour port perspective. See Orea and Wall (2016) for a comprehensive discussion on these topics.
Table 5 shows the efficiency scores of the ten European ports, according to the following techniques: the VRS output orientation (DEA_VRS_OUT); the VRS input orientation (DEA_VRS_IN); the CRS (DEA_CRS); and translog SFA (SFA_TR_LG).
Table 5 DEA and SFA efficiency scores for each port
Spearman correlations have been calculated by considering the DEA and SFA efficiency scores to verify whether the ports' ranks are (approximately) the same. The results are provided in Table 6. As can be seen in the table, all the ranking correlations among the diverse techniques are positive, and the Spearman correlation between SFA_TR_LG and DEA_CRS appears to be relatively high. According to the results of the statistical analysis, even though the efficiency scores slightly differ among the different techniques, the efficiency scores do not present conflicting results.
Table 6 Spearman rank correlations among the efficiency scores obtained by different techniques
Figures 1a-b-c show the (positive) relationship between SFA_TR_LG and the DEA efficiency estimates.
Relationship between the parametric and non-parametric efficiency scores
Table 5 indicates that in the DEA-VRS approach three ports (Le Havre, Marseille and Rotterdam) are on the efficient frontier using both the input and output orientations. This result is consistent with the distinction between the input and output DEA orientations which reflect the different ways of reaching the efficient production frontier. Two ports, Le Havre and Marseille, also show the best score in the DEA-CRS approach, while Rotterdam decreases its performance marginally. Rotterdam and Amsterdam present the best score in the translog SFA technique. Several other ports operate at a high level of efficiency, for instance Antwerp and Algeciras. Bremerhaven appears to be inefficient in both SFA and DEA approaches. Compared to the other ports, Marseille has very low values of the terminal quay length, while Le Havre presents the minimum value of number of employees and Rotterdam has the maximum value of total gross weight of goods handled. In the DEA approach, since the efficient ports determine the technology set, the consequences are (1) that at least one port has its efficiency equal to 1 and (2) the number of inputs and outputs used in the model determines the number of efficient ports. In contrast, in the SFA approach, ports have efficiency equal to 1 only when u = 0. Therefore, there are firms with a DEA efficiency equal to 1 but have much lower SFA efficiency scores. The port policy actions aimed at inefficient ports should be considered by referring to these efficient ports to improve operational performance. Specifically, these policies could refer to the efficient 'peer' ports, since the DEA 'principle of dominance' assumes that an inefficient DMU is dominated by one (or more) peer(s) that presents the best practices.
Diverse relevant features are missing in the present analysis: the peer weights (or benchmarks), the mathematical derivation of the slacks for the efficient ports (in radial model), hypothesis tests of the different CRS/VRS technologies, and extensive explorations of the causes of the variation in the efficiency (and of the validation of the approaches proposed). Similarly, in the SFA approach, several control variables could be considered because they can have an impact on the estimated efficiency values. In fact, in addition to the estimation of the efficiency, the SFA analyses factors determining the variations in the level of efficiency. These features are beyond the aim of this paper. Concerning the indicators' assumptions needed to perform the competitive SFA and DEA, since the models can both be consistently wrong and both could report the same erroneous results, the fact that DEA and SFA appear to provide consistent results does not validate the assumptions of the model. As a consequence, the selection of the dimensions, the choice over the NACE codes, the signs of the values and the results of past studies require an appropriate corresponding analysis and further investigation.
In regard to the SFA's maximum-likelihood estimates, Table 7 indicates the corresponding results.
Table 7 Standard and translog SFA estimates
These maximum likelihood values can be also reported in equation form to estimate the translog production frontier, as follows in eq. (4).
$$ \mathit{\ln}(TGW)=80.75+3.09\mathit{\ln}(NOE)-17.46\mathit{\ln}(CTL)--1.22\mathit{\ln}\left({NOE}^2\right)-0.19\mathit{\ln}\left({CTL}^2\right)+2.16\mathit{\ln}(NOE)\mathit{\ln}(CTL)+\left(v-u\right) $$
The results indicate the β coefficients have different sign and size between standard and translog SFA. The number of employees is positively affecting output and it is statistically significant. This positive correlation with the port's output results consistent with the authors assumption. Nevertheless, the sign of 'square of number of employees' is negative, which indicates that the output increases but in decreasing manner. The negative and statistically significant coefficient of 'terminal quay length' suggests that the higher the terminal quay length, the smaller the output. This finding is coherent with the involvement of a dimension (the terminal quay length) that is a more neutral measurement compared to the container quay length since each port activity is no longer limited to just the containers' handling. Concerning the parameters of the technical efficiency model, the signs of the determinants need to be analysed as well to verify if they result in an increase or in a decrease of the inefficiency of the ports.
The conceptual framework presented in this paper is empirically analysed considering the port of Antwerp in the different scenarios. The port of Antwerp represents the most extensive port area in the world and several recent research contributions focused on this port in empirical analysis (Haezendonck and Langenus 2019; Leloup 2019). Among others, Esser et al. (2019) discussed the importance of this port as a job generator for the province of Antwerp. This port experienced exceptional economic growth in the last decade, and it has been selected as a case study because DEA and SFA techniques offer significantly different performance estimations among the diverse scenarios.
It is important to underline that the set of ten ports analysed in the paper presents heterogeneous features which should be taken into account, even though this investigation is not discussed in the present paper. For instance, Table 8 provides the (significantly different) distribution of firms by NACE codes and container ports, while Table 9 shows the distribution of firms by NACE codes and firm sizes.
Table 8 Distribution of firms by NACE codes and ports
Table 9 Port of Antwerp: Distribution of firms by NACE codes and firm size
Furthermore, present work does not consider additional features connected—for instance—to the diverse company's financial characteristics, the standardized legal form, the full-time (or part-time) prevalent jobs structure, etc. In addition to Antwerp, the present case study considers the port of Rotterdam to compare the results. The remaining eight ports are excluded from the analysis even though each of them presents specific relevant characteristics. One might think to the port of Marseille that experienced (1) a recent port reform (Lacoste and Douet 2013) and (2) increasing investments to realize an efficient integration of this port with the hinterland (Cariou et al. 2014). Therefore, further research is required on different ports and contextual factors that could potentially affect the results.
Assuming that management and port authorities are able to influence port performance, the economic significance of the current model refers to the specific policies that can be used to stimulate ports' behaviour towards diverse topics. Current case study suggests the involvement of different combinations of NACE codes for each scenario, that result in significantly different clusters of firms (and employees) analysed. Ten different scenarios have been proposed considering DEA_CRS and SFA_TR_LG scores, since these approaches presented the highest Spearman correlation. The first scenario refers to the whole set of NACE codes quoted in Table 1. Differently, the second scenario includes only the 3011 code, while the third scenario adds to this code the number of workers belonging to 3012 code, and so on. Table 10 shows the step-by-step procedure and the different efficiency scores, while Fig. 2 visually indicates the relationship among these scores for each scenario.
Table 10 DEA and SFA efficiency scores: (CRS) DEA and SFA results
Relationship between the parametric and non parametric efficiency scores
Empirical results reveal that the port of Antwerp presents different efficiency values for each scenario, and it reaches very high performance values through both the SFA and DEA techniques. This finding confirms that the combination of the NACE categories significantly impacts the performance evaluation. Managers can choose which sectors to improve, which particular improvement strategies to support, which specific service to add, and so on. One example could refer to the services for passengers and/or the concessions of the ferry routes even though they often require political decisions (Wergeland 2016), and/or the measures to support operational costs caused by accidents and dangerous port occurrences (Antão et al. 2016). In general, it is very important to mitigate the potential inaccuracies of involving specific labour categories in the port handling sector. Nevertheless, several NACE codes involved in the analysis could be inadequate. For instance, categories such as ship repairs (NACE codes 3011, 3012, 3315) and passenger-related services (NACE codes 5010 and 5030) have limited relevance with container terminals. In the authors' opinion, the usage of a broader selection of the port activities appears to be appropriate in the present analysis, although additional proxies⎯and/or different NACE selection⎯can be considered in further investigation.
The indicators' assumptions proposed to perform the performance analysis represent the major concern for the benchmarking study. This article investigates the performances of ten European ports and it assumes that the widely debated labour indicator can be estimated by fixing the firms involved in the NACE codes and NUTS2 regions. Empirical results show that, on the one hand, this approach could be useful in avoiding the exclusion of this measurement due to difficulties in collecting labour data. On the other hand, since the NACE selection impacts on the benchmarking, it is important to address the issue connected to the usage of the labour force via a coherent and consistent model. Supplementary finding derives from the results of both SFA and DEA techniques that do not present conflicting results. The Spearman coefficients show positive ranking correlations (which is relatively high considering translog SFA and DEA CRS). The outcomes of the empirical study confirm that policy actions can refer to these techniques to verify the potential impact of specific measures. In particular, the result for the labour indicator provides evidence for the relevance of the assumptions connected to it, showing significant differences among the performance evaluation. Accordingly, since the number of workers can be used to verify the efficacy of employment policies, especially when the implementation of new policies concerns definite (NACE) labour categories, management can design policy actions throughout the model proposed in the current work. Furthermore, the involvement of the NUTS2 territorial districts should be relevant to define policy measures according to the peculiarities (heterogeneity) of a specific region, as well as the impact of the national law on the reorganisation process of each port. In fact, the role of the government has strategic importance in regard to interventions aimed at a specific business, and could incorporate components connected to ports handling multiple NUTS2 regions. Even though current empirical work analyses a limited set of indicators, the outcomes highlighted the importance of their selection, and confirm the critical role of the DEA and SFA approaches as tools to support management decisions since they allow to verify the consistency of the different efficiency estimations.
One of the main limits of the current research concerns the output, since port activity is no longer limited to just cargo handling. Further investigation on the involvement of additional criteria that can be considered when the NACE/NUTS levels appear to be not fully satisfactory. For instance, features connected to ports which are close to regional border handling multiple NUTS2 districts should be debated. Several contextual factors must also be considered in the benchmark analysis to detect whether they affect port efficiency.
CRS or CCR:
Constant returns to scale
CTL:
Terminal quay length
DEA:
DMU:
Decisions making unit
NACE:
European statistical classification of economic activ
NIRS:
Non-increasing returns to scale
NUTS:
Nomenclature of territorial units for statistics
SFA:
Stochastic frontier analysis
TE:
Technical efficiency
TGW:
Total gross weight of goods handled in each port
VRS or BCC:
Variable returns to scale
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The authors are grateful to Professor Manolis Kavussanos and two anonymous referees for their helpful reviews and suggestions at the IAME 2019 conference where an earlier version of this paper was presented. We are also grateful to Professor Kee-Hung Lai and two different anonymous referees for their comments during the submission procedure of this article to Journal of Shipping and Trade.
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Department of Management and Quantitative Studies, University of Naples 'Parthenope', Naples, Italy
Claudio Quintano, Paolo Mazzocchi & Antonella Rocca
Claudio Quintano
Paolo Mazzocchi
Antonella Rocca
All authors have directly contributed to the planning, analysis, and writing of the paper. The authors have read and approved the final manuscript.
Claudio Quintano is Emeritus Professor of Economic Statistics. E-mail: [email protected]
Paolo Mazzocchi is Associate Professor of Economic Statistics. E-mail: [email protected]
Antonella Rocca is an Assistant Professor of Economic Statistics. E-mail: [email protected]
Correspondence to Paolo Mazzocchi.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Quintano, C., Mazzocchi, P. & Rocca, A. A competitive analysis of EU ports by fixing spatial and economic dimensions. J. shipp. trd. 5, 18 (2020). https://doi.org/10.1186/s41072-020-00075-x | CommonCrawl |
\begin{document}
\maketitle
\begin{abstract} We obtain spectral inequalities and asymptotic formulae for the discrete spectrum of the operator $\frac12\, \log(-\Delta)$ in an open set $\Omega\in\Bbb R^d$, $d\ge2$, of finite measure with Dirichlet boundary conditions. We also derive some results regarding lower bounds for the eigenvalue $\lambda_1(\Omega)$ and compare them with previously known inequalities. \end{abstract}
\thispagestyle{empty} \parindent=0pt \parskip=5pt
\section{Introduction} \label{sec:introduction}
In the present paper, we study spectral estimates for the logarithmic Laplacian
$L_{\text{\tiny $\Delta \,$}}\!= \log (-\Delta)$, which is a (weakly) singular integral operator with Fourier symbol $2\log |\eta|$ and arises as formal derivative $\partial_s \Big|_{s=0} (-\Delta)^s$ of fractional Laplacians at $s= 0$. The study of $L_{\text{\tiny $\Delta \,$}}\!$ has been initiated recently in \cite{HW}, where its relevance for the study of asymptotic spectral properties of the family of fractional Laplacians in the limit $s \to 0^+$ has been discussed. A further motivation for the study of $L_{\text{\tiny $\Delta \,$}}\!$ is given in \cite{jarohs-saldana-weth}, where it has been shown that this operator allows to characterize the $s$-dependence of solution to fractional Poisson problems for the full range of exponents $s \in (0,1)$. The logarithmic Laplacian also arises in the geometric context of the $0$-fractional perimeter, which has been studied recently in \cite{DNP}.
For matters of convenience, we state our results for the operator $\mathcal H= \frac{1}{2}L_{\text{\tiny $\Delta \,$}}\!$ which corresponds to the quadratic form \begin{equation} \label{log-quadratic}
\varphi \mapsto (\varphi,\varphi)_{log} := \frac{1}{(2\pi)^{d}} \, \int_{\Bbb R^d} \log(|\xi|)\, |\widehat{\varphi}(\xi)|^2\, d\xi. \end{equation} Here and in the following, we let $\widehat{\varphi}$ denote the Fourier transform $$ \xi \mapsto \widehat{\varphi}(\xi)= \int_{{\mathbb R}^d} e^{-ix \xi} \varphi(x)\,dx $$ of a function $\varphi\in L^2({\mathbb R}^d)$. Let $\Omega\subset \Bbb R^d$ be an open set of finite measure, and let ${\mathbb H}(\Omega)$ denote the closure of $C^\infty_c(\Omega)$ with respect to the norm \begin{equation}
\label{eq:def-norm--star}
\varphi \mapsto \|\varphi\|_{*}^2:= \int_{\Bbb R^d} \log(e + |\xi|)\, |\widehat{\varphi}(\xi)|^2\, d\xi. \end{equation} Then $(\cdot,\cdot)_{log}$ defines a closed, symmetric and semibounded quadratic form with domain ${\mathbb H}(\Omega) \subset L^2(\Omega)$, see Section~\ref{sec:prel-basic-prop} below. Here and in the following, we identify $L^2(\Omega)$ with the space of functions $u \in L^2({\mathbb R}^d)$ with $u \equiv 0$ on ${\mathbb R}^d \setminus \Omega$. Let $$ \mathcal H : {\mathcal D}(\mathcal H) \subset L^2(\Omega) \to L^2(\Omega) $$ be the unique self-adjoint operator associated with the quadratic form. The eigenvalue problem for $\mathcal H$ then writes as \begin{equation}\label{D} \left\{
\begin{aligned} \mathcal H \varphi &= \lambda \varphi, &&\qquad \text{in $\Omega$,}\\ \varphi &= 0, &&\qquad \text{on ${\mathbb R}^d \setminus \Omega$.}
\end{aligned} \right. \end{equation} We understand (\ref{D}) in weak sense, i.e. $$ \varphi \in {\mathbb H}(\Omega) \quad \text{and}\quad (\varphi,\psi)_{log}= \lambda \int_{\Omega}\varphi(x)\psi(x)\,dx \quad \text{for all $\psi \in {\mathbb H}(\Omega)$.} $$ As noted in \cite[Theorem 1.4]{HW}, there exists a sequence of eigenvalues $$ \lambda_1(\Omega)< \lambda_2(\Omega) \le \dots, \qquad \lim_{k \to \infty} \lambda_k(\Omega) = \infty $$
and a corresponding complete orthonormal system of eigenfunctions. We note that the discreteness of the spectrum is a consequence of the fact that the embedding ${\mathbb H}(\Omega) \hookrightarrow L^2(\Omega)$ is compact. In the case of bounded open sets, the compactness of this embedding follows easily by Pego's criterion~\cite{Pego}. In the case of unbounded open sets of finite measure, the compactness can be deduced from \cite[Theorem 1.2]{jarohs-weth} and estimates for $\|\cdot\|_*$, see Corollary~\ref{cor-compact-embedding} below.
In Section \ref{sec:prel-basic-prop}, using the results from \cite{HW} and \cite{FKV}, we discuss properties of functions from $\mathcal D(\mathcal H)$. In particular, we show that $e^{ix\xi}\big|_{x\in\Omega} \in \mathcal D(\mathcal H)$, $\xi\in\Bbb R^d$, provided $\Omega$ is an open bounded sets with Lipschitz boundary.
In Section \ref{sec:deriving-an-upper-1} we obtain a sharp upper bound for the Riesz means and for the number of eigenvalues $N(\lambda)$ of the operator $\mathcal H$ below $\lambda$. Here we use technique developed in papers \cite{Bz1}, \cite{Bz2}, \cite{LY} and \cite{L}. In \cite{Lap} it was noticed that such technique could be applied for a class of pseudo-differential operators with Dirichlet boundary conditions in domains of finite measure without any requirements on the smoothness of the boundary.
We discuss lower bounds for $\lambda_1(\Omega)$ in Section \ref{sec:lower-bound-lambd}. In Theorem \ref{lower-bound-lambda_1-first} we present an estimate that is valid for arbitrary open sets of finite measure. For sets with Lipschitz boundaries, H.Chen and T.Weth \cite{HW} have proved a Faber-Krahn inequality for the operator $\mathcal H$ that reduces the problem to the estimate of
$\lambda_1(B)$, where $B$ is a ball satisfying $|B| = |\Omega|$, see Corollary \ref{cor-faber-krahn}. In Theorem \ref{lower-bound-lambda-1-second} we find an estimate for $\lambda_1(B_d)$, where $B_d$ is the unit ball, that is better in lower dimensions than the one obtained in Theorem \ref{lower-bound-lambda_1-first}. We also compare our results with bounds resulting from previously known spectral inequalities obtained in \cite{BK} and \cite{B}.
In Section \ref{LowB1} we obtain asymptotic lower bounds using the coherent states transformation approach given in \cite{G}. It allows us to derive, in Section \ref{Weyl}, asymptotics for the Riesz means of eigenvalues in Theorem \ref{3.1} and for $N(\lambda)$ in Corollary \ref{3.2}. Here $\Omega \subset {\mathbb R}^N$ is an arbitrary open set of finite measure without any additional restrictions on the boundary.
Finally in Section \ref{LowB2} we obtain uniform bounds on the Riesz means of the eigenvalues using the fact that for bounded open sets with Lipschitz boundaries we have $e^{ix\xi}\big|_{x\in\Omega} \in \mathcal D(\mathcal H)$.
\section{Preliminaries and basic properties of eigenvalues} \label{sec:prel-basic-prop}
As before, let $(\cdot,\cdot)_{log}$ denote the quadratic form defined in (\ref{log-quadratic}), and let, for an open set $\Omega \subset {\mathbb R}^d$,
${\mathbb H}(\Omega)$ denote the closure of $C^\infty_c(\Omega)$ with respect to the norm $\|\cdot\|_*$ defined in (\ref{eq:def-norm--star}).
\begin{lem} \label{closed-semibounded} Let $\Omega \subset {\mathbb R}^d$ be an open set of finite measure. Then $(\cdot,\cdot)_{log}$ defines a closed, symmetric and semibounded quadratic form with domain ${\mathbb H}(\Omega) \subset L^2(\Omega)$. \end{lem}
\begin{proof} Obviously, the form $(\cdot,\cdot)_{log}$ is symmetric. For functions $\varphi \in C^\infty_c(\Omega)$, we have \begin{equation}
\label{eq:basic-fourier-ineq}
(2\pi)^{d} \|\varphi\|_2^2 =\|\widehat \varphi\|_2^2 \le \|\varphi\|_*^2. \end{equation} Moreover, with $c_1:= \log (e+2)+ \sup \limits_{t \ge 2}\frac{\log (e+t)}{\log t}$ we have \begin{align}
\frac{\|\varphi\|_{*}^2}{c_1} &\le \| \widehat \varphi\|_{2}^2+ \int_{|\xi| \ge 2}\ln |\xi| |\widehat{\varphi}(\xi)|^2 \, d\xi \nonumber\\
&\le (2\pi)^d \bigl( \|\varphi\|_{2}^2 + (\varphi,\varphi)_{log}\bigr)
- \int_{|\xi| \le 2}\ln |\xi| |\widehat{\varphi}(\xi)|^2 \, d\xi \nonumber \\
&\le (2\pi)^d \bigl( \|\varphi\|_{2}^2 + (\varphi,\varphi)_{log}\bigr)
+ \bigl\| \ln |\cdot| \bigr\|_{L^1(B_2(0))} \|\widehat{\varphi}\|_\infty^2 \label{closed-semibounded-est-1} \end{align} while \begin{equation}
\label{eq:closed-semibounded-est-2}
\|\widehat{\varphi}\|_\infty^2 \le \|\varphi\|_1^2 \le |\Omega|\, \|\varphi\|_2^2. \end{equation} Consequently, \begin{align}
(\varphi,\varphi)_{log} &\ge \frac{\|\varphi\|_*^2}{(2\pi)^d c_1}-
\left(1+ \frac{|\Omega|\, \bigl\| \ln |\cdot| \bigr\|_{L^1(B_2(0))}}{(2\pi)^{d}}\right)\|\varphi\|_2^2 \label{intermediate-est}\\
&\ge \left(\frac{1}{c_1}\,-\,1\,-\,\frac{|\Omega|\, \bigl\| \ln |\cdot| \bigr\|_{L^1(B_2(0))}}{(2\pi)^{d}}\right)\|\varphi\|_2^2. \nonumber \end{align}
In particular, $(\varphi,\varphi)_{log}$ is semibounded. Moreover, it follows from (\ref{intermediate-est}) and the completeness of $({\mathbb H}(\Omega),\|\cdot\|_*)$ that the form $(\varphi,\varphi)_{log}$ is closed on ${\mathbb H}(\Omega)$. \end{proof}
\begin{lem} \label{equivalent-norms} Let $\Omega \subset {\mathbb R}^d$ be an open set of finite measure. Then \begin{equation}
\label{eq:def-norm-double-star}
\varphi \mapsto \|\varphi\|_{**}^2:=
\int \!\!\! \int_{|x-y|\le 1} \frac{(\varphi(x)-\varphi(y))^2}{|x-y|^d}\,dxdy \end{equation}
defines an equivalent norm to the norm $\|\cdot\|_*$ defined in (\ref{eq:def-norm--star}) on $C^\infty_c(\Omega)$. \end{lem}
\begin{proof} Let $\varphi \in C^\infty_c(\Omega)$. By \cite[Lemma 2.7]{FKV}, we have \begin{equation}
\label{eq:FKV-lemma}
\|\varphi\|_2 \le c_2 \|\varphi\|_{**} \qquad \text{with a constant $c_2>0$ independent of $\varphi$.} \end{equation}
In particular, $\|\cdot\|_{**}$ defines a norm on $C^\infty_c(\Omega)$. Next we note that, by \cite[Theorem 1.1(ii) and Eq. (3.1)]{HW}, $$
(\varphi,\varphi)_{log} = \frac{1}{2}\int_{{\mathbb R}^d}[L_{\text{\tiny $\Delta \,$}}\! \varphi(x)]\varphi(x)\,dx = \kappa_d \|\varphi\|_{**}^2 - \int_{\Bbb R^d} [j * \varphi] \varphi \,dx + \zeta_d \|\varphi\|_2^2
$$ with \begin{equation}
\label{eq:def-zeta_d} \kappa_d:= \frac{\pi^{- \frac{d}{2}} \Gamma(d/2)}{4}, \qquad \zeta_d:= \log 2 + \frac{1}{2}\left(\psi\left(d/2\right) -\gamma\right) \end{equation} and $$
j: {\mathbb R}^d \setminus \{0\} \to {\mathbb R}, \qquad j(z)= 2 \kappa_d 1_{{\mathbb R}^d \setminus B_d}(z)|z|^{-d}. $$ Here $\psi:= \frac{\Gamma'}{\Gamma}$ is the Digamma function and $\gamma= -\Gamma'(1)$ is the Euler-Mascheroni constant. Consequently, we have \begin{align}
\Bigl|(\varphi,\varphi)_{log}- \kappa_d \|\varphi\|_{**}^2\Bigr| &\le
\|j\|_\infty \|\varphi\|_1^2 + \zeta_d \|\varphi\|_2^2 \nonumber\\
&\le \Bigl(\|j\|_\infty |\Omega| +\zeta_d\Bigr)\|\varphi\|_2^2. \label{modulus-ineq-quad-form} \end{align} As a consequence of (\ref{eq:basic-fourier-ineq}) and (\ref{modulus-ineq-quad-form}), we find that \begin{align*}
\|\varphi\|_{**}^2 &\le \frac{1}{\kappa_d}\Bigl[ (\varphi,\varphi)_{log} +
\bigl(\|j\|_\infty |\Omega|+\zeta_d\bigr) \|\varphi\|_2^2 \Bigr]\\
&\le \frac{1}{(2\pi)^d \kappa_d}\Bigl(1 + \|j\|_\infty |\Omega|+\zeta_d\Bigr)\|\varphi\|_*^2. \end{align*} Moreover, by (\ref{closed-semibounded-est-1}), (\ref{eq:closed-semibounded-est-2}), (\ref{eq:FKV-lemma}) and (\ref{modulus-ineq-quad-form}) we have \begin{align*}
&\frac{\|\varphi\|_{*}^2}{c_1}
\le (2\pi)^d \bigl(\|\varphi\|_{2}^2 + (\varphi,\varphi)_{log}\bigr)
+ \bigl\| \ln |\cdot| \bigr\|_{L^1(B_2(0))} |\Omega| \|\varphi\|_2^2\\
&\le (2\pi)^d \Bigl( \kappa_d \|\varphi\|_{**}^2
+ \bigl(1+ \|j\|_\infty |\Omega| +\zeta_d\bigr)\|\varphi\|_2^2 \Bigr)
+ \bigl\| \ln |\cdot| \bigr\|_{L^1(B_2(0))} |\Omega| \|\varphi\|_2^2\\
&\le c_3 \|\varphi\|_{**}^2 \end{align*}
with $c_3 = (2\pi)^d \kappa_d + c_2\bigl[(2\pi)^d \bigl(1+ \|j\|_\infty |\Omega|+\zeta_d\bigr) +\bigl\| \ln |\cdot| \bigr\|_{L^1(B_2(0))}|\Omega| \bigr]$. Hence the norms $\|\cdot\|_{*}$ and $\|\cdot\|_{**}$ are equivalent on $C^\infty_c(\Omega)$. \end{proof}
\begin{cor} \label{cor-compact-embedding} Let $\Omega \subset {\mathbb R}^d$ be an open set of finite measure. Then the embedding ${\mathbb H}(\Omega) \hookrightarrow L^2(\Omega)$ is compact. \end{cor}
\begin{proof} Let $\tilde {\mathbb H}(\Omega)$ be defined as the space of functions $\varphi \in L^2({\mathbb R}^d)$ with $\varphi \equiv 0$ on ${\mathbb R}^d \setminus \Omega$ and $$
\int \!\!\! \int_{|x-y|\le 1} \frac{(\varphi(x)-\varphi(y))^2}{|x-y|^d}\,dxdy <\infty. $$
By \cite[Theorem 1.2]{jarohs-weth}, the Hilbert space $(\tilde {\mathbb H}(\Omega),\|\cdot\|_{**})$ is compactly embedded in $L^2(\Omega)$.
Since, by Lemma~\ref{equivalent-norms}, the norms $\|\cdot\|_*$ and $\|\cdot\|_{**}$ are equivalent on $C^\infty_c(\Omega)$, the space ${\mathbb H}(\Omega)$ is embedded in $\tilde {\mathbb H}(\Omega)$. Hence the claim follows. \end{proof}
\begin{cor} \label{space-equivalence} Let $\Omega \subset {\mathbb R}^d$ be a bounded open set with Lipschitz boundary. \begin{enumerate} \item[(i)] The space ${\mathbb H}(\Omega)$ is equivalently given as the set of functions $\varphi \in L^2({\mathbb R}^d)$ with $\varphi \equiv 0$ on ${\mathbb R}^d \setminus \Omega$ and
\begin{equation}
\label{eq:kernel-finiteness-cond}
\int \!\!\! \int_{|x-y|\le 1} \frac{(\varphi(x)-\varphi(y))^2}{|x-y|^d}\,dxdy <\infty.
\end{equation} \item[(ii)] ${\mathbb H}(\Omega)$ contains the characteristic function $1_\Omega$ of $\Omega$ and also the restrictions of exponentials $x \mapsto 1_\Omega(x) \, e^{ix \xi}$, $\xi \in {\mathbb R}^d$. \end{enumerate} \end{cor}
\begin{proof}
(i) Let, as in the proof of Corollary~\ref{cor-compact-embedding}, $\tilde {\mathbb H}(\Omega)$ be the space of functions $\varphi \in L^2({\mathbb R}^d)$ with $\varphi \equiv 0$ on ${\mathbb R}^d \setminus \Omega$ and with (\ref{eq:kernel-finiteness-cond}), endowed with the norm $\|\cdot\|_{**}$. Since $\Omega \subset {\mathbb R}^d$ be a bounded open set with Lipschitz boundary, it follows from \cite[Theorem 3.1]{HW} that $C_0^\infty(\Omega) \subset \tilde {\mathbb H}(\Omega)$ is dense. Hence the claim follows from Lemma~\ref{equivalent-norms}.
(ii) follows from (i) and a straightforward computation. \end{proof}
Next we note an observation regarding the scaling properties of the eigenvalues $\lambda_k(\Omega)$. \begin{lem} \label{lemma-scaling-properties} Let $\Omega \subset {\mathbb R}^d$ be a bounded open set with Lipschitz boundary, and let $$ R\Omega:= \{R x\::\: x \in \Omega\}. $$ Then we have $$ \lambda_k(R \Omega) = \lambda_k(\Omega) - \log R \qquad \text{for all $k \in {\mathbb N}$.} $$ \end{lem}
\begin{proof} Since $C_0^\infty(\Omega) \subset {\mathbb H}(\Omega)$ is dense, it suffices to note that \begin{equation}
\label{eq:scaling-test-functions}
(\varphi_R,\varphi_R)_{log} = (\varphi,\varphi)_{log} - \log R \|\varphi\|_{L^2({\mathbb R}^d)}^2 \qquad \text{for $\varphi \in C^\infty_c({\mathbb R}^d)$} \end{equation}
with $\varphi_R \in C^\infty_c({\mathbb R}^d)$ defined by $\varphi_R(x)= R^{-\frac{d}{2}}\varphi(\frac{x}{R})$, whereas $\|\varphi_R\|_{L^2({\mathbb R}^d)}= \|\varphi\|_{L^2({\mathbb R}^d)}$. Since $$ \widehat{\varphi_R}= R^{\frac{d}{2}} \widehat{\varphi}(R \,\cdot \,) $$ we have \begin{align*} &(\varphi_R,\varphi_R)_{log}\\
&= \frac{1}{(2\pi)^{d}} \, \int_{\Bbb R^d} \log(|\xi|)\, |\widehat{\varphi_R}(\xi)|^2\, d\xi = \frac{R^{d}}{(2\pi)^{d}} \, \int_{\Bbb R^d} \log(|\xi|)\, |\widehat{\varphi}(R \xi)|^2\, d\xi\\
&= \frac{1}{(2\pi)^{d}} \, \int_{\Bbb R^d} \bigl(\log(|\xi|)-\log R\bigr)\, |\widehat{\varphi}(\xi)|^2\, d\xi =(\varphi,\varphi)_{log} - \log R \|\varphi\|_{L^2({\mathbb R}^d)}^2, \end{align*} as stated in (\ref{eq:scaling-test-functions}).\\ \end{proof}
\section{An upper trace bound} \label{sec:deriving-an-upper-1}
Throughout this section, we let $\Omega \subset {\mathbb R}^d$ denote an open set of finite measure. Let $\{\varphi_k\}$ and $\{\lambda_k\}$ be the orthonormal in $L^2(\Omega)$ system of eigenfunctions and the eigenvalues of the operator $\mathcal H$ respectively. In what follows we denote $$ (\lambda - t)_+ = \begin{cases} \lambda - t, & {\rm if} \quad t <\lambda, \\ 0, \quad & {\rm if} \quad t \ge \lambda. \end{cases} $$ Then we have
\begin {thm}\label{1.1} For the eigenvalues of the problem \eqref{D} and any $\lambda\in \Bbb R$ we have
\begin{equation}\label{BU}
\sum_{k}(\lambda - \lambda_k)_+ \le \frac{1}{(2\pi)^{d}}\, |\Omega|\, e^{d\lambda} \, |B_d|\, d^{-1}, \end{equation}
where $|B_d|$ is the measure of the unit ball in $\Bbb R^d$.
\end{thm}
\begin{proof} Extending the eigenfunction $\varphi_k$ by zero outside $\Omega$ and using the Fourier transform we find
\begin{multline*} \sum_{k}(\lambda - \lambda_k)_+ = \sum_{k}\left(\lambda (\varphi_k, \varphi_k) - (\mathcal H\varphi_k, \varphi_k) \right)_+ \\
= \frac{1}{(2\pi)^{d}}\, \left(\sum_k \int_{\Bbb R^d} \left(\lambda - \log(|\xi|) \right) \, |\widehat{\varphi_k}(\xi)|^2 \, d\xi \right)_+\\ \le
\frac{1}{(2\pi)^{d}}\, \int_{\Bbb R^d} \left(\lambda - \log(|\xi|) \right)_+ \, \sum_k |\widehat{\varphi_k}(\xi)|^2 \, d\xi. \end{multline*}
Using that $\{\varphi_k\}$ is an orthonormal basis in $L^2(\Omega)$ and denoting $e_\xi = e^{-i (\cdot,\xi)}$we have $$
\sum_k |\widehat{\varphi_k}(\xi)|^2 = \sum_k |(e_\xi, \varphi_k)|^2 = \|e_\xi\|^2_{L^2(\Omega)} = |\Omega|, $$ and finally obtain \begin{align*}
\sum_{k}(\lambda - \lambda_k)_+ & \le \frac{1}{(2\pi)^{d}}\, |\Omega|\, \int_{\Bbb R^d} \left(\lambda - \log(|\xi|) \right)_+ \\
& = \frac{1}{(2\pi)^{d}}\, |\Omega|\, e^{d\lambda} \, \int_{|\xi|\le 1} \log(|\xi|^{-1}) \, d\xi. \end{align*} We complete the proof by computing the last integral. \end{proof}
\noindent Let $\eta >\lambda$ and let us consider the function $$ \psi_\lambda(t) = \frac{1}{\eta - \lambda} (\eta - t)_+. $$ Denote by $\chi$ the step function $$ \chi_\lambda (t) = \begin{cases} 1, \quad & {\rm if} \quad t<\lambda,\\ 0,\quad & {\rm if} \quad t \ge \lambda, \end{cases} $$ and let $$ N(\lambda) = \# \{k:\, \lambda_k<\lambda\}, $$ be the number of the eigenvalues below $\lambda$ of the operator $\mathcal H$.
Then by using the previous statement we have $$ N(\lambda) \le \frac{1}{\eta - \lambda} \, \sum_k (\eta - \lambda_k)_+ \le
\frac{1}{\eta - \lambda} \, \frac{1}{(2\pi)^{d}}\, |\Omega|\, e^{d\eta} \, |B_d|\, d^{-1}. $$ Minimising the right hand side w.r.t. $\eta$ we find $\eta = \lambda + \frac1d $ and thus obtain the following
\begin{cor} \label{cor-N-lambda} For the number $N(\lambda)$ of the eigenvalues of the operator $\mathcal H$ below $\lambda$ we have
\begin{equation}\label{Numb} N(\lambda) \le
e^{\lambda d +1}
\frac{1}{(2\pi)^{d}}\, |\Omega| \, |B_d|. \end{equation}
\end{cor}
\section{A lower bound for $\lambda_1(\Omega)$} \label{sec:lower-bound-lambd}
In this section, we focus on lower bounds for the first eigenvalue $\lambda_1= \lambda_1(\Omega)$. From Corollary~\ref{cor-N-lambda}, we readily deduce the following bound.
\begin{thm} \label{lower-bound-lambda_1-first} Let $\Omega \subset {\mathbb R}^d$ be an open set of finite measure. Then we have \begin{equation}
\label{eq:est-lambda_1-first}
\lambda_1(\Omega) \ge \frac{1}{d} \log \frac{(2\pi)^{d}}{e |\Omega| \, |B_d|}. \end{equation}
In particular, if $|\Omega| \le \frac{(2\pi)^{d}}{e\, |B_d|}$, then the operator $\mathcal H$ does not have negative eigenvalues. \end{thm}
\begin{proof}
If $\lambda < \frac{1}{d} \log \frac{(2\pi)^{d}}{e |\Omega| \, |B_d|}$, then
$N(\lambda)<1$ by (\ref{Numb}), and therefore $N(\lambda)=0$. Consequently, $\mathcal H$ does not have eigenvalues below $\frac{1}{d} \log \frac{(2\pi)^{d}}{e |\Omega| \, |B_d|}$. \end{proof}
\begin{rem} Note that the inequalities \eqref{BU}, \eqref{Numb} and \eqref{eq:est-lambda_1-first} hold for any open set $\Omega$ of finite measure without any additional conditions on its boundary. \end{rem}
In the following, we wish to improve the bound given in Theorem~\ref{lower-bound-lambda_1-first} in low dimensions $d$ for open boundary sets with Lipschitz boundary. We shall use the following Faber-Krahn type inequality.
\begin{thm} (\cite[Corollary 1.6]{HW})\\ \label{sec:faber-Krahn-main}
Let $\rho>0$. Among all bounded open sets $\Omega$ with Lipschitz boundary and $|\Omega| = \rho$, the ball $B=B_r(0)$ with $|B|=\rho$ minimizes $\lambda_1(\Omega)$. \end{thm}
\begin{cor} \label{cor-faber-krahn} For every open bounded sets $\Omega$ with Lipschitz boundary we have \begin{equation}
\label{eq:sharp-lower-bound}
\lambda_1(\Omega) \ge \lambda_1(B_d) + \frac{1}{d}\log \frac{|B_d|}{|\Omega|}, \end{equation} and equality holds if $\Omega$ is a ball. \end{cor}
\begin{proof} The result follows by combining Theorem~\ref{sec:faber-Krahn-main} with the identity $$ \lambda_1(B_r(0)) = \lambda_1(B_d) + \log \frac{1}{r}\qquad \text{for $r>0$,} $$ which follows from the scaling property of $\lambda_1$ noted in Lemma~\ref{lemma-scaling-properties}. \end{proof}
Corollary~\ref{cor-faber-krahn} gives a sharp lower bound, but it contains the unknown quantity $\lambda_1(B_d)$. By Theorem~\ref{lower-bound-lambda_1-first}, we have \begin{align}
\lambda_1(B_d) &\ge \frac{1}{d} \log \frac{(2\pi)^{d}}{e |B_d|^2} =
\log (2\pi) -\frac{1}{d}\bigl(1+ 2 \log |B_d|\bigr) \nonumber \\ &= \frac{2}{d} \log \Gamma\left(d/2\right) + \log 2 + \frac{2}{d} \log \frac{d}{2} -\frac{1}{d}. \label{lower-bound-lambda-1-first} \end{align} The following theorem improves this lower bound in low dimensions $d \ge 2$.
\begin{thm} \label{lower-bound-lambda-1-second} For $d \ge 2$, we have \begin{equation}
\label{eq:est-lambda-1-first} \lambda_1(B_d) \ge \log \bigl(2 \sqrt{d +2}\bigr) -
\frac{2^{d+1} |B_d|^2 (d +2)^{\frac{d}2} }{d (2\pi)^{2d}}. \end{equation} \end{thm}
\begin{proof}
Let $u \in L^2(B_d)$ be radial with $\|u\|_{L^2}=1$. Then $\widehat u$ is also radial, and \begin{align*}
|\widehat u(\xi)|&=|\widehat u(s)|= s^{1-\frac{d}{2}}\left|\int_{0}^{1} u(r)J_{\frac{d}{2}-1}(rs)r^{\frac{d}{2}}dr\right| \\
&\le s^{1-\frac{d}{2}} \left( \int_0^{1}r^{d-1} u^2(r)\,dr\right)^{1/2} \left(\int_0^{1}rJ_{\frac{d}{2}-1}^2(sr) \,dr\right)^{1/2}\\
&=\frac{s^{1-\frac{d}{2}}}{\sqrt{|S^{d-1}|}} \left(s^{-2} \int_0^{s} \tau J_{\frac{d}{2}-1}^2(\tau) \,d\tau\right)^{1/2} \\
&=\frac{s^{-\frac{d}{2}}}{\sqrt{|S^{d-1}|}}
\left(\int_0^{s} \tau J_{\frac{d}{2}-1}^2(\tau) \,d\tau\right)^{1/2}\qquad \text{for $\xi \in {\mathbb R}^d$ with $s = |\xi|$.} \end{align*} Consequently, $$
|S^{d-1}| |\widehat u(s)|^2 \le s^{-d} \int_0^{s} \tau J_{\frac{d}{2}-1}^2(\tau) \,d\tau. $$ In the case where, in addition, $u$ is a radial eigenfunction of (\ref{D}) corresponding to $\lambda_1$ in $\Omega= B_d$, it follows that, for every $\lambda \in {\mathbb R}$, \begin{align*}
&(2\pi)^{d}[\lambda-\lambda_1] = \int_{{\mathbb R}^d} (\lambda -\ln |\xi|)|\widehat u(\xi)|^2\,d\xi \le \int_{{\mathbb R}^d} (\lambda -\ln |\xi|)_+|\widehat u(\xi)|^2\,d\xi\\
&=|S^{d-1}|
\int_0^\infty s^{d-1} (\lambda -\ln s)_+|\widehat u(s)|^2\,ds \le \int_0^\infty \frac{(\lambda -\ln s)_+}{s} \int_0^{s} \!\!\!\tau J_{\frac{d}{2}-1}^2(\tau) \,d\tau \,ds\\ &= \int_0^\infty \tau J_{\frac{d}{2}-1}^2(\tau) \int_{\tau}^\infty \frac{(\lambda -\ln s)_+}{s} \,ds d\tau= \int_0^{e^\lambda} \tau J_{\frac{d}{2}-1}^2(\tau) \int_{\tau}^{e^\lambda} \frac{\lambda -\ln s}{s} \,ds d\tau\\ &= \int_0^{e^\lambda} \tau J_{\frac{d}{2}-1}^2(\tau) \int_{\ln \tau}^{\lambda}(\lambda - s) \,ds d\tau= \int_0^{e^\lambda} \tau J_{\frac{d}{2}-1}^2(\tau) \int_{0}^{\lambda- \ln \tau} s \,ds d\tau\\ &= \frac{1}{2} \int_0^{e^\lambda} \tau J_{\frac{d}{2}-1}^2(\tau) \bigl(\lambda- \ln \tau \bigr)^2 \,d\tau = \frac{e^{2\lambda}}{2} \int_0^{1} \tau J_{\frac{d}{2}-1}^2(e^\lambda \tau) \ln^2 \tau \,d\tau. \end{align*} We now use the following estimate for Bessel functions of the first kind: \begin{equation}
\label{eq:bessel-est-proof} J_\nu(x) \le \frac{x^\nu}{2^\nu \Gamma(\nu+1)} \quad \text{for $\:\nu > \sqrt{3}-2$, $\:0 \le x < 2 \sqrt{2(\nu+2)}$.} \end{equation} A proof of this elementary estimate is given in the Appendix. We wish to apply (\ref{eq:bessel-est-proof}) with $\nu = \frac{d}{2}-1$. This gives $$ e^{2\lambda} J_{\frac{d}{2}-1}^2(r_0 e^\lambda \tau) \le e^{d\lambda} \frac{\tau^{d-2}}{2^{d-2}\Gamma^2 (\frac{d}{2})}=
\frac{d^2 |B_d|^2e^{d\lambda}}{(2\pi)^{d}} \tau^{d-2}
\qquad \text{for $\tau \in [0,1]$} $$ if $d \ge 2$ and $e^\lambda \le 2 \sqrt{d +2}$, i.e., if \begin{equation} \label{condition-proof} d \ge 2 \quad \text{and}\quad \lambda \le \log \bigl(2 \sqrt{d +2}\bigr). \end{equation}
Here we used that $|B_d|= \frac{2}{d} \frac{\pi^{\frac{d}{2}}}{\Gamma(d/2)}$. Consequently, if (\ref{condition-proof}) holds, we find that $$
(2\pi)^{d}[\lambda-\lambda_1] \le \frac{d^2 |B_d|^2e^{d\lambda}}{(2\pi)^{d}}\int_0^{1}
\tau^{d-1} \ln^2 \tau \,d\tau, $$ where $$ \int_0^{1} \tau^{d-1} \ln^2 \tau d\tau = - \frac{2}{d} \int_0^1 \tau^{d-1} \ln \tau d\tau = \frac{2}{d^2} \int_{0}^1 \tau^{d-1}d\tau = \frac{2}{d^3}. $$ Hence $$
(2\pi)^{d}[\lambda -\lambda_1] \le \frac{2|B_d|^2}{d (2\pi)^{d}}e^{d\lambda}, \quad \text{i.e.,}\quad \lambda_1 \ge \lambda- \frac{2|B_d|^2 }{d (2\pi)^{2d}} e^{d\lambda}. $$ Inserting the value $\lambda = \log \bigl(2 \sqrt{d +2}\bigr)$ from (\ref{condition-proof}), we deduce that $$
\lambda_1= \lambda_1(B_d) \ge \log \bigl(2 \sqrt{d +2}\bigr) - \frac{2^{d+1} |B_d|^2 (d +2)^{\frac{d}2} }{d(2\pi)^{2d}}, $$ as claimed. \end{proof}
\begin{rem}{\rm \label{rem-comparison-of-other-bounds} It seems instructive to compare the lower bounds given in (\ref{lower-bound-lambda-1-first}) and (\ref{eq:est-lambda-1-first}) with other bounds obtained from spectral estimates which are already available in the literature. We first mention Beckner's logarithmic estimate of uncertainty \cite[Theorem 1]{B}, which implies that\footnote{We note here that a different definition of Fourier transform is used in \cite{B} and therefore the inequality looks slightly different} \begin{equation*}
(\varphi,\varphi)_{log} \ge \int_{{\mathbb R}^d} \left[\psi\left(d/4\right)+ \log \frac{2}{|x|}\right]\varphi^2(x) dx \ge \left[\psi\left(d/4\right)+ \log 2\right]\|\varphi\|_2^2 \end{equation*} for functions $\varphi \in C^\infty_c(B_d)$ and therefore \begin{equation}
\label{eq:beckner-lambda-1-est} \lambda_1(B_d) \ge \psi\left(d/4\right)+ \log 2 . \end{equation} Here, as before, $\psi = \frac{\Gamma'}{\Gamma}$ denotes the Digamma function. Next we state a further lower bound for $(\varphi,\varphi)_{log}$ which follows from \cite[Proposition 3.2 and Lemma 4.11]{HW}. We have \begin{equation} \label{cw-inequality}
(\varphi,\varphi)_{log} \ge \zeta_d \|\varphi\|_2^2 \qquad \text{for $\varphi \in C^\infty_c(B_d)$,} \end{equation} where $\zeta_d$ is given in (\ref{eq:def-zeta_d}), i.e., $$ \zeta_d = \log 2 + \frac{1}{2}\left( \psi(d/2)-\gamma\right) = \left\{
\begin{aligned}
&- \gamma + \sum_{k=1}^{\frac{d-1}{2}} \frac{1}{2k-1},&&\qquad \text{$d$ odd,}\\
&\log 2 - \gamma + \sum_{k=1}^{\frac{d-2}{2}} \frac{1}{k},&&\qquad \text{$d$ even.}
\end{aligned} \right. $$ Inequality (\ref{cw-inequality}) implies that \begin{equation} \label{cw-lambda-1-bound} \lambda_1(B_d) \ge \zeta_d. \end{equation} The latter inequality can also be derived from a lower bound of Ba$\rm{\tilde{n}}$uelos and Kulczycki for the first Dirichlet eigenvalue $\lambda_1^\alpha(B_d)$ of the fractional Laplacian $(-\Delta)^{\alpha/2}$ in $B_d$. In \cite[Corollary 2.2]{BK}, it is proved that $$ \lambda_1^\alpha(B_d) \ge 2^\alpha \frac{\Gamma(1+\frac{\alpha}{2}) \Gamma(\frac{d+\alpha}{2})}{\Gamma(\frac{d}{2})}\qquad \text{for $\alpha \in (0,2)$.} $$ Combining this inequality with the characterization of $\lambda_1(B_d)$ given in \cite[Theorem 1.5]{HW}, we deduce that $$
\lambda_1(B_d)= \lim_{\alpha \to 0^+}\frac{\lambda_1^\alpha(B_d)-1}{\alpha}\ge \frac{d}{d\alpha}\Big|_{\alpha=0}\, 2^\alpha \frac{\Gamma(1+\frac{\alpha}{2}) \Gamma(\frac{d+\alpha}{2})}{\Gamma(\frac{d}{2})} = \zeta_d, $$ as stated in (\ref{cw-lambda-1-bound}).
We briefly comment on the quality of the lower bounds obtained here in low and high dimensions. In low dimensions $d \ge 2$, (\ref{eq:est-lambda-1-first}) is better than the bounds (\ref{lower-bound-lambda-1-first}), (\ref{eq:beckner-lambda-1-est}) and (\ref{cw-lambda-1-bound}). In dimension $d=1$ where the bound (\ref{eq:est-lambda-1-first}) is not available, the bound (\ref{lower-bound-lambda-1-first}) yields the best value. The following table shows numerical values of the bounds $b_1(d)$, $b_2(d)$, $b_3(d)$ resp. $b_4(d)$ given by (\ref{lower-bound-lambda-1-first}), (\ref{eq:est-lambda-1-first}), (\ref{eq:beckner-lambda-1-est}), (\ref{cw-lambda-1-bound}), respectively.
\begin{center} {\tiny \renewcommand{1}{1.6}
\begin{tabular}{ l | l | l | l | l |l | l | l | l | l | l |}
$d$ & 1 & 2 & 3 & 4&5&6&7&8&9&10\\ \hline
$b_1(d)$ & $-0,55$ & $0,19$ & $0,55$ & $0,79$ &$0,97$&$1,12$ & $1,25$ & $1,36$ & $1,46$ & $1,55$ \\ \hline $b_2(d)$ & $\quad/$& $1,28 $& $1,48 $ & $1,59 $& $1,67$&$1,73$ &$1,79$ & $1,84$& $1,89$ & $1,94$\\ \hline $b_3(d)$ &$-3.53$ & $-1,27$ & $-0,39$ &$0,12$&$0,47$ & $0,73$&$0,94$& $1,12$ &$1,27$ & $1,40$ \\ \hline $b_4(d)$ &$-0,58$& $0,12$ &$0,42$ &$0,62$& $0,76$ &$0,87$ &$0,96$ & $1,03$ & $1,10$ & $1,16$ \end{tabular} \renewcommand{1}{1} } \end{center}
To compare the bounds in high dimensions, we consider the asymptotics as $d \to \infty$. Since $\frac{\log \Gamma(t)}{t} = \log t - 1 + o(t)$ as $t \to \infty$, the bound (\ref{lower-bound-lambda-1-first}) yields \begin{equation} \label{lower-bound-lambda-1-first-asymptotics} \lambda_1(B_d) \ge \log d - 1 + o(1) \qquad \text{as $d \to \infty$,} \end{equation} whereas (\ref{eq:est-lambda-1-first}) obviously gives \begin{equation} \label{lower-bound-est-lambda-1-first-asymptotics} \lambda_1(B_d) \ge \log \sqrt{d+2} + \log 2 + o(1) \qquad \text{as $d \to \infty$,} \end{equation} Moreover, from (\ref{eq:beckner-lambda-1-est}) and the fact that \begin{equation}
\label{eq:Digamma-asymptotics} \psi(t) = \log t + o(1)\qquad\text{as $t \to \infty$,} \end{equation} we deduce that \begin{equation}
\label{eq:beckner-lambda-1-est-asymptotics} \lambda_1(B_d) \ge \log d - \log 2 + o(1) \qquad \text{as $d \to \infty$,} \end{equation} Finally, (\ref{cw-inequality}) and (\ref{eq:Digamma-asymptotics}) yield \begin{equation} \label{cw-lambda-1-bound-asymptotics} \lambda_1(B_d) \ge \log \sqrt{d} + \log 2 -\frac{\gamma}{2} + o(1) \qquad \text{as $d \to \infty$.} \end{equation} So (\ref{eq:beckner-lambda-1-est-asymptotics}) provides the best asymptotic bound as $d \to \infty$.
Numerical computations indicate that the bound (\ref{eq:est-lambda-1-first}) is better than the other bounds for $2 \le d \le 21$, and (\ref{eq:beckner-lambda-1-est}) is the best among these bounds for $d \ge 22$. }
\end{rem}
\section{An asymptotic lower trace bound}\label{LowB1}
Throughout this section, we let $\Omega \subset {\mathbb R}^d$ denote an open set of finite measure. In this section we prove the following asymptotic lower bound. A similar statement was obtained in \cite{G} for the Dirichlet boundary problem for a fractional Laplacian.
\begin {thm}\label{2.1} For the eigenvalues of the problem \eqref{D} and any $\lambda\in \Bbb R$ we have
\begin{equation}\label{BLow}
\liminf_{\lambda\to\infty} e^{-d\lambda} \sum_{k}(\lambda - \lambda_k)_+ \ge \frac{1}{(2\pi)^{d}}\, |\Omega|\, \, |B_d|\, d^{-1}. \end{equation}
\end{thm}
\begin{proof} Let us fix $\delta>0$ and consider $$ \Omega_\delta = \{ x\in \Omega: \, {\rm dist}(x, \Bbb R^d \setminus \Omega) >\delta\}. $$
Since $\delta$ is arbitrary it suffices to show the lower bound \eqref{BLow}, where $\Omega$ is replaced by $\Omega_\delta$. Let $g\in C_0^\infty(\Bbb R^d)$ be a real-valued even function, $\|g\|_{L^2(\Bbb R^d)} = 1$ with support in $\{x\in \Bbb R^d: \, |x| \le \delta/2\}$. For $\xi\in \Bbb R^d$ and $x\in \Omega_\delta$ we introduce the \lq\lq coherent state" $$ e_{\xi,y}(x) = e^{-i\xi x} g(x-y).
$$
Note that $\|e_{\xi,y}\|_{L^2(\Bbb R^d)} = 1$. Using the properties of coherent states \cite[Theorem 12.8]{LL} we obtain $$ \sum_{k}(\lambda - \lambda_k)_+ \ge
\frac{1}{(2\pi)^{d}}\, \int_{\Bbb R^d} \int_{\Omega_\delta} (e_{\xi,y}, (\lambda - \mathcal H)_+ e_{\xi,y})_{L^2(\Omega)} \, dy d\xi. $$ Since $t \mapsto (\lambda-t)_+$ is convex then applying Jensen's inequality to the spectral measure of $\mathcal H$ we obtain
\begin{equation}\label{jensen} \sum_{k}(\lambda - \lambda_k)_+ \ge \frac{1}{(2\pi)^{d}}\, \int_{\Bbb R^d} \int_{\Omega_\delta} \left(\lambda - (\mathcal H e_{\xi,y}, e_{\xi,y})_{L^2(\Omega)} \right)_+ \, dy d\xi. \end{equation}
Next we consider the quadratic form
\begin{multline*}
\left(\mathcal H e_{\xi,y}, e_{\xi,y} \right)_{L^2(\Omega)} = \frac{1}{(2\pi)^d} \, \int_{\Bbb R^d} \int_{\Omega} \int_{\Omega} e^{i(x-z)(\eta-\xi)} g(x-y)g(z-y) \log(|\eta|) \, dz dx d\eta \\ = \frac{1}{(2\pi)^d} \, \int_{\Bbb R^{d}} \int_{\Omega} \int_{\Omega}
e^{i(x-z)\rho} g(x-y)g(z-y) \log(|\xi-\rho|) \, dz dx d\rho\\ = \frac{1}{(2\pi)^d} \, \int_{\Bbb R^{d}} \int_{\Omega} \int_{\Omega}
e^{i(x-z)\rho} g(x-y)g(z-y) \left( \log|\xi| + \log \left(\left|\xi -\rho\right|/|\xi| \right)\right) \, dz dx d\rho\\
= \log|\xi| + R(y,\xi). \end{multline*}
Since $g\in C_0^\infty(\Bbb R^d)$ we have for any $M>0$
\begin{multline*} R(y,\xi) = \\ \frac{1}{(2\pi)^d} \, \int_{\Bbb R^{d}} \int_{\Omega} \int_{\Omega}
e^{i(x-y)\rho} g(x-y) e^{i(y-z)\rho} g(z-y) \log \left(\left|\xi -\rho\right|/|\xi| \right) \, dz dx d\rho\\
= \int_{\Bbb R^{d}} |\widehat{g}|^2\, \log \left(\left|\xi -\rho\right|/|\xi| \right) \, d\rho
\le C_M\,
\int_{\Bbb R^{d}} (1+ |\rho|)^{-M} \log \left(\left|\xi -\rho\right|/|\xi| \right) \, d\rho\\
\le C\, |\xi|^{-1}. \end{multline*}
Therefore from \eqref{jensen} we find
\begin{equation}\label{below1}
\sum_{k}(\lambda - \lambda_k)_+ \ge (2\pi)^{-d}\, |\Omega_\delta| \, \int_{\Bbb R^d} (\lambda - \log|\xi| - C|\xi|^{-1})_+ \, d\xi. \end{equation}
Let us redefine the spectral parameter $\lambda = \ln \mu$. Then introducing polar coordinates we find
\begin{multline}\label{below22}
\int_{\Bbb R^d} (\lambda - \log|\xi| - C|\xi|^{-1})_+ \, d\xi = \left|\Bbb S^{d-1} \right| \, \int_0^\infty \left(\ln \frac{\mu}{r} - \frac{C}{r}\right)_+ \, r^{d-1}dr\\ =
\mu^d\, \left|\Bbb S^{d-1} \right| \, \int_0^\infty \left(\ln \frac{1}{r} - \frac{C}{\mu r}\right)_+ \, r^{d-1}dr \end{multline}
The expression in the latter integral is positive if $ - r\ln r > C\mu^{-1}$. The function $ -r\ln r $ is concave.
\qquad\qquad\qquad \qquad \qquad{\centering {\includegraphics[scale=.4]{concave.png}} }
\noindent Its maximum is achieved at $r=1/e$ at the value $1/e$. The equation $ - r\ln r = C\mu^{-1}$ has two solutions $r_1(\mu)$ and $r_2(\mu)$ such that $r_1(\mu) \to 0$ and $r_2(\mu)\to 1$ as $\mu \to\infty$ Therefore
\begin{multline}\label{below3} \int_0^\infty \left(\ln \frac{1}{r} - \frac{C}{\mu r}\right)_+ \, r^{d-1}dr \ge \int_{r_1(\mu)}^{r_2(\mu)} \left(\ln \frac{1}{r} - \frac{C}{\mu r}\right) \, r^{d-1}dr \\
=-\frac1d\, r^d \ln r \Big|_{r_1(\mu)}^{r_2(\mu)}+ \frac{C}{\mu(d+1)} r^{d+1} \Big|_{r_1(\mu)}^{r_2(\mu)}
+ \frac{1}{d^2}r^d \Big|_{r_1(\mu)}^{r_2(\mu)} \to \frac{1}{d^2} \quad {\rm as} \quad \mu\to\infty. \end{multline}
Putting together \eqref{below1}, \eqref{below22} and \eqref{below3} and using $\mu = e^\lambda$ we obtain $$
\liminf_{\lambda\to\infty} e^{-d\lambda} \sum_{k}(\lambda - \lambda_k)_+ \ge \frac{1}{(2\pi)^{d}}\, |\Omega_\delta|\, \, |B_d|\, d^{-1}. $$ Since $\delta>0$ is arbitrary we complete the proof of Theorem \ref{2.1}.
\end{proof}
\section{Weyl asymptotics}\label{Weyl}
\noindent Throughout this section, we let $\Omega \subset {\mathbb R}^d$ denote an open set of finite measure. Combining Theorems \ref{1.1} and \ref{2.1} we have
\begin {thm}\label{3.1} The Riesz means of the eigenvalues of the Dirichlet boundary value problem \eqref{D} satisfy the following asymptotic formula
\begin{equation}\label{Weyl1}
\lim_{\lambda\to\infty} e^{-d\lambda}\, \sum_{k}(\lambda - \lambda_k)_+ = \frac{1}{(2\pi)^{d}}\, |\Omega|\, |B_d|\, d^{-1}. \end{equation}
\end{thm} As a corollary we can obtain asymptotics of the number of the eigenvalues of the operator $\mathcal H$.
\begin{cor} \label{3.2} The number of the eigenvalues $N(\lambda)$ of the Dirichlet boundary value problem \eqref{D} below $\lambda$ satisfies the following asymptotic formula
\begin{equation}\label{Weyl2}
\lim_{\lambda\to\infty} e^{-d\lambda} \, N(\lambda) = \frac{1}{(2\pi)^{d}}\, |\Omega|\, |B_d|. \end{equation}
\end{cor}
\begin{proof} In order to prove \eqref{Weyl2} we use two simple inequalities. If $h>0$, then
\begin{equation}\label{Nabove} \frac{(\lambda + h - \lambda_k)_+ - (\lambda - \lambda_k)_+}{h} \ge 1_{\text{\tiny $(-\infty,\lambda)$}}(\lambda_k) \end{equation}
and
\begin{equation}\label{Nbelow} \frac{(\lambda - \lambda_k)_+ - (\lambda - h- \lambda_k)_+}{h}
\le 1_{\text{\tiny $(-\infty,\lambda)$}}(\lambda_k) \end{equation}
The inequality \eqref{Nabove} implies, together with Theorems \ref{1.1} and~\ref{2.1}, that \begin{align*} &\limsup_{\lambda \to \infty}e^{-d\lambda}N(\lambda) \le \limsup_{\lambda \to \infty}e^{-d\lambda} \sum_{k}\frac{(\lambda + h - \lambda_k)_+ - (\lambda - \lambda_k)_+}{h}\\ &\le \frac{1}{h}\Bigl[e^{dh} \limsup_{\lambda \to \infty}e^{-d(\lambda+h)} \sum_{k}(\lambda + h - \lambda_k)_+ -\liminf_{\lambda \to \infty}e^{-d\lambda} \sum_{k}(\lambda - \lambda_k)_+\Bigr]\\
&\le \frac{|\Omega| |B_d|}{d(2\pi)^d}\:\frac{e^{dh}-1}{h} \qquad \text{for every $h>0$} \end{align*} and thus \begin{equation}
\label{eq:limsup-N-lambda-ineq}
\limsup_{\lambda\to\infty}e^{-d\lambda}N(\lambda) \le \frac{|\Omega| |B_d|}{d(2\pi)^d}\lim_{h \to 0^+}\frac{e^{dh}-1}{h}= \frac{|\Omega|\, |B_d|}{(2\pi)^{d}}. \end{equation} Moreover, \eqref{Nabove} implies, together with Theorems \ref{1.1} and~\ref{2.1}, that \begin{align*} &\liminf_{\lambda \to \infty}e^{-d\lambda}N(\lambda) \ge \liminf_{\lambda \to \infty}e^{-d\lambda} \sum_{k}\frac{(\lambda - \lambda_k)_+ - (\lambda -h - \lambda_k)_+}{h}\\ &\ge \frac{1}{h}\Bigl[e^{dh} \liminf_{\lambda \to \infty}e^{-d \lambda} \sum_{k}(\lambda - \lambda_k)_+ -e^{-dh}\limsup_{\lambda \to \infty} e^{-d(\lambda-h)} \sum_{k}(\lambda -h - \lambda_k)_+\Bigr]\\
&\ge \frac{|\Omega| |B_d|}{d(2\pi)^d}\:\frac{1-e^{-dh}}{h} \qquad \text{for every $h>0$} \end{align*} and therefore \begin{equation}
\label{eq:liminf-N-lambda-ineq}
\liminf_{\lambda\to\infty}e^{-d\lambda}N(\lambda) \ge \frac{|\Omega| |B_d|}{d(2\pi)^d}\lim_{h \to 0^+}\frac{1-e^{-dh}}{h}= \frac{|\Omega|\, |B_d|}{(2\pi)^{d}}. \end{equation} The claim follows by combining (\ref{eq:limsup-N-lambda-ineq}) and (\ref{eq:liminf-N-lambda-ineq}). \end{proof}
\section{An exact lower trace bound}\label{LowB2}
In this section we prove the following exact lower bound in the case of bounded open sets with Lipschitz boundary.
\begin {thm}\label{2.1-new-lower-bound} Let $\Omega \subset {\mathbb R}^d$, $N \ge 2$ be an open bounded set with Lipschitz boundary, let $\tau \in (0,1)$, and let \begin{equation} \label{def-C-Omega}
C_{\Omega,\tau} := \frac{1}{|\Omega|(2\pi)^d} \int_{\Bbb R^d}(1+|\rho|)^\tau \log(1+|\rho|) |\widehat{1_\Omega}(\rho)|^2\,d\rho, \end{equation} where $1_\Omega$ denotes the indicator function of $\Omega$.
For any $\lambda \ge 2 C_{\Omega,\tau}$, we have
\begin{equation}\label{BL} \sum_{k}(\lambda - \lambda_k)_+
\ge \frac{|\Omega|\, |B_d|}{(2\pi)^{d}\,d} \Bigl[e^{d\lambda} \,- \,a_\tau\, C_{\Omega,\tau}\,e^{(d-\tau)\lambda} \,- \,b_\tau\, C_{\Omega,\tau}^2\, e^{(d-2\tau)\lambda} \,-\, (d \lambda + 1) \Bigr] \nonumber \end{equation} with $a_\tau:= \frac{d(d-\tau)-1}{d-\tau}$ and $b_\tau := 4d \tau$. \end{thm}
\begin{rem} In the definition of $C_{\Omega,\tau}$, we need $\tau<1$, otherwise the integral might not converge. In particular, if $\Omega=B_d$ is the unit ball in ${\mathbb R}^d$, we have $$
\widehat{1_\Omega}(\rho)= (2\pi)^{\frac{d}{2}} |\rho|^{-\frac{d}{2}}J_{\frac{d}{2}}(|\rho|) $$ where $J_{\frac{d}{2}}(r)= O(\frac{1}{\sqrt{r}})$ as $r \to \infty$. Hence the integral defining $C_{\Omega,\tau}$ converges if $\tau <1$. A similar conclusion arises for cubes or rectangles, where $$ \widehat{1_\Omega}(\rho) = f_1(\rho_1) \cdot \dots \cdot f_d(\rho_d) $$
and $f_j(s) = O(\frac{1}{s})$ as $|s| \to \infty$, $j=1,\dots,d$.
On the other hand, if $\Omega \subset {\mathbb R}^d$ is an open bounded set with Lipschitz boundary, we have \begin{equation}
\label{eq:C-Omega-tau-finiteness} C_{\Omega,\tau}<\infty \qquad \text{for $\tau \in (0,1)$.} \end{equation} Indeed, in this case, $\Omega$ has finite perimeter, i.e., $1_\Omega \in BV({\mathbb R}^d)$. Therefore, as noted e.g. in \cite[Theorem 2.14]{Lombardini}, $\Omega$ also has finite fractional perimeter $$
P_\tau(\Omega)= \int_{\Omega}\int_{{\mathbb R}^d \setminus \Omega} |x-y|^{-d-\tau}\,dxdy = \frac{1}{2} \int \!\! \int_{{\mathbb R}^{2d}}\frac{(1_\Omega(x)-1_\Omega(y))^2}{|x-y|^{d+\tau}}\,dxdy $$ for every $\tau \in (0,1)$. Moreover, $P_\tau(\Omega)$ coincides, up to a constant, with the integral $$
\int_{\Bbb R^d}|\rho|^\tau |\widehat{1_\Omega}(\rho)|^2\,d\rho $$ which therefore is also finite for every $\tau \in (0,1)$. Since moreover $1_\Omega$ and therefore also $\widehat{1_\Omega}$ are functions in $L^2(\Bbb R^d)$ and for every $\varepsilon>0$ there exists $C_\varepsilon>0$ with $$
(1+|\rho|)^\tau \log(1+|\rho|) \le C_\varepsilon \bigl(1 + |\rho|^{\tau+\varepsilon}\bigr) \qquad \text{for $\rho \in {\mathbb R}^d$,} $$ it follows that (\ref{eq:C-Omega-tau-finiteness}) holds. \end{rem}
In the proof of Theorem~\ref{2.1-new-lower-bound}, we will use the following elementary estimate.
\begin{lem} \label{elem-lemma} For $r \ge 0$, $s>0$ and $\tau \in (0,1)$, we have \begin{equation}
\label{eq:first-elem-ineq} \log\left(1 + \frac{r}{s}\right) \le \frac{1}{s} \log(1+r) \qquad \text{if $s \in (0,1)$} \end{equation} and \begin{equation}
\label{eq:second-elem-ineq} \log\left(1 + \frac{r}{s}\right) \le \frac{(1+r)^\tau}{s^\tau} \log(1+r) \qquad \text{if $s \ge 1$.} \end{equation} In particular, $$ \log\left(1 + \frac{r}{s}\right) \le \max \left\{\frac{1}{s}, \frac{1}{s^\tau} \right\} (1+r)^\tau\log(1+r) \qquad \text{for $r,s>0$.} $$ \end{lem}
\begin{rem} The obvious bound $\log(1 + \frac{r}{s}) \le \frac{r}{s}$ will not be enough for our purposes. We need an upper bound of the form $g(s)h(r)$ where $h$ grows less than linearly in $r$. \end{rem}
\begin{proof}[Proof of Lemma~\ref{elem-lemma}] Let first $s \in (0,1)$. Since $$
\log\left(1 + \frac{r}{s}\right)\Big|_{r=0} = 0 = \frac{1}{s} \log(1+r)\Big|_{r=0} $$ and, for every $r>0$, $$ \frac{d}{dr} \log\left(1 + \frac{r}{s}\right) = \frac{1}{s+r} \le \frac{1}{s+ sr} = \frac{d}{dr} \frac{1}{s} \log(1+r), $$ inequality (\ref{eq:first-elem-ineq}) follows. To see (\ref{eq:second-elem-ineq}), we fix $s>1$, and we note that $$
\log\left(1 + \frac{r}{s}\right)\Big|_{r=0} = 0 = \frac{(1+r)^\tau}{s^\tau} \log(1+r)\Big|_{r=0}. $$ Moreover, for $0 < r \le s-1$, we have \begin{align*} &\frac{d}{dr} \frac{(1+r)^\tau}{s^\tau} \log(1+r)= \frac{(1+r)^{\tau-1}}{s^\tau}(1+ \tau \log(1+r))\\ &\ge \frac{(1+r)^{\tau-1}}{s^\tau} \ge \frac{1}{s}\ge \frac{1}{s+r}= \frac{d}{dr} \log\left(1 + \frac{r}{s}\right), \end{align*} so the inequality holds for $r \le s-1$. If, on the other hand, $r \ge s-1$, we have obviously $$ \log\left(1 + \frac{r}{s}\right) \le \log(1 + r) \le \frac{(1+r)^\tau}{s^\tau} \log(1+r). $$ \end{proof}
We may now complete the
\begin{proof}[Proof of Theorem~\ref{2.1-new-lower-bound}]
For $\xi \in {\mathbb R}^d$, we define $f_\xi \in L^2(\Bbb R^d)$ by $f_\xi(x)= \frac{1}{\sqrt{|\Omega|}}1_{\Omega} e^{-i x \xi}$. Note that $\|f_{\xi}\|_{L^2(\Bbb R^d)} = 1$ for any $\xi \in \Bbb R^d$. We write \begin{align*}
\sum_{k}(\lambda - \lambda_k)_+ &= \sum_{k}(\lambda - \lambda_k)_+ \|\varphi_k\|_{L^2(\Omega)}^2 = \frac{1}{(2\pi)^{d}} \sum_{k}(\lambda - \lambda_k)_+
\|\widehat{\varphi_k}\|_{L^2(\Bbb R^{d})}^2 \\
&=\frac{|\Omega|}{(2\pi)^{d}} \sum_{k}(\lambda - \lambda_k)_+ \int_{\Bbb R^{d}} |\langle f_\xi,\varphi_k \rangle|^2\,d\xi \\
&=\frac{|\Omega|}{(2\pi)^{d}} \int_{\Bbb R^{d}} \sum_{k}(\lambda - \lambda_k)_+ |\langle f_\xi,\varphi_k \rangle|^2\,d\xi. \end{align*}
Since $\sum \limits_{k} |\langle f_\xi,\varphi_k \rangle|^2 = \|f_{\xi}\|_{L^2(\Bbb R^d)}^2 = 1$ for $\xi \in \Bbb R^d$, Jensen's inequality gives \begin{align} \sum_{k}(\lambda - \lambda_k)_+ &\ge
\frac{|\Omega|}{(2\pi)^{d}} \int_{\Bbb R^{d}} \Bigl( \lambda \sum_{k}|\langle f_\xi,\varphi_k \rangle|^2 - \sum_k \lambda_k |\langle f_\xi,\varphi_k \rangle|^2\Bigr)_+\,d\xi \nonumber\\
&=\frac{|\Omega|}{(2\pi)^{d}} \int_{\Bbb R^{d}} \Bigl( \lambda - \sum_k \lambda_k |\langle f_\xi,\varphi_k \rangle|^2\Bigr)_+\,d\xi \nonumber\\
&=\frac{|\Omega|}{(2\pi)^{d}} \int_{\Bbb R^{d}} \Bigl( \lambda - ( \mathcal H f_\xi, f_\xi ) \Bigr)_+\,d\xi. \label{jensen-new} \end{align} Here, since $$
\sqrt{|\Omega|}\, \widehat {f_\xi}(\eta-\xi)= \int_{\Omega}e^{-i(\eta-\xi) x}e^{-ix \xi}\,dx = \int_{\Omega}e^{-i \eta x} \,dx = \widehat{1_\Omega}(\eta) $$ for $\eta, \xi \in {\mathbb R}^d$, we have \begin{align}
&|\Omega|(2\pi)^d \left(\mathcal H f_{\xi}, f_{\xi} \right) = |\Omega| \int_{\Bbb R^d}
\log |\eta| |\widehat {f_\xi}(\eta)|^2d \eta = |\Omega| \int_{\Bbb R^d}
\log |\eta-\xi| |\widehat {f_\xi}(\eta-\xi)|^2d \eta \nonumber\\
&=\int_{\Bbb R^d} \log|\eta-\xi| |\widehat{1_\Omega}(\eta)|^2\,d\eta \le \int_{\Bbb R^d}
\left[ \log |\xi| +\log (1+|\eta|/|\xi|)\right] |\widehat{1_\Omega}(\eta)|^2\,d\eta \nonumber\\
&\le \log |\xi| \int_{\Bbb R^d} |\widehat{1_\Omega}(\eta)|^2\,d\eta+ \max \left\{ \frac{1}{|\xi|}, \frac{1}{|\xi|^\tau}\right\}
\int_{\Bbb R^d}(1+|\eta|)^{\tau} (\log(1+|\eta|)|\widehat{1_\Omega}(\eta)|^2\,d\eta \nonumber\\
&= |\Omega|(2\pi)^d \Bigl(\log |\xi| + \max \left\{ \frac{1}{|\xi|}, \frac{1}{|\xi|^\tau}\right\} C_{\Omega,\tau} \Bigr)\qquad \text{for $\xi \in {\mathbb R}^d$,}\label{jensen-new-compl} \end{align} where $C_{\Omega,\tau}$ is defined in (\ref{def-C-Omega}). Here we used Lemma~\ref{elem-lemma}. Combining (\ref{jensen-new}) and (\ref{jensen-new-compl}), we get \begin{equation} \sum_{k}(\lambda - \lambda_k)_+ \ge
\frac{|\Omega|}{(2\pi)^{d}} \int_{\Bbb R^{d}} \Bigl(\lambda - \log |\xi| -
\max \left\{ \frac{1}{|\xi|}, \frac{1}{|\xi|^\tau}\right\} C_{\Omega,\tau}\Bigr)_+d\xi. \label{intermediate-new} \end{equation} Let us redefine the spectral parameter $\lambda = \log \mu$ again. Then we find \begin{align}
&\int_{\Bbb R^{d}} \Bigl(\lambda - \log |\xi| -
\max \left\{ \frac{1}{|\xi|}, \frac{1}{|\xi|^\tau} \right\} C_{\Omega,\tau} \Bigr)_+d\xi \nonumber \\
&= \left|\Bbb S^{d-1} \right| \, \int_0^\infty \left(\log \frac{\mu}{r} - \max \left\{ \frac{1}{r}, \frac{1}{r^\tau}\right\} C_{\Omega,\tau} \right)_+ \, r^{d-1}dr \nonumber \\ &=
\mu^d\, \left|\Bbb S^{d-1} \right| \, \int_0^\infty \left(-\log r - \max \left\{ \frac{1}{\mu^{1-\tau} r}, \frac{1}{r^\tau}\right\} \frac{C_{\Omega,\tau}}{\mu^\tau} \right)_+ \, r^{d-1}dr
\nonumber \\ &\ge
\mu^d\, \left|\Bbb S^{d-1} \right| \, \int_{\frac{1}{\mu}}^\infty \left(-\log r - \frac{1}{r^\tau}\frac{C_{\Omega,\tau}}{\mu^\tau} \right)_+ \, r^{d-1}dr. \label{below2} \end{align} For the last inequality, we used the fact that $\frac{1}{\mu^{1-\tau} r} \le \frac{1}{r^\tau}$ for $r \ge \frac{1}{\mu}$.
Next we note that the function $r \mapsto f_\mu(r) = -\log r - \frac{1}{r^\tau}\frac{C_{\Omega,\tau}}{\mu^\tau}$ satisfies \begin{equation}
\label{eq:boundary-conditions} f_\mu(r)<0 \quad \text{for $r \ge 1$}\qquad \text{and}\qquad \lim_{r \to 0^+}f_\mu(r)= -\infty. \end{equation} Moreover, this function has two zeros $r_1(\mu), r_2(\mu)$ with $0<r_1(\mu)< \frac{1}{\mu} < r_2(\mu)<1$ and $$ f_\mu(r)\ge 0 \qquad \text{if and only if}\quad r_1(\mu) \le r \le r_2(\mu). $$ To see this, we write $$ f_\mu(r)= \frac{1}{r^\tau}g(r^\tau) \qquad \text{with}\qquad g(s)=-\frac{s}{\tau} \log s - \frac{C_{\Omega,\tau}}{\mu^\tau} $$ and note that $g$ is strictly concave since $s \mapsto g'(s)= -\frac{1}{\tau} - \log s$ is strictly decreasing. Consequently, $g$ has at most two zeros, and the same is true for $f$. Combining this with (\ref{eq:boundary-conditions}) and the fact that $$ f(1/\mu) = \log \mu - C_{\Omega,\tau} > 0 $$ since $\lambda \ge 2 C_{\Omega,\tau} >C_{\Omega,\tau}$ by assumption, the claim above follows. From (\ref{below2}), we thus obtain the lower bound \begin{align}
\label{eq:r-2-mu-est}
\int_{\Bbb R^{d}} &\Bigl(\lambda - \log |\xi| -
\max \bigl\{ \frac{1}{|\xi|}, \frac{1}{|\xi|^\tau} \bigl\} C_{\Omega,\tau} \Bigr)_+d\xi \\
&\ge \mu^d\, \left|\Bbb S^{d-1} \right| \, \int_{\frac{1}{\mu}}^{r_2(\mu)} \left(-\log r - \frac{1}{r^\tau}\frac{C_{\Omega,\tau}}{\mu^\tau} \right)_+ \, r^{d-1}dr. \nonumber \end{align} Next, we claim that \begin{equation} \label{r-2-mu-lower-bound} r_2(\mu) \ge r_3(\mu):= e^{\frac{1}{2\tau}\bigl(\sqrt{1- \frac{4\tau C_{\Omega,\tau}}{\mu^\tau}}\;-1\bigr)}. \end{equation} Here we note that $\frac{4\tau C_{\Omega,\tau}}{\mu^\tau}=\frac{4\tau C_{\Omega,\tau}}{e^{\tau \lambda}} <1$ since $\lambda \ge 2 C_{\Omega,\tau}$ by assumption. To see (\ref{r-2-mu-lower-bound}), we write $$ r_3(\mu)= e^{- c_\mu \frac{C_{\Omega,\tau}}{\mu^\tau}}\qquad \text{with}\qquad c_\mu = \frac{\mu^\tau}{2 \tau C_{\Omega,\tau}}\Bigl(1 - \sqrt{1- \frac{4\tau C_{\Omega,\tau}}{\mu^\tau}}\Bigr), $$ noting that $$ \frac{\tau C_{\Omega,\tau}}{\mu^\tau} c_\mu^2 -c_\mu +1= 0 $$ and therefore \begin{align*} &f(r_3(\mu)) = f(e^{-\frac{c_\mu C_{\Omega,\tau}}{\mu^\tau}})=\frac{c_\mu C_{\Omega,\tau}}{\mu^\tau} - \frac{1}{e^{-\tau \frac{c_\mu C_{\Omega,\tau}}{\mu^\tau}}} \frac{C_{\Omega,\tau}}{\mu^\tau}\\ &=\frac{C_{\Omega,\tau}}{\mu^\tau e^{-\tau \frac{c_\mu C_{\Omega,\tau}}{\mu^\tau}}} \Bigl( c_\mu e^{-\tau \frac{c_\mu C_{\Omega,\tau}}{\mu^\tau}} - 1\Bigr) h\ge \frac{C_{\Omega,\tau}}{\mu^\tau e^{-\tau \frac{c_\mu C_{\Omega,\tau}}{\mu^\tau}}}\Bigl( c_\mu \bigl(1 - \tau \frac{c_\mu C_{\Omega,\tau}}{\mu^\tau}\bigr)-1\Bigr) = 0. \end{align*} This proves (\ref{r-2-mu-lower-bound}). As a consequence of the inequality $\sqrt{1-a} \ge 1-\frac{a}{2} -\frac{a^2}{2}$ for $0 \le a \le 1$, we also have $$ r_3(\mu) \ge e^{- \bigl(\frac{C_{\Omega,\tau}}{\mu^\tau}+\frac{4\tau C_{\Omega,\tau}^2}{\mu^{2\tau}}\bigr)} = : r_4(\mu). $$ Consequently, \begin{align*}
\int_{\Bbb R^{d}} &\Bigl(\lambda - \log |\xi| -
\max \bigl\{ \frac{1}{|\xi|}, \frac{1}{|\xi|^\tau} \bigl\} C_{\Omega,\tau} \Bigr)_+d\xi \\
&\ge \mu^d\, \left|\Bbb S^{d-1} \right| \, \int_{\frac{1}{\mu}}^{r_4(\mu)} \left(-\log r - \frac{1}{r^\tau}\frac{C_{\Omega,\tau}}{\mu^\tau} \right)_+ \, r^{d-1}dr\\ &=
\mu^d\, \left|\Bbb S^{d-1} \right| \, \Bigl[-\frac{r^d}{d} \log r + \frac{1}{d^2}r^d - \frac{C_{\Omega,\tau}}{\mu^\tau(d-\tau)}r^{d-\tau} \Bigr]_{\frac{1}{\mu}}^{r_4(\mu)}\\ &=
\mu^d\, \left|\Bbb S^{d-1} \right| \,\Bigl( \Bigl[-\frac{r_4(\mu)^d}{d} \log r_4(\mu) + \frac{1}{d^2}r_4(\mu)^d - \frac{C_{\Omega,\tau}}{\mu^\tau(d-\tau)}r_4(\mu)^{d-\tau} \Bigr]\\ &- \Bigl[\frac{\mu^{-d}}{d} \log \mu + \frac{1}{d^2}\mu^{-d} - \frac{C_{\Omega,\tau}}{\mu^\tau(d-\tau)}\mu^{\tau-d} \Bigr]\Bigr), \end{align*} which implies that \begin{align*}
\int_{\Bbb R^{d}} &\Bigl(\lambda - \log |\xi| -
\max \bigl\{ \frac{1}{|\xi|}, \frac{1}{|\xi|^\tau} \bigl\} C_{\Omega,\tau} \Bigr)_+d\xi \\ &\ge
\mu^d\, \left|\Bbb S^{d-1} \right| \,\Bigl(\frac{1}{d^2}r_4(\mu)^d - \frac{C_{\Omega,\tau}}{\mu^\tau(d-\tau)}r_4(\mu)^{d-\tau}- \frac{\mu^{-d}}{d} \log \mu - \frac{1}{d^2}\mu^{-d} \Bigr)\\ &=
\mu^d\, \left|\Bbb S^{d-1} \right| \,\Bigl(\frac{1}{d^2}e^{- d\bigl(\frac{C_{\Omega,\tau}}{\mu^\tau}+\frac{4\tau C_{\Omega,\tau}^2}{\mu^{2\tau}}\bigr)}
- \frac{C_{\Omega,\tau}}{\mu^\tau(d-\tau)}e^{- (d-\tau)\bigl(\frac{C_{\Omega,\tau}}{\mu^\tau}+\frac{4\tau C_{\Omega,\tau}^2}{\mu^{2\tau}}\bigr)}\\ &- \frac{\mu^{-d}}{d} \log \mu - \frac{1}{d^2}\mu^{-d} \Bigr). \end{align*} Since $$ e^{- d\bigl(\frac{C_{\Omega,\tau}}{\mu^\tau}+\frac{4\tau C_{\Omega,\tau}^2}{\mu^{2\tau}}\bigr)} \ge 1- d\bigl(\frac{C_{\Omega,\tau}}{\mu^\tau}+\frac{4\tau C_{\Omega,\tau}^2}{\mu^{2\tau}}\bigr) $$ and $$ e^{- (d-\tau)\bigl(\frac{C_{\Omega,\tau}}{\mu^\tau}+\frac{4\tau C_{\Omega,\tau}^2}{\mu^{2\tau}}\bigr)} \le 1, $$ we conclude that \begin{align*}
\int_{\Bbb R^{d}} &\Bigl(\lambda - \log |\xi| -
\max \bigl\{ \frac{1}{|\xi|}, \frac{1}{|\xi|^\tau} \bigl\} C_{\Omega,\tau} \Bigr)_+d\xi \\ &\ge
\mu^d\, \frac{\left|\Bbb S^{d-1} \right|}{d^2} \,\Bigl(1- d\bigl(\frac{C_{\Omega,\tau}}{\mu^\tau}+\frac{4\tau C_{\Omega,\tau}^2}{\mu^{2\tau}}\bigr)
- \frac{C_{\Omega,\tau}}{\mu^\tau(d-\tau)}- \mu^{-d}(d \log \mu + 1) \Bigr)\\ &=
\frac{\left|B_d \right|}{d} \,\Bigl(\mu^d - C_{\Omega,\tau}(d-\frac{1}{d-\tau})\mu^{d-\tau} - 4d\tau C_{\Omega,\tau}^2 \mu^{d-2\tau} - (d \log \mu + 1) \Bigr)\\ &=
\frac{\left|B_d \right|}{d} \,\Bigl(e^{d \lambda} - \frac{d(d-\tau)-1}{d-\tau} C_{\Omega,\tau}e^{(d-\tau)\lambda} - 4d\tau C_{\Omega,\tau}^2 e^{(d-2\tau)\lambda} - (d \lambda + 1) \Bigr). \end{align*} Combining the last estimate with (\ref{intermediate-new}), we get the asserted lower bound. \end{proof}
\section{Appendix: Note on a bound for Bessel functions} \label{sec:appendix:-note-bound}
The following elementary bound might be known but seems hard to find in this form.
\begin{lem} For $\nu \ge \sqrt{3}-2$ and $0 \le x \le 2 \sqrt{2(\nu+2)}$ we have $$
|J_\nu(x)| \le \frac{x^\nu}{2^\nu \Gamma(\nu+1)}. $$ \end{lem}
\begin{proof} We use the representation $$ J_\nu(x)= \Bigl(\frac{x}{2}\Bigr)^{\nu} \sum_{m=0}^\infty \frac{(-1)^m}{m! \Gamma(m+\nu + 1)} \Bigl(\frac{x}{2}\Bigr)^{2m}. $$ For $0 \le x \le 2 \sqrt{2(\nu+2)}$ and $m \ge 1$, we have $$ \Bigl(\frac{x}{2}\Bigr)^{2} \le (m+1)(m+\nu+1) = \frac{(m+1) \Gamma(m+\nu + 2)}{\Gamma(m+\nu + 1)} $$ and therefore \begin{equation}
\label{eq:bessel-proof-1} \frac{\Gamma(\nu+1)}{(m+1)! \Gamma(m+\nu + 2)} \Bigl(\frac{x}{2}\Bigr)^{2(m+1)} \le \frac{\Gamma(\nu+1)}{m! \Gamma(m+\nu + 1)} \Bigl(\frac{x}{2}\Bigr)^{2m}. \end{equation} Consequently, \begin{align*} J_\nu(x) &= \frac{x^\nu}{2^\nu \Gamma(\nu+1)}\Bigl[1 + \sum_{m=1}^\infty \frac{(-1)^m \Gamma(\nu+1)}{m! \Gamma(m+\nu + 1)} \Bigl(\frac{x}{2}\Bigr)^{2m}\Bigr]\le \frac{x^\nu}{2^\nu \Gamma(\nu+1)}. \end{align*} From (\ref{eq:bessel-proof-1}) we also deduce that \begin{align*} J_\nu(x) &\ge \frac{x^\nu}{2^\nu \Gamma(\nu+1)} \Bigl[1 - \frac{\Gamma(\nu+1)}{\Gamma(\nu + 2)} \Bigl(\frac{x}{2}\Bigr)^{2}+ \frac{\Gamma(\nu+1)}{2\Gamma(\nu + 3)} \Bigl(\frac{x}{2}\Bigr)^{4}- \frac{\Gamma(\nu+1)}{6\Gamma(\nu + 4)} \Bigl(\frac{x}{2}\Bigr)^{6}\Bigr]\\ &= \frac{x^\nu}{2^\nu \Gamma(\nu+1)} \Bigl[1-\frac{1}{\nu + 1}f\bigl(\bigl(\frac{x}{2}\bigr)^{2} \bigr)\Bigr] \end{align*} with $f: {\mathbb R} \to {\mathbb R}$ given by $f(t)= t - \frac{t^2}{2(\nu+2)}+ \frac{t^3}{6(\nu+2)(\nu+3)}$. Since $$ f'(t)= 1- \frac{t}{\nu+2} + \frac{t^2}{2(\nu+2)(\nu+3)}, \qquad \text{and}\qquad f''(t)= \frac{1}{\nu+2}\bigl(\frac{t}{\nu+3}- 1\bigr) $$ we have $$ f'(t) \ge f'(\nu+3) = 1- \frac{\nu+3 }{\nu+2} + \frac{\nu+3}{2(\nu+2)} = 1 - \frac{1}{2}\frac{\nu+3 }{\nu+2} \ge 0 \quad \text{for $t \in {\mathbb R}$ if $\nu \ge -1$} $$ and therefore $$ f(t) \le f(2(\nu +2))= 2(\nu+2) - \frac{[2(\nu+2)]^2}{2(\nu+2)}+ \frac{[2(\nu+2)]^3}{6(\nu+2)(\nu+3)}= \frac{4(\nu+2)^2}{3(\nu+3)} $$ for $t \le 2(\nu+2)$ if $\nu \ge -1$. Since $\frac{4(\nu+2)^2}{3(\nu+3)} \le \frac{2}{\nu+1}$ for $\nu \ge \sqrt{3}-2$, we conclude that $$ J_\nu(x) \ge \frac{x^\nu}{2^\nu \Gamma(\nu+1)} \Bigl[1-\frac{1}{\nu + 1}f\bigl(\bigl(\frac{x}{2}\bigr)^{2} \bigr)\Bigr]\ge - \frac{x^\nu}{2^\nu \Gamma(\nu+1)}. $$ for $\nu \ge \sqrt{3}-2$ and $0 \le x \le 2 \sqrt{2(\nu+2)}$. The claim thus follows. \end{proof}
\noindent {\it Acknowledgements}. AL was supported by the RSF grant 19-71-30002.
\end{document} | arXiv |
\begin{document}
\title{Real-analytic AbC constructions on the torus}
\begin{abstract} In this article we demonstrate a way to extend the AbC (approximation by conjugation) method invented by Anosov and Katok from the smooth category to the category of real-analytic diffeomorphisms on the torus. We present a general framework for such constructions and prove several results. In particular, we construct minimal but not uniquely ergodic diffeomorphisms and nonstandard real-analytic realizations of toral translations. \end{abstract}
\tableofcontents
\section{Introduction} An important question in Smooth Ergodic Theory asks if there are smooth versions to the objects and concepts of abstract ergodic theory. One of the most powerful tools of constructing volume preserving $C^{\infty}$-diffeomorphisms with prescribed ergodic or topological properties on any compact connected manifold $M$ of dimension $m\geq 2$ admitting a non-trivial circle action $\mathcal{S} = \left\{\phi^t\right\}_{t \in \mathbb{S}^1}$ is the so called approximation by conjugation-method developed by D.V. Anosov and A. Katok in their fundamental paper \cite{AK}. These diffeomorphisms are constructed as limits of conjugates $T_n = H^{-1}_n \circ \phi^{\alpha_{n}} \circ H_n$, where $\alpha_{n} = \frac{p_n}{q_n}= \alpha_{n-1} + \frac{1}{s_{n-1} \cdot k_{n-1} \cdot l_{n-1} \cdot q^2_{n-1}} \in \mathbb{Q}$, $H_n = h_n \circ H_{n-1}$ and $h_n$ is a measure-preserving diffeomorphism satisfying $\phi^{\alpha_{n-1}} \circ h_n = h_n \circ \phi^{\alpha_{n-1}}$. In each step the conjugation map $h_n$ and the parameters $k_{n-1}, l_{n-1}$ are chosen such that the diffeomorphism $f_n$ imitates the desired property with a certain precision. Then the parameter $s_{n-1}$ is chosen large enough to guarantee closeness of $f_{n}$ to $f_{n-1}$ in the $C^{\infty}$-topology and so the convergence of the sequence $\left(f_n\right)_{n \in \mathbb{N}}$ to a limit diffeomorphism is provided. This method enables the construction of smooth diffeomorphisms with specific ergodic properties (e. g. weak mixing ones in \cite[section 5]{AK}) or non-standard smooth realizations of measure-preserving systems (e. g. \cite[section 6]{AK} and \cite{FSW}). See also the very interesting survey article \cite{FK} for more details and other results of this method.
Unfortunately, there are great challenging differences in the real-analytic category as discussed in \cite[section 7.1]{FK}: Since maps with very large derivatives in the real domain or its inverses are expected to have singularities in a small complex neighbourhood, for a real analytic family $S_t$, $0 \leq t \leq t_0$, $S_0 = \text{id}$, the family $h^{-1} \circ S_t \circ h$ is expected to have singularities very close to the real domain for any $t>0$. So, the domain of analycity for maps of our form $f_n = H^{-1}_n \circ \phi^{\alpha_{n}} \circ H_n$ will shrink at any step of the construction and the limit diffeomorphism will not be analytic. Thus, it is necessary to find conjugation maps of a special form which may be inverted more or less explicitly in such a way that one can guarantee analycity of the map and its inverse in a large complex domain. Using very explicit conjugation maps Saprykina was able to construct examples of volume-preserving uniquely ergodic real-analytic diffeomorphims on $\mathbb{T}^2$ (\cite{S}). Fayad and Katok designed such examples on any odd-dimensional sphere in \cite{FK-ue}.
The goal of this article is to reproduce some examples of smooth dynamical systems obtained by the AbC (approximation by conjugation) scheme in the category of real-analytic diffeomorphisms on the torus $\mathbb{T}^d$, $d \geq 2$. For this purpose, we introduce the concept of block-slide type maps on the torus and demonstrate that these maps in a certain sense can be approximated well enough by measure preserving real-analytic diffeomorphisms. This allows us to carry out many AbC constructions in the real-analytic category. We briefly summarise certain previously past results and prove several new ones using real-analytic approximation of block-slide type maps. Note that all constructions in this article are done on the torus. Real-analytic AbC constructions on arbitrary real-analytic manifolds continue to remain an intractable problem.
Throughout this article $\mathbb T^d$ will denote the $d$ dimensional torus and $\mu $ will stand for the usual Lebesgue measure on $\mathbb T^d$. We will use $\text{Diff }^\omega_\rho(\mathbb T^d\, \mu )$ to denote the set of real-analytic $\mu$-measure preserving diffeomorphisms of the $d$ dimensional torus whose lift can be extended holomorphically to a complex neighbourhood of diameter at least $\rho$.
First we tackle the problem of non-standard realizations (i.e. to find a diffeomorphism which is metrically but not smoothly isomorphic to a given measure-preserving transformation). The first author used the AbC method and the concept of real-analytic approximation of block-slide type maps to find examples of measure preserving real-analytic, ergodic diffeomorphisms on the torus that are metrically isomorphic to some irrational rotation of the circle. The precise theorem can be stated as follows:
\begin{theorem}[\cite{Ba-Ns}] \label{nsr circle rotation} For any $\rho>0$ and any integer $d\geq 1$, there exist real-analytic diffeomorphisms $T\in\text{Diff }^\omega_\rho(\mathbb T^d\, \mu )$ which are metrically isomorphic to some irrational rotations of the circle. \end{theorem}
The diffeomorphisms constructed by Anosov and Katok in \cite[section 4]{AK} realized circle rotations smoothly with Liouvillean rotation numbers. However, it was not clear from this construction which Liouvillean rotations were realized. Later, Fayad, Saprykina and Windsor extended this result in \cite{FSW} and proved that any Liouvillean rotation of the circle can be realized. In the analytic category, we can not expect realization of every Liouvillean rotation of the circle but we can give a precise description of a subset of some of the Liouvillean rotations we realize. We introduce the set $\mathcal{L}_{\ast}$ of numbers contained in the set of Non-Brjuno numbers: $\mathcal{\alpha} \in \mathbb{R}$ is in $\mathcal{L}_{\ast}$ if for every $k \in \mathbb{N}$ there is $\left( p,q\right) \in \mathbb{Z} \times \mathbb{N}$ with $p,q$ relatively prime satisfying \begin{align}
\Big|{\alpha - \frac{p}{q}}\Big| < \frac{1}{\mathrm{e}^{\mathrm{e}^{k^q}}}. \end{align} In a later section we will examine this set of numbers. In particular, we will show that $\mathcal{L}_{\ast}$ is a dense $G_{\delta}$-subset of $\mathbb{R}$. We will prove,
\begin{maintheorem} \label{nsr circle rotation estimated} For any $\rho>0$ and every $\alpha \in \mathcal{L}_{\ast}$ there exists a real-analytic diffeomorphism $T\in\text{Diff }^\omega_\rho(\mathbb T^2\, \mu )$ which is metrically isomorphic to the rotation $S_\alpha$ of the circle. \end{maintheorem}
In the realm of non-standard realizations, there is another set of questions dedicated to the realization of ergodic translations of a torus on another manifold. In the original paper Anosov and Katok showed that certain ergodic translations on a $d$ dimensional torus can be realized as measure preserving smooth diffeomorphisms on any smooth manifold admitting an effective $\mathbb T^1$ action (see \cite[section 6]{AK}). We should note that this result was further improved by Benhenda and it was shown that one can realize any ergodic translation on $\mathbb T^d$ with one arbitrary Liouvillean coordinate (see \cite{Mb-ts}).
It appears that the block-slide type maps allow enough flexibility for us to realize some of these ergodic translations analytically on another torus of arbitrary dimension. We prove,
\begin{maintheorem} \label{theorem nsr total translations} For any $\rho>0$ and any two integers, $h\geq 1$ and $d\geq 2$, there exists an ergodic real-analytic diffeomorphism $T\in\text{Diff }^\omega_\rho(\mathbb T^d)$ which is metrically isomorphic to an ergodic translation of $\mathbb T^h$. \end{maintheorem}
And the obvious corollary follows:
\begin{maintheorem} For any $\rho>0$ and any two integers, $h\geq 1$ and $d\geq 2$, there exists an ergodic real-analytic diffeomorphism $T\in\text{Diff }^\omega_\rho(\mathbb T^d)$ such that $T$ has a discrete spectrum generated (over $\mathbb Z$) by $h$ linearly independent eigenvalues. \end{maintheorem}
There is a conjecture of Kolmogorov in \cite{Kol} stating that on a $d$ dimensional real-analytic manifold an ergodic real-analytic diffeomorphism preserving an analytic measure may have a discrete spectrum with only $d$ distinct eigenvalues. Our result falsifies this conjecture.
Another aspect of the approximation by conjugation scheme deals with the problem of finding diffeomorphisms with a prescribed dynamical property. Originally Anosov and Katok produced examples of measure preserving smooth diffeomorpshims that are weakly mixing on any manifold admitting an effective $\mathbb T^1$ action. Later Fayad and Saprykina constructed weakly mixing diffeomorphisms in the restricted space $\mathcal{A}_{\alpha}\left(M\right)= \overline{\left\{h \circ R_{\alpha} \circ h^{-1} \ : h \in \text{Diff}^{\infty}\left(M, \mu \right)\right\}}^{C^{\infty}}$ for every Liouvillean number $\alpha$ (\cite{FS}) in case of dimension $2$. In case of the disc $\mathbb{D}^2$ and the annulus $\mathbb{A}$ this gives the dichotomy that a number is Diophantine if and only if there is not ergodic $C^{\infty}$-diffeomorphism with that rotation number. In that paper \cite{FS}, the authors were even able to construct examples of weakly mixing real analytic diffeomorphims of the two dimensional torus for rotation numbers $\alpha$ that satisfy a condition of similar type as our above one (namely that for some $\delta>0$ the equation $\abs{\alpha- \frac{p}{q}} < \exp(-q^{1+\delta})$ has an infinite number of relatively prime integer solution $p,q$). We should note that the method of reparametrization of linear flows as in \cite{F} is more appropriate to get weakly mixing analytic diffeomorphisms on $\mathbb T^d$.
Using the AbC method and the concept of real-analytic approximation of block-slide type maps, the second author showed that on a torus of any dimension greater than one there are examples of weakly mixing real-analytic diffeomorphims preserving a measurable Riemannian metric.
\begin{theorem}[\cite{Ku-Wm}] For any $\rho > 0$ and any integer $d \geq 2$, there are weakly mixing real-analytic diffeomorphisms $T \in \text{Diff }^\omega_\rho(\mathbb T^d,\mu)$ preserving a measurable Riemannian metric. \end{theorem}
This result solved \cite{GK}, Problem 3.9., about the existence of real-analytic volume-preserving IM-diffeomorphisms (i. e. diffeomorphisms preserving an absolutely continuous probability measure and a measurable Riemannian metric) in the case of tori. In this before-mentioned paper \cite{GK}, Gunesch and Katok constructed volume-preserving weakly mixing $C^{\infty}$-diffeomorphisms preserving a measurable Riemannian metric (see also \cite{KG} for the same result in the restricted spaces $\mathcal{A}_{\alpha}(M)$ for arbitrary Liouville number $\alpha$) and gave a comprehensive consideration of IM-diffeomorphisms and IM-group actions. In particular, the existence of a measurable invariant metric for a diffeomorphism is equivalent to the existence of an invariant measure for the projectivized derivative extension which is absolutely continuous in the fibers. Recently, the second author examined the ergodic behaviour of the derivative extension with respect to such a measure (\cite{Ku-Der}). It provides the only known examples of measure-preserving diffeomorphisms whose differential is ergodic with respect to a smooth measure in the projectivization of the tangent bundle. It is an interesting open problem to exhibit such an examination in the real-analytic case.
In another version (called ``toplogical version'' in \cite{FK}) of the AbC-method one tries to exercise control over every orbit of the initial $\mathbb T^1$ action, while the original construction by Anosov and Katok was only able to exercise control over almost every orbit of the $\mathbb T^1$ action. Such topological constructions deal with minimality and the number of ergodic invariant measures (e. g. unique ergodicity) for intance (see e. g. \cite{FH}, \cite{FSW}, \cite{Win}). We can use such techniques and prove that the limiting diffeomorphims obtained in Theorem \ref{nsr circle rotation} are in fact uniquely ergodic with respect to the Lebesgue measure (\cite[Theorem 1.1.]{Ba-Ns}): For any $\rho>0$ , there exist uniquely ergodic real-analytic diffeomorphisms $T\in\text{Diff }^\omega_\rho(\mathbb T^2, \mu )$ which are metrically isomorphic to some irrational rotations of the circle. We note that minor modifications will extend the above result to the $d$ dimensional torus but we do not do so in this article. Instead we can produce more exotic examples. We show that there are minimal but not uniquely ergodic measure preserving real-analytic diffeomorphisms.
\begin{maintheorem}\label{theorem prescribed no of measures} For any $\rho > 0$, and any natural number $r$, there exists a real-analytic diffeomorphism $T \in \text{Diff }^\omega_\rho(\mathbb T^2,\mu)$ which is minimal and has exactly $r$ ergodic invariant measures each of which are absolutely continuous with respect to the Lebesgue measure. \end{maintheorem}
This result parallels a result of Windsor in the smooth category (\cite{Win}). While conversely a uniquely ergodic transformation on a compact metric space preserving a Borel measure is minimal on the support of the measure (e.g. \cite{KH}, Proposition 4.1.18), the first example that minimality does not imply unique ergodicity is due to Markov (see \cite{NS}, section 9.35.). In the analytic category Furstenberg constructed skew-products admitting uncountably many ergodic measures (\cite{Fu} or see \cite{KH}, Corollary 12.6.4.). In fact, these counterexamples bear a great meaning in the history of Ergodic Theory: They showed that the so-called \textit{quasi-ergodic hypothesis} (i. e. each orbit is dense in each surface of constant energy) does not imply the equality of space means and time means and so helped to find the right notion of ergodicity.
In a forthcoming paper the authors use the AbC-method with the real-analytic approximation of block-slide type maps to construct $T \in \text{Diff}^{\omega}_{\rho}\left( \mathbb T^d,\mu\right)$ with disjoint convolutions and a homogeneous spectrum of multiplicity $2$ for its Cartesian square $T \times T$. In \cite{Ku-Dc}, $C^{\infty}$-diffeomorphisms in $\mathcal{A}_{\alpha}(M)$ with these properties were constructed.
\section{Preliminaries}
Here we introduce the basic concepts and establish notations that we will use for the rest of this article.
For a natural number $d$, we will denote the $d$ dimensional torus by $\mathbb T^d:=\mathbb R^d/\mathbb Z^d$. The standard Lebesgue measure on $\mathbb T^d$ will be denoted by $\mu$. We define $\phi$, a measure preserving $\mathbb T^1$ action on the torus $\mathbb T^d$ as follows: \begin{align} \phi^t(x_1,\ldots, x_d)=(x_1+t,x_2,\ldots, x_d) \end{align}
\subsection{The topology of real-analytic diffeomorphisms on the torus}
We give a description of the space of diffeomorphisms that are interesting to us. Any real-analytic diffeomorphism on $\mathbb T^d$ homotopic to the identity admits a lift to a map from $\mathbb R^d$ to $\mathbb R^d$ and has the following form \begin{align} F(x_1,\ldots , x_d)=(x_1+f_1(x_1,\ldots, x_d),\ldots,x_d+f_d(x_1,\dots,x_d)) \end{align} where $f_i:\mathbb R^d\to \mathbb R$ are $\mathbb Z^d$-periodic real-analytic functions. Any real-analytic $\mathbb Z^d$-periodic function defined on $\mathbb R^d$ can be extended to some complex neighbourhood \footnote{we identify $\mathbb R^d$ inside $\mathbb C^d$ via the natural inclusion $(x_1,\ldots , x_d)\mapsto (x_1+i0,\ldots ,x_d+i0)$.} of $\mathbb R^d$ as a holomorphic (complex analytic) function. For a fixed $\rho>0$, let \begin{align}
\Omega_\rho:=\{(z_1,\ldots,z_d)\in\mathbb C^d:|\text{Im}(z_1)|<\rho ,\ldots, |\text{Im}(z_d)|<\rho\} \end{align}
and for a function $f$ defined on this set, put
\begin{align}
\|f\|_\rho:=\sup_{(z_1,\ldots, z_d)\in\Omega_\rho}|f(z_1,\ldots, z_d)|
\end{align}
We define $C^\omega_\rho(\mathbb T^d)$ to be the space of all $\mathbb Z^d$-periodic real-analytic functions on $\mathbb R^d$ that extends to a holomorphic function on $\Omega_\rho$ and $\|f\|_\rho<\infty$.
We define, $\text{Diff }^\omega_\rho(\mathbb T^d,\mu)$ to be the set of all measure preserving real-analytic diffeomorphisms of $\mathbb T^d$ homotopic to the identity, whose lift $F(x)=(x_1+f_1(x),\ldots,x_d+f_d(x))$ to $\mathbb R^d$ satisfies $f_i\in C^\omega_\rho(\mathbb T^d)$ and we also require the lift $\tilde{F}(x)=(x_1+\tilde{f}_1(x),\ldots,x_d+\tilde{f}_d(x))$ of its inverse to $\mathbb R^d$ to satisfies $\tilde{f}_i\in C^\omega_\rho(\mathbb T^d)$. The metric $d$ in $\text{Diff }^\omega_\rho(\mathbb T^d,\mu)$ is defined by \begin{align*}
d_\rho(f,g)=\max\{\tilde{d}_\rho(f,g),\tilde{d}_\rho(f^{-1},g^{-1})\} \qquad\text{where}\qquad \tilde{d}_\rho(f,g)=\max_{i=1,\ldots, d}\{\inf_{n\in\mathbb Z}\|f_i-g_i+n\|_\rho\} \end{align*} Let $F=(F_1,\ldots, F_d)$ be the lift of a diffeomorphism in $\text{Diff }^\omega_\rho(\mathbb T^d,\mu)$, we define the norm of the total derivative \begin{align*}
\|DF\|_\rho:=\max_{\substack{i=1,\ldots, d\\j=1,\ldots, d}}\Big\|\frac{\partial F_i}{\partial x_j}\Big\|_\rho \end{align*}
Next, with some abuse of notation, we define the following two spaces \begin{align} C^\omega_\infty (\mathbb T^d) := & \cap_{n=1}^\infty C^\omega_n(\mathbb T^d) \label{6.789} \\ \text{Diff }^\omega_\infty (\mathbb T^d,\mu) := & \cap_{n=1}^\infty \text{Diff }^\omega_n(\mathbb T^d,\mu) \label{4.569} \end{align}
Note that the functions in \ref{6.789} can be extended to $\mathbb C^n$ as entire functions. We also note that $\text{Diff }^\omega_\infty (\mathbb T^d,\mu)$ is closed under composition. To see this, let $f,g\in \text{Diff }^\omega_\infty (\mathbb T^d,\mu)$ and $F$ and $G$ be their corresponding lifts. Then note that $F\circ G$ is the lift of $f\circ g$ (with $\pi:\mathbb R^2\to\mathbb T^2$ as the natural projection, $\pi\circ F\circ G=f\circ\pi\circ G=f\circ g\circ \pi$). Now for the complexification of $F$ and $G$ note that the composition $F\circ G(z)=(z_1+g_1(z)+f_1(G(z)),\ldots, z_d+g_d(z)+f_d(G(z)) )$. Since $g_i\in C^\omega_\infty (\mathbb T^d) $, we have for any $\rho,$ $\sup_{z\in\Omega_\rho}|\text{Im}(G(z))|\leq \max_i (\sup_{z\in\Omega_\rho}|\text{Im}(z_i)+\text{Im}(g_i(z))|) \leq \max_i (\sup_{z\in\Omega_\rho}|\text{Im}(z_i)|+\sup_{z\in\Omega_\rho}|\text{Im}(g_i(z))|)\leq \rho + \max_i (\sup_{z\in\Omega_\rho}|g_i(z)|)<\rho + const<\rho'<\infty$ for some $\rho'$. So, $\sup_{z\in\Omega_\rho}|z_i+g_i(z)+f_i(G(z))|\leq |z_i|+|g_i(z)|+|f_i(G(z))|<\infty$ since $z\in\Omega_\rho, g_i\in C^\omega_\infty (\mathbb T^d)$ and $G(z)\in \Omega_{\rho'}$. An identical treatment gives the result for the inverse.
All intermediate diffeomorphisms constructed during the AbC method in this paper will belong to this category. \footnote{We note that the existence of such real-analytic functions whose complexification is entire or as in this case, the complexification of their lift is entire is central to a real-analytic AbC method. As of now we only know how to construct such functions on the torus, odd dimensional spheres and certain homogeneous spaces. }
This completes the description of the analytic topology necessary for our construction. Also throughout this paper, the word ``diffeomorphism" will refer to a real-analytic diffeomorphism. Also, the word ``analytic topology" will refer to the topology of $\text{Diff }^\omega_\rho(\mathbb T^d,\mu)$ described above. See \cite{S} for a more extensive treatment of these spaces.
\subsection{Some partitions of the torus}
First we recall a few definitions. A sequence of partitions $\{\mathcal{P}_n\}_n$ of a \emph{Lebesgue space}\footnote{Also known as a \emph{standard probability space} or a \emph{Lebesgue-Rokhlin space}. We consider those spaces which are isomorphic mod $0$ to the unit interval with the usual Lebesgue measure.} $(M,\mu)$ is called \emph{generating} if there exists a measurable subset $M'$ of full measure such that $\{x\}=\cap_{n=1}^\infty\mathcal{P}_n(x)\;\;\forall x\in M'$. We say that the sequence $\{\mathcal{P}_n\}_n$ is \emph{monotonic} if $\mathcal{P}_{n+1}$ is a refinement of $\mathcal{P}_n$.
There are some partitions of $\mathbb T^d$ that are of special interest to us. They appear repeatedly in this article and we summarize them here.
Assume that we are given three natural numbers $l,k,q$ and a function $a:\{0,1,\ldots,k-1\}\to\{0,1,\ldots,q-1\}$. We define the following three partitions of $\mathbb T^d$: \begin{align} & \mathcal{T}_{q}:=\Big\{\Delta_{i,q}:=\big[\frac{i}{q},\frac{i+1}{q}\big)\times \mathbb T^{d-1}: i = 0,1,\ldots,q-1\Big\}\label{partition T}\\ & \mathcal{G}_{l,q}:=\Big\{\big[\frac{i_1}{lq},\frac{i_1+1}{lq}\big)\times\big[\frac{i_2}{l},\frac{i_2+1}{l}\big)\times\ldots\times\big[\frac{i_d}{l},\frac{i_d+1}{l}\big):i_1 = 0,1,\ldots,lq-1,\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (i_2,\ldots,i_{d}) \in \{0,1,\ldots,l-1\}^{d-1}\Big\}\label{partition G}\\ & \mathcal{G}_{j,l,q}:=\Big\{\big[\frac{i_1}{l^jq},\frac{i_1+1}{l^jq}\big)\times\big[\frac{i_2}{l},\frac{i_2+1}{l}\big)\times\ldots\times\big[\frac{i_{d-j+1}}{l},\frac{i_{d-j+1}+1}{l}\big)\times\mathbb T^{j-1}:i_1 = 0,1,\ldots,l^jq-1,\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (i_2,\ldots,i_{d-j+1}) \in \{0,1,\ldots,l-1\}^{d-j+1}\Big\}\label{partition Gj}\\ &\mathcal{R}_{a,k,q}:=\Big\{R_{j,q}:=\phi^{j/q}\Big(\bigcup_{i=0}^{k-1}\Delta_{a(i)k+i,kq}\Big), j=0,\ldots,q-1\Big\} \label{partition R} \end{align} We note $\phi^\alpha$ acts on the partitions \ref{partition T}, \ref{partition G}, \ref{partition Gj} and \ref{partition R} as a permutation for any choice of $p$ when $\alpha=p/q$.
\subsection{Block-slide type maps and their analytic approximations}
We recall that a \emph{step function} on the unit interval is a finite linear combination of indicator functions on intervals. We define for $1\leq i,j\leq d$ and $i\neq j$, the following piecewise continuous map on the $d$ dimensional torus, \begin{align} \mathfrak{h}:\mathbb T^d\to\mathbb T^d\qquad\text{defined by}\qquad\mathfrak{h}(x_1,\ldots,x_d):=(x_1,\ldots, x_i + s(x_j)\mod 1,\ldots, x_d) \end{align} where $s$ is a step function on the unit interval. We refer to any finite composition of maps of the above kind as a \emph{block-slide type of map} on the torus. The nomenclature is motivated from the fact that a finite composition of maps of the above kind has the effect of sliding solid blocks on the torus much like the game of nine.
Inspired by \cite{BK} the purpose of the section is to demonstrate that a block-slide type of map can be approximated extremely well by measure preserving real analytic diffeomorphisms outside a set of arbitrarily small measure. This can be achieved because step function can be approximated well by real analytic functions whose complexification is entire.
\begin{lemma} \label{lemma approx} Let $k$ and $N$ be two positive integer and $\beta=(\beta_0,\ldots,\beta_{k-1})\in [0,1)^k$. Assume $k$ is even. Consider a step function of the form \begin{align*} \tilde{s}_{\beta,N}:[0,1)\to \mathbb R\quad\text{ defined by}\quad \tilde{s}_{\beta,N}(x)=\sum_{i=0}^{kN-1}\tilde{\beta}_i\chi_{[\frac{i}{kN},\frac{i+1}{kN})}(x) \end{align*} Here $\tilde{\beta}_i:=\beta_j$ where $j:=i\mod k$. Then, given any $\varepsilon>0$ and $\delta>0$, there exists a periodic real-analytic function $s_{\beta,N}:\mathbb R\to\mathbb R$ satisfying the following properties: \begin{enumerate} \item Entirety: The complexification of $s_{\beta,N}$ extends holomorphically to $\mathbb C$. \item Proximity criterion: $s_{\beta,N}$ is $L^1$ close to $\tilde{s}_{\beta,N}$. We can say more, \begin{align}\label{nearness}
\sup_{x\in[0,1)\setminus F}|s_{\beta,N}(x)-\tilde{s}_{\beta,N}(x)|<\varepsilon \end{align} \item Periodicity: $s_{\beta,N}$ is $1/N$ periodic. More precisely, the complexification will satisfy, \begin{align}\label{boundedness} s_{\beta,N}(z+n/N)=s_{\beta,N}(z)\qquad\forall\; z\in\mathbb C\text{ and }n\in\mathbb Z \end{align} \item Bounded derivative: The derivative is small outside a set of small measure, \begin{align} \label{derivative bound}
\sup_{x\in[0,1)\setminus F}|s_{\beta,N}'(x)|<\varepsilon \end{align} \end{enumerate} Where $F=\cup_{i=0}^{kN-1}I_i\subset [0,1)$ is a union of intervals centred around $\frac{i}{kN},\;i=1,\ldots, kN-1$ and $I_0=[0,\frac{\delta}{2kN}]\cup[1-\frac{\delta}{2kN},1)$ and $\lambda(I_i)=\frac{\delta}{kN}\;\forall\; i$. \end{lemma}
\begin{proof} See \cite[Lemma 4.7]{Ba-Ns} and \cite[Lemma 3.6]{Ku-Wm}. \end{proof}
Note that the condition \ref{boundedness} in particular implies \begin{align*} \sup_{z: \text{Im}(z)<\rho}s_{\beta,N}(z)<\infty\quad\forall\; \rho>0 \end{align*}
Indeed, for any $\rho>0$, put $\Omega'_\rho=\{z=x+iy:x\in [0,1], |y|<\rho\}$ and note that entirety of $s_{\beta,N}$ combined with compactness of $\overline{\Omega'_\rho}$ implies $\sup_{z\in\Omega'_\rho}|s_{\beta,N}(z)|<C$ for some constant $C$. Periodicity of $s_{\alpha,N}$ in the real variable and the observation $\Omega_\rho=\cup_{n\in\mathbb Z}\left(\Omega'_\rho + n\right)$ implies that $\sup_{z\in\Omega_\rho}|s_{\beta,N}(z)|<C$. We have essentially concluded that $s_{\beta,N}\in C^\omega_\infty(\mathbb T^1)$.
We also make the observation that the condition $k$ is even is not really a necessary one. One can drop the condition after replacing $F$ with a different error set.
In order to prove convergence with a prescribed rotation number we require the following refinement of the lemma:
\begin{lemma} \label{lem:approx} Let $l, N \in \mathbb{N}$, $l$ even, and $\beta=\left(\beta_0,...,\beta_{l-1}\right) \in \left[0,1\right]^l$. We consider a step function of the form \begin{equation*} \tilde{s}_{\beta,N}:\left[0,1\right) \rightarrow \mathbb{R} \text{ defined by } \tilde{s}_{\beta,N}(x) = \sum^{lN-1}_{i=0} \tilde{\beta}_i \cdot \chi_{\left[\frac{i}{lN},\frac{i+1}{lN}\right)}(x), \end{equation*} where $\tilde{\beta}_i \coloneqq \beta_j$ in case of $j \equiv i \mod l$. Given any $\varepsilon \in \left( 0, \frac{1}{8} \right)$ and $\delta\in (0,1)$ let the number $A>0$ fulfil the conditions \begin{equation} \tag{A1} \label{eq:app1} A > - \frac{2l}{\pi \cdot \delta} \cdot \ln \left(- \ln \left( 1-\frac{\varepsilon}{8} \right) \right) \end{equation} and \begin{equation} \tag{A2} \label{eq:app2} A> \frac{2l}{\pi \cdot \delta} \cdot \ln \left(-\ln\left(\frac{\varepsilon}{2l}\right)\right). \end{equation} Then the $\frac{1}{N}$-periodic real entire function $s_{\beta, N, \varepsilon, \delta}$ given by \begin{align*} & s_{\beta, N, \varepsilon, \delta}(z) = \\ & \left(\sum^{\frac{l}{2}-1}_{i=0} \beta_i \cdot \left(\mathrm{e}^{-\mathrm{e}^{-A \cdot \sin\left(2\pi\left(Nz- \frac{i}{l}\right)\right)}} - \mathrm{e}^{-\mathrm{e}^{-A \cdot \sin\left(2\pi\left(Nz- \frac{i+1}{l}\right)\right)}}\right)\right) \cdot \mathrm{e}^{-\mathrm{e}^{-A \cdot \sin\left(2\pi Nz\right)}} \\ & +\left(\sum^{l-1}_{i=\frac{l}{2}} \beta_{i} \cdot \left(\mathrm{e}^{-\mathrm{e}^{-A \cdot \sin\left(2\pi\left(Nz- \frac{i}{l}\right)\right)}} - \mathrm{e}^{-\mathrm{e}^{-A \cdot \sin\left(2\pi\left(Nz- \frac{i+1}{l}\right)\right)}}\right)\right) \cdot \mathrm{e}^{-\mathrm{e}^{A \cdot \sin\left(2\pi Nz\right)}}. \end{align*} satisfies \begin{equation} \label{eq:condapprox}
\sup_{x \in [0,1) \setminus F} \left| s_{\beta, N, \varepsilon, \delta}(x) - \tilde{s}_{\beta,N}(x) \right| < \varepsilon, \end{equation} where $F= \bigcup^{lN-1}_{i=0} I_i \subset [0,1)$ is a union of intervals centered around $\frac{i}{lN}$, $i=1,...,lN-1$, $I_0 = \left[0, \frac{\delta}{2lN}\right]\cup \left[1-\frac{\delta}{2lN},1\right)$ and $\lambda\left(I_i\right)= \frac{\delta}{lN}$ for every $i$. \end{lemma}
\begin{proof} First of all, we point out that $s_{\beta, N, \varepsilon, \delta}$ is a $\frac{1}{N}$-periodic real entire function. Let $x \in [0,1) \setminus F$, namely $x \in \left[ \frac{j}{lN}+\frac{\delta}{2lN}, \frac{j+1}{lN}-\frac{\delta}{2lN} \right]$ for some $j \in \mathbb{Z}$, $0 \leq j \leq lN-1$. We write $x=\frac{j}{lN}+ \Delta$. Exploiting the fact $\sin(x) > \frac{x}{2}$ for $0<x < \frac{\pi}{2}$ we get \begin{equation*} \mathrm{e}^{-\mathrm{e}^{-A \cdot \sin(2\pi N \Delta)}} \geq \mathrm{e}^{-\mathrm{e}^{-A \cdot \pi N \Delta}}. \end{equation*} Using equation \ref{eq:app1} this implies \begin{equation} \label{eq:est1} \mathrm{e}^{-\mathrm{e}^{-A \cdot \sin\left( 2 \pi \left( Nx- \frac{s}{l} \right) \right)}}> 1 - \frac{\varepsilon}{8} \end{equation} in case of $0 \leq j-s < \frac{l}{2}$ or $-l < j-s < -\frac{l}{2}$. On the other hand, we use the fact $\sin(x) < \frac{x}{2}$ for $-\frac{\pi}{2} < x < 0$ and get \begin{equation*} \mathrm{e}^{-\mathrm{e}^{-A \cdot \sin(-2\pi N \Delta)}} \leq \mathrm{e}^{-\mathrm{e}^{A \cdot \pi N \Delta}}. \end{equation*} By applying condition \ref{eq:app2} this yields \begin{equation} \label{eq:est2} \mathrm{e}^{-\mathrm{e}^{-A \cdot \sin \left( 2 \pi \left( Nx- \frac{s}{l}\right)\right)}}< \frac{\varepsilon}{2l} \end{equation} in case of $\frac{l}{2} \leq j-s < l$ or $-\frac{l}{2} \leq j-s <0$. By the above estimates in equation \ref{eq:est1} and \ref{eq:est2} we get \begin{equation*} s_{\beta, N, \varepsilon, \delta}(x) \geq \beta_j \cdot \left( \left( 1- \frac{\varepsilon}{8} \right) \cdot \left( 1- \frac{\varepsilon}{8} \right) - \frac{\varepsilon}{2l} \right) - (l -1 ) \cdot \frac{\varepsilon}{2l} \end{equation*} and \begin{equation*} s_{\beta, N, \varepsilon, \delta}(x) \leq \beta_j + (l -1 ) \cdot \frac{\varepsilon}{2l}. \end{equation*} Altogether, we conclude \begin{equation*}
\left| s_{\beta, N, \varepsilon, \delta}(x) - \beta_j \right| < \varepsilon. \end{equation*} \end{proof}
Finally we piece together everything and demonstrate how a block-slide type of map on the torus can be approximated by a measure preserving real-analytic diffeomorphism.
\begin{proposition} \label{proposition approximation} Let $\mathfrak{h}:\mathbb T^d\to\mathbb T^d$ be a block-slide type of map which commutes with $\phi^{1/q}$ for some natural number $q$. Then for any $\varepsilon>0$ and $\delta>0$, there exists a real-analytic diffeomorphism $h\in\text{Diff }^\omega_\infty(\mathbb T^d,\mu)$ satisfying the following conditions: \begin{enumerate}
\item Proximity property: There exists a set $E\subset\mathbb T^d$ such that $\mu(E)<\delta$ and $\sup_{x\in\mathbb T^d\setminus E}\|h(x)-\mathfrak{h}(x)\|<\varepsilon$. \item Commuting property: $h\circ\phi^{1/q}=\phi^{1/q}\circ h$ \end{enumerate} In this case we say the the diffeomorphism $h$ is $(\varepsilon,\delta)$-close to the block-slide type map $\mathfrak{h}$. \end{proposition}
\begin{proof} First we assume that for some step function $\tilde{s}_{\beta,1}$ and any integer $i$ with $1<i\leq d$, the block-slide map $\mathfrak{h}$ is of the following type: \begin{align} \mathfrak{h}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{h}(x)=(x_1+\tilde{s}_{\beta,1}(x_i),x_2,\ldots,x_d) \end{align} Then we define the following function using $s_{\beta,1}$ as in lemma \ref{lemma approx}: \begin{align} h:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad h(x)=(x_1+s_{\beta,1}(x_i),x_2,\ldots,x_d) \end{align} With $F$ as lemma \ref{lemma approx}, we put $E=\mathbb T^{i-1}\times F\times \mathbb T^{d-i}$ and observe that $h$ satisfies all the conditions of the proposition.
Similarly, if for some step function $\tilde{s}_{\beta,q}$ and any integer $i$ with $1<i\leq d$, we have a block-slide map $\mathfrak{h}$ of the following type: \begin{align} \mathfrak{h}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{h}(x)=(x_1,\ldots,x_{i-1},x_i+\tilde{s}_{\beta,q}(x_1),x_{i+1},\ldots,x_d) \end{align} Then we define the following function using $s_{\beta,q}$ as in lemma \ref{lemma approx}: \begin{align} h:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad h(x)=(x_1,\ldots,x_{i-1},x_i+s_{\beta,q}(x_1),x_{i+1},\ldots,x_d) \end{align} With $F$ as lemma \ref{lemma approx}, we put $E=F\times \mathbb T^{d-1}$ and observe that $h$ satisfies all the conditions of the proposition.
So for a general block-slide type map which is obtained by a composition of several maps of the above type, we just take a composition of the individual approximations and a union of all the component error sets. \end{proof}
\subsection{Analytic AbC method} \label{subsection abc method}
Our objective now is to recall the approximation by conjugation scheme developed by Anosov and Katok in \cite{AK}. Though we modify this scheme slightly to be more suitable for our purpose and fit the notations of our article we insist that the method presented here is almost identical to the original construction. In more modern works, this method is often presented in a less formal way (see \cite{FK}) avoiding most technicalities but for our purpose we find the original scheme to be most suitable and we stick close to it.
The AbC method is an inductive process where a sequence of diffeomorphisms $T_n\in\text{Diff }^\omega_\infty(\mathbb T^d,\mu)$ is constructed inductively. The diffeomorphisms $T_n$ converge to some diffeomorphism $T$ $\in$ $\text{Diff }^\omega_\rho(\mathbb T^d,\mu)$. Additionally $T_n$ s are chosen carefully so that they satisfy some finite version of the desired property of $T$.
We now give an explicit description. At the beginning of the construction we fix a constant $\rho>0$ and note that all parameters chosen will depend on this $\rho$.
Assume that the construction has been carried out up to the $n$ th stage and we have the following information available to us:
\begin{enumerate} \item We have sequences of natural numbers $\{p_m\}_{m=1}^n$, $\{q_m\}_{m=1}^n$, $\{k_m\}_{m=1}^{n-1}$, $\{l_m\}_{m=1}^{n-1}$, $\{s_m\}_{m=1}^{n-1}$, a sequence of functions $\{a_m:\{0,\ldots, k_m\}\to\{0,\ldots, q_m-1\}\}_{m=1}^{n-1}$ and a sequence of numbers $\{\varepsilon_m\}_{m=1}^n$ . They satisfy the following condition: \begin{align} p_{m}=s_{m-1}k_{m-1}l_{m-1}q_{m-1}p_{m-1} + 1\qquad\quad q_{m}=s_{m-1}k_{m-1}l_{m-1}q_{m-1}^2\qquad\quad \varepsilon_m< 2^{-q_m} \end{align}
\item The sequence of diffeomorphisms $\{T_m\}_{m=1}^n$ is constructed as conjugates of a periodic translation. More precisely, \begin{align} T_m:=H_m^{-1}\circ\phi^{\alpha_{m}}\circ H_m\qquad\qquad H_m:=h_m\circ H_{m-1}\qquad\qquad h_m\in\text{Diff }^{\omega}_\infty(\mathbb T^d,\mu) \end{align} The diffeomorphisms $\{h_{m}\}_{m=1}^n$ satisfy the following commuting condition: \begin{align} h_{m}\circ\phi^{\alpha_{m-1}}=\phi^{\alpha_{m-1}}\circ h_{m} \end{align}
\item For $m =1,\ldots, n$, the diffeomorphism $T_m$ preserves and permutes two sequences of partitions, namely, $H_m^{-1}\mathcal{R}_{a_m,k_m,q_m}$ and $\mathcal{F}_{q_m}:=H_m^{-1}\mathcal{T}_{q_m}$.
\item For $m =1,\ldots, n$, $\mu(h_{m}^{-1}R_{i,q_{m-1}}\triangle\Delta_{i,q_{m-1}})<\varepsilon_{m-1}$ for any $R_{i,q_{m-1}}\in\mathcal{R}_{a_{m-1},k_{m-1},q_{m-1}}$ and $\Delta_{i,q_{m-1}}\in \mathcal{T}_{q_{m-1}}$ with the same $i$.
\item For $m =1,\ldots, n$, $\text{diam} (\mathcal{F}_{q_m}\cap E_{m})<\varepsilon_m$ \footnote{ This means that the diameter of the intersection of any atom of $\mathcal{F}_{q_m}$ and $E_m$ is less that $\varepsilon_m$.} for some measurable set $E_m$ satisfying $\mu(E_m)>1-\varepsilon_m$. (Note that this means $\mathcal{F}_{q_m}$ is a generating but not necessarily monotonic sequence of partitions.)
\item For $m =1,\ldots, n$: $d_\rho(T_m,T_{m-1})<\varepsilon_m$ and $d_0 \left( T^i_m, T^i_{m-1} \right) < \frac{1}{2^{m-1}}$ for $0 \leq i < q_{m-1}-1$. \end{enumerate}
Now we show how to do the construction at the $n+1$ th stage of this induction process. We proceed in the following order: \begin{enumerate} \item We choose $k_n $ and our function $a_n:\{0,\ldots, k_n\}\to\{0,\ldots, q_n-1\}$. This choice will depend on the construction we are doing and the specific properties we are targeting to prove.
\item We choose $l_n$ to be a large enough integer so that the following condition is satisfied: \begin{align}\label{ln criterion}
l_n>2^n\|DH_n\|_0 \end{align}
\item Find a block-slide type map $\mathfrak{h}_{a_n,k_n,l_n,q_n}$ which commutes with $\phi^{\alpha_n}$, maps the partition $\mathcal{G}_{l_nk_n,q_n}$ to $\mathcal{T}_{l_n^dk_n^d,q_n}$ and it maps the partition $\mathcal{T}_{q_n}$ to the partition $\mathcal{R}_{a_n,k_n,q_n}$.
\item Use proposition \ref{proposition approximation} to construct $h_{n+1}$ which is $(\varepsilon_n/2^{l_nk_nq_n},\varepsilon_n/2^{l_nk_nq_n})$ close to $\mathfrak{h}_{a_n,k_n,l_n,q_n}$. Put $E_n$ to be the error set in proposition \ref{proposition approximation}.
\item Ensure $|\alpha_{n+1}-\alpha_n|$ is small enough to guarantee $d_\rho(T_{n+1},T_{n})<\varepsilon_m$ and $d_0 \left( T^i_{n+1}, T^i_{n} \right) < \frac{1}{2^{n}}$ for $0 \leq i < q_{n}-1$. If either $l_n$ or $k_n$ above is chosen to be very large and this condition is satisfied, we put $s_n=1$. If our choice of $l_n$ or $k_n$ is too restrictive then we choose $s_n$ to be large enough so that convergence is guaranteed.
\end{enumerate} This completes the construction at the $n+1$ th stage. Note that this way convergence of $T_n$ to some $T\in\text{Diff }^\omega_\rho(\mathbb T^d,\mu)$ is guaranteed.
\begin{remark} \label{close iterates} By $d_0 \left( T^i_{n+1}, T^i_{n} \right) < \frac{1}{2^{n}}$ for $0 \leq i < q_{n}-1$ and every $n \in {\mathbb N}$ we get $d_0 \left(T^i, T^i_{n+1} \right) < \frac{1}{2^n}$ for $0 \leq i < q_{n+1}-1$. \end{remark}
We need another important constructions which is very handy for some application. Note that the partitions $\mathcal{F}_{q_n}$ are not necessarily monotonic. However the following proposition shows that a generating monotonic partition which is cyclically permuted by $T_n$ can be constructed from $\mathcal{F}_{q_n}$. This is identical to proposition 3.1 in \cite{AK}.
\begin{proposition} \label{proposition monotonic generating cyclic partition} With notations as in the approximation by conjugation scheme, we can find a sequence of partitions $\mathcal{M}_n$ of $\mathbb T^d$ satisfying the following three properties: \begin{enumerate} \item Monotonicity condition: $\mathcal{M}_{n+1}>\mathcal{M}_n$ \item Cyclic permutaion: The diffeomorphims $T_n$ cyclically permutes the atoms of $\mathcal{M}_n$. \item Generating condition: $\mathcal{M}_n\to\varepsilon$ as $n\to\infty$. \end{enumerate} \end{proposition}
\begin{proof} For any $n$ we define a measurable map $\mathfrak{c}_{n+1,n}^{(\mathfrak{1})}:\mathbb T^d/\mathcal{T}_{q_n}\to \mathbb T^d/\mathcal{R}_{a_n,k_n,q_n}$ by $\mathfrak{c}_{n+1,n}^{(\mathfrak{1})}(\Delta_{i,q_n})=R_{i,q_n}$ and another measurable map $\mathfrak{c}_{n+1,n}^{(\mathfrak{2})}:\mathbb T^d/\mathcal{T}_{q_{n+1}}\to\mathbb T^d/\mathcal{R}_{a_n,k_n,q_n}$ by $\mathfrak{c}_{n+1,n}^{(\mathfrak{2})}(\Delta_{i,q_{n+1}})=R_{j,q_n}$ where $j$ is such that $\Delta_{i,q_{n+1}}\subset R_{j,q_n}$. So at the level of $\mathbb T^d$ the composition $\mathfrak{c}_{n+1,n}=(\mathfrak{c}_{n+1,n}^{(\mathfrak{2})})^{-1}\circ\mathfrak{c}_{n+1,n}^{(\mathfrak{1})}$ gives us a correspondence which associates each atom of $\mathcal{T}_{q_n}$ with a union of atoms of $\mathcal{T}_{q_{n+1}}$. So we can define a new partition $\mathcal{T}_{q_{n+1},q_{n}}:=\{\mathfrak{c}_{n+1,n}(\Delta_{i,q_n}): 0\leq i<q_n\}$. Continuing this procedure, we obtain for any $m>n$, a partition $\mathcal{T}_{q_m,q_n}:=\{\mathfrak{c}_{m,m-1}\circ\ldots\circ\mathfrak{c}_{n+2,n+1}\circ\mathfrak{c}_{n+1,n}(\Delta_{i,q_n}): 0\leq i<q_n\}$. We note that this partition satisfies the following three conditions for any three integers $n$, $l$ and $m$ with $m>l>n$: \begin{enumerate} \item Monotonicity condition: $\mathcal{T}_{q_{m},q_{n+1}}>\mathcal{T}_{q_m,q_n}$ \item Cyclic permutation: $\phi^{1/q_n}$ cyclically permutes the atoms of $\mathcal{T}_{q_m,q_n}$. \item $\mathfrak{c}_{m,l}\circ\mathfrak{c}_{l,n}=\mathfrak{c}_{m,n}$. \end{enumerate} Now we define two new partitions as follows: \begin{align} & \mathcal{F}_{q_m,q_n}:=H_m^{-1}\mathcal{T}_{q_m,q_n}\\ & \mathcal{F}_{q_n}:=H_n^{-1}\mathcal{T}_{q_n} \end{align} We define the correspondence $\mathfrak{p}_{m,n}:\mathcal{F}_{q_n}\to\mathcal{F}_{q_m,q_n}$ by $\mathfrak{p}_{m,n}(H_n^{-1}(\Delta_{i,q_n}))=H_m^{-1}(\mathfrak{c}_{m,n}(\Delta_{i,q_n}))$. Now note that for any three integers $m>l>n$ we have the following three properties: \begin{enumerate} \item Monotonicity: $\mathcal{F}_{q_{m},q_{n+1}}>\mathcal{F}_{q_m,q_n}$. \item Cyclic permutation: $T_n$ cyclically permutes the atoms of $\mathcal{F}_{q_m,q_n}$ (since $\phi^{1/q_n}$ commutes with $h_j$ for $j>n$). \item $\mathfrak{p}_{m,l}\circ\mathfrak{p}_{l,n}$ $=\mathfrak{p}_{m,n}$. Indeed, for any $\Delta_{i,q_n}$ we observe, \begin{align*} \mathfrak{p}_{m,l}\circ\mathfrak{p}_{l,n}(H_n^{-1}(\Delta_{i,q_n}))= &\; \mathfrak{p}_{m,l}(H_l^{-1}(\mathfrak{c}_{l,n}(\Delta_{i,q_n})))\\ = &\; H_m^{-1}(\mathfrak{c}_{m,l}(\mathfrak{c}_{l,n}(\Delta_{i,q_n}))))\\ = &\; H_m^{-1}(\mathfrak{c}_{m,n}(\Delta_{i,q_n}))\\ = &\; \mathfrak{p}_{m,n}(H_n^{-1}(\Delta_{i,q_n})) \end{align*} \end{enumerate} Our next goal is to prove that the limit \begin{align}\label{6.567} \mathfrak{p}_{\infty,n}(H_n^{-1}(\Delta_{i,q_n}))):=\lim_{m\to\infty}(\mathfrak{p}_{m,n}(H_n^{-1}(\Delta_{i,q_n}))) \end{align} exists for any $n$ and $i$. In order to see this we note at first that $\mathfrak{c}_{m,n}(\Delta_{i,q_n})=\cup_{l\in\sigma}\Delta_{l,q_m}$ where $\sigma$ is some indexing set of size $q_m/q_n$. This implies $\mathfrak{c}_{m+1,n}(\Delta_{i,q_n})=\cup_{l\in\sigma}R_{l,q_m}$. And hence $h_{m+1}^{-1}(\mathfrak{c}_{m+1,n}(\Delta_{i,q_n}))=\cup_{l\in\sigma}h_{m+1}^{-1}(R_{l,q_n})$. Now note the following estimates: \begin{align*} & \; \mu(h_{m+1}^{-1}(R_{l,q_m})\triangle \Delta_{l,q_m})<\varepsilon_m\\ \Rightarrow & \; \mu(\bigcup_{l\in\sigma}h_{m+1}^{-1}(R_{l,q_m})\triangle \Delta_{l,q_m})<\frac{q_m}{q_n}\varepsilon_m\\ \Rightarrow & \; \mu(\bigcup_{l\in\sigma}h_{m+1}^{-1}(R_{l,q_m})\triangle \bigcup_{l\in\sigma}\Delta_{l,q_m})<\frac{q_m}{q_n}\varepsilon_m\\ \Rightarrow & \; \mu(h_{m+1}^{-1}(\mathfrak{c}_{m+1,n}(\Delta_{i,q_n}))\triangle \mathfrak{c}_{m,n}(\Delta_{i,q_n})<\frac{q_m}{q_n}\varepsilon_m\\ \Rightarrow & \; \mu(H_{m+1}^{-1}(\mathfrak{c}_{m+1,n}(\Delta_{i,q_n}))\triangle H_m^{-1}(\mathfrak{c}_{m,n}(\Delta_{i,q_n}))<\frac{q_m}{q_n}\varepsilon_m\\ \Rightarrow & \; \mu(\mathfrak{p}_{m+1,n}(H_n^{-1}(\Delta_{i,q_n}))\triangle \mathfrak{p}_{m,n}(H_n^{-1}(\Delta_{i,q_n}))<\frac{q_m}{q_n}\varepsilon_m\\ \Rightarrow & \; \sum_{i=0}^{q_n-1}\mu(\mathfrak{p}_{m+1,n}(H_n^{-1}(\Delta_{i,q_n}))\triangle \mathfrak{p}_{m,n}(H_n^{-1}(\Delta_{i,q_n}))<q_m\varepsilon_m\\ \Rightarrow & \; \sum_{i=0}^{q_n-1}\mu(\mathfrak{p}_{m_1,n}(H_n^{-1}(\Delta_{i,q_n}))\triangle \mathfrak{p}_{m_2,n}(H_n^{-1}(\Delta_{i,q_n}))<\sum_{m=m_1}^{m_2}q_m\varepsilon_m \end{align*} This shows that the sequence $\mathcal{F}_{q_m,q_n}$ converges as $m$ goes to infinity and shows the existence of \ref{6.567}. So we define the partition \begin{align} \mathcal{M}_n:=\{\mathfrak{p}_{\infty,n}(H_n^{-1}(\Delta_{i,q_n}))):0\leq i<q_n\} \end{align} And we note that this partition has the required properties. \end{proof}
\section{Non standard analytic realization of some ergodic rotations of the circle}
Non-standard realization problems demonstrate how a dynamical system can live on a non native manifold. In particular we are interested in exploring when an ergodic rotation of the circle can be measure theoretically isomorphic to a measure preserving real-analytic ergodic diffeomorphisms on a torus.
\subsection{Some measure theory}
Our goal here is to prove an abstract lemma from measure theory which is a slight generalization of Lemma 4.1 in \cite{AK}. Essentially we formulate a finite version of the conjugacy we will eventually prove.
Lemma 4.1 from \cite{AK} gave us an easily checkable finite version of the conjugacy that one can use to prove the existence of a metric isomorphism of the limiting diffeomorphisms. Since the generating partitions used in the $C^\infty$ non-standard realization problem can easily made to be monotonic, this Lemma was sufficient. But in the real-analytic case, our construction is not flexible enough to guarantee monotonicity, so we need a modified version. Let us recall Lemma 4.1 from \cite{AK} since we will need it for our version.
\begin{lemma} \label{4.1} Let $\{M^{(i)},\mu^{(i)}\},\;i=1,2$ be two Lebesgue spaces. Let $\mathcal{P}_n^{(i)}$ be a \emph{monotonic} sequences of generating finite partitions of $M^{(i)}$. Let $T_n^{(i)}$ be a sequence of automorphisms of $M^{(i)}$ satisfying $T_n^{(i)}\mathcal{P}_n^{(i)}=\mathcal{P}_n^{(i)}$ and suppose $ \lim_{n\to\infty}T_n^{(i)}=T^{(i)}$ weakly. Suppose that there are metric isomorphisms $K_n:M^{(1)}/\mathcal{P}_n^{(1)}\to M^{(2)}/\mathcal{P}_n^{(2)}$ satisfying: \begin{align}
& K_n^{-1}T_n^{(2)}|_{\mathcal{P}_n^{(2)}}K_n=T_n^{(1)}|_{\mathcal{P}_n^{(1)}}\\ & K_{n+1}(\mathcal{P}_{n}^{(1)})=K_n(\mathcal{P}_{n}^{(1)}) \end{align} Then the automorphisms $T^{(1)}$ and $T^{(2)}$ are metrically isomorphic. \end{lemma} We would also like to point out at this point of time that the metric isomorphism $K$ in the proof was defined to be \begin{align}\label{K(x)} K(x):=\cap_{n=1}^\infty K_n(P_n^{(1)}(x))\qquad\text{a.e.}\;\; x\in M^{(1)} \end{align}
Now we prove the following variation which will allow us to accommodate a marginal ``twist" that will appear in our construction.
\begin{lemma} \label{lemma mtl} Let $\{M^{(i)},\mu^{(i)}\},\;i=1,2$ be two Lebesgue spaces. Let $\mathcal{P}_n^{(i)}$ be a sequence of generating finite partitions of $M^{(i)}$. Let $\{\varepsilon_n\}$ be a sequence of positive numbers satisfying $\sum_{n=1}^\infty\varepsilon_n<\infty$. In addition, assume that there exists a sequence of sets $\{E_n^{(i)}\}$ in $M^{(i)}$ satisfying: \begin{align} & \mu^{(1)}(E_n^{(1)})=\mu^{(2)}(E_n^{(2)})<\varepsilon_n\label{mtl 1}\\ & P_{n+1}^{(1)}(x)\setminus E_{n+1}^{(1)}\subset P_{n}^{(1)}(x)\quad\forall\; x\in M^{(1)}\setminus E_{n+1}^{(1)}\label{mtl 2}\\ & P_{n+1}^{(2)}(y)\subset P_{n}^{(2)}(y)\quad \forall\; y\in M^{(2)}\label{mtl 3} \end{align}
Let $T_n^{(1)}$ and $T_n^{(2)}$ be two sequences of automorphisms of the spaces $M^{(1)}$ and $M^{(2)}$ satisfying: \begin{align} & T_n^{(i)}\mathcal{P}_n^{(i)}=\mathcal{P}_n^{(i)}\quad\quad i=1,2\label{mtl 4}\\ & \lim_{n\to\infty}T_n^{(i)}=T^{(i)}\quad\quad i=1,2\label{mtl 5}\\ & T_n^{(i)}(\cup_{m=n}^\infty E_m^{(i)})=\cup_{m=n}^\infty E_m^{(i)}\quad\quad i=1,2\label{mtl 6} \end{align} Note that the limit in \ref{mtl 5} is taken in the weak topology. Suppose additionally that there exists a sequence of metric isomorphisms $K_n:M^{(1)}/\mathcal{P}_n^{(1)}\to M^{(2)}/\mathcal{P}_n^{(2)}$ satisfying: \begin{align}
& K_n^{-1}T_n^{(2)}|_{\mathcal{P}_n^{(2)}}K_n=T_n^{(1)}|_{\mathcal{P}_n^{(1)}}\label{mtl 7}\\ & K_{n+1}(\mathcal{P}_{n+1}^{(1)}(x))\subset K_{n}(\mathcal{P}_{n}^{(1)}(x))\quad\forall\; x\in M^{(1)}\setminus E_{n+1}^{(1)}\label{mtl 8} \end{align} Then the automorphisms $T^{(1)}$ and $T^{(2)}$ are metrically isomorphic. \end{lemma}
\begin{proof} Put $F^{(i)}_N:=\cup_{n=N}^\infty E_{n}^{(i)}$. Consider the sequence of Lebesgue spaces $M^{(i)}_N:=M^{(i)}\setminus F^{(i)}_N$.
\noindent \emph{Claim 1: $\exists$ a metric isomorphism $K_{(N)}:M^{(1)}_N\to M^{(2)}_N$, satisfying $K_{(N)}^{-1}T^{(2)}|_{M_N^{(2)}}K_{(N)}=T^{(1)}|_{M_N^{(1)}}$.}
We define $\mathcal{P}^{(i)}_{N,k}$, a finite measurable partition of $M^{(i)}_N$ by $\mathcal{P}^{(i)}_{N,k}(x):=\mathcal{P}^{(i)}_{N+k}(x)\setminus F^{(i)}_N$. We note that the sequence of partition $\{\mathcal{P}^{(i)}_{N,k}\}_{k}$ is generating because $\{\mathcal{P}^{(i)}_{k}\}_k$ is generating. Additionally, condition \ref{mtl 2} makes $\{\mathcal{P}^{(i)}_{N,k}\}_k$ a monotonic sequence of partition. We define $K_{N,k}(\mathcal{P}^{(1)}_{N,k}(x)):=K_{N+k}(\mathcal{P}^{(1)}_{N+k}(x))\setminus F^{(2)}_{N}$. With this definition we claim that $K_{N,k+1}(\mathcal{P}^{(1)}_{N,k}(x))=K_{N,k}(\mathcal{P}^{(1)}_{N,k}(x))$ $\forall x\in M^{(1)}_N$. (Indeed, from \ref{mtl 8} we get for a.e. $x\in M^{(1)}_N$, $K_{N,k+1}(\mathcal{P}^{(1)}_{N,k+1}(x))=K_{N+k+1}(\mathcal{P}^{(1)}_{N+k+1}(x))\setminus F^{(2)}_{N}\subset K_{N+k}(\mathcal{P}^{(1)}_{N+k}(x))\setminus F^{(2)}_N=K_{N,k}(\mathcal{P}^{(1)}_{N,k}(x))$. This with the fact that $K_{N,k+1}(\mathcal{P}^{(1)}_{N,k+1}(x))\in \mathcal{P}^{(2)}_{N,k+1}$ and $\{\mathcal{P}^{(2)}_{N,k}\}_k$ is a monotonic sequence of partitions helps us in concluding the claim).
Observe that \ref{mtl 4}, \ref{mtl 6} and \ref{mtl 7} guarantees that $K_{N,k}^{-1}T_{N+k}^{(2)}|_{\mathcal{P}_{N,k}^{(2)}}K_{N,k}=T_{N+k}^{(1)}|_{\mathcal{P}_{N,k}^{(1)}}$. So we can apply Lemma \ref{4.1} to guarantee the existence of a metric isomorphism $K_{(N)}:M^{(1)}_N\to M^{(2)}_N$ defined for a.e. $x\in M^{(1)}_N$ by $K_{(N)}(x)=\cap^\infty_{k=1} K_{N,k}(\mathcal{P}^{(1)}_{N,k}(x))$. This finishes claim 1.
\noindent\emph{Claim 2: $K_{(N+1)}(x)=K_{(N)}(x)$ for a.e. $x\in M^{(1)}_{N}$}
Follows from the definition of $K_{(N)}$. Indeed, note that $K_{(N+1)}(x)=\cap^\infty_{k=1} K_{N+1,k}(\mathcal{P}^{(1)}_{N+1,k}(x))=\cap^\infty_{k=1}K_{N+k+1}(\mathcal{P}^{(1)}_{N+k+1}(x))\setminus F^{(2)}_{N+1}=\cap^\infty_{k=0} K_{N+k+1}(\mathcal{P}^{(1)}_{N+k+1}(x))\setminus F^{(2)}_{N+1}$. The last equality follows from \ref{mtl 8}.
\noindent\emph{Claim 3: There exists a metric isomorphism $K:M^{(1)}\to M^{(2)}$ satisfying $K^{-1} T^{(2)} K=T^{(1)}$}
Note that condition \ref{mtl 1} implies that a.e. $x\in E_n^{(1)}$ for at most finitely many $n$. Indeed, if $E=\{x:x\in E_n\text{ for infinitely many } n\}$, then $E\subset\cap_{n=m}^\infty E_n\;\forall\; m$ and $\lim_{m\to\infty}\mu^{(1)}(\cap_{n=m}^\infty E_n)=0$. So we can define for a.e. $x\in M^{(1)}$, $K(x):=K_{(N)}(x)$ if $x\in M^{(1)}_N$. Now claim 3 easily follows from claim 2. \end{proof}
\subsection{Construction of the conjugation map} \label{constr nsr} Let $\alpha \in \mathcal{L}_{\ast}$. We construct successively a sequence of measure-preserving diffeomorphisms $T_n = H^{-1}_n \circ \phi^{\alpha_{n}} \circ H_n$, where the conjugation maps $H_n = h_n \circ H_{n-1}$ and rational numbers $\alpha_{n} = \frac{p_{n}}{q_{n}} \in \mathbb Q$ are chosen in such a way that the functions $T_n$ converge to a diffeomorphism in $\text{Diff}^{\omega}_{\rho}\left( \mathbb T^2, \mu\right)$ with the desired properties. We present step $n+1$ of the construction, i.\,e. we assume that we have already defined $H_{n}$ and the numbers $\alpha_1,...,\alpha_{n-1}$. Additionally, we choose an even integer $l_n \in \mathbb{N}$ satisfying the condition \begin{equation} \label{eq:l1}
l_n > 2^{n+1} \cdot \|DH_{n}\|_{\rho_{n}+1}, \end{equation}
where $\rho_{n}= \| H_n \|_{\rho}$. This condition on $l_n$ will be used to ensure that the sequence of partitions we construct later is generating (see the proof of [Ba15], Proposition 6.3). In this step of the construction we have to define the conjugation map $h_{n+1}$ and to choose $\alpha_n$. \begin{figure}\label{figure generating}
\end{figure}
We start by describing the main combinatorial idea behind the proof of theorems \ref{nsr circle rotation} and \ref{nsr circle rotation estimated}. Given any two integers $l$ and $q$, there exists a block-slide type map which allows us to break down the partition $\mathcal{T}_{l^dq}$ and reform it into a partition $\mathcal{G}_{l,q}$ whose atoms have diameter less than $d/l$.
First we consider the following three step functions: \begin{align} &\psi_{l,q}^{(\mathfrak{1})}:[0,1)\to \mathbb R &\text{ defined by}\quad &\psi_{l,q}^{(\mathfrak{1})}(x)=\sum_{i=1}^{l-1}\frac{l-i}{l^2q}\chi_{[\frac{i}{l},\frac{i+1}{l}]}(x)\nonumber\\ &\psi_{l,q}^{(\mathfrak{2})}:[0,1)\to \mathbb R &\text{ defined by}\quad &\psi_{l,q}^{(\mathfrak{2})}(x)=\sum_{i=0}^{l^2q-1}\(\frac{i}{l}-\Big\lfloor\frac{i}{l}\Big\rfloor\)\chi_{[\frac{i}{l^2q},\frac{i+1}{l^2q}]}(x)\nonumber\\ &\psi_{l,q}^{(\mathfrak{3})}:[0,1)\to \mathbb R &\text{ defined by}\quad &\psi_{l,q}^{(\mathfrak{3})}(x)=\sum_{i=0}^{l-1}\frac{i}{l^2q}\chi_{[\frac{i}{l},\frac{i+1}{l}]}(x)\label{tilde psi} \end{align} Then we define the following three types of block slide map: \begin{align*} & \mathfrak{g}_{i,l,q}^{(\mathfrak{1})}:\mathbb T^d\to\mathbb T^d\qquad\text{defined by}\qquad\mathfrak{g}_{i,l,q}^{(\mathfrak{1})}\big((x_1,\ldots,x_d)\big)=(x_1+\psi_{l,q}^{(\mathfrak{1})}(x_i),x_2,\ldots,x_d)\\ & \mathfrak{g}_{i,l,q}^{(\mathfrak{2})}:\mathbb T^d\to\mathbb T^d\qquad\text{defined by}\qquad\mathfrak{g}_{i,l,q}^{(\mathfrak{2})}\big((x_1,\ldots,x_d)\big)=(x_1,\ldots,x_{i-1},x_i+\psi_{l,q}^{(\mathfrak{2})}(x_1),x_i,\ldots,x_d)\\ & \mathfrak{g}_{i,l,q}^{(\mathfrak{3})}:\mathbb T^d\to\mathbb T^d\qquad\text{defined by}\qquad\mathfrak{g}_{i,l,q}^{(\mathfrak{3})}\big((x_1,\ldots,x_d)\big)=(x_1-\psi_{l,q}^{(\mathfrak{3})}(x_i),x_2,\ldots,x_d) \end{align*} Note that the composition \begin{align} \mathfrak{g}_{i,l,q}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{g}_{i,l,q}=\mathfrak{g}_{i,l,q}^{(\mathfrak{3})}\circ\mathfrak{g}_{i,l,q}^{(\mathfrak{2})}\circ\mathfrak{g}_{i,l,q}^{(\mathfrak{1})} \end{align} maps the partition $\mathcal{G}_{l^{i},q}$ to $\mathcal{G}_{l^{i+1},q}$. Where \begin{align} & \mathcal{G}_{l^{j},q}:=\Big\{\big[\frac{i_1}{l^{j}q},\frac{i_1+1}{l^{j}q}\big)\times\big[\frac{i_2}{l},\frac{i_2+1}{l}\big)\times\ldots\times\big[\frac{i_{d-j+1}}{l},\frac{i_{d-j+1}+1}{l}\big)\times\mathbb T^{j-1}:i = 0,1,\ldots,lq-1,\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (i_2,\ldots,i_d) \in \{0,1,\ldots,l-1\}^{d-1}\Big\} \end{align} So the composition \begin{align} \mathfrak{g}_{l,q}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{g}_{l,q}=\mathfrak{g}_{2,l,q}\circ\ldots\circ\mathfrak{g}_{d,l,q} \end{align} maps the partition $\mathcal{G}_{l,q}$ to $\mathcal{T}_{l^dq}$.
Let $3 \varepsilon_n= \delta_n = \frac{1}{2^{n+1}}$. With the aid of Lemma \ref{lem:approx} we construct the following entire functions \begin{align*} &\psi_{1,n+1} = s_{\beta^{(1)}, N^{(1)}, \varepsilon_n, \delta_n}, \text{ where } \beta^{(1)}_0=0, \ \beta^{(1)}_i = \frac{l_n-i}{l^{2}_n \cdot q_n} \text{ for } i=1,...,l_n-1, \ N^{(1)}=1 \\ &\psi_{2,n+1} = s_{\beta^{(2)}, N^{(2)}, \varepsilon_n, \delta_n}, \text{ where } \beta^{(2)}_i=\frac{i}{l_n} \text{ for } i=0,..., l_n-1, \ N^{(2)}= l_nq_n \\ & \psi_{3,n+1} = s_{\beta^{(3)}, N^{(3)}, \varepsilon_n, \delta_n}, \text{ where } \beta^{(3)}_i = \frac{i}{l^{2}_n \cdot q_n} \text{ for } i=0,...,l_n-1, \ N^{(3)}=1 \end{align*} Let $A_{i,n+1}$ denote the corresponding number in the construction of $\psi_{i,n+1}$ from Lemma \ref{lem:approx}. Using these functions we define the conjugation maps \begin{align*} h_{1,n+1}(x_1,x_2) & = \left( x_1 + \psi_{1,n+1}(x_2) \mod 1, x_2 \right) \\ h_{2,n+1}(x_1,x_2) & = \left( x_1, x_2+ \psi_{2,n+1}(x_1) \mod 1 \right) \\ h_{3,n+1}(x_1,x_2) & = \left( x_1 - \psi_{3,n+1}(x_2) \mod 1, x_2 \right) \end{align*} approximating the maps $\mathfrak{g}_{2,l_n,q_n}^{(\mathfrak{j})}$ form above. Finally, we put \begin{equation*} h_{n+1} = h_{3,n+1} \circ h_{2,n+1} \circ h_{1,n+1}. \end{equation*} We point out that $\phi^{\alpha_n} \circ h_{n+1} = h_{n+1} \circ \phi^{\alpha_n}$.
\begin{lemma} \label{lem:A} Let $l_n \geq 4$. In the concrete situation of our constructions we can choose \begin{equation*} A_{i,n+1} = 2^{2n+5} \cdot l^2_n. \end{equation*} \end{lemma}
\begin{proof} Using Taylor expansion and the notation $x=\frac{\varepsilon_n}{2l_n}$ we calculate \begin{equation*} -\ln(-\ln(1-x))=-\ln\left( x + \frac{x^2}{2}+\frac{x^3}{3} + O(x^4) \right) \leq - \ln(x) = \ln(x^{-1}). \end{equation*} By our explicit definition of the number $\varepsilon_n$ we get \begin{equation*} \ln \left(2l_n \cdot \varepsilon^{-1}_n \right) = \ln \left( 2 l_n \cdot 3 \cdot 2^{n+1} \right) < \ln \left( 2^{n+4} \cdot l_n \right). \end{equation*} Then condition \ref{eq:app1} yields the requirement \begin{equation*} A_{i,n+1} \geq 2^{n+1} \cdot l_n \cdot \ln \left( 2^{n+4} \cdot l_n \right). \end{equation*} We note that condition \ref{eq:app2} is satisfied automatically. \end{proof}
\subsection{Proof of Convergence} Let $\rho>0$ be arbitrary. We want to prove convergence of $\left(T_n\right)_{n \in {\mathbb N}}$ in Diff$^{\omega}_{\rho}\left(\mathbb{T}^m\right)$. For this purpose, we introduce the numbers \begin{equation*}
\rho_k = \|H_k\|_{\rho} \text { for any } k \in {\mathbb N}. \end{equation*} Using the definitions of the conjugation maps we compute \begin{align*} & h^{-1}_{n+1} \circ \phi^{\alpha_{n+1}} \circ h_{n+1} \left(x_1,x_2 \right) = \Bigg{(} x_1+\alpha_{n+1}+\psi_{1,n+1}(x_2) - \\ &\qquad\qquad\qquad \psi_{1,n+1} \bigg{(} x_2+ \psi_{2,n+1}\left( x_1 + \psi_{1,n}(x_2) \right) - \psi_{2,n+1} \Big{(} x_1 + \alpha_{n+1} + \psi_{1,n+1}(x_2)\Big{)} \bigg{)} , \\ & \qquad\qquad\qquad\qquad\qquad \ x_2 + \psi_{2,n+1}\left( x_1 + \psi_{1,n+1}(x_2) \right)- \psi_{2,n+1} \bigg{(} x_1 + \alpha_{n+1}+ \psi_{1,n+1}(x_2) \bigg{)} \Bigg{)} \end{align*} We recall that $\psi_{2,n}$ is $\frac{1}{q_n}$-periodic and get \begin{align*} & h^{-1}_{n+1} \circ \phi^{\alpha_{n+1}} \circ h_{n+1} \left(x_1,x_2 \right) = \Bigg{(} x_1+\alpha_{n+1}+\psi_{1,n+1}(x_2) - \\ &\qquad\qquad\qquad \psi_{1,n+1} \bigg{(} x_2+ \psi_{2,n+1}\left( x_1 + \alpha_n + \psi_{1,n+1}(x_2) \right) - \psi_{2,n+1} \Big{(} x_1 + \alpha_{n+1} + \psi_{1,n+1}(x_2)\Big{)} \bigg{)} , \\ &\qquad\qquad\qquad\qquad\qquad \ x_2 + \psi_{2,n+1}\left( x_1 + \alpha_n+ \psi_{1,n+1}(x_2) \right)- \psi_{2,n+1} \bigg{(} x_1 + \alpha_{n+1}+ \psi_{1,n+1}(x_2) \bigg{)} \Bigg{)} \end{align*} Hereby, we conclude \begin{align*} & \left( h^{-1}_{n+1} \circ \phi^{\alpha_{n+1}} \circ h_{n+1} - \phi^{\alpha_n}\right) \left(x_1,x_2 \right) = \Bigg{(} \alpha_{n+1} - \alpha_n +\psi_{1,n+1}(x_2) -\\ &\qquad\qquad\qquad \psi_{1,n+1} \bigg{(} x_2+ \psi_{2,n+1}\left( x_1 + \alpha_n + \psi_{1,n+1}(x_2) \right) - \psi_{2,n+1} \Big{(} x_1 + \alpha_{n+1} + \psi_{1,n+1}(x_2)\Big{)} \bigg{)} , \\ &\qquad\qquad\qquad\qquad\qquad \ \psi_{2,n+1}\left( x_1 + \alpha_n+ \psi_{1,n+1}(x_2) \right)- \psi_{2,n+1} \bigg{(} x_1 + \alpha_{n+1}+ \psi_{1,n+1}(x_2) \bigg{)} \Bigg{)} \end{align*}
In the next step, we exploit the closeness of $h^{-1}_{n+1} \circ \phi^{\alpha_{n+1}} \circ h_{n+1} \left(x_1,x_2 \right)$ to $\phi^{\alpha_n}$ and find \begin{align*} d_{\rho} \left(f_{n+1}, f_{n} \right) = & d_{\rho} \left(H^{-1}_{n}\circ h^{-1}_{n+1} \circ \phi^{\alpha_{n+1}} \circ h_{n+1} \circ H_{n}, H^{-1}_{n}\circ \phi^{\alpha_{n}} \circ H_{n} \right) \\
\leq & \|DH_{n}\|_{\rho_{n}+1} \cdot \|h^{-1}_{n+1} \circ \phi^{\alpha_{n+1}} \circ h_{n+1} - \phi^{\alpha_n}\|_{\rho_{n}}. \end{align*}
In order to estimate $\|h^{-1}_{n+1} \circ \phi^{\alpha_{n+1}} \circ h_{n+1} - \phi^{\alpha_n}\|_{\rho_{n}}$ we will use the subsequent result:
\begin{lemma} \label{lem:est}
Let $\rho>0$ and $B^{\rho}= \left\{ z \in \mathbb{C} \; : \; \left|\text{im}\left(z\right)\right|< \rho\right\}$. Then we have \begin{equation*} \sup_{z \in B^{\rho}} \abs{s_{\beta, N, \varepsilon, \delta}(z)} \leq 2\pi \cdot N \cdot A \cdot \mathrm{e}^{2 \mathrm{e}^{A \cdot \mathrm{e}^{2\pi N \rho}} +A \cdot \mathrm{e}^{2\pi N \rho} + 2\pi N \rho} \end{equation*} and \begin{equation*} \sup_{z_1,z_2 \in B^{\rho}} \abs{s_{\beta, N, \varepsilon, \delta}(z_1)-s_{\beta, N, \varepsilon, \delta}(z_2)} \leq C \cdot A \cdot l \cdot N \cdot \mathrm{e}^{4 \cdot \mathrm{e}^{A \cdot \mathrm{e}^{2 \pi N \rho}}} \cdot \abs{z_1-z_2}, \end{equation*} where $C$ is a constant independent of $n$, $l$ and $N$. \end{lemma}
\begin{proof} First of all, we observe \begin{equation*} \frac{\mathrm{d}}{\mathrm{d}z}\mathrm{e}^{-\mathrm{e}^{-A \cdot \sin\left(2\pi\left(Nz- \frac{i}{l}\right)\right)}} = \mathrm{e}^{-\mathrm{e}^{-A \cdot \sin\left(2\pi\left(Nz- \frac{i}{l}\right)\right)}-A \cdot \sin\left(2\pi\left(Nz- \frac{i}{l}\right)\right)} \cdot 2 \pi \cdot A \cdot N \cdot \cos \left(2\pi \left(Nx-\frac{i}{l}\right) \right) \end{equation*} Using the mean value theorem this yields \begin{align*} \sup_{z \in B^{\rho}} \abs{s_{\beta, N, \varepsilon, \delta}(z)} & \leq l \cdot \mathrm{e}^{\mathrm{e}^{A \cdot \mathrm{e}^{2\pi N \rho}}} \cdot 2\pi \cdot N \cdot A \cdot \mathrm{e}^{A \cdot \mathrm{e}^{2\pi N \rho}} \cdot \mathrm{e}^{2\pi N \rho} \cdot \frac{1}{l} \cdot \mathrm{e}^{\mathrm{e}^{A \cdot \mathrm{e}^{2\pi N \rho}}} \\ & \leq 2\pi \cdot N \cdot A \cdot \mathrm{e}^{2 \mathrm{e}^{A \cdot \mathrm{e}^{2\pi N \rho}} +A \cdot \mathrm{e}^{2\pi N \rho} + 2\pi N \rho} \end{align*} Additionally, we get \begin{align*}
& \|Ds_{\beta,N,\varepsilon, \delta}\|_{\rho} \\ \leq & l \cdot \mathrm{e}^{2 \mathrm{e}^{A\cdot \mathrm{e}^{2\pi N \rho}}} \cdot 4\pi \cdot A \cdot N \cdot \mathrm{e}^{A\cdot \mathrm{e}^{2\pi N \rho}} \cdot \mathrm{e}^{2\pi \rho N} + 2\pi \cdot A \cdot N \cdot \mathrm{e}^{2 \mathrm{e}^{A\cdot \mathrm{e}^{2\pi N \rho}}} \cdot \mathrm{e}^{2A\cdot \mathrm{e}^{2\pi N \rho}} \cdot \mathrm{e}^{4\pi \rho N} \\ \leq & 6\pi \cdot A \cdot l \cdot N \cdot \mathrm{e}^{4 \cdot \mathrm{e}^{A \cdot \mathrm{e}^{2 \pi N \rho}}}. \end{align*} By applying the mean value theorem we obtain the second statement of the Lemma. \end{proof}
In addition to the before mentioned conditions we require the number $l_n$ to satisfy \begin{equation} \label{eq:l2} l_n > \mathrm{e}^{2\pi \cdot (\rho_{n}+1)}. \end{equation} Finally, we are able to prove convergence of the sequence $\left( f_n \right)_{n \in \mathbb{N}}$:
\begin{lemma} \label{lem:conv} Fix $\rho >0$. Then there is a sequence $\left(\alpha_n\right)_{n \in {\mathbb N}}$ of rational numbers converging to $\alpha$ monotonically such that the sequence $\left( T_n \right)_{n \in \mathbb{N}}$ converges to $T$ in Diff$^{\omega}_{\rho}\left(\mathbb{T}^2\right)$. \end{lemma}
\begin{proof} First of all, we introduce the number \begin{equation*} \rho^{\prime}_n = \rho_{n}+2\pi \cdot A_{1,n+1} \cdot \mathrm{e}^{2 \mathrm{e}^{A_{1,n+1} \cdot \mathrm{e}^{2\pi \rho_{n}}} +A_{1,n+1} \cdot \mathrm{e}^{2\pi \rho_{n}} + 2\pi \rho_{n}} \end{equation*} We recall that $C$ as well as the requirements on $l_n$ in equations \ref{eq:l1} and \ref{eq:l2} are independent of $q_n$. Hence, we can state the subsequent condition on $q_n$: \begin{equation} \label{eq:condq} q_n \geq 2 C^2 \cdot l_n \cdot \mathrm{e}^{4 \cdot \mathrm{e}^{2^{2n+5}l^3_n}} \end{equation} Under this restriction on the number $q_n$ we find $\alpha_n = \frac{p_n}{q_n}$ with $p_n,q_n$ relatively prime such that \begin{equation*} \abs{\alpha - \alpha_n } < \frac{1}{\mathrm{e}^{\mathrm{e}^{\left(2^{2n+6} \cdot l^3_n \cdot \mathrm{e}^{2 \pi \cdot \rho^{\prime}_n} \right)^{q_n}}}}, \end{equation*} because $\alpha \in \mathcal{L}_{\ast}$. By using $\rho^{\prime}_n$ and by applying Lemma \ref{lem:est} twice we get \begin{align*} & \sup_{(x_1,x_2) \in A^{\rho_{n}}} \abs{\psi_{2,n+1}\left( x_1 + \alpha_n + \psi_{1,n+1}(x_2) \right) - \psi_{2,n} \Big{(} x_1 + \alpha_{n+1} + \psi_{1,n+1}(x_2)\Big{)} } \\ \leq & \sup_{ y \in B^{\rho^{\prime}_n}} \abs{ \psi_{2,n+1}\left( y + \alpha_n\right) - \psi_{2,n+1} \left( y + \alpha_{n+1} \right) } \\ \leq & C \cdot \mathrm{e}^{\ln\left( A_{2,n+1} \cdot l_n \cdot q_n\right) + 4 \cdot \mathrm{e}^{A_{2,n+1} \cdot \mathrm{e}^{2 \pi \cdot q_n \cdot \rho^{\prime}_n}}} \cdot \abs{\alpha_{n+1} - \alpha_n} \end{align*} Under our conditions on the numbers $\alpha_k$ its value is less than $1$. Hereby, we conclude using equation \ref{eq:l2} \begin{align*}
& \|h^{-1}_{n+1} \circ \phi^{\alpha_{n+1}} \circ h_{n+1} - \phi^{\alpha_n}\|_{\rho_{n}} \leq \abs{\alpha_{n+1}-\alpha_n} +\\ & \qquad\qquad C \cdot \mathrm{e}^{\ln\left( A_{1,n+1} \cdot l_n\right) + 4 \cdot \mathrm{e}^{A_{1,n+1} \cdot \mathrm{e}^{2 \pi \cdot (\rho_{n}+1)}}} \cdot C \cdot \mathrm{e}^{\ln\left( A_{2,n+1} \cdot l_n \cdot q_n\right) + 4 \cdot \mathrm{e}^{A_{2,n+1} \cdot \mathrm{e}^{2 \pi \cdot q_n \cdot \rho^{\prime}_n}}} \cdot \abs{\alpha_{n+1} - \alpha_n} \\ & \qquad\qquad\qquad \leq 2 C^2 \cdot 2^{2n+5} \cdot l^3_n \cdot \mathrm{e}^{4 \cdot \mathrm{e}^{2^{2n+5}l^3_n}} \cdot \mathrm{e}^{\ln\left( 2^{2n+5} \cdot l^3_n \cdot q_n\right) + 4 \cdot \mathrm{e}^{2^{2n+5} \cdot l^2_n \cdot \mathrm{e}^{2 \pi \cdot q_n \cdot \rho^{\prime}_n}}} \cdot \abs{\alpha_{n+1} - \alpha_n} \end{align*} With the aid of condition \ref{eq:condq} we can continue the former estimates in the following way: \begin{align*}
& 2^{n+1} \cdot \|DH_{n}\|_{\rho_{n}+1} \cdot \|h^{-1}_{n+1} \circ \phi^{\alpha_{n+1}} \circ h_{n+1} - \phi^{\alpha_n} \|_{\rho_{n}} \\ \leq & \mathrm{e}^{2 \cdot \ln\left( 2^{2n+5} \cdot l^3_n \cdot q_n\right) + 4 \cdot \mathrm{e}^{2^{2n+5} \cdot l^2_n \cdot \left(\mathrm{e}^{2 \pi \cdot \rho^{\prime}_n}\right)^{q_n}}} \cdot \abs{\alpha_{n+1} - \alpha_n} \\ \leq & \mathrm{e}^{\mathrm{e}^{2^{2n+6} \cdot l^3_n \cdot \left(\mathrm{e}^{2 \pi \cdot \rho^{\prime}_n}\right)^{q_n}}} \cdot \abs{\alpha_{n+1} - \alpha_n} \\ \leq & \mathrm{e}^{\mathrm{e}^{\left(2^{2n+6} \cdot l^3_n \cdot \mathrm{e}^{2 \pi \cdot \rho^{\prime}_n} \right)^{q_n}}} \cdot \abs{\alpha_{n+1} - \alpha_n} \\ \end{align*} Using the above estimates we conclude: \begin{align*}
d_{\rho}\left(T_{n+1}, T_{n} \right) & \leq \| DH_{n}\|_{\rho_{n}+1} \cdot \|h^{-1}_{n+1} \circ \phi^{\alpha_{n+1}} \circ h_{n+1} - \phi^{\alpha_n}\|_{\rho_{n}} \\ & \leq \mathrm{e}^{\mathrm{e}^{\left(2^{2n+6} \cdot l^3_n \cdot \mathrm{e}^{2 \pi \cdot \rho^{\prime}_n} \right)^{q_n}}} \cdot \frac{1}{2^{n+1}} \abs{\alpha_{n+1} - \alpha_n} \\ & \leq \mathrm{e}^{\mathrm{e}^{\left(2^{2n+6} \cdot l^3_n \cdot \mathrm{e}^{2 \pi \cdot \rho^{\prime}_n} \right)^{q_n}}} \cdot \frac{1}{2^{n+1}} \cdot 2 \cdot \abs{\alpha-\alpha_n} \\ & < \frac{1}{2^n}. \end{align*} Hence, $\left(T_n\right)_{n \in \mathbb{N}}$ is a Cauchy sequence in Diff$^{\omega}_{\rho}\left( \mathbb{T}^2 \right)$. Since this is a complete space, we obtain convergence of $\left(T_n\right)_{n \in \mathbb{N}}$ to a real-analytic diffeomorphism $T \in \text{Diff}^{\omega}_{\rho}\left( \mathbb{T}^2 \right)$. \end{proof}
\subsection{Proof of conjugacy of \texorpdfstring{$T$}{TEXT} to the rotation \texorpdfstring{$R_{\alpha}$}{TEXT} of the circle}
This section is identical to section 6 of [Ba15] and we omit very detailed proofs which are available in that paper. We have a sequence of real-analytic diffeomorphisms $T_n$ converging to a real-analytic diffeomorphism $T$, a generating sequence of partitions $\mathcal{F}_{q_n}$ of $\mathbb T^2$ (see subsection \ref{subsection abc method}) and each $\mathcal{F}_{q_n}$ is cyclically permuted by $T_n$. We also know the convergence of $\left(\alpha_n\right)_{n \in \mathbb{N}}$ to the prescribed number $\alpha$.
On the other hand we approximate an irrational rotation of the circle by rational rotations. Let $\alpha_n=\frac{p_n}{q_n}$ be as in the approximation by conjugation scheme described above and consider a sequence of partitions of the circle as follows: \begin{align*} \mathcal{C}_{q_n}:=\Big\{\Gamma_{i,q_n}:=\Big[\frac{i}{q_n},\frac{i+1}{q_n}\Big):\; i=0,1,\ldots q_n-1\Big\} \end{align*} Clearly this is a sequence of partitions are monotonic and generating. We also, define a sequence of maps: \begin{align*} R_{\alpha_n}:S^1\to S^1,\quad\quad \text{ defined by }x\mapsto x+\alpha_n \end{align*} So, we have $R_{\alpha_n}\to R_\alpha$. We also define \begin{align*} \tilde{E}_{{n+1}}:=\bigcup_{i=0}^{q_{n+1}}\Big[\frac{i}{l_n^2q_n}-\frac{\mu(E_{{n+1}})}{2l_n^2q_n},\frac{i}{l_n^2q_n}+\frac{\mu(E_{{n+1}})}{2l_n^2q_n}\Big] \end{align*}
Following the notation of lemma \ref{lemma mtl} we let $M^{(1)}:=\mathbb T^2,\mu^{(1)}:=\mu,\mathcal{P}^{(1)}_n:=\mathcal{F}_{q_n}, E^{(1)}_n:=E_{n}, M^{(2)}:=\mathbb T^1,\mu^{(2)}:=\lambda, \mathcal{P}^{(2)}_{n}:=\mathcal{C}_{q_n}$ and $E^{(2)}_n:=\tilde{E}_{{n+1}}$. Finally we define the conjugacy $K_n$ by $K_n(H_n^{-1}\Delta_{i,q_n})=\Gamma_{i,q_n}$. This gives us that $\mathcal{P}^{(i)}_n$ is generating and conditions \ref{mtl 1}, \ref{mtl 2}, \ref{mtl 4} and \ref{mtl 5} in lemma \ref{lemma mtl}. Conditions \ref{mtl 7} and \ref{mtl 8} follows from the definition. Now note that $\phi^{\alpha_n}$ preserves $E_{{n+1}}^{(v)}$ and $E_{{n+1}}^{(d)}$ and hence $T_{n+1}$ preserves $E_{{n+1}}$. This gives us \ref{mtl 6} and completes the proof of Theorem \ref{nsr circle rotation estimated}.
\subsection{Set of numbers \texorpdfstring{$\mathcal{L}_{\ast}$}{TEXT}} \label{subsection:numbers} As announced in the introduction we examine the set of obtained rotation numbers. By well known arguments for spaces like $\mathcal{L}_{\ast}$ (e.\,g. \cite{Br}, Appendix A.2) we prove \begin{lemma} $\mathcal{L}_{\ast}$ is a dense $G_{\delta}$-subset of $\mathbb{R}$. \end{lemma}
\begin{proof} For each pair $\left(p,q\right) \in \mathbb{Z} \times \mathbb{N}$ with $p,q$ relatively prime and every $k \in \mathbb{N}$ we define the following open set \begin{equation*} O_{k}(p,q) = \Meng{ x \in \mathbb R}{ 0 < \abs{x - \frac{p}{q}} < \frac{1}{\mathrm{e}^{\mathrm{e}^{k^q}}}}. \end{equation*} Then the countable union \begin{equation*} U_k = \bigcup_{\left(p,q\right) \in \mathbb{Z} \times \mathbb{N} \text{ with } p,q \text{ relatively prime}} O_{k}(p,q) \end{equation*} is also open in $\mathbb{R}$ for every $k \in \mathbb{N}$. In the next step, we fix $k \in \mathbb{N}$. Obviously, each rational number $\omega \in \mathbb{Q}$ written in its lowest form $\omega= \frac{p}{q}$ lies in the closure of $O_{k}(p,q)$. Hence, each rational number lies in the closure of $U_k$. Since the rational numbers are dense in $\mathbb{R}$, $U_k$ is dense in $\mathbb{R}$. This applies to all $k \in \mathbb{N}$. Moreover, we observe \begin{equation*} \mathcal{L}_{\ast} = \bigcap_{k \in \mathbb{N}} U_k. \end{equation*} By the Baire category Theorem, $\mathcal{L}_{\ast}$ as a countable intersection of open dense sets is dense in $\mathbb{R}$. \end{proof}
\section{Non standard analytic realization of some ergodic translations of the torus}
Our goal in this section is to produce a proof of theorem \ref{theorem nsr total translations}. The proof of this theorem in the smooth category was done by Anosov and Katok in \cite{AK}. Later Benhenda in \cite{Mb-ts} produced an estimated version of this result showing that every ergodic toral translation with one arbitrary Liouvillian coordinate can be realized smoothly on any manifold admitting a circle action.
In our article we prove a real analytic version of this result. Unfortunately we do not have the techniques to prove results like theorem 1.1 and theorem 1.2 in \cite{AK} for arbitrary real-analytic manifolds. However the concept of block-slide type maps and their real-analytic approximations allow us just about enough flexibility on the torus to produce pretty much any combinatorial picture the approximation by conjugation scheme requires. So we prove that there exist real-analytic diffeomorphisms on $\mathbb T^d$ which are metrically conjugated to ergodic translations on $\mathbb T^h$ for arbitrary $d\geq 2$ and $h\geq 1$.
As of now we do not know if this concept of block-slide type maps can be successfully generalized to other types of real-analytic manifolds. It is possible that these maps can be generalized to odd dimensional spheres using transitive flows like those described in \cite{FK-ue}. The main obstruction to this generalizations seems to be the fact that the analytic flows they use commutes with the Hopf fibration but not with each other.
\subsection{Description of the required combinatorics} \label{constr transl}
Our objective here is to demonstrate that one can reproduce the approximation by conjugation scheme as described in \cite{AK} in its full generality in the real-analytic category on $\mathbb T^d$, $d \geq 2$. We show three basic kind of rearrangement techniques in this section. Together, these three kind of rearrangements will be sufficient to produce all constructions done in \cite{AK}.
\subsubsection*{Periodic interchange of two consecutive atoms}
\begin{figure}\label{figure interchange}
\end{figure}
Fix any two integers $k$ and $q$. Our objective here is to show that one can interchance two consecutive atoms of $\mathcal{T}_{kq}$ periodically inside each atom of $\mathcal{T}_q$. More precisely we show that there exist a block-slide type map $\mathfrak{f}_{k,q}$ that interchanges the atom $\Delta_{ik,kq}$ with the atom $\Delta_{ik+1,kq}$ for $i=0,\ldots,q-1$ and leaves all other atoms of $\mathcal{T}_{kq}$ unchanged.
We begin by considering the following step functions (or more appropriately piecewise constant functions): \begin{align} & \sigma^{(\mathfrak{1})}_{kq}:(0,1]\to\mathbb R\qquad\qquad\text{defined by}\qquad \sigma^{(\mathfrak{1})}_{kq}(t)=\begin{cases} 0 & \quad\text{if } t \in (0,1/2] \\ 1/(kq) & \quad\text{if } t\in (1/2,1]\end{cases} \\ & \sigma^{(\mathfrak{2})}_{kq}:(0,1]\to\mathbb R\qquad\qquad\text{defined by}\qquad \sigma^{(\mathfrak{2})}_{kq}(t)=\begin{cases} 1/(kq) & \quad\text{if } t \in (0,1/2] \\ 0 & \quad\text{if } t \in (1/2,1]\end{cases}\\ & \sigma^{(\mathfrak{3})}_{kq}:(0,1]\to\mathbb R\qquad\qquad\text{defined by}\qquad \sigma^{(\mathfrak{3})}_{kq}(t)=\begin{cases} 0 & \quad\text{if } qt\mod 1 \in (0,\frac{1}{k}]\\ 1/2 & \quad\text{if } qt\mod 1 \in (\frac{1}{k},1]\end{cases}\\ & \sigma^{(\mathfrak{4})}_{kq}:(0,1]\to\mathbb R\qquad\qquad\text{defined by}\qquad \sigma^{(\mathfrak{4})}_{kq}(t)=\begin{cases} 0 & \quad\text{if } qt\mod 1 \in (0,\frac{2}{k}]\\ 1/2 & \quad\text{if } qt\mod 1 \in (\frac{2}{k},1]\end{cases} \end{align} Note that $\sigma^{(\mathfrak{3})}_{kq}$ is $1/q$ periodic and we can define the following piecewise continuous functions on $\mathbb T^d$: \begin{align} & \mathfrak{f}_{kq}^{(\mathfrak{1})}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{f}_{kq}^{(\mathfrak{1})}\big((x_1,\ldots,x_d)\big)=(x_1-\sigma_{kq}^{(\mathfrak{1})}(x_2),x_2,\ldots,x_d)\\ & \mathfrak{f}_{kq}^{(\mathfrak{2})}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{f}_{kq}^{(\mathfrak{2})}\big((x_1,\ldots,x_d)\big)=(x_1,x_2+\sigma_{kq}^{(\mathfrak{3})}(x_1),\ldots,x_d)\\ & \mathfrak{f}_{kq}^{(\mathfrak{3})}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{f}_{kq}^{(\mathfrak{3})}\big((x_1,\ldots,x_d)\big)=(x_1+\sigma_{kq}^{(\mathfrak{2})}(x_2),x_2,\ldots,x_d)\\ & \mathfrak{f}_{kq}^{(\mathfrak{4})}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{f}_{kq}^{(\mathfrak{4})}\big((x_1,\ldots,x_d)\big)=(x_1,x_2+1/2,\ldots,x_d)\\ & \mathfrak{f}_{kq}^{(\mathfrak{5})}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{f}_{kq}^{(\mathfrak{5})}\big((x_1,\ldots,x_d)\big)=(x_1-\sigma_{kq}^{(\mathfrak{2})}(x_2),x_2,\ldots,x_d)\\ & \mathfrak{f}_{kq}^{(\mathfrak{6})}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{f}_{kq}^{(\mathfrak{6})}\big((x_1,\ldots,x_d)\big)=(x_1,x_2+\sigma_{kq}^{(\mathfrak{3})}(x_1),\ldots,x_d)\\ & \mathfrak{f}_{kq}^{(\mathfrak{7})}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{f}_{kq}^{(\mathfrak{7})}\big((x_1,\ldots,x_d)\big)=(x_1-\sigma_{kq}^{(\mathfrak{1})}(x_2),x_2,\ldots,x_d)\\ & \mathfrak{f}_{kq}^{(\mathfrak{8})}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{f}_{kq}^{(\mathfrak{8})}\big((x_1,\ldots,x_d)\big)=(x_1,x_2+\sigma_{kq}^{(\mathfrak{4})}(x_2),\ldots,x_d) \end{align} We compose all the functions above into the following function: \begin{align} \mathfrak{f}_{k,q}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad \mathfrak{f}_{k,q}:=\mathfrak{f}_{kq}^{(\mathfrak{8})}\circ\mathfrak{f}_{kq}^{(\mathfrak{7})}\circ\mathfrak{f}_{kq}^{(\mathfrak{6})}\circ\mathfrak{f}_{kq}^{(\mathfrak{5})}\circ\mathfrak{f}_{kq}^{(\mathfrak{4})}\circ\mathfrak{f}_{kq}^{(\mathfrak{3})}\circ\mathfrak{f}_{kq}^{(\mathfrak{2})}\circ\mathfrak{f}_{kq}^{(\mathfrak{1})} \end{align}
\subsubsection*{Periodic rearrangement of atoms}
\begin{figure}\label{figure walking}
\end{figure}
Now we show that for for any given $l$ and $i$ with $0\leq l< k$ and $0\leq i<q$, there exists a map of the block-slide kind which will allow us to rearrange the atoms of $\mathcal{T}_{k,q}$ so that for any $i$ with $0\leq i<q$, the atom $\Delta_{l+kj,kq}$ is moved to $\Delta_{l+k(j+i),kq}$ while any atom that is not of the form $\Delta_{l+kj',kq}$ is left invariant.
Now we describe this map. Consider the following block-slide type map, \begin{align} & \mathfrak{w}_{l,k,q}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{w}_{l,k,q}=\phi^{l/(kq)}\circ\mathfrak{f}_{k,q}\circ\phi^{-l/(kq)} \end{align} and consider the following composition, \begin{align} & \mathfrak{w}_{i,l,k,q}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{w}_{i,l,k,q}=\phi^{1/(kq)}\circ\mathfrak{w}_{l+ki-2,k,q}\circ\ldots\mathfrak{w}_{l+1,k,q}\circ\mathfrak{w}_{l,k,q} \end{align}
Note that the above is the map we desired at the begining.
\subsubsection*{Recovering $\mathcal{T}_{l^dq}$ from a generating partition}
We show that given any two integers $l$ and $q$, there exists a block-slide type map which allows us to break down the partition $\mathcal{T}_{l^dq}$ and reform it into a partition $\mathcal{G}_{l,q}$ whose atoms have diamter less than $d/l$.
We consider the following three types of block slide map: \begin{align*} & \mathfrak{g}_{i,l,q}^{(\mathfrak{1})}:\mathbb T^d\to\mathbb T^d\qquad\text{defined by}\qquad\mathfrak{g}_{i,l,q}^{(\mathfrak{1})}\big((x_1,\ldots,x_d)\big)=(x_1+\psi_{l,q}^{(\mathfrak{1})}(x_i),x_2,\ldots,x_d)\\ & \mathfrak{g}_{i,l,q}^{(\mathfrak{2})}:\mathbb T^d\to\mathbb T^d\qquad\text{defined by}\qquad\mathfrak{g}_{i,l,q}^{(\mathfrak{2})}\big((x_1,\ldots,x_d)\big)=(x_1,\ldots,x_{i-1},x_i+\psi_{l,q}^{(\mathfrak{2})}(x_1),x_{ i+1},\ldots,x_d)\\ & \mathfrak{g}_{i,l,q}^{(\mathfrak{3})}:\mathbb T^d\to\mathbb T^d\qquad\text{defined by}\qquad\mathfrak{g}_{i,l,q}^{(\mathfrak{3})}\big((x_1,\ldots,x_d)\big)=(x_1-\psi_{l,q}^{(\mathfrak{3})}(x_i),x_2,\ldots,x_d) \end{align*} with the maps $\psi_{l,q}^{(\mathfrak{i})}$ defined as above. Note that the composition \begin{align} \mathfrak{g}_{i,l,q}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{g}_{i,l,q}=\mathfrak{g}_{i,l,q}^{(\mathfrak{3})}\circ\mathfrak{g}_{i,l,q}^{(\mathfrak{2})}\circ\mathfrak{g}_{i,l,q}^{(\mathfrak{1})} \end{align} maps the partition $\mathcal{G}_{j,l,q}$ to $\mathcal{G}_{j+1,l,q}$, where \begin{align} & \mathcal{G}_{j,l,q}:=\Big\{\big[\frac{i_1}{l^{j}q},\frac{i_1+1}{l^{j}q}\big)\times\big[\frac{i_2}{l},\frac{i_2+1}{l}\big)\times\ldots\times\big[\frac{i_{d-j+1}}{l},\frac{i_{d-j+1}+1}{l}\big)\times\mathbb T^{j-1}:i_1 = 0,1,\ldots,lq-1,\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (i_2,\ldots,i_{d-j+1}) \in \{0,1,\ldots,l-1\}^{d-j}\Big\} \end{align} So the composition \begin{align} \mathfrak{g}_{l,q}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{g}_{l,q}=\mathfrak{g}_{d-1,l,q}\circ\ldots\circ\mathfrak{g}_{1,l,q} \end{align} maps the partition $\mathcal{G}_{l,q}$ to $\mathcal{T}_{l^dq}=\mathcal{G}_{l^dq}$.
\subsubsection*{Piecing everything together}
Our objective now is to demonstrate that there is a $\frac{1}{q}$-periodic block-slide type of map which maps the partition $\mathcal{G}_{l,q}$ to $\mathcal{R}_{a,k,q}$. Such a map is obtained after taking a composition of some of the maps defined above:
Consider the following composition: \begin{align} \mathfrak{h}_{a,k,q}^{(\mathfrak{1})}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{h}_{a,k,q}^{(\mathfrak{1})}=\mathfrak{w}_{a(k-1),k-1,k,q}\circ\ldots\circ\mathfrak{w}_{a(1),1,k,q}\circ\mathfrak{w}_{a(0),0,k,q} \end{align} and note that the above map maps the partition $\mathcal{R}_{a,k,q}$ to the decomposition $\mathcal{T}_{q}$. Next we define: \begin{align} \mathfrak{h}_{a,k,q}^{(\mathfrak{2})}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{h}_{l,q}^{(\mathfrak{2})}=\mathfrak{g}_{l,q} \end{align} and note that the above map maps the partition $\mathcal{G}_{l,q}$ to $\mathcal{T}_{l^dq}$. So the composition: \begin{align} \mathfrak{h}_{a,k,l,q}:\mathbb T^d\to\mathbb T^d\qquad\qquad\text{defined by}\qquad\mathfrak{h}_{a,k,l,q}=(\mathfrak{h}_{a,k,q}^{(\mathfrak{1})})^{-1}\circ\mathfrak{h}_{kl,q}^{(\mathfrak{2})} \end{align} and note that the above map satisfies the following properties: \begin{enumerate} \item $\mathfrak{h}_{a,k,l,q}^{-1}(\mathcal{R}_{a,k,q})=\mathcal{T}_q$. \item $\mathfrak{h}_{a,k,l,q}^{-1}(\mathcal{T}_{l^dk^dq})=\mathcal{G}_{lk,q}$. \item { $\mathfrak{h}_{a,k,l,q}\circ\phi^{\alpha}=\phi^{\alpha}\circ\mathfrak{h}_{a,k,l,q}$} for any $p$ and $\alpha=p/q$. \end{enumerate}
\subsection{Periodic approximation of ergodic translations of the torus}
This entire section is identical to section 6 in \cite{AK}. So we only recall the portions that we need. For exact proofs, one may refer to the original article.
\begin{lemma} \label{8.90} There exists sequences $\alpha_n=(\alpha_n^{(1)},\ldots,\alpha_n^{(h)})$ and $\gamma_n=(\gamma_n^{(1)},\ldots,\gamma^{(h)}_n)$ satisfying the following properties: \begin{enumerate} \item $\gcd(\gamma_n^{(1)},\ldots,\gamma_n^{(h)})=1$ \item There exists integers $p_n,q_n$ such that $\gcd(p_n,q_n)=1$ and $\alpha_n=(p_n/q_n)\gamma_n$. \item There exists integers $r_n$ such that $q_n=r_n\gamma_{n-1}^{(h)}$. \item There exists integers $s_n$ such that $\gamma_{n+1}^{(h)}=s_n\gamma_{n}^{(h)}$. \item $\gamma_{n+1}^{(i)}\equiv \gamma^{(i)}_n\mod q_{n}$, for $i=1,\ldots, h$. \item There exists integers $m_n$ such that \begin{align} \frac{p_{n+1}}{q_{n+1}}=\frac{p_n}{q_n}+\frac{1}{m_ns_nq_n^2} \end{align} \item Let $\Gamma_n'\subset \mathbb T^{h-1}\times\{0\}\subset \mathbb T^h$ be a fundamental domain of the flow $T^{t \gamma_n}$. Let $d_n:=\text{diam}(\Gamma_n),\;$ $\sigma_{n}=\mu_{h-1}(\partial(\Gamma_n))$. Then $d_{n+1}<1/(2^{n}\gamma_{n}^{(h)}\sigma_n)$. \item \begin{align}
\Big|\frac{\gamma_{n+1}}{\gamma_{n+1}^{(h)}}-\frac{\gamma_{n}}{\gamma_{n}^{(h)}}\Big|<\frac{1}{2^n\sigma_nq_n} \end{align} \end{enumerate} \end{lemma}
With $\alpha_n,p_n,q_n,\gamma_n,r_n,s_n,m_n$ and $\Gamma_n'$ as in lemma \ref{8.90}, we construct the following two sequences of partitions of $\mathbb T^{h-1}\times\{0\}\subset\mathbb T^h$: \begin{align} &\tilde{\mathcal{F}}_{q_n}':=\big\{\Gamma_{i,q_n}':\Gamma_{i,q_n}':=T^{i\gamma_n/\gamma_n^{(h)}}\Gamma_n', i=0,\ldots, q_n-1\big\} \\ &\tilde{\mathcal{F}}_{q_n,q_{n+1}}':=\big\{\Gamma_{i,q_n,q_{n+1}}':\Gamma_{i,q_n,q_{n+1}}':=T^{i\gamma_n/\gamma_n^{(h)}}(\cup\{\Gamma_{j,q_{n+1}}':T^{j\gamma_{n+1}/\gamma_{n+1}^{(h)}}(0)\in\Gamma_n', \nonumber\\ & \hspace{200pt} j=0,\ldots, q_{n+1}-1, i=0,\ldots, q_n-1\})\big\} \end{align} Note that $\mathcal{F}'_{q_n}>\mathcal{F}'_{q_n,q_{n+1}}$. and they are both generating sequence of partitions. We construct the following two sequence of partitions of $\mathbb T^h$ from the above two partitions: \begin{align} &\tilde{\mathcal{F}}_{q_n}:=\big\{\Gamma_{i,q_n}:\Gamma_{i,q_n}:=T^{i\gamma_n/q_n}(\cup\{T^{t\gamma_n}\Gamma_n':0\leq t<1/q_n\}),\; i=0,\ldots, q_n-1\big\}\\ & \tilde{\mathcal{F}}_{q_n,q_{n+1}}:=\big\{\Gamma_{i,q_n,q_{n+1}}:\Gamma_{i,q_n,q_{n+1}}:=T^{i\gamma_n/q_n}(\cup\{T^{t\gamma_{n+1}}\Gamma_{0,q_n,q_{n+1}}':0\leq t<1/(r_n\gamma_{n+1}^{(h)})\}),\nonumber\\ &\hspace{280pt} i=0,\ldots, q_n-1\big\} \end{align}
The following proposition summarizes certain properties of the above partitions. For a proof one can refer to page 28-29 of \cite{AK}. \begin{proposition}\label{proposition cyclic approximation of translations} With $\alpha_n,p_n,q_n,\gamma_n,r_n,s_n,m_n$ as in lemma \ref{8.90} we can conclude the following: \begin{enumerate} \item The sequence of partitions $\tilde{\mathcal{F}}_{q_n,q_{n+1}}$ and $\tilde{\mathcal{F}}_{q_n}$ are respectively preserved and permuted by $T^{\alpha_n}$. \item $\mu_h(\Gamma_{i,q_n,q_{n+1}} \triangle \Gamma_{i,q_n})<1/(2^{n-3}q_n)$ \footnote{Here $\mu_h$ denotes the standard Lebesgue measure on $\mathbb T^h$ } for any $\Gamma_{i,q_n,q_{n+1}}\in \tilde{\mathcal{F}}_{q_n,q_{n+1}}$ and $\Gamma_{i,q_n}\in \tilde{\mathcal{F}}_{q_n}$ with the same $i$. \item The sequence of periodic translations $T^{\alpha_n}:\mathbb T^h\to\mathbb T^h$ converges to an ergodic translation $T^{\alpha}:\mathbb T^h\to\mathbb T^h$. \end{enumerate} \end{proposition} Next proposition is identical to lemma 6.2 in \cite{AK}. \begin{proposition} \label{proposition monotonic generating cyclic partition 2} Under the same hypothesis as proposition \ref{proposition cyclic approximation of translations} we can find a sequence of partitions $\tilde{\mathcal{M}}_n$ of $\mathbb T^h$ satisfying the following three properties: \begin{enumerate} \item Monotonicity condition: $\tilde{\mathcal{M}}_{n+1}>\tilde{\mathcal{M}}_n$ \item Cyclic permutaion: The diffeomorphims $T^{\alpha_n}$ cyclically permutes the atoms of $\tilde{\mathcal{M}}_n$. \item Generating condition: $\tilde{\mathcal{M}}_n\to\varepsilon$ as $n\to\infty$. \end{enumerate} \end{proposition}
\begin{proof} We use a method similar to the proof of \ref{proposition monotonic generating cyclic partition}. We define for any $n$, the following three maps \begin{align} & \mathfrak{q}_{n+1,n}^{(\mathfrak{1})}:\mathbb T^h/\tilde{\mathcal{F}}_{q_n}\to\mathbb T^h/\tilde{\mathcal{F}}_{q_n,q_{n+1}}\qquad\text{defined by}\quad \mathfrak{q}_{n+1,n}^{(\mathfrak{1})}(\Gamma_{i,q_n}):=\Gamma_{i,q_n,q_{n+1}}\\ & \mathfrak{q}_{n+1,n}^{(\mathfrak{2})}:\mathbb T^h/\tilde{\mathcal{F}}_{q_{n+1}}\to\mathbb T^h/\tilde{\mathcal{F}}_{q_n,q_{n+1}}\quad\text{defined by}\nonumber\\ &\hspace{150pt}\mathfrak{q}_{n+1,n}^{(\mathfrak{2})}(\Gamma_{i,q_{n+1}}):=\Gamma_{j,q_n,q_{n+1}}\;\text{if}\;\Gamma_{i,q_{n+1}}\subset\Gamma_{j,q_n,q_{n+1}}\\ & \mathfrak{q}_{n+1,n}:\mathbb T^h/\tilde{\mathcal{F}}_{q_n}\to\mathbb T^h/\tilde{\mathcal{F}}_{q_{n+1}}\qquad\text{defined by}\quad \mathfrak{q}_{n+1,n}:=(\mathfrak{q}_{n+1,n}^{(\mathfrak{2})})^{-1}\circ\mathfrak{q}_{n,n+1}^{(\mathfrak{1})} \end{align} and more generally we define the composition map for any $m$ and $n$ with $m>n$ as follows: \begin{align}
\mathfrak{q}_{m,n}:\mathbb T^h/\tilde{\mathcal{F}}_{q_n}\to\mathbb T^h/\tilde{\mathcal{F}}_{q_{m}}\qquad\text{defined by}\quad \mathfrak{q}_{n,m}=\mathfrak{q}_{m,m-1}\circ\ldots\circ\mathfrak{q}_{n+1,n} \end{align}
We now define the partition $\tilde{\mathcal{M}}=\{\lim_{m\to\infty}\mathfrak{q}_{m,n}(\Gamma_{i,q_n}): 0\leq i< q_n\}$ (see proposition \ref{proposition monotonic generating cyclic partition}) and finally the correspondence: \begin{align} \mathfrak{q}_{\infty,n}:\mathbb T^h/\tilde{\mathcal{F}}_{q_n}\to\mathbb T^h/\tilde{\mathcal{M}}\qquad\text{defined by}\quad \mathfrak{q}_{\infty,n}=\lim_{m\to\infty}\mathfrak{q}_{m,n} \end{align} The rest of the proof involves proving that $\tilde{\mathcal{M}}$ is indeed a partition satisfying the required conditions. This part can be completed identical to proposition \ref{proposition monotonic generating cyclic partition} and we do not repeat it again. \end{proof}
\subsection{Analytic diffeomorphisms metrically isomorphic to a shift on a Torus}
Our goal in this section is to prove theorem \ref{theorem nsr total translations}:
\begin{proof} First we introduce the following two correspondences \begin{align} & K_n:\mathbb T^h/\tilde{\mathcal{F}}_{q_n}\to\mathbb T^d/\mathcal{F}_{q_n}\qquad\qquad\text{defined by}\qquad K_n(\Gamma_{i,q_n})=H_n^{-1}(\Delta_{i,q_n})\\ & \tilde{K}_n:\mathbb T^h/\tilde{\mathcal{M}}_n\to\mathbb T^d/\mathcal{M}_n\qquad\qquad\text{defined by}\qquad \tilde{K}_n(\mathfrak{q}_{\infty,n}(\Gamma_{i,q_n}))=\mathfrak{p}_{\infty,n}(H_n^{-1}(\Delta_{i,q_n})) \label{4.5124} \end{align} Clearly the above two maps satisfy $\tilde{K}_n=\mathfrak{p}_{n,\infty}\circ K_n\circ\mathfrak{q}_{n,\infty}^{-1}$. We claim that we can choose parameters carefully so that the following condition can be satisfied: \begin{align} \label{5.436}
K_{n+1}|_{\tilde{\mathcal{F}}_{q_n,q_{n+1}}}=\mathfrak{p}_{n+1,n}\circ K_n\circ\mathfrak{q}_{n+1,n}^{-1} \end{align} Before we do that, we make some observation about $K_n$ and $\tilde{K}_n$. First note that we have the following relationship: \begin{align} \tilde{K}_n\circ T^{\alpha_n}=T_n\circ \tilde{K}_n \end{align} Indeed, observe that using proposition \ref{proposition monotonic generating cyclic partition 2}, \ref{4.5124}, proposition \ref{proposition monotonic generating cyclic partition} we get \begin{align*} \tilde{K}_n(T^{\alpha_n}(\mathfrak{q}_{\infty,n}(\Gamma_{i,q_n})))= & \;\tilde{K}_n(\mathfrak{q}_{\infty,n}(\Gamma_{p_n+i\mod q_n,q_n}))\\ = &\; \mathfrak{p}_{\infty,n}(H_n^{-1}(\Delta_{p_n+i\mod q_n,q_n}))\\ = &\;\mathfrak{p}_{\infty,n}(T_n(H_n^{-1}(\Delta_{i,q_n})))\\ = &\; T_n(\mathfrak{p}_{\infty,n}(H_n^{-1}(\Delta_{i,q_n})))\\ = &\;T_n(\tilde{K}_n(\mathfrak{q}_{n,\infty}(\Gamma_{i,q_n}))) \end{align*}
The next observation we need is \begin{align}
\tilde{K}_{n+1}|_{\mathbb T^h/\tilde{\mathcal{M}}_n}=\tilde{K}_n \end{align} Indeed observe that using \ref{4.5124}, \begin{align*} \tilde{K}_{n+1}(\mathfrak{q}_{\infty,n}(\Gamma_{i,q_n}))= & \; \tilde{K}_{n+1}(\mathfrak{q}_{\infty,n+1}\circ\mathfrak{q}_{n+1,n}(\Gamma_{i,q_n}))\\ =&\; \mathfrak{p}_{\infty,n+1}(K_{n+1}(\mathfrak{q}_{n+1,n}(\Gamma_{i,q_n})))\\ =&\; \mathfrak{p}_{\infty,n+1}(\mathfrak{p}_{n,n+1}(K_n(\Gamma_{i,q_n})))\\ =&\; \mathfrak{p}_{\infty,n}(K_n(\Gamma_{i,q_n}))\\ =&\; \tilde{K}_n(\mathfrak{q}_{n,\infty}(\Gamma_{i,q_n})) \end{align*}
Our job now is to choose parameters correctly in proposition \ref{proposition cyclic approximation of translations} and the analytic approximation by conjugation scheme and simultaneously construct $K_n$ s satisfying condition \ref{5.436}.
The construction is by induction and assume that we have selected parameters $\alpha_j,p_j,q_j,\gamma_j,r_j,s_j$ for $j=1,2, \ldots, n $ in proposition \ref{proposition cyclic approximation of translations} so that these satisfy all the conditions in lemma \ref{8.90}. We recall that the parameter $m_n$ in lemma \ref{8.90} can be chosen to be arbitrarily large. We will use this freedom to make the approximation by conjugation scheme work. Assume the approximation by conjugation scheme has been successfully carried out up to the $n$ th stage.
At the $n+1$ th stage, we choose parameters and proceed with our construction in the following order: \begin{enumerate}
\item Choose integer vector $\gamma_{n+1}^h$ and integers $r_n, s_n$ as in lemma \ref{8.90}.
\item Choose the parameter $k_n:=s_n\gamma_{n+1}^{(h)}$ for the approximation by conjugation scheme. Next we define the partition: \begin{align} &\tilde{\mathcal{F}}_{k_nq_n}\coloneqq\{\Gamma_{i,k_nq_n}:\Gamma_{i,k_nq_n}\coloneqq T^{i/(k_nq_n)}(\cup\{T^{t\gamma_{n+1}}\Gamma_{n+1}: 0\leq t<1/(k_nq_n)\}),\nonumber\\ &\hspace{280pt} i=0,\ldots, k_nq_n-1\} \end{align} Note that with this choice of $k_n$ we have $\tilde{\mathcal{F}}_{k_nq_n}>\tilde{\mathcal{F}}_{q_n,q_{n+1}}$.
\item Choose the functions $a_n$ in the approximation by conjugation scheme. In order to do so, we define the following correspondence: \begin{align} \hat{K}_n:\mathbb T^h/\tilde{\mathcal{F}}_{k_nq_n}\to\mathbb T^d/\mathcal{T}_{k_nq_n}\qquad\qquad\text{defined by}\qquad\hat{K}_n(\Gamma_{i,k_nq_n})=\Delta_{i,k_nq_n} \end{align} Now we choose a function $a_n:\{0,\ldots,k_n-1\}\to\{0,\ldots,q_n-1\}$ so that the following equality holds: \begin{align} R_{0,q_n}:=\bigcup_{i=0}^{k_n-1}\Delta_{a_n(i)k_n+i,k_nq_n}=\hat{K}_n(\Gamma_{0,q_n,k_nq_{n}}) \end{align} This allows us to define the partition \begin{align} \mathcal{R}_{a_n,k_n,q_n}:=\big\{R_{i,q_n}:R_{i,q_n}:=\phi^{i/q_n}R_{0,q_n}\big\} \end{align} Note that $\hat{K}_n(\Gamma_{i,q_n,q_{n+1}})=R_{i,q_n}$ and hence we conclude that $\hat{K}_n\circ T^{\gamma_n/q_n}=\phi^{1/q_n}\circ\hat{K}_n$.
\item Choose the parameter $l_n$ large enough so that the analytic approximation by conjugation scheme works.
\item we choose the parameter $m_n$ in proposition \ref{proposition cyclic approximation of translations}. We require that $m_n=\gamma_n^{(h)}l_n$.
\end{enumerate}
So, after having chosen all the parameters, we can define the partition $\tilde{\mathcal{F}}_{q_{n+1}}$ of $\mathbb T^h$ and note that $\tilde{\mathcal{F}}_{q_{n+1}}>\tilde{\mathcal{F}}_{q_n,k_nq_n}$. Now we define the following correspondence: \begin{align} \bar{K}_n:\mathbb T^h/\tilde{\mathcal{F}}_{q_{n+1}}\to\mathbb T^d/\mathcal{T}_{q_{n+1}}\qquad\qquad\text{defined by}\qquad\bar{K}_n(\Gamma_{i,q_{n+1}})=\Delta_{i,q_{n+1}} \end{align} and observe that $\bar{K}_n(\Gamma_{i,k_nq_n})=\hat{K}_n(\Gamma_{i,k_nq_n})$. All that remains is to verify that $K_{n+1}$ satisfy \ref{5.436}. So we calculate using definitions and facts from proposition \ref{proposition monotonic generating cyclic partition} and its proof: \begin{align*} \mathfrak{p}_{n+1,n}\circ K_{n}\(\Gamma_{i,q_n}\)= &\; \mathfrak{p}_{n+1,n}\circ H_{n}^{-1}\(\Delta_{i,q_n}\)\\ = &\; H_{n+1}^{-1}\circ\mathfrak{c}_{n+1,n}\(\Delta_{i,q_n}\)\\ = &\; H_{n+1}^{-1}\(\bigcup_{j:\Delta_{j,q_{n+1}}\subset R_{i,q_n}}\Delta_{j, q_{n+1}}\)\\ = &\; H_{n+1}^{-1}\(\hat{K}_n(\Gamma_{i,q_n,q_{n+1}})\)\hspace{100pt}\ldots(\text{see item 3 above})\\ = &\; H_{n+1}^{-1}\(\hat{K}_n(\bigcup_{j:\Gamma_{j,k_nq_{n}}\subset \Gamma_{i,q_n,q_{n+1}}}\Gamma_{j,k_nq_n})\)\\ = &\; H_{n+1}^{-1}\(\bigcup_{j:\Gamma_{j,k_nq_{n}}\subset \Gamma_{i,q_n,q_{n+1}}}\hat{K}_n(\Gamma_{j,k_nq_n})\)\\ = &\; H_{n+1}^{-1}\(\bigcup_{j:\Gamma_{j,k_nq_{n}}\subset \Gamma_{i,q_n,q_{n+1}}}\bar{K}_n(\Gamma_{j,k_nq_n})\)\\ = &\; H_{n+1}^{-1}\circ\bar{K}_n\(\Gamma_{i,q_n,q_{n+1}}\)\\ = &\; H_{n+1}^{-1}\circ\bar{K}_n\circ\mathfrak{q}_{n,n+1}\(\Gamma_{i,q_n}\)\\ = &\; {K}_{n+1}\circ\mathfrak{q}_{n,n+1}\(\Gamma_{i,q_n}\) \end{align*} This completes the proof \end{proof}
\section{Minimal diffeomorphisms with a prescribed number of ergodic invariant measures.} Let $\rho>0$ and $r \in {\mathbb N}$. In order to prove Theorem \ref{theorem prescribed no of measures} we aim at constructing a minimal $T \in\text{Diff }^\omega_\rho(\mathbb T^2\, \mu )$ with exactly $r$ ergodic invariant measures. We fix an arbitrary countable set $\Xi= \left\{\rho_i\right\}_{i\in {\mathbb N}}$ of Lipschitz functions that is dense in $C\left(\mathbb T^2, \mathbb R\right)$. In addition to our usual assumptions we require the number $l_n$ to satisfy \begin{equation} \label{cond l birk}
l_n > n^2 \cdot \|DH^{-1}_n \|_0 \cdot \max_{i=1,\ldots, n} \text{Lip}(\rho_i), \end{equation} where $\text{Lip}(\rho)$ is the Lipschitz constant of $\rho$.
First of all, we show that a permutation $\Pi$ of the partition $\mathcal{S}_{kq,l} $ which commutes with $\phi^{1/q}$ is a block slide type of map. This property will be required in the construction of our conjugation map in subsection \ref{subsec:constrmin}: $h_n = h_{\mathfrak{1},n} \circ h_{\mathfrak{2},n}$. This time there are different parts of the torus $\mathbb{T}^2$ introduced with distinct aims. On the one hand, we will divide it into $r$ sets $N_t$ by requirements on the $x_2$-coordinate. Each set naturally supports an absolutely continuous probability measure $\mu_t$ given by the normalized restriction of the Lebesgue measure $\mu$. These will enable us to build the ergodic invariant measures as the limits $\xi_t$ of the sequence $\xi^n_t \coloneqq \left(H_n\right)^{\ast} \mu_t$. \\ On the other hand, we will use stripes corresponding to small parts of the $x_1$-axis on which the conjugation map $h^{-1}_{\mathfrak{2},n}$ will intermingle the sets $\tilde{N}_t$ to prove minimality of the limit diffeomorphism $T$. These parts are measure theoretically insignificant because the measure of these sets will converge to zero as $n \rightarrow \infty$. \\ In order to achieve these aims we need the so-called trapping map $h^{-1}_{\mathfrak{1},n}$ introduced in subsection \ref{subsec:constrmin}. On the ``minimality'' - part, this map captures parts of every orbit $\left\{\phi^{\alpha_{n}} \circ H_n\left(x\right)\right\}_{k=0,...,q_{n}-1}$ so that the conjugation map $h^{-1}_{\mathfrak{2},n}$ can spread it over the almost whole manifold. Then we can prove minimality in chapter \ref{min} by arguing that every element in a family of sufficiently small cubes covering the whole manifold is met by the orbit $\left\{h^{-1}_{n} \circ \phi^{k \alpha_{n}} \circ H_n\left(x\right)\right\}_{k=0,...,q_{n}-1}$ and the image of any cube under $H^{-1}_{n-1}$ has a small diameter, which converges to $0$ as $n\rightarrow \infty$. In addition the trapping map is used to gain control of almost everything of every orbit $\left\{H^{-1}_n \circ \phi^{k\alpha_{n}} \left(x\right)\right\}_{k=0,...,q_{n}-1}$. This allows us to prove a convergence result on Birkhoff sums (see Lemma \ref{lem:birk}), which in turn enables us to exclude the existence of further ergodic invariant measures besides the previously mentioned $\xi_t$.
\subsection{Approximation of arbitrary permutations}
Suppose we have three natural numbers $l$, $k$ and $q$, and a permutation $\Pi$ of the partition $\mathcal{S}_{kq,l} $ which commutes with $\phi^{1/q}$. Our objective here is to show that $\Pi$ is a block slide type of map. This will be achieved in two steps. In the first step we show that there exists a product of two 2-cycles and then we will prove that all transpositions are block-slide type of maps:
\subsubsection*{Product of two 2-cycles}
\begin{figure}\label{2 cycle}
\end{figure}
We now show that for any choice of natural numbers $l,k$ and $q$ there exists a block-slide type of map which has the same effect as the product of two 2-cycles in the symmetric group of $lkq$ elements.
In order to make this precise we need the following notation. For any $i=0,\ldots, kq-1$ and $j=0,\ldots, s-1$ we define \begin{align} \mathcal{S}_{kq,l}:=\{S_{i,j}^{kq,l}:=[i/(kq),(i+1)/(kq))\times[j/l,(j+1)/l), 0\leq i< kq, 0\leq j< l\} \end{align}
First we define the following step function: \begin{align} & \sigma^{(\mathfrak{4})}_{kq}:(0,1]\to\mathbb R\qquad\qquad\text{defined by}\qquad \sigma^{(\mathfrak{4})}_{kq}(t)=\begin{cases} 2/(kq) & \quad\text{if } t \in ((l-1)/l,1] \\ 0 & \quad\text{if } t\in (0,(l-1)/l]\end{cases} \end{align}
And then the following two maps of block-slide type: \begin{align} & \mathfrak{g}_{kq,l}^{(\mathfrak{1})}:\mathbb T^2\to\mathbb T^2\qquad\qquad\text{defined by}\qquad\mathfrak{g}_{kq,l}^{(\mathfrak{1})}\big((x_1,x_2)\big)=(x_1 - \sigma^{(\mathfrak{4})}(x_2),x_2)\\ & \mathfrak{g}_{kq,l}^{(\mathfrak{2})}:\mathbb T^2\to\mathbb T^2\qquad\qquad\text{defined by}\qquad\mathfrak{g}_{kq,l}^{(\mathfrak{2})}\big((x_1,x_2)\big)=(x_1 + \sigma^{(\mathfrak{4})}(x_2),x_2) \end{align} Finally we piece everything together and define the following block-slide type of map \begin{align} & \mathfrak{g}_{k,q,l}:\mathbb T^2\to\mathbb T^2\qquad\qquad\text{defined by}\qquad\mathfrak{g}_{k,q,l}=\mathfrak{g}_{k,q,l}^{(\mathfrak{2})}\circ\mathfrak{f}_{0,k,q}\circ\mathfrak{g}_{k,q,l}^{(\mathfrak{1})}\circ\mathfrak{f}_{0,k,q} \end{align} using the map $\mathfrak{f}_{0,k,q}$ from section \ref{constr transl}.
We end this section after noting that the above block-slide type of map takes $S_{0,l-1}^{kq,l}\to S_{1,l-1}^{kq,l}$, $S_{1,l-1}^{kq,l}\to S_{0,l-1}^{kq,l}$, $S_{2,l-1}^{kq,l}\to S_{3,l-1}^{kq,l}$, $S_{3,l-1}^{kq,l}\to S_{2,l-1}^{kq,l}$ and acts as an identity everywhere else. This is the same as the product of two $2$-cycles in the symmetric group on a set of $k\times l$ elements.
\subsubsection*{Transposition}
\begin{figure}\label{transposition}
\end{figure}
Finally we show that there exists a block-slide on the torus which switches two blocks and leaves all other invariant. Unfortunately if we work with the partition $\mathcal{S}_{kq,l}$, we do not know if such a map exists. The way we circumnavigate this problem is to go to a finer partition, namely $\mathcal{S}_{kq,2l}$ and show that with some care, a product of two 2-cycles in $\mathcal{S}_{kq,2l}$ is a transposition in $\mathcal{S}_{kq,l}$
First we define the following two step functions: \begin{align} & \sigma^{(\mathfrak{5})}_{kq,l}:(0,1]\to\mathbb R\qquad\qquad\text{defined by}\qquad \sigma^{(\mathfrak{5})}_{kq,l}(t)=\begin{cases} 0 & \quad\text{if } t \in ((2l-1)/(2l),1] \\ 2/(kq) & \quad\text{if } t \in ((2l-2)/(2l),(2l-1)/(2l)] \\ 0 & \quad\text{if } t\in (0,(2l-2)/(2l)]\end{cases}\\ & \sigma^{(\mathfrak{6})}_{kq,l}:(0,1]\to\mathbb R\qquad\qquad\text{defined by}\qquad \sigma^{(\mathfrak{6})}_{kq,l}(t)=\begin{cases} 0 & \quad\text{if } qt\mod 1 \in (0,2/k] \\ 1/(2l) & \quad\text{if } qt\mod 1 \in (2/k,4/k] \\ 0 & \quad\text{if } qt\mod 1 \in (4/k,1]\end{cases} \end{align} And then the following four block-slide type of map: \begin{align} & \mathfrak{h}_{kq,l}^{(\mathfrak{1})}:\mathbb T^2\to\mathbb T^2\qquad\qquad\text{defined by}\qquad\mathfrak{h}_{kq,l}^{(\mathfrak{1})}\big((x_1,x_2)\big)=(x_1 + \sigma^{(\mathfrak{5})}_{kq,l}(x_2),x_2)\\ & \mathfrak{h}_{kq,l}^{(\mathfrak{2})}:\mathbb T^2\to\mathbb T^2\qquad\qquad\text{defined by}\qquad\mathfrak{h}_{kq,l}^{(\mathfrak{2})}\big((x_1,x_2)\big)=(x_1 - \sigma^{(\mathfrak{5})}_{kq,l}(x_2),x_2)\\ & \mathfrak{h}_{kq,l}^{(\mathfrak{3})}:\mathbb T^2\to\mathbb T^2\qquad\qquad\text{defined by}\qquad\mathfrak{h}_{kq,l}^{(\mathfrak{3})}\big((x_1,x_2)\big)=(x_1, x_2 + \sigma^{(\mathfrak{6})}_{kq,l}(x_1))\\ & \mathfrak{h}_{kq,l}^{(\mathfrak{4})}:\mathbb T^2\to\mathbb T^2\qquad\qquad\text{defined by}\qquad\mathfrak{h}_{kq,l}^{(\mathfrak{4})}\big((x_1,x_2)\big)=(x_1, x_2 - \sigma^{(\mathfrak{6})}_{kq,l}(x_1)) \end{align} Finally we piece everything together and define the following block-slide type of map \begin{align} & \mathfrak{h}_{k,q,l}:\mathbb T^2\to\mathbb T^2\qquad\qquad\text{defined by}\qquad\mathfrak{h}_{k,q,l} \coloneqq \mathfrak{h}_{k,q,l}^{(\mathfrak{2})}\circ\mathfrak{h}_{k,q,l}^{(\mathfrak{4})}\circ\mathfrak{g}_{k,q,2l}\circ\mathfrak{h}_{k,q,l}^{(\mathfrak{3})}\circ\mathfrak{h}_{k,q,l}^{(\mathfrak{1})} \end{align}
More generally we can define for any $(\mathfrak{i},\mathfrak{j})\neq (0,l-1)$, the following block-slide type of map: \begin{align} & \mathfrak{h}_{k,q,l}^{(\mathfrak{i},\mathfrak{j})}:\mathbb T^2\to\mathbb T^2\qquad\text{defined by}\\ &
\qquad\mathfrak{h}_{k,q,l}^{(\mathfrak{i},\mathfrak{j})} \coloneqq \phi^{(\mathfrak{i}-1)/(kq)}\circ(\mathfrak{h}_{kq,l}^{(\mathfrak{4})})^{2(l-1-\mathfrak{j})}\circ\mathfrak{h}_{k,q,l}\circ(\mathfrak{h}_{k,q,l}^{(\mathfrak{3})})^{2(l-1-\mathfrak{j})}\circ\phi^{-(\mathfrak{i}-1)/(kq)} \end{align}
We end this section by observing that $\mathfrak{h}_{k,q,l}^{(\mathfrak{i},\mathfrak{j})}$ maps $S_{0,l-1}^{kq,l}\to S_{\mathfrak{i},\mathfrak{j}}^{kq,l}$, $S_{\mathfrak{i},\mathfrak{j}}^{kq,l}\to S_{0,l-1}^{kq,l}$ and acts as identity everywhere else. So we obtained all transpositions of the form $(1,n)$ in the symmetric group on a set of $kl$ elements.
\subsubsection*{All permutations are block-slide type of maps}
We now show that any permutation which commutes with $\phi^{1/q}$ is a block-slide type of map.
\begin{maintheorem} \label{permutation = block-slide} Let $\Pi$ be any permutation of $kql$ elements. We can naturally consider $\Pi$ to be a permutation of the partition $\mathcal{S}_{kq,l}$ of the torus $\mathbb T^2$. Assume that $\Pi$ which commutes with $\phi^{1/q}$. Then $\Pi$ is a block-slide type of map. \end{maintheorem}
\begin{proof} Follows from the fact that all permutations are generated by transpositions. \end{proof}
\subsection{Description of the required combinatorics} \label{subsec:constrmin}
Here we prove theorem \ref{theorem prescribed no of measures}. We begin by describing the combinatorics we need at the $n+1$ th stage of the induction process abstractly.
For $t=0,\ldots, r-1$, we consider the following subsets of $\mathbb T^2$: \begin{align}
N_t\coloneqq \mathbb T^1\times \Big[\frac{t}{r},\frac{t+1}{r}\Big) \end{align} We denote the restriction of the Lebesgue measure $\mu$ to $N_t$ by $\mu_t$.
\begin{figure}\label{step function}
\end{figure}
For natural numbers $n,l$ and $q$ we define the following partition of the torus $\mathbb T^2$: \begin{align}
\mathcal{G}_{l^3q}\coloneqq \Big\{G_{i,j, l^3q}:G_{i,j,l^3q}\coloneqq \Big[\frac{i}{l^3q}, \frac{i+1}{l^3q}\Big)\times\Big[\frac{j}{lr},\frac{j+1}{lr}\Big), 0\leq i< l^3q, 0\leq j<lr\Big\} \end{align} We define the following permutation of the above partition: \begin{align}
\mathfrak{h}^{(\mathfrak{2})}:\mathbb T^2\to\mathbb T^2 \end{align} which acts on the atoms of partition $\mathcal{G}_{l^3q}$ that are contained in $[0,1/(lq)) \times \mathbb T^1$ in the following way (for $t=0, \ldots,r-1$)
\begin{align}
&\text{If } 0\leq i <l: \quad (\mathfrak{h}^{(\mathfrak{2})})^{-1}\Big(G_{i,j,l^3q}\Big) = G_{i',j',l^3q}, \qquad \text{where } i'= \lfloor \frac{j}{r} \rfloor, \ j'=r \cdot i +j \mod r, \\
&\text{if } l \leq i <l^2: \quad (\mathfrak{h}^{(\mathfrak{2})})^{-1}\Big(G_{i,tl+j,l^3q}\Big) = G_{i',tl+j',l^3q}, \qquad \text{where } i'= \lfloor \frac{i}{l} \rfloor \cdot l + j, \ j'=i \mod l. \end{align} We extend this permutation to the whole of $\mathbb T^2$ equivariantly. Since the above description is not very clear, we give a somewhat imprecise but more demonstrative description of the above map. Note that the following rectangles get mapped in the following way: \begin{align*}
& (\mathfrak{h}^{(\mathfrak{2})})^{-1} \Big(\Big[\frac{i}{l^3q},\frac{i+1}{l^3q}\Big)\times\Big[0,1\Big)\Big)= \Big[0,\frac{1}{l^2q}\Big)\times\Big[\frac{i}{l},\frac{i+1}{l}\Big)\qquad\text{if}\quad 0\leq i<l\\
& (\mathfrak{h}^{(\mathfrak{2})})^{-1} \Big(\Big[\frac{i}{l^3q},\frac{i+1}{l^3q}\Big)\times\Big[\frac{t}{r},\frac{t+1}{r}\Big)\Big)= \Big[\frac{i'}{l^2q},\frac{i'+1}{l^2q}\Big)\times\Big[\frac{tl+j'}{lr},\frac{(t+1)l+j'+1}{lr}\Big)\quad\text{if}\quad l\leq i<l^2, \end{align*} where $i'= \lfloor \frac{i}{l} \rfloor$ and $j'= i \mod l$. Notice that in the first region narrow rectangular stripes of full height get squished and are distributed over the full height of the torus which will allow us to prove minimality. While all other rectangles are mapped to rectangles of small diameter but they remain within the horizontal strip $N_t$ on the torus. These stripes will form the support of a preimage of the invariant measures. \\ By the previous subsection we know that this is a block slide type of map and hence allows good analytic approximations by Proposition \ref{proposition approximation}. We denote this $(\varepsilon, \delta)$-approximation by $h^{(\mathfrak{2})}$, the corresponding ``bad set'' by $E$ and set $F=\mathbb T^2 \setminus E$.
\begin{figure}\label{minimal with specified number of measures}
\end{figure}
For this number $\delta$ and given natural numbers $q,l,n$, we define the following step function: \begin{align}
\tilde{\kappa}^{(\mathfrak{1})}:[0,\frac{1}{l^2q})\to\mathbb R\qquad\text{defined by}\qquad \tilde{\kappa}^{(\mathfrak{1})}(x)=\begin{cases} 0 &\quad\text{if}\quad x\in [0,\frac{1}{n^2l^3q})\\\frac{\delta}{lr} &\quad\text{if}\quad x\in [\frac{1}{n^2l^3q},\frac{2}{n^2l^3q})\\
2\frac{\delta}{lr} &\quad\text{if}\quad x\in [\frac{1}{n^2l^3q},\frac{2}{n^2l^3q})\\
\ldots &\quad \ldots\\\ldots & \quad \ldots\\
(\lfloor \frac{n^2}{2}\rfloor -2)\frac{\delta}{lr} &\quad\text{if}\quad x\in [\frac{\lfloor \frac{n^2}{2}\rfloor-1}{n^2l^3q},\frac{\lfloor \frac{n^2}{2}\rfloor}{n^2l^3q})\\
(\lfloor \frac{n^2}{2}\rfloor -1)\frac{\delta}{lr} &\quad\text{if}\quad x\in [\frac{\lfloor \frac{n^2}{2}\rfloor}{n^2l^3q},\frac{\lfloor \frac{n^2}{2}\rfloor+1}{n^2l^3q})\\
(\lfloor \frac{n^2}{2}\rfloor -2)\frac{\delta}{lr} &\quad\text{if}\quad x\in [\frac{\lfloor \frac{n^2}{2}\rfloor+1}{n^2l^3q},\frac{\lfloor \frac{n^2}{2}\rfloor + 2}{n^2l^3q})\\
\ldots &\quad \ldots\\\ldots & \quad \ldots\\
\frac{\delta}{lr} &\quad\text{if}\quad x\in [\frac{n^2-2}{n^2l^3q},\frac{n^2-1}{n^2l^3q})\\
0 &\quad\text{if}\quad x\in [\frac{n^2-1}{n^2l^3q},\frac{1}{l^3q})\end{cases} \end{align} Let $\kappa^{\mathfrak{1}}$ be the $\frac{1}{l^3q}$-periodic real-analytic $\left( \tilde{\varepsilon}, \tilde{\delta} \right)$-approximation of $\tilde{\kappa}^{(\mathfrak{1})}$. With the aid of this we define \begin{align}
h^{(\mathfrak{1})}:\mathbb T^2\to\mathbb T^2\qquad\text{defined by}\qquad h^{(\mathfrak{1})}(x_1,x_2)\coloneqq (x_1, x_2+\kappa^{(\mathfrak{1})}(x_1)) \end{align} and we often refer to the above map as ``trapping map''. The purpose of this map is to capture a large portion of \emph{every} $\phi$ orbit. \\ Hereby we introduce the so-called ``trapping zones'' (for $t=0, \dots,r-1$ and $s=0, \dots, lq-1$)
\begin{align}
& A_{s,i} = h^{(\mathfrak{1})}\Big( \phi^{\frac{s}{lq}}\big(\bigcup_{j=0}^{lr-1}G_{i,j,l^3q}\big) \cap F \Big) \qquad\text{if}\quad 0\leq i<l\\
& B^t_{s,i} = h^{(\mathfrak{1})}\Big(\phi^{\frac{s}{lq}}\big(\bigcup_{j=0}^{l-1}G_{i,tl+j,l^3q} \big) \cap F\Big) \qquad\text{if}\quad l\leq i<l^2 \end{align}
In our specific constructions we define $h_{n+1}=h_{\mathfrak{1},n+1} \circ h_{\mathfrak{2},n+1}$ using the parameters $q=q_n$, $l=l_n$, $\varepsilon<\frac{\varepsilon_n}{2^{l_nq_n}}$, $\delta =\delta_n < \frac{1}{n^4 \cdot 2^{l_nq_n}}$, $\tilde{\varepsilon} = \tilde{\varepsilon}_n < \frac{\delta_n}{2^{l_nq_n}}$ and $\tilde{\delta}= \tilde{\delta}_n < \frac{\delta_n}{2^{l_nq_n}}$.
As announced we have the following trapping property:
\begin{lemma} \label{lem trap} Let $x \in \mathbb T^2$ be arbitrary. Then the orbit $\left\{ \phi^{k \alpha_{n+1}}(x) \right\}_{k=0, \ldots, q_{n+1}}$ meets every set $A_{s,i}$. Moreover, for every $B^t_{s,i}$ at least $\omega^n_t(x) \cdot \frac{\left(1-\frac{8}{n^2}\right) \cdot q_{n+1}}{l^3_n q_n}$ iterates of the orbit $\left\{ \phi^{k \alpha_{n+1}}(x) \right\}_{k=0, \ldots, q_{n+1}}$ lie in $B^t_{s,i}$, where $\omega^n_t(x)$ does not depend on $s,i$. On the contrary at most $\frac{10}{n^2} \cdot q_{n+1}$ iterates are not captured by the collection of sets $B^t_{s,i}$. \end{lemma}
\begin{proof} Let $x=\left(x_1, x_2 \right) \in \mathbb T^2$ and $i \in \left\{0,\ldots, l^3q-1\right\}$ be arbitrary. Note that \begin{equation} G_{i,j,l^3q} \cap F \supseteq \left[ \frac{i + \frac{\delta}{2}}{l^3q}, \frac{i + 1- \frac{\delta}{2}}{l^3q} \right] \times \left[ \frac{j+\frac{\delta}{2}}{lr}, \frac{j+1-\frac{\delta}{2}}{lr} \right] \end{equation} by our approximation Proposition \ref{proposition approximation}. Due to $\frac{n^2 \cdot \delta_n}{l_nr}< \frac{1}{n^2 l_n r}$ and our choice of the approximative step function $\kappa^{(\mathfrak{1})}$ there are at most four sections $\left[ \frac{i}{l^3q}+ \frac{u+\frac{\tilde{\delta}}{2}}{n^2l^3q}, \frac{i}{l^3q}+ \frac{u+1-\frac{\tilde{\delta}}{2}}{n^2l^3q} \right]$, where $u \in \left\{1,\ldots,n^2-2\right\}$, on an arbitrary $\left[ \frac{i + \frac{\delta}{2}}{l^3q}, \frac{i + 1- \frac{\delta}{2}}{l^3q} \right]$-section such that $x_2$ does not belong to any of the $h^{(\mathfrak{1})}\Big( \big(G_{i,j,l^3q}\big) \cap F \Big)$-domains for $j=0,\ldots, lr-1$. Since $\left\{ k \cdot \alpha_{n+1} \right\}_{k=0,\ldots, q_{n+1}-1}$ is equidistributed on $\mathbb{S}^1$, the number of iterates $k$, such that $\left\{ \phi^{k \alpha_{n+1}}(x) \right\}_{k=0, \ldots, q_{n+1}-1}$ is captured by one of these domains is at least \begin{equation} \lfloor \frac{\left(1-\frac{6}{n^2}\right) \cdot \left(1-\tilde{\delta}_n\right) \cdot q_{n+1}}{l^3_n q_n} \rfloor \geq \frac{\left(1-\frac{8}{n^2}\right) \cdot q_{n+1}}{l^3_n q_n}. \end{equation} Depending on the point $x \in \mathbb T^2$ there is a portion $\omega^n_t(x)$ of these iterates spent in trapping regions $B^t_{s,i}$ belonging to $N_t$. This portion does not depend on the indices $s,i$. Since there are $l_nq_n \cdot \left(l^2_n-l_n \right)$ such indices, the last claim follows. \end{proof}
\subsection{Proof of minimality} \label{min}
\subsubsection*{Criterion for minimality} We recall the notion of a minimal dynamical system: \begin{definition} Let $X$ be a topological space and $f: X \rightarrow X$ be a continuous transformation. The map $f$ is called minimal if for every $x \in X$ the orbit $\left\{f^i\left(x\right)\right\}_{i \in \mathbb{N}}$ is dense in $X$. \end{definition} Equivalently $f$ is minimal if for every $x \in X$ and every non-empty open set $U \subseteq X$ there is $i \in \mathbb{N}$ such that $f^i\left(x\right) \in U$. In the case of $X$ being a metric space every open set contains an $\gamma$-ball for $\gamma$ sufficiently small. Thus, $f$ is minimal if for every $x \in X$, every $\gamma >0$ and for every $\gamma$-ball $B_{\gamma}$ there is $i \in \mathbb{N}$ such that $f^i\left(x\right) \in B_{\gamma}$. Hereby, we can deduce the subsequent criterion of minimality in the setting of our constructions: \begin{lemma} \label{lem:critmin} Suppose that the set of iterates $\left\{h^{-1}_{n+1} \circ \phi^{i \cdot \alpha_{n+1}} \circ H_{n+1}\left(x\right)\right\}_{i=0,...,q_{n+1}-1}$ meets every set of the form $\left[ \frac{j_1}{l_nq_n}, \frac{j_1 + 1}{l_nq_n}\right] \times \left[ \frac{j_2}{l_n}, \frac{j_2 + 1}{l_n}\right]$ for every $x \in \mathbb{T}^2$. Moreover, we assume that the sequence $\left(T_n\right)_{n \in \mathbb{N}}$ constructed as in section \ref{subsec:constrmin} converges to a diffeomorphism $T$ in the Diff$^{\omega}_{\rho}$-topology and satisfies $d_0\left(T^i,T^i_{n+1}\right)<\frac{1}{2^n}$ for all $i=0,...,q_{n+1}-1$. Then $T = \lim_{n\rightarrow \infty} T_n$ is minimal. \end{lemma}
\begin{proof} At first we observe that \begin{equation*}
\text{diam} \left(H^{-1}_{n}\left(\left[ \frac{j_1}{l_nq_n}, \frac{j_1 + 1}{l_nq_n}\right] \times \left[ \frac{j_2}{l_n}, \frac{j_2 + 1}{l_n}\right]\right)\right) \leq \left\|DH_{n}\right\|_0 \cdot \frac{2}{l_n}, \end{equation*}
which converges to $0$ as $n\rightarrow \infty$ (because of $\left\|DH_{n} \right\|_0 < \frac{l_n}{2^n}$ by equation \ref{ln criterion}), and that the family of sets $\left[ \frac{j_1}{l_nq_n}, \frac{j_1 + 1}{l_nq_n}\right] \times \left[ \frac{j_2}{l_n}, \frac{j_2 + 1}{l_n}\right]$ covers the whole space $\mathbb T^2$. Hence, for every $\varepsilon >0$ and $y \in \mathbb{T}^m$ there is $M_1 \in \mathbb{N}$ such that for every $n \geq M_1$ there exists a set $H^{-1}_{n}\left(\left[ \frac{j_1}{l_nq_n}, \frac{j_1 + 1}{l_nq_n}\right] \times \left[ \frac{j_2}{l_n}, \frac{j_2 + 1}{l_n}\right]\right) \subseteq B_{\frac{\varepsilon}{2}}\left(y\right)$. \\ Let $x \in \mathbb T^2$, $\varepsilon >0$ and an $\varepsilon$-ball $B_{\varepsilon}\left(y\right)$, at which $y \in \mathbb{T}^2$, be arbitrary. Since $d_0\left(T^i,T^i_{n+1}\right)<\frac{1}{2^n}$ for all $i=0,...,q_{n+1}-1$ there is $M_2 \in \mathbb{N}$ such that $d_0\left(T^i,T^i_{n+1}\right)<\frac{\varepsilon}{2}$ for all $i=0,...,q_{n+1}-1$ and $n\geq M_2$. \\ We consider $n\geq \tilde{N} \coloneqq \max\left\{M_1,M_2\right\}$. Then there is a set $H^{-1}_{n}\left(\left[ \frac{j_1}{l_nq_n}, \frac{j_1 + 1}{l_nq_n}\right] \times \left[ \frac{j_2}{l_n}, \frac{j_2 + 1}{l_n}\right]\right)\subseteq B_{\frac{\varepsilon}{2}}\left(y\right)$ and by assumption an $i < q_{n+1}$ such that $T^i_{n+1}\left(x\right) \in H^{-1}_{n}\left(\left[ \frac{j_1}{l_nq_n}, \frac{j_1 + 1}{l_nq_n}\right] \times \left[ \frac{j_2}{l_n}, \frac{j_2 + 1}{l_n}\right]\right)\subseteq B_{\frac{\varepsilon}{2}}\left(y\right)$. By the triangle inequality we obtain \begin{equation*} d\left(T^i\left(x\right), y\right) \leq d\left(T^i\left(x\right), T^i_{n+1}\left(x\right)\right) + d\left(T^i_{n+1}\left(x\right), y\right) \leq d_0\left(T^i, T^i_{n+1}\right) + \frac{\varepsilon}{2} < \varepsilon. \end{equation*} Thus, we conclude $T^i\left(x\right) \in B_{\varepsilon}\left(y\right)$. Hence, $T$ is minimal. \end{proof}
\subsubsection*{Application of the criterion} The conditions on the convergence of the sequence $\left(T_n\right)_{n \in \mathbb{N}}$ and proximity $d_0\left(T^i,T^i_{n+1}\right)<\frac{1}{2^n}$ for all $i=0,...,q_{n+1}-1$ are fulfilled by Remark \ref{close iterates}. Let $x \in \mathbb{T}^2$ and $\left[ \frac{j_1}{l_nq_n}, \frac{j_1 + 1}{l_nq_n}\right] \times \left[ \frac{j_2}{l_n}, \frac{j_2 + 1}{l_n}\right]$ be arbitrary. We have to show that the orbit $\left\{h^{-1}_{n+1} \circ \phi^{i\alpha_{n+1}} \circ H_{n+1}\left(x\right)\right\}_{i=0,...,q_{n+1}-1}$ meets $\left[ \frac{j_1}{l_nq_n}, \frac{j_1 + 1}{l_nq_n}\right] \times \left[ \frac{j_2}{l_n}, \frac{j_2 + 1}{l_n}\right]$. For this purpose, we note that there is $i \in \left\{0,...,q_{n+1}-1\right\}$ with $\phi^{i\alpha_{n+1}}\circ H_{n+1}\left(x\right) \in A_{j_1,j_2}$ by Lemma \ref{lem trap}. Then we compute \begin{align*} h^{-1}_{n+1}\left(A_{j_1,j_2} \right) & = h^{-1}_{2,n+1} \left(\phi^{\frac{j_1}{l_nq_n}}\big(\bigcup_{j=0}^{l_nr-1}G_{j_2,j,l^3_nq_n}\big) \cap F_n\right) = \phi^{\frac{j_1}{l_nq_n}} \circ h^{-1}_{2,n+1} \left(\bigcup_{j=0}^{l_nr-1}G_{j_2,j,l^3_nq_n} \cap F_n\right) \\ & \subset \phi^{\frac{j_1}{l_nq_n}} \left( \left[0, \frac{1}{l^2_nq_n}\right] \times \left[\frac{j_2}{l_n}, \frac{j_2 + 1}{l_n}\right] \right) \subset \left[\frac{j_1}{l_nq_n}, \frac{j_1+1}{l_nq_n}\right] \times \left[\frac{j_2}{l_n}, \frac{j_2 + 1}{l_n}\right] \end{align*} and we can apply Lemma \ref{lem:critmin} to prove the minimality of $T$.
\subsection{The ergodic invariant measures}
\subsubsection*{Construction of the measures} \label{subsubsec:constrm} As announced we will construct the ergodic invariant measures with the aid of the normalized restrictions $\mu_t$ of the Lebesgue measure on the sets $N_t$, i.e. $\mu_t\left(A\right) = \frac{\mu\left(A \cap N_t\right)}{\mu\left(N_t\right)}$ for any measurable set $A \subseteq \mathbb{T}^2$. Since each set $N_t$ is $\phi^{\beta}$-invariant for any $\beta \in \mathbb{S}^1$, we have $\left(\phi^{\beta}\right)_{\ast} \mu_t = \mu_t$. With these we define the measures $\xi^n_t \coloneqq \left(H^{-1}_n\right)_{\ast} \mu_t$ and can prove their $T_n$-invariance: \begin{equation*} \left(T_n\right)_{\ast} \xi^n_t = \left(T_n\right)_{\ast} \left( \left(H^{-1}_n\right)_{\ast} \mu_t\right) = \left(T_n \circ H^{-1}_n\right)_{\ast} \mu_t = \left( H^{-1}_n \circ \phi^{\alpha_{n+1}}\right)_{\ast} \mu_t = \left(H^{-1}_n\right)_{\ast} \left(\phi^{\alpha_{n+1}}\right)_{\ast} \mu_t = \xi^n_t. \end{equation*} Here we used the relation $f_{\ast}g_{\ast} \mu = \left(f \circ g\right)_{\ast} \mu$ for maps $f,g$. This holds because we have for any measurable set $A$: \begin{equation*} f_{\ast}g_{\ast} \mu\left(A\right) = g_{\ast}\mu\left(f^{-1}\left(A\right)\right) = \mu\left(g^{-1}\left(f^{-1}\left(A\right)\right)\right) = \mu\left(\left(f \circ g\right)^{-1}\left(A\right)\right) = \left(f \circ g \right)_{\ast} \mu\left(A\right). \end{equation*} In the next step we want to estimate $\mu\left(H^{-1}_{n+1}\left(N_t\right) \triangle H^{-1}_{n}\left(N_t\right)\right)$. For this purpose, we have to examine which parts of the set $N_t$ are not mapped back to $N_t$ under $h^{-1}_{n+1} = h^{-1}_{\mathfrak{2},n+1} \circ h^{-1}_{\mathfrak{1},n+1}$. The measure difference is composed of our error set $E_n \cap N_t$, the part, where the conjugation map $h^{-1}_{n+1}$ is constructed to prove minimality (i.e. on the $l_nq_n$ sets $\left[\frac{k}{l_nq_n}, \frac{k}{l_nq_n}+\frac{1}{l^2_n \cdot q_n}\right] \times \mathbb{T}$ for $k=0,...,l_nq_n-1$), and the part that is not mapped back to $N_t$ under $h^{-1}_{\mathfrak{1},n+1}$. The last one is caused by the translation about at most $\frac{n^2 \delta_n}{2l_n}$ in the $x_2$-coordinate produced by $h^{-1}_{\mathfrak{1},n+1}$. Altogether, we obtain: \begin{equation} \label{eq:cauchy} \mu\left(H^{-1}_{n+1}\left(N_t\right) \triangle H^{-1}_{n}\left(N_t\right)\right) = \mu\left(h^{-1}_{n+1}\left(N_t\right) \triangle N_t\right) \leq \frac{1}{l_n} + \frac{n^2 \delta_n}{l_n} +\mu(E_{n+1}) \leq \frac{1}{l_n}. \end{equation} Now we can use the same approach as in \cite{Win}, chapter 7: \\ By equation \ref{eq:cauchy} the sequence $\left\{H^{-1}_n\left(N_t\right)\right\}_{n \in \mathbb{N}}$ is a Cauchy sequence in the metric on the associated measure algebra. Since this space is complete (e.g. \cite{Pet}, Proposition 1.4.3.), there exists a limit $B_t \coloneqq \lim_{n \rightarrow \infty} H^{-1}_n \left(N_t\right)$ in the measure algebra. For this limit we have $\mu\left(B_t\right) = \mu\left(N_t\right)$, because $H^{-1}_n$ is measure-preserving. The sets $B_t$ and $B_s$ are measurably disjoint due to the disjointness of the sets $N_t$ and $N_s$. Moreover, we have weak convergence of the measures $\left(\xi^n_t\right)_{n \in \mathbb{N}}$ to a measure $\xi_t$, where $\xi_t\left(A\right) = \frac{\mu\left(A \cap B_t\right)}{\mu\left(B_t\right)}$ for any measurable set $A \subseteq \mathbb{T}^2$. For this absolutely continuous measure $\xi_t$ we conclude $\lim_{n \rightarrow \infty}\left(T_n\right)_{\ast} \xi^n_t\left(A\right) = T_{\ast} \xi_t\left(A\right)$ due to the triangel inequality \begin{align*} \mu\left(H_n\left(T^{-1}_n A\right) \cap N_t\right) & = \mu\left(T^{-1}_n A \cap H^{-1}_n\left(N_t\right)\right) \leq \mu\left(T^{-1}_n A \cap B_t\right) + \mu\left(H^{-1}_n\left(N_t\right) \triangle B_t\right) \\ & \leq \mu\left(T^{-1}_n A \triangle T^{-1}A\right) + \mu\left(T^{-1} A \cap B_t\right) + \mu\left(H^{-1}_n\left(N_t\right) \triangle B_t\right) \end{align*} (where the first summand converges to $0$ as $n \rightarrow \infty$ because of $T_n \rightarrow T$). So we obtain \begin{equation*} \xi_t = \lim_{n \rightarrow \infty} \xi^n_t = \lim_{n \rightarrow \infty} \left(T_n\right)_{\ast} \xi^n_t = T_{\ast} \xi_t \end{equation*} using the shown $T_n$-invariance of the measure $\xi^n_t$. Thus, the measures $\xi_t$ are $T$-invariant. \\ Furthermore, these measures $\xi_t$ are linearly independent because the sets $B_1,...,B_d$ are measurably disjoint as noted before. Since any non-ergodic invariant measure can be written as a linear combination of ergodic measures (\cite{Wa}, Theorem 5.15), there cannot be less than $d$ ergodic measures.
\subsubsection*{Estimates on Birkhoff sums} \label{trappropmin} In this subsection we show that the measures $\xi_t$ are the only possible ergodic measures for $T$. For this purpose, we will prove a result on the Birkhoff sums (see Lemma \ref{lem:birk2}) and have to gain control over almost everything of every $\phi$-orbit. In this connection the following sets are useful: In case of $0 \leq s < l_n q_n$, $0 < j_1 < l_n$ and $0 \leq j_2 < l_n$ we introduce \begin{equation*} \Delta^t_{s,j_1,j_2} = \left[\frac{s}{l_nq_n} + \frac{j_1}{l^2_n\cdot q_n}, \frac{s}{l_nq_n} + \frac{j_1+1}{l^2_n\cdot q_n} \right] \times \left[\frac{t}{r} + \frac{j_2}{l_nr}, \frac{t}{r} + \frac{j_2+1}{l_nr} \right] \end{equation*} Note that there are $l^3_n q_n \cdot \left(1-\frac{1}{l_n}\right)$ such sets $\Delta^t_{s, j_1, j_2}$ on $N_t$. We denote the family of these sets by $\Omega^t_n$ as well as the union of these sets by $\tilde{\Omega}^t_n$. Then $\mu\left(N_t \setminus \tilde{\Omega}^t_n\right) = \frac{1}{l_nr}$, i.e. $\mu_t\left(N_t \setminus \tilde{\Omega}^t_n\right)\leq \frac{1}{l_n}$.\\
We observe that diam$\left(H^{-1}_{n} \left(\Delta^t_{s,j_1,j_2}\right)\right) < \|DH^{-1}_n\|_0 \cdot \frac{1}{l_n}$. By the requirements on the number $l_n$ in equation \ref{cond l birk} we obtain \begin{equation*}
\left| \rho_i\left(H^{-1}_{n} \left(x\right)\right) - \rho_i\left(H^{-1}_{n} \left(y\right)\right) \right| \leq \textnormal{Lip}\left(\rho_i\right) \cdot \textnormal{diam}\left(H^{-1}_{n} \left(\Delta^t_{s,j_1,j_2}\right)\right) < \frac{1}{n^2} \end{equation*} for every $x,y \in \Delta^t_{s,j_1,j_2}$ and the function $\rho_i \in \Xi$ in case of $i=1,...,n$. Averaging over all $y \in \Delta^t_{s,j_1,j_2}$ we obtain: \begin{equation} \label{eq:block}
\left| \rho_i \left(H^{-1}_{n}\left(x\right)\right) - \frac{1}{\xi^n_t\left(H^{-1}_{n} \left(\Delta^t_{s,j_1,j_2}\right)\right)} \int_{H^{-1}_{n} \left(\Delta^t_{s,j_1,j_2}\right)} \rho_i \: d\xi^n_t \right| < \frac{1}{n^2}. \end{equation} Furthermore, we recall that the image of the trapping region $B^t_{s,i}$ under $h^{-1}_{n+1}$ is contained in $\Delta^t_{s,\lfloor \frac{i}{l_n} \rfloor, i \mod l_n}$. Vice versa, $B^t_{s, j_1 \cdot l_n + j_2}$ is the unique trapping region that is mapped into $\Delta^t_{s,j_1,j_2}$. Hence, we can estimate the number of $i \in \left\{0,...,q_{n+1}-1\right\}$ such that $h^{-1}_{n+1} \circ \phi^{i \cdot \alpha_{n+1}} \left(x\right)$ is contained in $\Delta^t_{s, j_1, j_2}$ by $\varpi^n_t\left(x\right)\cdot \frac{\left(1-\frac{8}{n^2}\right) \cdot q_{n+1}}{l^3_n q_n} $ for arbitrary $x \in \mathbb{T}^2$ using Lemma \ref{lem trap}.
\begin{lemma} \label{lem:birk} Let $\rho_i \in \Xi$ and $i=1,...,n$. Then for every $y \in \mathbb{T}^2$ we have \begin{equation*}
\inf_{\xi^n \in \Theta_n} \left| \frac{1}{q_{n+1}} \sum^{q_{n+1}-1}_{k=0} \rho_i\left(T^k_{n+1}y\right) - \int \rho_i \: d\xi^n \right| < \frac{20}{n^2} \cdot \left\|\rho_i\right\|_0 + \frac{1}{n^2}, \end{equation*} where $\Theta_n$ is the simplex generated by $\left\{ \xi^n_0,..., \xi^n_{r-1}\right\}$. \end{lemma}
\begin{proof} Let $x \in \mathbb{T}^2$ be arbitrary. We introduce the measure $\xi^n_x \coloneqq \sum^{r-1}_{t=0} \varpi^n_t\left(x\right) \cdot \xi^n_t \in \Theta_n$. \\
The set of numbers $k \in \left\{0,1,...,q_{n+1}-1\right\}$ such that the iterates $\phi^{k \cdot \alpha_{n+1}}\left(x\right)$ are not contained in one of the trapping regions of the second kind is denoted by $I_{a}$. Referred to Lemma \ref{lem trap} there are at most $\frac{10}{n^2} \cdot q_{n+1}$ numbers in $I_{a}$. We obtain $\left|\sum_{k \in I_a} \rho_i\left( H^{-1}_{n+1} \circ \phi^{k \cdot \alpha_{n+1}} \left(x\right)\right)\right| \leq \left\|\rho_i\right\|_0 \cdot \frac{10}{n^2} \cdot q_{n+1}$. \\ Moreover, we denote the set of $k \in \left\{0,1,...,q_{n+1}-1\right\}$ such that the iterate $h^{-1}_{n+1} \circ \phi^{k\alpha_{n+1}}\left(x\right)$ is contained in the corresponding trapping region $\Delta \in \Omega^t_n$ by $I_{\Delta}$. By the above considerations there are at least $\varpi^n_t\left(x\right) \cdot \frac{\left(1-\frac{8}{n^2}\right) \cdot q_{n+1}}{l^3_n q_n}= \varpi^n_t\left(x\right) \cdot q_{n+1} \cdot \left(1-\frac{8}{n^2}\right) \cdot \mu_t\left(\Delta\right)$ and at most $\varpi^n_t\left(x\right) \cdot q_{n+1} \cdot \mu_t\left(\Delta\right)$ many numbers in $I_{\Delta}$ for an arbitrary $\Delta \in \Omega^t_n$. Thus, we obtain for an arbitrary $\Delta \in \Omega^t_n$ using equation \ref{eq:block}: \begin{align*}
& \left|\frac{1}{q_{n+1}}\sum_{j \in I_{\Delta}} \rho_i \left( H^{-1}_{n+1} \circ \phi^{j\alpha_{n+1}}\left(x\right)\right) - \int_{H^{-1}_{n} \left(\Delta\right)} \rho_i \: d\left(\varpi^n_t\left(x\right)\xi^n_t\right) \right| \\
\leq & \frac{\left(\varpi^n_t\left(x\right)\mu_t\right)\left(\Delta\right)}{n^2} + \frac{8}{n^2} \cdot \int_{H^{-1}_{n}\left(\Delta\right)} \left|\rho_i \right| \: d\left(\varpi^n_t\left(x\right)\xi^n_t\right) \leq \left(\varpi^n_t\left(x\right)\mu_t\right)\left(\Delta\right) \cdot \left(\frac{1}{n^2} + \frac{8}{n^2} \cdot \left\|\rho_i\right\|_0 \right). \end{align*} Altogether, we conclude \begin{align*}
& \left| \frac{1}{q_{n+1}} \sum^{q_{n+1}-1}_{k=0} \rho_i\left(H^{-1}_{n+1} \circ \phi^{k\alpha_{n+1}}x\right) - \int \rho_i \:d\xi^n_x \right| = \Bigg{|} \frac{1}{q_{n+1}} \sum^{q_{n+1}-1}_{k=0} \rho_i\left(H^{-1}_{n+1} \circ \phi^{k\alpha_{n+1}}(x)\right) \\
&\qquad\qquad\qquad\qquad\qquad - \sum^{r-1}_{t=0} \left( \sum_{\Delta \in \Omega^t_n} \int_{H^{-1}_{n} \left(\Delta\right)} \rho_i \: d\left(\varpi^n_t\left(x\right)\xi^n_t\right) + \int_{H^{-1}_{n} \left(N_t \setminus \tilde{\Omega}^t_n\right)} \rho_i \: d\left(\varpi^n_t\left(x\right)\xi^n_t\right)\right) \Bigg{|} \\
& \qquad\qquad\qquad \leq \left| \sum^{r-1}_{t=0} \sum_{\Delta \in \Omega^t_n} \left(\frac{1}{q_{n+1}}\sum_{j \in I_{\Delta}} \rho_i \left( H^{-1}_{n+1} \circ \phi^{j\alpha_{n+1}}\left(x\right)\right) - \int_{H^{-1}_{n} \left(\Delta\right)} \rho_i d\left(\varpi^n_t\left(x\right)\xi^n_t\right) \right)\right| \\
& \qquad\qquad\qquad\qquad\qquad\qquad + \frac{1}{q_{n+1}} \cdot \left\|\rho_i\right\|_0 \cdot \frac{10}{n^2} \cdot q_{n+1} + \left\|\rho_i\right\|_0 \cdot \sum^{r-1}_{t=0} \left(\varpi^n_t\left(x\right)\mu_t\right)\left(N_t \setminus \tilde{\Omega}^t_n\right) \\
& \qquad\qquad\qquad \leq \frac{1}{n^2} + \frac{8}{n^2} \cdot \left\|\rho_i\right\|_0 + \left\|\rho_i\right\|_0 \cdot \frac{10}{n^2}+ \frac{2 \cdot \left\|\rho_i\right\|_0}{l_n} = \frac{20}{n^2} \cdot \left\|\rho_i\right\|_0 + \frac{1}{n^2}. \end{align*} With $x= H_{n+1}\left(y\right)$ we obtain the claim. \end{proof}
We point out that the measure $\xi^n_x$ used in the above proof was dependent on the point $x$, but independent of the function $\rho \in \Xi$. \begin{lemma} \label{lem:birk2} For every $\rho \in \Xi$ and $y \in \mathbb{T}^2$ we have \begin{equation*}
\inf_{\xi^n \in \Theta_n} \left| \frac{1}{q_{n+1}} \sum^{q_{n+1}-1}_{k=0} \rho\left(T^k\left(y\right)\right) - \int \rho \: d\xi^n \right| \rightarrow 0 \ \ \text{ as } n\rightarrow \infty, \end{equation*} where $\Theta_n$ is the simplex generated by $\left\{ \xi^n_0,..., \xi^n_{r-1}\right\}$. \end{lemma}
\begin{proof} By Remark \ref{close iterates} we have \begin{equation*} d^{\left(q_{n+1}\right)}_0\left(T,T_{n+1}\right) \coloneqq \max_{i=0,1,...,q_{n+1}-1} d_0\left(T^i, T^i_{n+1}\right) \stackrel{n \rightarrow \infty}{\rightarrow} 0. \end{equation*}
Then for every $\rho \in \Xi$ we have $\left| \rho\left(T^i\left(x\right)\right)-\rho\left(T^i_{n+1}\left(x\right)\right)\right| \stackrel{n \rightarrow \infty}{\rightarrow} 0$ uniformly for $i=0,1,...,q_{n+1}-1$, because every continuous function on the compact space $\mathbb{T}^2$ is uniformly continuous. Thus, we get: $\left\| \frac{1}{q_{n+1}} \sum^{q_{n+1}-1}_{i=0} \rho\left(T^i\left(x\right)\right) - \frac{1}{q_{n+1}} \sum^{q_{n+1}-1}_{i=0} \rho\left(T^i_n\left(x\right)\right)\right\|_0 \stackrel{n \rightarrow \infty}{\rightarrow} 0$. Applying the previous Lemma \ref{lem:birk} we obtain the claim. \end{proof}
Since the family $\Xi$ is dense in $C\left(\mathbb{T}^2, \mathbb{R}\right)$, the convergence holds for every continuous function by an approximation argument. \\ Now we can prove that the measures $\xi_0,...,\xi_{r-1}$ are the only possible ergodic ones: Assume that there is another ergodic invariant probability measure $\xi$. By the Birkhoff Ergodic Theorem we have for every $\rho \in C\left(\mathbb{T}^2, \mathbb{R}\right)$ \begin{equation*} \lim _{n \rightarrow \infty} \frac{1}{n}\sum^{n-1}_{k=0} \rho\left(T^k\left(x\right)\right) = \int_{\mathbb{T}^2} \rho \; d\xi \;\;\; \text{ for $\xi$-a.e. } x \in \mathbb{T}^2. \end{equation*} With the aid of Lemma \ref{lem:birk2} we obtain for every $\rho \in C\left(\mathbb{T}^2, \mathbb{R}\right)$ and $x$ in a set of $\xi$-full measure: \begin{equation*} \int_{\mathbb{T}^2} \rho \; d\xi = \lim _{n \rightarrow \infty} \frac{1}{n}\sum^{n-1}_{k=0} \rho\left(T^k\left(x\right)\right) = \lim _{n \rightarrow \infty} \frac{1}{q_{n+1}}\sum^{q_{n+1}-1}_{k=0} \rho\left(T^k\left(x\right)\right) = \lim _{n \rightarrow \infty} \int_{\mathbb{T}^2} \rho \;d\xi^n, \end{equation*} where $\xi^n$ is in the simplex generated by $\left\{\xi^n_0,...,\xi^n_{r-1}\right\}$. As noted this measure does not depend on the function $\rho$. Thus, we have for every $\rho \in C\left(\mathbb{T}^2, \mathbb{R}\right)$: $\lim _{n \rightarrow \infty} \int_{\mathbb{T}^2} \rho \;d\xi^n=\int_{\mathbb{T}^2} \rho \; d\xi$. Since the simplex generated by $\left\{\xi_0,...,\xi_{r-1}\right\}$ is weakly closed, this implies that $\xi$ is in this simplex. We recall that ergodic measures are the extreme points in the set of invariant Borel probability measures (see \cite{Wa}, Theorem 5.15.). Then $\xi$ has to be one of the measures $\left\{\xi_0,...,\xi_{r-1}\right\}$ and we obtain a contradiction. Hence, the measures $\xi_t$, $t=0,\ldots, r-1$ are the only possible ergodic ones. Since we have already observed that these are linearly independent and any non-ergodic invariant measure can be written as a linear combination of ergodic ones, we conclude that the $\xi_t$ are exactly the ergodic measures of $T$.
\subsection{Possible Generalizations}
\begin{remark} By putting the combinatorics from \cite[section 5]{AK} on the rectangles $\left[ \frac{i}{l^3q}, \frac{i+1}{l^3q} \right) \times \left[ \frac{t}{r}, \frac{t+1}{r} \right)$ that are not contained in the minimality region, we can construct the diffeomorphism $T$ to be even weakly mixing with respect to each measure $\xi_t$. We can realize these combinatorics with the aid of Theorem \ref{permutation = block-slide}. \end{remark}
\begin{remark} With some additional technical and notational effort it is possible to generalize Theorem \ref{theorem prescribed no of measures} and the previous Remark to any torus $\mathbb T^d$, $d \geq 2$. Indeed, we construct the $r$ ergodic invariant measures on $N_t = \mathbb{S}^1 \times \left[ \frac{t}{r}, \frac{t+1}{r} \right) \times \mathbb T^{d-2}$ and consider partition elements \begin{equation} \left[ \frac{i_1}{l^{d+1}q}, \frac{i_1+1}{l^{d+1}q} \right) \times \left[ \frac{i_2}{lr}, \frac{i_2+1}{lr} \right) \times \left[\frac{i_3}{l},\frac{i_3+1}{l} \right) \times \ldots \times \left[\frac{i_d}{l}, \frac{i_d+1}{l} \right) \end{equation} as building blocks in the description of the combinatorics. Then we use the combinatorics from the beginning of section \ref{constr nsr} to map sets of the form $\left[ \frac{i}{l^{d+1}q}, \frac{i+1}{l^{d+1}q} \right) \times \left[ \frac{j}{lr}, \frac{j+1}{lr} \right) \times [0,1)^{d-2}$ to sets of the form $\left[ \frac{i_1}{l^{3}q}, \frac{i_1+1}{l^{3}q} \right) \times \left[ \frac{j}{lr}, \frac{j+1}{lr} \right) \times \left[\frac{i_3}{l},\frac{i_3+1}{l} \right) \times \ldots \times \left[\frac{i_d}{l}, \frac{i_d+1}{l} \right)$. \end{remark}
\begin{remark} In this Remark we present modifications in order to prove the existence of a real-analytic diffeomorphism $T \in \text{Diff }^\omega_\rho(\mathbb T^2,\mu)$ which is minimal and has countable many ergodic invariant measures. By weak*-convergence there must be at least one singular ergodic measure. Indeed, we have precisely one singular measure and the other invariant measures are absolutely continuous with respect to Lebesgue measure.
This time we are going to construct the invariant measures on sets $N_t$, $t \in \mathbb Z$. For $t \in {\mathbb N}$ we define $N_t = \mathbb{S}^1 \times \left[ \frac{t}{t+1}, \frac{ t+1}{t+2} \right) \subset \mathbb T^2$ with $x_2$-length $\frac{1}{(t+2) \cdot (t+1)}$. For each $n \in {\mathbb N}$ we choose $t_n \in {\mathbb N}$ such that \begin{equation} \label{eq t}
\frac{1}{(t_n + 1) \cdot t_n } < \frac{\delta_{n-1}}{2^n \cdot \|DH^{-1}_n \|_0 \cdot \max_{i=1, \ldots, n} \text{Lip}(\rho_i)}. \end{equation} Hereby, we define the further sets \begin{align*}
& N_0 = \mathbb{S}^1 \times \left[ \frac{1}{(t_1 + 1) \cdot t_1 }, \frac{1}{2} \right), \\
& N_{-n} = \mathbb{S}^1 \times \left[ \frac{1}{\left( t_{n+1}+1 \right) \cdot t_{n+1}}, \frac{1}{\left(t_n +1 \right) \cdot t_n} \right) \text{ for every } n \in {\mathbb N}. \end{align*} Additionally, for every $n \in {\mathbb N}$ we will use sets \begin{align*}
& \bar{N}_{t_n} = \mathbb{S}^1 \times \left[ \frac{t_n}{t_n +1}, 1 \right) = \bigcup_{t \geq t_n} N_t \\
& N^{(1)}_{-n} = \mathbb{S}^1 \times \left[ 0, \frac{1}{\left( t_n+1 \right) \cdot t_n } \right). \end{align*} This time we choose $l_n = \left(t_n + 1 \right)!$. Note that by equation \ref{eq t} the conditions \ref{ln criterion} and \ref{cond l birk} are satisfied. Moreover, this choice of $l_n$ allows us to consider building blocks $\left[ \frac{i}{l^3_n q_n}, \frac{i+1}{l^3_n q_n} \right) \times \left[ \frac{j}{l_n}, \frac{j+1}{l_n} \right)$ for the $\frac{1}{l_nq_n}$-equivariant combinatorics of $h^{-1}_{\mathfrak{2},n+1}$ to be contained in $\bar{N}_{t_n}$, $N^{(1)}_{-n}$ and $N_t$ for $-n< t < t_n$. As before, $h^{-1}_{\mathfrak{2}, n+1}$ is supposed to map long stripes $\left[ \frac{i}{l^3_n q_n}, \frac{i+1}{l^3_n q_n} \right) \times \left[0,1 \right)$ to $\left[0,\frac{1}{l^2_n q_n} \right) \times \left[ \frac{i}{l_n}, \frac{i+1}{l_n} \right)$ for $0 \leq i < l_n$. For $l_n \leq i < l^2_n$ $h^{-1}_{\mathfrak{2}, n+1}$ maps stripes of width $\frac{1}{l^3_nq_n}$ and full height in the particular set $N^{(1)}_{-n}$ or $N_t$ for $-n< t < t_n$ to sets with height $\frac{1}{l_n}$, while on $\bar{N}_{t_n}$ $h^{-1}_{\mathfrak{2}, n+1}$ acts approximately as the identity on the building blocks. For the trapping map $h^{(\mathfrak{1})}$ the step function is constructed with steps of seize $\frac{\delta}{l}$ in our modification. \end{remark}
\section{Future Work}
Finally we note that Theorem \ref{permutation = block-slide} can be used to upgrade many constructions from the smooth category to the analytic category on the torus. We list some results here.
\subsection{Real-analytic diffeomorphisms with homogeneous spectrum and disjointness of convolutions}
The second author in \cite{Ku-Dc} was able to show that on any smooth compact connected manifold M of dimension $m \geq 2$ admitting a smooth non-trivial circle action, there exists a smooth diffeomorphism $f \in A_\alpha = \{h \circ \phi^\alpha \circ h^{-1} : h \in \text{Diff}^\infty (M,\mu)\}$ for every Liouvillian number $\alpha$ which admits a good approximation of type $(h, h + 1)$, a maximal spectral type disjoint with its convolutions and a homogeneous spectrum of multiplicity two for the Cartesian square $f \times f$. Its is possible to generalize this result to the analytic category for some Liouvillian numbers.
\begin{theorem} For any $\rho>0$, there exist real-analytic diffeomorphisms $T\in \text{Diff }_\rho^\omega (\mathbb T^2, \mu)$ that have a maximal spectral type disjoint with its convolutions, a homogeneous spectrum of multiplicity 2 for $T \times T$ and admit a good approximation of type $(h, h + 1)$. \end{theorem}
\subsection{Coding untwisted AbC diffeomorphisms and the anti-classification problem}
This work is motivated from a series of pioneering work done by Belezney, Foreman, Hjorth, Rudolph and Weiss on the interface of ergodic theory and foundations of mathematics. They were able to show that the conjugacy problem in abstract ergodic theory is non Borel. Later Foreman and Weiss found a method to code a `large' class of smooth diffeomorphisms constructed on $\mathbb T^2$ or the annulus or the disk by an untwisted version of the AbC method into some symbolic systems known as \emph{uniform circular systems}. This in particular shows that the measure isomorphism relation among pairs $(S,T)$ of measure preserving diffeomorphisms of $M$ is not a Borel set with respect to the $C^\infty$ topology.
The first author was able to show that the constructions we do in the real-analytic category on $\mathbb T^2$ are robust enough to construct a large family of untwisted AbC diffeomorphisms measure theoretically isomorphic to uniform circular systems. Loosely this can be summarized into the following theorem:
\begin{theorem}[\cite{Ba-Sr}] Let $T$ be an ergodic transformation on a standard measure space. Then the following are equivalent: \begin{enumerate} \item $T$ is measure theoretically isomorphic to a real-analytic (untwisted) AbC diffeomorphism (satisfying some requirements). \item $T$ is isomorphic to a uniform circular system (with `fast' growing parameters). \end{enumerate} \end{theorem}
This along with some additional works of Foreman and Weiss would imply an anti-classification result for measure preserving real-analytic diffeomorphisms. More precisely, \begin{theorem} The measure-isomorphism relation among pairs $(S,T)\in\text{Diff }^\omega_\rho(\mathbb T^2,\mu)\times \text{Diff }^\omega_\rho(\mathbb T^2,\mu)$ is not a Borel set with respect to the $\text{Diff }^\omega_\rho(\mathbb T^2,\mu)$ topology. \end{theorem}
\noindent\emph{Acknowledgement:} The authors would like to thank Anatole Katok for numerous discussions and constant encouragements. Additionally, the second author would like to thank the Center for Dynamics and Geometry at Penn State for hospitality and financial support at a visit in November 2016 when large parts of this paper were completed. He also acknowledges financial support by the ``Forschungsfonds'' at the Department of Mathematics, University of Hamburg.
\end{document} | arXiv |
Qualia Mind, meanwhile, combines more than two dozen ingredients that may support brain and nervous system function – and even empathy, the company claims – including vitamins B, C and D, artichoke stem and leaf extract, taurine and a concentrated caffeine powder. A 2014 review of research on vitamin C, for one, suggests it may help protect against cognitive decline, while most of the research on artichoke extract seems to point to its benefits to other organs like the liver and heart. A small company-lead pilot study on the product found users experienced improvements in reasoning, memory, verbal ability and concentration five days after beginning Qualia Mind.
One often-cited study published in the British Journal of Pharmacology looked at cognitive function in the elderly and showed that racetam helped to improve their brain function.19 Another study, which was published in Psychopharmacology, looked at adult volunteers (including those who are generally healthy) and found that piracetam helped improve their memory.20
Modafinil is a prescription smart drug most commonly given to narcolepsy patients, as it promotes wakefulness. In addition, users indicate that this smart pill helps them concentrate and boosts their motivation. Owing to Modafinil, the feeling of fatigue is reduced, and people report that their everyday functions improve because they can manage their time and resources better, as a result reaching their goals easier.
Want to try a nootropic stack for yourself? Your best bet is to buy Smart Drugs online. You can get good prices and have the supplements delivered to your home. This means no hassle for you. And after you get them in the mail, you can start to see the benefits for yourself. If you're going to order smart drugs on the internet, it's important to go with one of the top manufacturers so that you get the best product possible.
…The first time I took supplemental potassium (50% US RDA in a lot of water), it was like a brain fog lifted that I never knew I had, and I felt profoundly energized in a way that made me feel exercise was reasonable and prudent, which resulted in me and the roommate that had just supplemented potassium going for an hour long walk at 2AM. Experiences since then have not been quite so profound (which probably was so stark for me as I was likely fixing an acute deficiency), but I can still count on a moderately large amount of potassium to give me a solid, nearly side effect free performance boost for a few hours…I had been doing Bikram yoga on and off, and I think I wasn't keeping up the practice because I wasn't able to properly rehydrate myself.
We included studies of the effects of these drugs on cognitive processes including learning, memory, and a variety of executive functions, including working memory and cognitive control. These studies are listed in Table 2, along with each study's sample size, gender, age and tasks administered. Given our focus on cognition enhancement, we excluded studies whose measures were confined to perceptual or motor abilities. Studies of attention are included when the term attention refers to an executive function but not when it refers to the kind of perceptual process taxed by, for example, visual search or dichotic listening or when it refers to a simple vigilance task. Vigilance may affect cognitive performance, especially under conditions of fatigue or boredom, but a more vigilant person is not generally thought of as a smarter person, and therefore, vigilance is outside of the focus of the present review. The search and selection process is summarized in Figure 2.
Both nootropics startups provide me with samples to try. In the case of Nootrobox, it is capsules called Sprint designed for a short boost of cognitive enhancement. They contain caffeine – the equivalent of about a cup of coffee, and L-theanine – about 10 times what is in a cup of green tea, in a ratio that is supposed to have a synergistic effect (all the ingredients Nootrobox uses are either regulated as supplements or have a "generally regarded as safe" designation by US authorities)
70 pairs is 140 blocks; we can drop to 36 pairs or 72 blocks if we accept a power of 0.5/50% chance of reaching significance. (Or we could economize by hoping that the effect size is not 3.5 but maybe twice the pessimistic guess; a d=0.5 at 50% power requires only 12 pairs of 24 blocks.) 70 pairs of blocks of 2 weeks, with 2 pills a day requires (70 \times 2) \times (2 \times 7) \times 2 = 3920 pills. I don't even have that many empty pills! I have <500; 500 would supply 250 days, which would yield 18 2-week blocks which could give 9 pairs. 9 pairs would give me a power of:
I took 1.5mg of melatonin, and went to bed at ~1:30AM; I woke up around 6:30, took a modafinil pill/200mg, and felt pretty reasonable. By noon my mind started to feel a bit fuzzy, and lunch didn't make much of it go away. I've been looking at studies, and users seem to degrade after 30 hours; I started on mid-Thursday, so call that 10 hours, then 24 (Friday), 24 (Saturday), and 14 (Sunday), totaling 72hrs with <20hrs sleep; this might be equivalent to 52hrs with no sleep, and Wikipedia writes:
As shown in Table 6, two of these are fluency tasks, which require the generation of as large a set of unique responses as possible that meet the criteria given in the instructions. Fluency tasks are often considered tests of executive function because they require flexibility and the avoidance of perseveration and because they are often impaired along with other executive functions after prefrontal damage. In verbal fluency, subjects are asked to generate as many words that begin with a specific letter as possible. Neither Fleming et al. (1995), who administered d-AMP, nor Elliott et al. (1997), who administered MPH, found enhancement of verbal fluency. However, Elliott et al. found enhancement on a more complex nonverbal fluency task, the sequence generation task. Subjects were able to touch four squares in more unique orders with MPH than with placebo.
Low-tech methods of cognitive enhancement include many components of what has traditionally been viewed as a healthy lifestyle, such as exercise, good nutrition, adequate sleep, and stress management. These low-tech methods nevertheless belong in a discussion of brain enhancement because, in addition to benefiting cognitive performance, their effects on brain function have been demonstrated (Almeida et al., 2002; Boonstra, Stins, Daffertshofer, & Beek, 2007; Hillman, Erickson, & Kramer, 2008; Lutz, Slagter, Dunne, & Davidson, 2008; Van Dongen, Maislin, Mullington, & Dinges, 2003).
Natural nootropic supplements derive from various nutritional studies. Research shows the health benefits of isolated vitamins, nutrients, and herbs. By increasing your intake of certain herbal substances, you can enhance brain function. Below is a list of the top categories of natural and herbal nootropics. These supplements are mainstays in many of today's best smart pills.
My first impression of ~1g around 12:30PM was that while I do not feel like running around, within an hour I did feel like the brain fog was lighter than before. The effect wasn't dramatic, so I can't be very confident. Operationalizing brain fog for an experiment might be hard: it doesn't necessarily feel like I would do better on dual n-back. I took 2 smaller doses 3 and 6 hours later, to no further effect. Over the following weeks and months, I continued to randomly alternate between potassium & non-potassium days. I noticed no effects other than sleep problems.
Either way, if more and more people use these types of stimulants, there may be a risk that we will find ourselves in an ever-expanding neurological arm's race, argues philosophy professor Nicole Vincent. But is this necessarily a bad thing? No, says Farahany, who sees the improvement in cognitive functioning as a social good that we should pursue. Better brain functioning would result in societal benefits, she argues, "like economic gains or even reducing dangerous errors."
More than once I have seen results indicating that high-IQ types benefit the least from random nootropics; nutritional deficits are the premier example, because high-IQ types almost by definition suffer from no major deficiencies like iodine. But a stimulant modafinil may be another such nootropic (see Cognitive effects of modafinil in student volunteers may depend on IQ, Randall et al 2005), which mentions:
"We stumbled upon fasting as a way to optimize cognition and make yourself into a more efficient human being," says Manuel Lam, an internal medicine physician who advises Nootrobox on clinical issues. He and members of the company's executive team have implanted glucose monitors in their arms — not because they fear diabetes but because they wish to track the real-time effect of the foods they eat.
What worries me about amphetamine is its addictive potential, and the fact that it can cause stress and anxiety. Research says it's only slightly likely to cause addiction in people with ADHD, [7] but we don't know much about its addictive potential in healthy adults. We all know the addictive potential of methamphetamine, and amphetamine is closely related enough to make me nervous about so many people giving it to their children. Amphetamines cause withdrawal symptoms, so the potential for addiction is there.
This continued up to 1 AM, at which point I decided not to take a second armodafinil (why spend a second pill to gain what would likely be an unproductive set of 8 hours?) and finish up the experiment with some n-backing. My 5 rounds: 60/38/62/44/5023. This was surprising. Compare those scores with scores from several previous days: 39/42/44/40/20/28/36. I had estimated before the n-backing that my scores would be in the low-end of my usual performance (20-30%) since I had not slept for the past 41 hours, and instead, the lowest score was 38%. If one did not know the context, one might think I had discovered a good nootropic! Interesting evidence that armodafinil preserves at least one kind of mental performance.
Capsule Connection sells 1000 00 pills (the largest pills) for $9. I already have a pill machine, so that doesn't count (a sunk cost). If we sum the grams per day column from the first table, we get 9.75 grams a day. Each 00 pill can take around 0.75 grams, so we need 13 pills. (Creatine is very bulky, alas.) 13 pills per day for 1000 days is 13,000 pills, and 1,000 pills is $9 so we need 13 units and 13 times 9 is $117.
And yet aside from anecdotal evidence, we know very little about the use of these drugs in professional settings. The Financial Times has claimed that they are "becoming popular among city lawyers, bankers, and other professionals keen to gain a competitive advantage over colleagues." Back in 2008 the narcolepsy medication Modafinil was labeled the "entrepreneur's drug of choice" by TechCrunch. That same year, the magazine Nature asked its readers whether they use cognitive-enhancing drugs; of the 1,400 respondents, one in five responded in the affirmative.
Also known as Arcalion or Bisbuthiamine and Enerion, Sulbutiamine is a compound of the Sulphur group and is an analog to vitamin B1, which is known to pass the blood-brain barrier easily. Sulbutiamine is found to circulate faster than Thiamine from blood to brain. It is recommended for patients suffering from mental fatigue caused due to emotional and psychological stress. The best part about this compound is that it does not have most of the common side effects linked with a few nootropics.
Stimulants are drugs that accelerate the central nervous system (CNS) activity. They have the power to make us feel more awake, alert and focused, providing us with a needed energy boost. Unfortunately, this class encompasses a wide range of drugs, some which are known solely for their side-effects and addictive properties. This is the reason why many steer away from any stimulants, when in fact some greatly benefit our cognitive functioning and can help treat some brain-related impairments and health issues.
So, I thought I might as well experiment since I have it. I put the 23 remaining pills into gel capsules with brown rice as filling, made ~30 placebo capsules, and will use the one-bag blinding/randomization method. I don't want to spend the time it would take to n-back every day, so I will simply look for an effect on my daily mood/productivity self-rating; hopefully Noopept will add a little on average above and beyond my existing practices like caffeine+piracetam (yes, Noopept may be as good as piracetam, but since I still have a ton of piracetam from my 3kg order, I am primarily interested in whether Noopept adds onto piracetam rather than replaces). 10mg doses seem to be on the low side for Noopept users, weakening the effect, but on the other hand, if I were to take 2 capsules at a time, then I'd halve the sample size; it's not clear what is the optimal tradeoff between dose and n for statistical power.
In general, I feel a little bit less alert, but still close to normal. By 6PM, I have a mild headache, but I try out 30 rounds of gbrainy (haven't played it in months) and am surprised to find that I reach an all-time high; no idea whether this is due to DNB or not, since Gbrainy is very heavily crystallized (half the challenge disappears as you learn how the problems work), but it does indicate I'm not deluding myself about mental ability. (To give a figure: my last score well before I did any DNB was 64, and I was doing well that day; on modafinil, I had a 77.) I figure the headache might be food related, eat, and by 7:30 the headache is pretty much gone and I'm fine up to midnight.
Following up on the promising but unrandomized pilot, I began randomizing my LLLT usage since I worried that more productive days were causing use rather than vice-versa. I began on 2 August 2014, and the last day was 3 March 2015 (n=167); this was twice the sample size I thought I needed, and I stopped, as before, as part of cleaning up (I wanted to know whether to get rid of it or not). The procedure was simple: by noon, I flipped a bit and either did or did not use my LED device; if I was distracted or didn't get around to randomization by noon, I skipped the day. This was an unblinded experiment because finding a randomized on/off switch is tricky/expensive and it was easier to just start the experiment already. The question is simple too: controlling for the simultaneous blind magnesium experiment & my rare nicotine use (I did not use modafinil during this period or anything else I expect to have major influence), is the pilot correlation of d=0.455 on my daily self-ratings borne out by the experiment?
The blood half-life is 12-36 hours; hence two or three days ought to be enough to build up and wash out. A week-long block is reasonable since that gives 5 days for effects to manifest, although month-long blocks would not be a bad choice either. (I prefer blocks which fit in round periods because it makes self-experiments easier to run if the blocks fit in normal time-cycles like day/week/month. The most useless self-experiment is the one abandoned halfway.)
By which I mean that simple potassium is probably the most positively mind altering supplement I've ever tried…About 15 minutes after consumption, it manifests as a kind of pressure in the head or temples or eyes, a clearing up of brain fog, increased focus, and the kind of energy that is not jittery but the kind that makes you feel like exercising would be the reasonable and prudent thing to do. I have done no tests, but feel smarter from this in a way that seems much stronger than piracetam or any of the conventional weak nootropics. It is not just me – I have been introducing this around my inner social circle and I'm at 7/10 people felt immediately noticeable effects. The 3 that didn't notice much were vegetarians and less likely to have been deficient. Now that I'm not deficient, it is of course not noticeable as mind altering, but still serves to be energizing, particularly for sustained mental energy as the night goes on…Potassium chloride initially, but since bought some potassium gluconate pills… research indicates you don't want to consume large amounts of chloride (just moderate amounts).
(I was more than a little nonplussed when the mushroom seller included a little pamphlet educating one about how papaya leaves can cure cancer, and how I'm shortening my life by decades by not eating many raw fruits & vegetables. There were some studies cited, but usually for points disconnected from any actual curing or longevity-inducing results.)
Studies show that B vitamin supplements can protect the brain from cognitive decline. These natural nootropics can also reduce the likelihood of developing neurodegenerative diseases. The prevention of Alzheimer's and even dementia are among the many benefits. Due to their effects on mental health, B vitamins make an excellent addition to any smart drug stack.
That study is also interesting for finding benefits to chronic piracetam+choline supplementation in the mice, which seems connected to a Russian study which reportedly found that piracetam (among other more obscure nootropics) increased secretion of BDNF in mice. See also Drug heuristics on a study involving choline supplementation in pregnant rats.↩
There are hundreds of cognitive enhancing pills (so called smart pills) on the market that simply do NOT work! With each of them claiming they are the best, how can you find the brain enhancing supplements that are both safe and effective? Our top brain enhancing pills have been picked by sorting and ranking the top brain enhancing products yourself. Our ratings are based on the following criteria.
Rabiner et al. (2009) 2007 One public and one private university undergraduates (N = 3,390) 8.9% (while in college), 5.4% (past 6 months) Most common reasons endorsed: to concentrate better while studying, to be able to study longer, to feel less restless while studying 48%: from a friend with a prescription; 19%: purchased it from a friend with a prescription; 6%: purchased it from a friend without a prescription
Table 4 lists the results of 27 tasks from 23 articles on the effects of d-AMP or MPH on working memory. The oldest and most commonly used type of working memory task in this literature is the Sternberg short-term memory scanning paradigm (Sternberg, 1966), in which subjects hold a set of items (typically letters or numbers) in working memory and are then presented with probe items, to which they must respond "yes" (in the set) or "no" (not in the set). The size of the set, and hence the working memory demand, is sometimes varied, and the set itself may be varied from trial to trial to maximize working memory demands or may remain fixed over a block of trials. Taken together, the studies that have used a version of this task to test the effects of MPH and d-AMP on working memory have found mixed and somewhat ambiguous results. No pattern is apparent concerning the specific version of the task or the specific drug. Four studies found no effect (Callaway, 1983; Kennedy, Odenheimer, Baltzley, Dunlap, & Wood, 1990; Mintzer & Griffiths, 2007; Tipper et al., 2005), three found faster responses with the drugs (Fitzpatrick, Klorman, Brumaghim, & Keefover, 1988; Ward et al., 1997; D. E. Wilson et al., 1971), and one found higher accuracy in some testing sessions at some dosages, but no main effect of drug (Makris et al., 2007). The meaningfulness of the increased speed of responding is uncertain, given that it could reflect speeding of general response processes rather than working memory–related processes. Aspects of the results of two studies suggest that the effects are likely due to processes other than working memory: D. E. Wilson et al. (1971) reported comparable speeding in a simple task without working memory demands, and Tipper et al. (2005) reported comparable speeding across set sizes.
Not all drug users are searching for a chemical escape hatch. A newer and increasingly normalized drug culture is all about heightening one's current relationship to reality—whether at work or school—by boosting the brain's ability to think under stress, stay alert and productive for long hours, and keep track of large amounts of information. In the name of becoming sharper traders, medical interns, or coders, people are taking pills typically prescribed for conditions including ADHD, narcolepsy, and Alzheimer's. Others down "stacks" of special "nootropic" supplements.
The question of whether stimulants are smart pills in a pragmatic sense cannot be answered solely by consideration of the statistical significance of the difference between stimulant and placebo. A drug with tiny effects, even if statistically significant, would not be a useful cognitive enhancer for most purposes. We therefore report Cohen's d effect size measure for published studies that provide either means and standard deviations or relevant F or t statistics (Thalheimer & Cook, 2002). More generally, with most sample sizes in the range of a dozen to a few dozen, small effects would not reliably be found.
Do note that this isn't an extensive list by any means, there are plenty more 'smart drugs' out there purported to help focus and concentration. Most (if not all) are restricted under the Psychoactive Substances Act, meaning they're largely illegal to sell. We strongly recommend against using these products off-label, as they can be dangerous both due to side effects and their lack of regulation on the grey/black market.
I stayed up late writing some poems and about how [email protected] kills, and decided to make a night of it. I took the armodafinil at 1 AM; the interesting bit is that this was the morning/evening after what turned out to be an Adderall (as opposed to placebo) trial, so perhaps I will see how well or ill they go together. A set of normal scores from a previous day was 32%/43%/51%/48%. At 11 PM, I scored 39% on DNB; at 1 AM, I scored 50%/43%; 5:15 AM, 39%/37%; 4:10 PM, 42%/40%; 11 PM, 55%/21%/38%. (▂▄▆▅ vs ▃▅▄▃▃▄▃▇▁▃)
You'll find several supplements that can enhance focus, energy, creativity, and mood. These brain enhancers can work very well, and their benefits often increase over time. Again, nootropics won't dress you in a suit and carry you to Wall Street. That is a decision you'll have to make on your own. But, smart drugs can provide the motivation boost you need to make positive life changes.
Since dietary supplements do not require double-blind, placebo-controlled, pharmaceutical-style human studies before going to market, there is little incentive for companies to really prove that something does what they say it does. This means that, in practice, nootropics may not live up to all the grandiose, exuberant promises advertised on the bottle in which they come. The flip side, though? There's no need to procure a prescription in order to try them out. Good news for aspiring biohackers—and for people who have no aspirations to become biohackers, but still want to be Bradley Cooper in Limitless (me).
An entirely different set of questions concerns cognitive enhancement in younger students, including elementary school and even preschool children. Some children can function adequately in school without stimulants but perform better with them; medicating such children could be considered a form of cognitive enhancement. How often does this occur? What are the roles and motives of parents, teachers, and pediatricians in these cases? These questions have been discussed elsewhere and deserve continued attention (Diller, 1996; Singh & Keller, 2010).
The methodology would be essentially the same as the vitamin D in the morning experiment: put a multiple of 7 placebos in one container, the same number of actives in another identical container, hide & randomly pick one of them, use container for 7 days then the other for 7 days, look inside them for the label to determine which period was active and which was placebo, refill them, and start again.
28,61,36,25,61,57,39,56,23,37,24,50,54,32,50,33,16,42,41,40,34,33,31,65,23,36,29,51,46,31,45,52,30, 50,29,36,57,60,34,48,32,41,48,34,51,40,53,73,56,53,53,57,46,50,35,50,60,62,30,60,48,46,52,60,60,48, 47,34,50,51,45,54,70,48,61,43,53,60,44,57,50,50,52,37,55,40,53,48,50,52,44,50,50,38,43,66,40,24,67, 60,71,54,51,60,41,58,20,28,42,53,59,42,31,60,42,58,36,48,53,46,25,53,57,60,35,46,32,26,68,45,20,51, 56,48,25,62,50,54,47,42,55,39,60,44,32,50,34,60,47,70,68,38,47,48,70,51,42,41,35,36,39,23,50,46,44,56,50,39
On the other hand, sometimes you'll feel a great cognitive boost as soon as you take a pill. That can be a good thing or a bad thing. I find, for example, that modafinil makes you more of what you already are. That means if you are already kind of a dick and you take modafinil, you might act like a really big dick and regret it. It certainly happened to me! I like to think that I've done enough hacking of my brain that I've gotten over that programming… and that when I use nootropics they help me help people. | CommonCrawl |
Tagged: matrix representation
by Yu · Published 01/21/2018
Find the Matrix Representation of $T(f)(x) = f(x^2)$ if it is a Linear Transformation
For an integer $n > 0$, let $\mathrm{P}_n$ denote the vector space of polynomials with real coefficients of degree $2$ or less. Define the map $T : \mathrm{P}_2 \rightarrow \mathrm{P}_4$ by
\[ T(f)(x) = f(x^2).\]
Determine if $T$ is a linear transformation.
If it is, find the matrix representation for $T$ relative to the basis $\mathcal{B} = \{ 1 , x , x^2 \}$ of $\mathrm{P}_2$ and $\mathcal{C} = \{ 1 , x , x^2 , x^3 , x^4 \}$ of $\mathrm{P}_4$.
Read solution
The Rank and Nullity of a Linear Transformation from Vector Spaces of Matrices to Polynomials
Let $V$ be the vector space of $2 \times 2$ matrices with real entries, and $\mathrm{P}_3$ the vector space of real polynomials of degree 3 or less. Define the linear transformation $T : V \rightarrow \mathrm{P}_3$ by
\[T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = 2a + (b-d)x – (a+c)x^2 + (a+b-c-d)x^3.\]
Find the rank and nullity of $T$.
The Matrix Representation of the Linear Transformation $T (f) (x) = ( x^2 – 2) f(x)$
Let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis.
Let $T : \mathrm{P}_3 \rightarrow \mathrm{P}_{5}$ be the map defined by, for $f \in \mathrm{P}_3$,
\[T (f) (x) = ( x^2 – 2) f(x).\]
Determine if $T(x)$ is a linear transformation. If it is, find the matrix representation of $T$ relative to the standard basis of $\mathrm{P}_3$ and $\mathrm{P}_{5}$.
Linear Transformation $T:\R^2 \to \R^2$ Given in Figure
Let $T:\R^2\to \R^2$ be a linear transformation such that it maps the vectors $\mathbf{v}_1, \mathbf{v}_2$ as indicated in the figure below.
Find the matrix representation $A$ of the linear transformation $T$.
Matrix Representation, Rank, and Nullity of a Linear Transformation $T:\R^2\to \R^3$
Let $T:\R^2 \to \R^3$ be a linear transformation such that
\[T\left(\, \begin{bmatrix}
\end{bmatrix} \,\right)
\end{bmatrix} \text{ and }
T\left(\, \begin{bmatrix}
4\\
(a) Find the matrix representation of $T$ (with respect to the standard basis for $\R^2$).
(b) Determine the rank and nullity of $T$.
(The Ohio State University, Linear Algebra Midterm)
Linear Transformation that Maps Each Vector to Its Reflection with Respect to $x$-Axis
Let $F:\R^2\to \R^2$ be the function that maps each vector in $\R^2$ to its reflection with respect to $x$-axis.
Determine the formula for the function $F$ and prove that $F$ is a linear transformation.
Eigenvalues and Eigenvectors of The Cross Product Linear Transformation
We fix a nonzero vector $\mathbf{a}$ in $\R^3$ and define a map $T:\R^3\to \R^3$ by
\[T(\mathbf{v})=\mathbf{a}\times \mathbf{v}\] for all $\mathbf{v}\in \R^3$.
Here the right-hand side is the cross product of $\mathbf{a}$ and $\mathbf{v}$.
(a) Prove that $T:\R^3\to \R^3$ is a linear transformation.
(b) Determine the eigenvalues and eigenvectors of $T$.
The Matrix for the Linear Transformation of the Reflection Across a Line in the Plane
Let $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself which is the reflection across a line $y=mx$ for some $m\in \R$.
Then find the matrix representation of the linear transformation $T$ with respect to the standard basis $B=\{\mathbf{e}_1, \mathbf{e}_2\}$ of $\R^2$, where
\[\mathbf{e}_1=\begin{bmatrix}
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
A Linear Transformation Preserves Exactly Two Lines If and Only If There are Two Real Non-Zero Eigenvalues
Let $T:\R^2 \to \R^2$ be a linear transformation and let $A$ be the matrix representation of $T$ with respect to the standard basis of $\R^2$.
Prove that the following two statements are equivalent.
(a) There are exactly two distinct lines $L_1, L_2$ in $\R^2$ passing through the origin that are mapped onto themselves:
\[T(L_1)=L_1 \text{ and } T(L_2)=L_2.\]
(b) The matrix $A$ has two distinct nonzero real eigenvalues.
Differentiating Linear Transformation is Nilpotent
Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less.
Consider the differentiation linear transformation $T: P_n\to P_n$ defined by
\[T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).\]
(a) Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a basis of $P_2$. Find the matrix representation $A$ of the linear transformation $T$ with respect to the basis $B$.
(b) Compute $A^3$, where $A$ is the matrix obtained in part (a).
(c) If you computed $A^3$ in part (b) directly, then is there any theoretical explanation of your result?
(d) Now we consider the general case. Let $B$ be any basis of the vector space of $P_n$ and let $A$ be the matrix representation of the linear transformation $T$ with respect to the basis $B$.
Prove that without any calculation that the matrix $A$ is nilpotent.
Null Space, Nullity, Range, Rank of a Projection Linear Transformation
Let $\mathbf{u}=\begin{bmatrix}
\end{bmatrix}$ and $T:\R^3 \to \R^3$ be the linear transformation
\[T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}} \,\right)\mathbf{u}.\]
(a) Calculate the null space $\calN(T)$, a basis for $\calN(T)$ and nullity of $T$.
(b) Only by using part (a) and no other calculations, find $\det(A)$, where $A$ is the matrix representation of $T$ with respect to the standard basis of $\R^3$.
(c) Calculate the range $\calR(T)$, a basis for $\calR(T)$ and the rank of $T$.
(d) Calculate the matrix $A$ representing $T$ with respect to the standard basis for $\R^3$.
(e) Let
\[B=\left\{\, \begin{bmatrix}
\end{bmatrix}, \begin{bmatrix}
\end{bmatrix} \,\right\}\] be a basis for $\R^3$.
Calculate the coordinates of $\begin{bmatrix}
x \\
y \\
\end{bmatrix}$ with respect to $B$.
(The Ohio State University, Linear Algebra Exam Problem)
Subspace Spanned By Cosine and Sine Functions
Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$.
Define the map $f:\R^2 \to \calF[0, 2\pi]$ by
\[\left(\, f\left(\, \begin{bmatrix}
\alpha \\
\beta
\end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta \sin x.\] We put
\[V:=\im f=\{\alpha \cos x + \beta \sin x \in \calF[0, 2\pi] \mid \alpha, \beta \in \R\}.\]
(a) Prove that the map $f$ is a linear transformation.
(b) Prove that the set $\{\cos x, \sin x\}$ is a basis of the vector space $V$.
(c) Prove that the kernel is trivial, that is, $\ker f=\{\mathbf{0}\}$.
(This yields an isomorphism of $\R^2$ and $V$.)
(d) Define a map $g:V \to V$ by
\[g(\alpha \cos x + \beta \sin x):=\frac{d}{dx}(\alpha \cos x+ \beta \sin x)=\beta \cos x -\alpha \sin x.\] Prove that the map $g$ is a linear transformation.
(e) Find the matrix representation of the linear transformation $g$ with respect to the basis $\{\cos x, \sin x\}$.
(Kyoto University, Linear Algebra exam problem)
Differentiation is a Linear Transformation
Let $P_3$ be the vector space of polynomials of degree $3$ or less with real coefficients.
(a) Prove that the differentiation is a linear transformation. That is, prove that the map $T:P_3 \to P_3$ defined by
\[T\left(\, f(x) \,\right)=\frac{d}{dx} f(x)\] for any $f(x)\in P_3$ is a linear transformation.
(b) Let $B=\{1, x, x^2, x^3\}$ be a basis of $P_3$. With respect to the basis $B$, find the matrix representation of the linear transformation $T$ in part (a).
Restriction of a Linear Transformation on the x-z Plane is a Linear Transformation
Let $T:\R^3 \to \R^3$ be a linear transformation and suppose that its matrix representation with respect to the standard basis is given by the matrix
1 & 0 & 2 \\
0 &3 &0 \\
4 & 0 & 5
(a) Prove that the linear transformation $T$ sends points on the $x$-$z$ plane to points on the $x$-$z$ plane.
(b) Prove that the restriction of $T$ on the $x$-$z$ plane is a linear transformation.
(c) Find the matrix representation of the linear transformation obtained in part (b) with respect to the standard basis
\[\left\{\, \begin{bmatrix}
\end{bmatrix} \,\right\}\] of the $x$-$z$ plane.
Find Matrix Representation of Linear Transformation From $\R^2$ to $\R^2$
Let $T: \R^2 \to \R^2$ be a linear transformation such that
\end{bmatrix} \,\right)=\begin{bmatrix}
\end{bmatrix}, T\left(\, \begin{bmatrix}
\end{bmatrix}.\] Then find the matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for every $\mathbf{x}\in \R^2$, and find the rank and nullity of $T$.
Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$
\[ T(\mathbf{e}_1)=\begin{bmatrix}
\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
\end{bmatrix},\] where $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ are the standard basis of $\R^3$.
Then find the rank and the nullity of $T$.
Find a General Formula of a Linear Transformation From $\R^2$ to $\R^3$
Suppose that $T: \R^2 \to \R^3$ is a linear transformation satisfying
\end{bmatrix}\,\right)=\begin{bmatrix}
\end{bmatrix} \text{ and } T\left(\, \begin{bmatrix}
\end{bmatrix}.\] Find a general formula for
x_1 \\
x_2
\end{bmatrix} \,\right).\]
(The Ohio State University, Linear Algebra Math 2568 Exam Problem)
Give a Formula For a Linear Transformation From $\R^2$ to $\R^3$
Let $\{\mathbf{v}_1, \mathbf{v}_2\}$ be a basis of the vector space $\R^2$, where
\[\mathbf{v}_1=\begin{bmatrix}
\end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix}
\end{bmatrix}.\] The action of a linear transformation $T:\R^2\to \R^3$ on the basis $\{\mathbf{v}_1, \mathbf{v}_2\}$ is given by
T(\mathbf{v}_1)=\begin{bmatrix}
\end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}
\end{bmatrix}.
Find the formula of $T(\mathbf{x})$, where
\[\mathbf{x}=\begin{bmatrix}
\end{bmatrix}\in \R^2.\]
Linear Transformation $T(X)=AX-XA$ and Determinant of Matrix Representation
Let $V$ be the vector space of all $n\times n$ real matrices.
Let us fix a matrix $A\in V$.
Define a map $T: V\to V$ by
\[ T(X)=AX-XA\] for each $X\in V$.
(a) Prove that $T:V\to V$ is a linear transformation.
(b) Let $B$ be a basis of $V$. Let $P$ be the matrix representation of $T$ with respect to $B$. Find the determinant of $P$.
Idempotent Linear Transformation and Direct Sum of Image and Kernel
Let $A$ be the matrix for a linear transformation $T:\R^n \to \R^n$ with respect to the standard basis of $\R^n$.
We assume that $A$ is idempotent, that is, $A^2=A$.
Then prove that
\[\R^n=\im(T) \oplus \ker(T).\]
Properties of Nonsingular and Singular Matrices
Irreducible Polynomial $x^3+9x+6$ and Inverse Element in Field Extension
Determine the Number of Elements of Order 3 in a Non-Cyclic Group of Order 57
If Eigenvalues of a Matrix $A$ are Less than $1$, then Determinant of $I-A$ is Positive
Eigenvalues of Squared Matrix and Upper Triangular Matrix | CommonCrawl |
\begin{document}
\makeabstracttitle \begin{abstract} This review discusses methods for learning parameters for image reconstruction problems using bilevel formulations. Image reconstruction typically involves optimizing a cost function to recover a vector of unknown variables that agrees with collected measurements and prior assumptions. State-of-the-art image reconstruction methods learn these prior assumptions from training data using various machine learning techniques, such as bilevel methods.
One can view the bilevel problem as formalizing hyperparameter optimization, as bridging machine learning and cost function based optimization methods, or as a method to learn variables best suited to a specific task. More formally, bilevel problems attempt to minimize an upper-level loss function, where variables in the upper-level loss function are themselves minimizers of a lower-level cost function.
This review contains a running example problem of learning tuning parameters and the coefficients for sparsifying filters used in a regularizer. Such filters generalize the popular total variation regularization method, and learned filters are closely related to convolutional neural networks approaches that are rapidly gaining in popularity. Here, the lower-level problem is to reconstruct an image using a regularizer with learned sparsifying filters; the corresponding upper-level optimization problem involves a measure of reconstructed image quality based on training data.
This review discusses multiple perspectives to motivate the use of bilevel methods and to make them more easily accessible to different audiences. We then turn to ways to optimize the bilevel problem, providing pros and cons of the variety of proposed approaches. Finally we overview bilevel applications in image reconstruction.
\end{abstract}
\footnotetext{The final publication is available from now publishers via \url{http://dx.doi.org/10.1561/2000000111}.}
\tableofcontents
\chapter{Introduction} \label{chap: intro}
Methods for image recovery aim to estimate a good-quality image from noisy, incomplete, or indirect measurements. Such methods are also known as computational imaging. For example, image denoising and image deconvolution attempt to recover a clean image from a noisy and/or blurry input image, and image inpainting tries to complete missing measurements from an image. Medical image reconstruction aims to recover images that humans can interpret from the indirect measurements recorded by a system like a Magnetic Resonance Imaging (MRI) or Computed Tomography (CT) scanner. Such image reconstruction applications are a type of inverse problem \cite{engl:96}.
New methods for image reconstruction attempt to lower complexity, decrease data requirements, or improve image quality for a given input data quality. For example, in CT, one goal is to provide doctors with information to help their patients while reducing radiation exposure \cite{mccollough:17:ldc}. To achieve these lower radiation doses, the CT system must collect data with lower beam intensity or fewer views. Similarly, in MRI, collecting fewer k-space samples can reduce scan times. Such \dquotes{undersampling} leads to an under-determined problem, with fewer knowns (measurements from a scanner) than unknowns (pixels in the reconstructed image), requiring advanced image reconstruction methods.
Existing reconstruction methods make different assumptions about the characteristics of the images being recovered. Historically, the assumptions are based on easily observed (or assumed) characteristics of the desired output image, such as a tendency to have smooth regions with few edges or to have some form of sparsity \cite{eldar:12:cs}. More recent machine learning approaches use training data to discover image characteristics. These learning-based methods often outperform traditional methods, and are gaining popularity in part because of increased availability of training data and computational resources \citep{wang:16:apo,hammernik:2020:machinelearningimage}.
There are many design decisions in learning-based reconstruction methods. How many parameters should be learned? What makes a set of parameters \dquotes{good?} How can one learn these good parameters? Using a bilevel methodology is one systematic way to address these questions.
Bilevel methods are so named because they involve two \dquotes{levels} of optimization: an upper-level loss function that defines a goal or measure of goodness (equivalently, badness) for the learnable parameters and a lower-level cost function that uses the learnable parameters, typically as part of a regularizer. The main benefits of bilevel methods are learning task-based hyperparameters in a principled approach and connecting machine learning techniques with image reconstruction methods that are defined in terms of optimizing a cost function, often called model-based image reconstruction methods. Conversely, the main challenge with bilevel methods is the computational complexity. However, like with neural networks, that complexity is highest during the training process, whereas deployment has lower complexity because it uses only the lower-level problem.
The methods in this review are broadly applicable to bilevel problems, but we focus on formulations and applications where the lower-level problem is an image reconstruction cost function that uses regularization based on analysis sparsity. The application of bilevel methods to image reconstruction problems is relatively new, but there are a growing number of promising research efforts in this direction. We hope this review serves as a primer and unifying treatment for readers who may already be familiar with image reconstruction problems and traditional regularization approaches but who have not yet delved into bilevel methods.
This review lies at the intersection of a specific machine learning method, bilevel, and a specific application, filter learning for image reconstruction. For overviews of machine learning in image reconstruction, see \citep{hammernik:2020:machinelearningimage,ravishankar:20:irf}. For an overview of image reconstruction methods, including classical, variational, and learning-based methods, see \citep{mccann:2019:biomedicalimagereconstruction}. Finally, for historical overviews of bilevel optimization and perspectives on its use in a wide variety of fields, see \citep{dempe:2003:annotatedbibliographybilevel,dempe:2020:bileveloptimizationadvances}. Within the image recovery field, bilevel methods have also been used, \eg, in learning synthesis dictionaries \citep{mairal:2012:taskdrivendictionarylearning}.
The structure of this review is as follows. The remainder of the introduction defines our notation and presents a running example bilevel problem. \cref{chap: image recon} provides background information on the lower-level image reconstruction cost function and analysis regularizers. \cref{chap: hpo} provides background information on the upper-level loss function, specifically loss function design and hyperparameter optimization strategies. These background sections \blue{provide motivation and context for the rest of the review;} they are not exhaustive overviews of these broad topics. \cref{chap: ift and unrolled} presents building blocks for optimizing a bilevel problem. \cref{chap: bilevel methods} uses these building blocks to discuss optimization methods for the upper-level loss function. \cref{chap: applications} discusses previous applications of the bilevel method in image recovery problems, including signal denoising, image inpainting, and medical image reconstruction. It also overviews bilevel formulations for blind learning and learning space-varying tuning parameters. Finally, \cref{chap: conclusion} offers summarizing commentary on the benefits and drawbacks of bilevel methods for computational imaging, connects and compares bilevel methods to other machine learning approaches, and proposes future directions for the field.
\section{Notation}
This review focuses on continuous-valued, discrete space signals. Some papers, \eg, \citep{calatroni:2017:bilevelapproacheslearning,delosreyes:2017:bilevelparameterlearning}, analyze signals in function space, arguing that the goal of high resolution imagery is to approximate a continuous space reality and that analysis in the continuous domain can yield insights and optimization algorithms that are resolution independent. However, the majority of bilevel methods are motivated and described in discrete space. The review does not include discrete-valued settings, such as image segmentation; those problems often require different techniques to optimize the lower-level cost function, although some recent work uses dual formulations to bridge this gap \citep{knobelreiter:2020:beliefpropagationreloaded,ochs:2016:techniquesgradientbasedbilevel}.
The literature is inconsistent in how it refers to variables in machine learning problems. For consistency within this document, we define the following terms: \begin{itemize}[noitemsep,topsep=0pt]
\item \textbf{Hyperparameters}:
Any adjustable
parameters that are part of a model.
Tuning parameters and model parameters are both sub-types of hyperparameters.
This document uses \params to denote a vector of hyperparameters.
\item \textbf{Tuning parameters}:
Scalar parameters that weight terms in a cost function
to determine the relative importance of each term.
This review uses $\beta$ to denote individual tuning parameters.
\item \textbf{Model parameters}:
Parameters, generally in vector or matrix form,
that are used in the structure of a cost or loss function,
typically as part of the regularization term.
In the running example in the next section,
the model parameters are typically filter coefficients,
denoted \vc. \end{itemize}
We write vectors as column vectors and use bold to denote matrices (uppercase letters) and vectors (lowercase letters). Subscripts index vector elements, so $x_i$ is the $i$th element in \vx. For functions that are applied element-wise to vectors, we use notation following the Julia programming language \cite{bezanson:17:jaf}, where $f.(\vx)$ denotes the function $f$ applied element wise to its argument: \[ \vx \in \F^\sdim \implies
f.(\vx) = \begin{bmatrix}
f(x_1) \\
\vdots \\
f(x_\sdim)
\end{bmatrix}
\in \F^\sdim .\] \blue{We will often use this notation in combination with a transposed vector of ones to sum the result of a function applied element-wise to a vector, \ie, \begin{equation}
\vone' f.(\vx) = \sum_{i=1}^N f(x_i) .\end{equation} For example, the standard Euclidean norm is equivalent to $\vone' f.(\vx)$ when $f(x) = \abs{x}^2$ and and the vector 1-norm can be similarly written when $f(x) = \abs{x}$. This notation is helpful for regularizers that do not correspond to norms. } The field \F can be either \R or \C, depending on the application.
\begin{table}[p]
\input{TablesAndAlgs/tab,variablelookup}
\iffigsatend \tabletag{1.1} \fi
\caption{ Overview of frequently used symbols in the review.}
\label{tab: variable lookup} \end{table}
Convolution between a vector, \vx, and a filter, \vc, is denoted as $\vc \conv \vx$. This review assumes all convolutions use circular boundary conditions. Thus, convolution is equivalent to multiplication with a square, circulant matrix: \[
\vc \conv \vx = \mC \vx .\] The conjugate mirror reversal of \vc is denoted as $\Tilde{\vc}$ and its application is equivalent to multiplying with the adjoint of \mC: \[
\Tilde{\vc} \conv \vx = \mC' \vx ,\] \blue{where the prime indicates the Hermitian transpose operation. }
Finally, for partial derivatives, we use the notation that \begin{align}
\nabla_\vx f(\vx,\vy) &= \frac{\partial f(\vx,\vy)}{\partial \vx} \in \F^N,
\nonumber \\
\nabla_{\vx \vy} f(\vx,\vy) &= \left[ \frac{\partial^2 f(\vx,\vy)}{\partial x_i \partial y_j} \right]
\in \F^{\sdim \times \ydim},
\label{eq:nabla-x-y} \text{ and }
\\
\nabla_{\vx \vy} f(\xhat,\hat{\vy}) &= \nabla_{\vx\vy} f(\vx,\vy) \evalat_{\vx=\xhat, \vy=\hat{\vy}} \in \F,
\nonumber \end{align} where $f : \F^N \by \F^M \rightarrow \F$.
\trefs{tab: variable lookup}{tab: function lookup} summarize our frequently used notation for variables and functions.
\begin{table}[ht]
\centering
\input{TablesAndAlgs/tab,variablelookupfunctions}
\iffigsatend \tabletag{1.2} \fi
\caption{ Overview of frequently used functions in the review. }
\label{tab: function lookup} \end{table}
\section{Defining a Bilevel Problem} \label{sec: bilevel set-up}
\blue{This section introduces a generic bilevel problem; the next presents a specific bilevel problem that serves as a running example throughout the review.} Later sections discuss many of the ideas presented here more thoroughly. Our hope is that an early introduction to the formal problem motivates readers and that this section acts as a quick-reference guide to our notation.
This review considers the image reconstruction problem where the goal is to form an estimate $\xhat \in \F^\sdim$ of a (vectorized) latent image, given a set of measurements $\vy \in \F^\ydim$. For denoising problems, $\sdim=\ydim$, but the two dimensions may differ significantly in more general image reconstruction problems. The forward operator, $\mA \in \F^{\ydim \by \sdim}$ models the physics of the system such that one would expect $\vy = \mA \vx$ in an ideal (noiseless) system. We focus on linear imaging systems here, but the concepts generalize readily to nonlinear forward models. When known (in a supervised training setting), we denote the true, underlying signal as $\xtrue \in \F^\sdim$. \blue{Most bilevel methods are supervised, but \sref{sec: prev results loss function} presents a few examples of unsupervised bilevel methods. }
We focus on model-based image reconstruction methods where the goal is to estimate \vx from \vy by solving an optimization problem of the form \begin{equation}
\xhat
= \xhat(\params)
= \argmin_{\vx \in \F^\sdim} \ofcn(\vx \, ; \params, \vy)
\label{eq: xhat definition}. \end{equation} To simplify notation, we drop \vy from the list of \ofcn arguments except where needed for clarity. The quality of the estimate \xhat can depend greatly on the choice of the hyperparameters \params. Historically there have been numerous approaches pursued for choosing \params, such as cross validation \cite{stone:78:cva}, generalized cross validation \cite{golub:79:gcv}, the discrepancy principle \cite{phillips:62:atf}, and Bayesian methods \cite{saquib:98:mpe}, among others.
Bilevel methods provide a framework for choosing hyperparameters. A bilevel problem for learning hyperparameters \params has the following \dquotes{double minimization} form: \begin{align}
\paramh =
\argmin_{\params \in \F^\paramsdim}
& \underbrace{\lfcn (\params \,;\, \xhat(\params))}_{\lfcn(\params)}
\text{ where } \label{eq: generic bilevel upper-level}
\tag{UL}\\
&\xhat(\params) = \argmin_{\vx \in \F^\sdim} \ofcn(\vx \, ; \params) \label{eq: generic bilevel lower-level}
\tag{LL}. \end{align} \fref{fig: generic bilevel} depicts a generic bilevel problem for image reconstruction. The upper-level (UL) loss function, $\lfcn : \R^\paramsdim \times \F^\sdim \mapsto \R$, quantifies how (not) good is a vector \params of learnable parameters. The upper-level depends on the solution to the lower-level (LL) cost function, \ofcn, which depends on \params. The upper-level can also be called the outer optimization, with the lower-level being the inner optimization. Another terminology is leader-follower, as the minimizer of the lower-level follows where the upper-level loss leads. We will also write the upper-level loss function with a single parameter as $\lfcn(\params) \blue{\defeq \lfcn(\params \,;\, \xhat(\params))}$.
\begin{figure}
\caption{
Depiction of a typical bilevel problem
for image reconstruction,
illustrated using XCAT phantom from \citep{segars:10:4xp}.
The upper box represents the training process,
with the upper-level loss
and lower-level cost function.
During training, one minimizes the upper-level loss
with respect to a vector of parameters, \params,
that are used in the image reconstruction task.
Once learned, \paramh
is typically deployed in the same
image reconstruction task,
shown in the lower box.
}
\label{fig: generic bilevel}
\end{figure}
We write the lower-level cost as an optimization problem with \dquotes{argmin} and thus implicitly assume that \ofcn has unique minimizer, \xhat. The lower-level is guaranteed to have a unique minimizer when $\ofcn$ is a strictly convex function of $\vx$. (See \cref{chap: ift and unrolled} for more discussion of this point). \blue{More generally, there may be a set of lower-level minimizers, each having some possibly distinct upper-level loss function value. } For more discussion, \citep{dempe:2003:annotatedbibliographybilevel} defines optimistic and pessimistic versions of the bilevel problem for the case of multiple lower-level solutions.
Bilevel methods typically use training data. Specifically, one often assumes that a given set of \Ntrue good quality images \( \xtrue_1, \ldots, \xtrue_{\Ntrue} \in \F^\sdim \) are representative of the images of interest in a given application. (For simplicity of notation we assume the training images have the same size, but they can have different sizes in practice.) We typically generate corresponding simulated measurements for each training image using the imaging system model: \begin{equation}
\vy_\ntrue
= \mA \xtrue_\ntrue + \vn_\ntrue
,\quad
\ntrue = 1,\ldots, \Ntrue
, \label{eq: y=Ax+n} \end{equation} where $\vn_\ntrue \in \F^\ydim$ denotes an appropriate random noise realization \footnote{
A more general system model allows the noise to depend on the data
and system model,
\ie, $\vn_j(\mA, \vx_j)$.
This generality is needed for applications with certain noise distributions
such as Poisson noise. }. In \eqref{eq: y=Ax+n}, we add one noise realization to each of the \Ntrue images; in practice one could add multiple noise realizations to each $\xtrue_\ntrue$ to augment the training data. We then use the training pairs $ (\xtrue_\ntrue, \vy_\ntrue) $ to learn a good value of \params. After those parameters are learned, we reconstruct subsequent test images using \eqref{eq: xhat definition} with the learned hyperparameters \paramh.
An alternative to the upper-level formulation \eqref{eq: generic bilevel upper-level} is the following stochastic formulation of bilevel learning: \begin{align}
\paramh =
\argmin_{\params \in \F^\paramsdim}
& \underbrace{\E{\lfcn(\params)}}_{
\approx \frac{1}{J} \sum_{\ntrue=1}^\Ntrue \lfcn (\params \,;\, \xhat_\ntrue(\params))}
\label{eq: stochastic bilevel upper-level}
\\
\text{ where }
&\xhat_\ntrue(\params) = \argmin_{\vx \in \F^\sdim} \ofcn(\vx \, ; \params, \vy_j) \label{eq: stochastic bilevel lower-level}. \end{align} The expectation, taken with respect to the training data and noise distributions, is typically approximated as a sample mean over $J$ training examples.
\blue{ The definition of bilevel methods used in \eqref{eq: generic bilevel upper-level} is not universal in the literature. In some works, bilevel methods refer to nested optimization problems with two levels, even when the two levels result from reformulating a single-level problem, \eg, \citep{poon:2021:smoothbilevelprogramming}. That definition is much more encompassing, and includes primal-dual reformulations, Lagrangian reformulations of constrained optimization problems, and alternating methods that introduce then minimize over an auxiliary variable. }
\blue{ Another term in the literature, sometimes used interchangeably with a bilevel problem, is a mathematical program with equilibrium constraints (MPEC). As shown in \cref{chap: ift and unrolled}, many bilevel optimization methods start by transforming the two-level problem into an equivalent single-level problem by replacing the lower-level optimization with a set of constraints based on optimally conditions. Bilevel problems are thus a subset of MPECs. MPECs are generally challenging due to their non-convex nature; even when the lower-level cost function is convex, the upper-level loss function is rarely convex. Importantly, $\lfcn(\cdot,\cdot)$ is often convex with respect to both arguments. However, $\lfcn(\params) = \lfcnargs$ is generally non-convex in \params due to how the lower-level minimizer depends on \params. There is a large literature on MPEC problems, \eg, \citep{fletcher:2002:numericalexperiencesolving,colson:2007:overviewbileveloptimization,dempe:2003:annotatedbibliographybilevel}, and on non-convex optimization more generally \cite{jain:17:nco}. Bilevel methods are one sub-field in this large literature. }
\section{Running Example}
To offer a concrete example, this review will frequently refer to the following running example \eqref{eq: bilevel for analysis filters}, a filter learning bilevel problem: \begin{align}
\paramh &= \argmin_{\params \in \F^\paramsdim} \onehalf \normrsq{\xhat(\params) - \xtrue}_2
, \text{ where } \nonumber \\
\xhat(\params) &= \argmin_{\vx \in \F^\sdim} \onehalf \norm{\mA \vx-\vy}^2_2
+ \ebeta{0} \sum_{k=1}^K \ebeta{k} \mat{1}' \sparsefcn.(\xmath{\vc_k} \conv \vx; \epsilon),
\label{eq: bilevel for analysis filters}
\tag{Ex} \end{align} where $\params \in \F^\paramsdim$ contains all variables that we wish to learn: the filter coefficients $\xmath{\vc_k} \in \F^\filterdim$ and tuning parameters $\beta_k \in \R $ for all $k \in [1,K]$. We include an auxiliary tuning parameter, $\beta_0 \in \R$, for easier comparison to other models. \fref{fig: running example for bilevel} depicts the running example \blue{and \fref{fig: vertbars simple bilevel filter example} shows example learned filters for a toy training image. Ref. \citep{effland:2020:variationalnetworksoptimal} demonstrates how a spectral analysis of learned filters and penalty functions can be interpreted to provide insight into real-world problems. }
\begin{figure}
\caption{Bilevel problem in \eqref{eq: bilevel for analysis filters}.
The vector of learnable hyperparameters, \params,
includes the tuning parameters, $\beta_k$,
and the filter coefficients, \xmath{\vc_k},
shown as example filters.
Although this review will generally
consider learning filters of a single size,
the figure depicts how the framework easily
extends to 2d filters of different sizes.}
\label{fig: running example for bilevel}
\end{figure}
\blue{ The learnable hyperparameters can also include the sparsifying function \sparsefcn, its corner rounding parameter $\epsilon$, the forward model \mA, or some aspect of the data-fit term. For example, \citep{haber:2003:learningregularizationfunctionals,effland:2020:variationalnetworksoptimal} learn the regularization functional and \citep{ehrhardt:2021:inexactderivativefreeoptimization,sherry:2020:learningsamplingpattern} learn part of the forward model. Such examples are relatively rare in the bilevel methods literature to date. }
\blue{ Unlike many learning problems (see examples in \sref{sec: filter constraints}), the running example \eqref{eq: bilevel for analysis filters} does not include any constraints on \params. Learned filters should be those that are best at the given task, where \dquotes{best} is defined by the upper-level loss function. Therefore, a zero mean or norm constraint is not generally required, though some authors have found such constraints helpful, \eg, \citep{kobler:2021:totaldeepvariation,chen:2014:insightsanalysisoperator}. Following previous literature, \eg, \citep{samuel:2009:learningoptimizedmap}, the tuning parameters in \eqref{eq: bilevel for analysis filters} are written in terms of an exponential function to ensure positivity. One could re-write \eqref{eq: bilevel for analysis filters} without this exponentiation \dquotes{trick} and then add a non-negativity constraint to the upper-level problem; most of the methods discussed in this review generalize to this common variation by substituting gradient methods for projected gradient methods. }
In \eqref{eq: bilevel for analysis filters}, we drop the sum over \Ntrue training images for simplicity; the methods easily extend to multiple training signals. For ease of notation, we further simplify by considering \xmath{\vc_k} to be of length \filterdim for all $k$, \eg, a 2D filter might be $\sqrt{\filterdim} \by \sqrt{\filterdim}$. In practice, the filters may be of different lengths with minimal impact on the methods presented in this review.
\begin{figure}
\caption{
Example learned filters for a simple training image,
normalized for easier visualization.
The true image is zero-mean and
repeats three columns
of signal value $\neg0.25$
and one column of signal value $0.75$.
(a) Noisy image.
The lower plot shows a profile
of one row of the image
(marked by a dotted line).
The signal-to-noise ratio, as defined in \eqref{eq: snr definition},
is given in parenthesis.
(b) The denoised image using learned filters
as in \eqref{eq: bilevel for analysis filters}.
(c) Randomly initialized filters for the bilevel method ($K=4$ and $S=4\cdot2$).
(d) Corresponding learned filters.
As expected based on the training image,
the learned filters primarily involve
vertical differences.
\apref{sec: vertbars}
provides further details
including the regularization strength of each learned filter.
}
\label{fig: vertbars simple bilevel filter example}
\end{figure}
The function \sparsefcn in \eqref{eq: bilevel for analysis filters} is a sparsity-promoting function. If we were to choose
$\phi(z) = |z|$, then the regularizer would involve 1-norm terms of the type common in compressed sensing formulations: \[ \mat{1}' \sparsefcn.(\xmath{\vc_k} \conv \vx) = \norm{\xmath{\vc_k} \conv \vx}_1 .\] However, to satisfy differentiability assumptions (see \cref{chap: ift and unrolled}), this review will often consider \sparsefcn to denote the following ``corner rounded'' 1-norm having the shape of a hyperbola with the corresponding first and second derivative: \begin{align}
\sparsefcn(z) &= \sqrt{z^2 + \epsilon^2} \label{eq: corner rounded 1-norm}
\tag{CR1N}
\\
\dsparsefcn(z) &= \frac{z}{\sqrt{z^2 + \epsilon^2}} \in [0,1) \nonumber \\
\ddsparsefcn(z) &= \frac{\epsilon^2}{\left( z^2 + \epsilon^2 \right)^{3/2}} \in (0,\frac{1}{\epsilon}] \nonumber ,\end{align} where $\epsilon$ is a small, relative to the expected range of $z$, parameter that controls the amount of corner rounding. (Here, we use a dot over the function rather than $\nabla$ to indicate a derivative because \sparsefcn has a scalar argument.)
\section{Conclusion}
Bilevel methods for selecting hyperparameters offer many benefits. Previous papers motivate them as a principled way to approach hyperparameter optimization \citep{holler:2018:bilevelapproachparameter,dempe:2020:bileveloptimizationadvances}, as a task-based approach to learning \citep{peyre:2011:learninganalysissparsity,haber:2003:learningregularizationfunctionals,delosreyes:2017:bilevelparameterlearning}, and/or as a way to combine the data-driven improvements from learning methods with the theoretical guarantees and explainability provided by cost function-based approaches \cite{chen:2021:learnabledescentalgorithm,calatroni:2017:bilevelapproacheslearning,kobler:2021:totaldeepvariation}. A corresponding drawback of bilevel methods are their computational cost; see \crefs{chap: ift and unrolled}{chap: bilevel methods} for further discussion.
The task-based nature of bilevel methods is a particularly important advantage; \sref{sec: hpo filter learning} exemplifies why by comparing the bilevel problem to single-level, non-task-based approaches for learning sparsifying filters. Task-based refers to the hyperparameters being learned based on how well they work in the lower-level cost function-- the image reconstruction task in our running example. The learned hyperparameters can also adapt to the training dataset and noise characteristics. The task-based nature yields other benefits, such as making constraints or regularizers on the hyperparameters generally unnecessary; \sref{sec: prev results loss function} presents some exceptions and \citep{dempe:2020:bileveloptimizationadvances} further discusses bilevel methods for applications with constraints.
There are three main elements to a bilevel approach. First, the lower-level cost function in a bilevel problem defines a goal, such as image reconstruction, including what hyperparameters can be learned, such as filters for a sparsifying regularizer. \cref{chap: image recon} provides background on this element specifically for image reconstruction tasks, such as the one in \eqref{eq: bilevel for analysis filters}. \sref{sec: prev results lower level} reviews example cost functions used in bilevel methods.
Second, the upper-level loss function determines how the hyperparameters should be learned. While the squared error loss function in the running example is a common choice, \cref{chap: hpo} discusses other loss functions based on supervised and unsupervised image quality metrics. \sref{sec: prev results loss function} then reviews example loss functions used in bilevel methods.
While less apparent in the written optimization problem, the third main element for a bilevel problem is the optimization approach, especially for the upper-level problem. \sref{sec: hyperparameter search strategies} briefly discusses various hyperparameter optimization strategies, then \crefs{chap: ift and unrolled}{chap: bilevel methods} present multiple gradient-based bilevel optimization strategies. Throughout the review, we refer to the running example to show how the bilevel optimization strategies apply.
\chapter{Background: Cost Functions and Image Reconstruction} \label{chap: image recon}
This review focuses on bilevel problems having image reconstruction as the lower-level problem. Image reconstruction involves undoing any transformations inherent in an imaging system, \eg, a camera or CT scanner, and removing measurement noise, \eg, thermal and shot noise, to realize an image that captures an underlying object of interest, \eg, a patient's anatomy. \fref{fig: image recon pipline} shows an example image reconstruction pipeline for CT data. The following sections formally define image reconstruction, discuss why regularization is important, and overview common approaches to regularization.
\section{Image Reconstruction \label{sec: image recon background}}
Although the true object is in continuous space, image reconstruction is almost always performed on sampled, discretized signals \cite{lewitt:03:oom}. Without going into detail of the discretization process, we define $\xtrue \in \F^\sdim$ as the \dquotes{true,} discrete signal. The goal of image reconstruction is to recover an estimate $\xhat \approx \xtrue$ given corrupted measurements $\vy \in \F^\ydim$. Although we define the signal as a one-dimensional vector for notational convenience, the mathematics generalize to arbitrary dimensions.
\begin{figure}
\caption{Example image reconstruction pipe-line,
illustrated using XCAT phantom from \citep{segars:10:4xp}.
Here $\mathcal{A}$ denotes the actual physical mapping
of the imaging system
and \mA denotes the numerical system matrix
used for reconstruction.
}
\label{fig: image recon pipline}
\end{figure}
To find \xhat, image reconstruction involves minimizing a cost function, $\ofcnargs$, with two terms: \begin{align}
\xhat = \argmin_{\vx \in \F^\sdim}
\underbrace{\overbrace{\dfcnargs}^{\text{Data-fit}} + \;\;\; \beta
\overbrace{\regfcn(\vx \, ; \params)}^{\text{Regularizer}}}_{\ofcnargs}
\label{eq: general data-fit plus reg} \end{align} The first term, \dfcnargs, is a data-fit term that captures the physics of the ideal (noiseless) system using the matrix $\mA \in \F^{\ydim \by \sdim}$; that matrix models the physical system such that we expect an observation, \vy, to be $\vy \approx \mA \vx$.
The most common data-fit term penalizes the square Euclidean norm of the \dquotes{measurement error,} $\dfcnargs = \normsq{\mA \vx - \vy}_2$. This intuitive data-fit term can be derived from a maximum likelihood perspective, assuming a white Gaussian noise distribution \citep{elad:07:avs}. \blue{Using the system model \eqref{eq: y=Ax+n} and assuming the noise is normally distributed with zero-mean and variance $\sigma^2$, the maximum likelihood estimate $\xhat_{\text{MLE}}$ is the image that is most likely given the observation \vy, \ie, \begin{align*}
\xhat_{\text{MLE}} &= \argmax_{\vx \in \F^\sdim} \text{Prob}(\vx \, ; \, \vy, \sigma^2) .\end{align*} Substituting the assumed Gaussian distribution (and ignoring constants independent of \vx), \begin{align*}
\xhat_{\text{MLE}}
&= \argmax_{\vx \in \F^\sdim} e^{\frac{\neg 1}{2\sigma^2} \norm{\mA \vx - \vy}^2 }
= \argmin_{\vx \in \F^\sdim} \onehalf \norm{\mA \vx - \vy}^2 = \mA^+ \vy ,\end{align*} where $\mA^+$ is the pseudo-inverse of \mA.}
\blue{ The regularization term in \eqref{eq: general data-fit plus reg} can be motivated by maximum \textit{a posteriori} probability (MAP) estimation \citep{elad:07:avs}. Rather than maximizing the likelihood of \vx, the MAP estimate $\xhat_{\text{MAP}}$ maximizes the conditional probability of \vx given the observation \vy \begin{align*}
\xhat_\text{MAP} &= \argmax_{\vx \in \F^\sdim} \text{Prob}(\vx | \vy)
\\
&= \argmax_{\vx \in \F^\sdim} \text{Prob}(\vy | \vx) \text{Prob}(\vx) \end{align*} by Bayes theorem. A MAP estimator requires assuming a prior distribution on \vx. Taking the logarithm and substituting the assumed Gaussian distribution for
$\text{Prob}(\vy|\vx \, ; \, \sigma^2)$ yields \begin{align*}
\xhat_\text{MAP}
&= \argmin_{\vx \in \F^\sdim} \frac{1}{2\sigma^2} \norm{\mA \vx - \vy}^2 - \log{\text{Prob}(\vx)} ,\end{align*} where the regularization term in \eqref{eq: general data-fit plus reg} comes from the log probability of \vx, \ie, the two are equivalent when one assumes the probability model $\text{Prob}(\vx) = \frac{1}{Z(\params)} \exp\{-R(\vx \, ; \, \params)\}$, where $Z(\params)$ is a scalar such that the probability integrates to one. The MLE estimate is equivalent to the MAP estimate when the prior on \vx is an (unbounded) ``uniform'' distribution.}
\blue{ While MAP estimation provides a useful perspective, common regularizers do not correspond to proper probability models. Further, the connection between the regularization perspective and the Bayesian perspective is simplest when the parameters \params are given. To learn \params, Bayesian formulations must consider the partition function $Z(\params)$; that complication is avoided for bilevel formulations using a regularized lower-level problem.}
Many image reconstruction problems have linear system models. In image denoising problems, one takes $\mA=\I$. For image inpainting, \mA is a diagonal matrix of 1's and 0's, where the 0's correspond to sample indices of missing data \cite{guillemot:14:iio}. In MRI, the system matrix is often approximated as a diagonal matrix times a discrete Fourier transform matrix, though more accurate models are often needed \cite{fessler:10:mbi}. In some settings, one can learn \mA \cite{golub:80:aao}, or at least parts of \mA \cite{ying:07:jir}, as part of the estimation process. Although the bilevel method generalizes to learning \mA, the majority of papers in the field assume \mA is known; \cref{chap: applications} discusses a few exceptions.
Using the system model \eqref{eq: y=Ax+n}, if \vn were known and \mA were invertible, we could simply compute $\xhat = \xtrue = \mA^{\neg 1}(\vy-\vn)$. However, \vn is random and, while we may be able to model its characteristics, we never know it exactly. Further, the system matrix, \mA, is often not invertible because the reconstruction problem is frequently under-determined, with fewer knowns than unknowns ($\ydim < \sdim$). Therefore, we must include prior assumptions about \xtrue to make the problem feasible. These assumptions about \xtrue are captured in the second, regularization term in \eqref{eq: general data-fit plus reg}, which depends on \params. The following section further discusses regularizers.
In sum, image reconstruction involves finding \xhat that matches the collected data \textit{and} satisfies a set of prior assumptions. The data-fit term encourages \xhat to be a good match for the data; without this term, there would be no need to collect data. The regularization term encourages \xhat to match the prior assumptions. Finally, the tuning parameter, $\beta$, controls the relative importance of the two terms. The cost function can be minimized using different optimization techniques depending on the form of each term.
This section is a very short overview of image reconstruction methods. See \citep{mccann:2019:biomedicalimagereconstruction} for a more thorough review of biomedical image reconstruction.
\section{Sparsity-Based Regularizers} \label{sec: sparsity based regularizers background}
The regularization, or prior assumption, term in \eqref{eq: general data-fit plus reg} often involves assumptions about sparsity \cite{eldar:12:cs,chambolle:2016:introductioncontinuousoptimization}. The basic idea behind sparsity-based regularization is that the true signal is sparse in some representation, while the noise or corruption is not. Thus, one can use the representation to separate the noise and signal, and then keep only the sparse signal component. \blue{In fact, a known sparsifying representation for a signal can help to \dquotes{reconstruct a signal from far fewer measurements than required by the Shannon-Nyquist sampling theorem} \citep{chambolle:2016:introductioncontinuousoptimization}. }
The regularization design problem therefore requires determining what representation best sparsifies the signal. There are two main types of sparsity-based regularizers corresponding to two representational assumptions: synthesis and analysis \cite{elad:07:avs,ravishankar:20:irf}; \fref{fig: analysis vs synthesis} depicts both. While both are popular, this review concentrates on analysis regularizers, which are more widely represented in the bilevel image reconstruction literature. This section briefly compares the analysis and synthesis formulations. Here we simplify the formulas by considering $\mA=\I$; the discussion generalizes to reconstruction by including \mA. For more thorough discussions of analysis and synthesis regularizers, see \citep{elad:07:avs,nam:2013:cosparseanalysismodel,ravishankar:20:irf}.
\begin{figure}
\caption{Depiction of synthesis and analysis sparsity.
Under the synthesis model of sparsity
(left),
\vx is a linear combination of a few
dictionary atoms.
The dictionary, \mD, is typically wide,
with more atoms (columns) than elements in \vx.
Under the analysis model of sparsity
(right),
\vx is orthogonal to many filters.
The filter matrix, \mOmega, is typically tall,
with more filters (rows) than elements in \vx.
}
\label{fig: analysis vs synthesis}
\end{figure}
\subsection{Synthesis Regularizers}
Synthesis regularizers model a signal being composed of building blocks, or \dquotes{atoms.} Small subsets of the atoms span a low dimensional subspace and the sparsity assumption is that the signal requires using only a few of the atoms. More formally, the synthesis model is $\vy = \vx + \vn$, where the signal $\vx = \mD \vz$ and \vz is a sparse vector. The columns of $\mD \in \F^{\sdim \by K}$ contain contain the $K$ dictionary atoms and form a low dimensional subspace for the signal. If \mD is a wide matrix ($\sdim < K$), the dictionary is over-complete and it is easier to represent a wide range of signals with a given number of dictionary atoms. The dictionary is complete when \mD is square (and full rank) and under-complete if \mD is tall (an uncommon choice).
Assuming one knows or has already learned \mD, one can use the sparsity synthesis assumption to denoise a noisy signal \vy by optimizing \begin{align}
\xhat = \mD \cdot \parenr{ \underbrace{\argmin_{\vz \in \F^K} \onehalf \normsq{\mD \vz - \vy} + \vone'\sparsefcn.(\vz)}_{\hat{\vz}} }
. \label{eq: strict synthesis} \end{align} The estimation procedure involves finding the sparse codes, $\hat{\vz}$, from which the image is synthesized via $\xhat = \mD \hat{\vz}$. Common sparsity-inducing functions, \sparsefcn, are \blue{the absolute value or a non-zero indicator function, equivalent to the} 1-norm and 0-norm \blue{respectively}. The 2-norm is occasionally used in the regularizer, but it does not yield true sparse codes and it over-penalizes large values \cite{elad:10}.
As written in \eqref{eq: strict synthesis}, the synthesis formulation constrains the signal, \vx, to be in the range of \mD. This \dquotes{strict synthesis} model can be undesirable in some applications, \eg, when one is not confident in the quality of the dictionary. An alternative formulation is \begin{align}
\xhat &= \argmin_{\vx \in \F^\sdim} \onehalf \normsq{\vx - \vy} + \beta \regfcn(\vx),
\nonumber \\
\regfcn(\vx) &= \min_{\vz \in \F^K} \onehalf \normsq{\vx - \mD \vz} + \vone' \sparsefcn.(\vz),
\label{eq:R-like-lasso} \end{align} which no longer constrains \vx to be exactly in the range of \mD. One can also learn \mD while solving \eqref{eq:R-like-lasso} \cite{peyre:11:aro}.
Both synthesis denoising forms have equivalent sparsity constrained versions; one can replace $\vone' \sparsefcn.(\vz)$ with a characteristic function that is 0 within some desired set and infinite outside it, \eg, \begin{align}
\psi(\vz) =
\begin{cases}
0 & \text{ if } \norm{\vz}_0 \leq \kappa \\
\infty & \text{ else, }
\label{eq: 0-norm characteristic function}
\end{cases} \end{align} for some sparsity constraint given by the hyperparameter $\kappa \in \mathbb{N}$.
See \citep{candes:2006:robustuncertaintyprinciples,elad:10} for discussions of when the synthesis model can guarantee accurate recovery of signals. The minimization problem in \eqref{eq:R-like-lasso} is called sparse coding and is closely related to the LASSO problem \citep{tibshirani:1996:regressionshrinkageselection}. One can think of the entire dictionary \mD as a hyperparameter that can be learned with a bilevel method \cite{zhou:17:bmb}.
\subsection{Analysis Regularizers}
Analysis regularizers model a signal as being sparsified when mapped into another vector space by a linear transformation, often represented by a set of filters. More formally, an analysis model assumes the signal satisfies $\mOmega \vx = \vz$ for a sparse coefficient vector \vz. Often the rows of the matrix $\mOmega \in \F^{K \by \sdim}$ are thought of as filters and the rows of \mOmega where $[\mOmega \vx]_k = 0$ span a subspace to which \vx is orthogonal. The analysis operator is called over-complete if \mOmega is tall ($\sdim < K$), complete if \mOmega is square (and full rank), and under-complete if \mOmega is wide.
A particularly common analysis regularizer is based on a discretized version of total variation (TV) \cite{rudin:92:ntv}, and uses finite difference filters (or, more generally, filters that approximate higher-order derivatives). The finite difference filters sparsify any piece-wise constant (flat) regions in the signal, leaving the edges that are often approximately sparse in natural images. Other common analysis regularizers include the discrete Fourier transform (DFT), curvelets, and wavelet transforms \citep{candes:2011:compressedsensingcoherent}.
The literature is less consistent in analysis regularizer vocabulary, and \mOmega has been called an analysis dictionary, an analysis operator, a filter matrix, and a cosparse operator. The term \dquotes{cosparse} comes from the sparsity holding in the codomain of the transformation \mbox{$T\{\vx\} = \mOmega \vx$}. The cosparsity of \vx with respect to \mOmega is the number of zeros in $\mOmega \vx$ or $K - \norm{\mOmega \vx}_0$ \citep{nam:2013:cosparseanalysismodel}. Correspondingly, \dquotes{cosupport} describes the indices of the rows where $\mOmega \vx = 0$. We find the phrase \dquotes{analysis operator} intuitive for general \mOmega's and \dquotes{filter matrix} more descriptive when referring to the specific (common) case when the rows of \mOmega are dictated by a set of convolutional filters.
Assuming one knows, or has already learned, \mOmega, one can use the analysis sparsity assumption to denoise a noisy signal, \vy, by optimizing \begin{equation}
\xhat = \argmin_{\vx \in \F^\sdim} \onehalf \normsq{\vx - \vy} + \beta \vone' \sparsefcn.(\mOmega \vx)
. \label{eq:analysis-with-one-term} \end{equation} An alternative version is \begin{align}
\xhat = \argmin_{\vx \in \F^\sdim} \onehalf \normsq{\vx - \vy} + \beta \regfcn(\vx) \label{eq: analysis opt function} \\
\regfcn(\vx) = \min_{\vz \in \F^K} \onehalf \normsq{\mOmega \vx - \vz} + \vone' \sparsefcn.(\vz). \nonumber \end{align} As in the synthesis case, both analysis formulations have equivalent sparsity-constrained forms using a characteristic function as in \eqref{eq: 0-norm characteristic function}.
See \citep{candes:2011:compressedsensingcoherent} for an error bound on the estimated signal \xhat when using a 1-norm as the regularization function.
\subsection{Comparing Analysis and Synthesis Approaches} \label{sec: analysis vs synthesis} \label{sec: define dual problem}
The analysis and synthesis models are equivalent when the dictionary and analysis operator are invertible, with $\mD = \mOmega^{\neg1}$ \citep{elad:07:avs}. Furthermore, in the denoising scenario where the system matrix \mA is identity, the two are almost equivalent in the under-complete case, with the lack of full equivalence stemming from the analysis form not constraining \vx to be in the range space \mD \cite{elad:07:avs}.
\blue{ As shown in \citep[Example 3.1]{chambolle:2016:introductioncontinuousoptimization}, the analysis model can more generally be related to a Lasso-like problem using Legendre-Fenchel conjugates and convex duality. \apref{sec: primal dual background} briefly reviews duality and the main results from primal-dual analysis used throughout this review. Considering the analysis operator learning problem \eqref{eq:analysis-with-one-term}, when the sparsity promoting function \sparsefcn is convex and $\sparsefcn(z) < \infty$ for some $z$, the dual problem corresponding to \eqref{eq:analysis-with-one-term} is \begin{equation*}
\hat{\xmath{\vd}} = \argmin_{\xmath{\vd} \in \F^K} \onehalf \normsq{\mOmega' \xmath{\vd} - \vy} + \sparsefcn^*(\xmath{\vd}) ,\end{equation*} where \xmath{\vd} is the dual variable and $\sparsefcn^*$ is the conjugate function of \sparsefcn. (The primal solution \xhat can be computed from $\hat{\xmath{\vd}}$ using \eqref{eq: primal dual minimizer relation}.) This dual problem is similar in form to the inner minimization in the strict synthesis formulation \eqref{eq: strict synthesis}. This relation between the analysis model and its dual formulation is limited to cases where \sparsefcn is convex. }
Whether analysis-based or synthesis-based regularizers are generally preferable is an open question, and the answer likely depends on the application and the relative importance of reconstruction accuracy and speed \citep{elad:07:avs}. Synthesis regularization is perhaps easier to interpret because of its \blue{generative nature. In contrast, bilevel analysis filter learning is a discriminative learning approach: the task-based filters must learn to distinguish \dquotes{good} and \dquotes{bad} image features. }
The synthesis approach used to be \dquotes{widely considered to provide superior results} \citep[950]{elad:07:avs}. However, \citep{elad:07:avs} goes on to show that an analysis regularizer produced more accurate reconstructed images in experiments on real images. Later analysis-based results also show competitive, if not superior, quality results when compared to similar synthesis models \citep{hawe:13:aol, ravishankar:2013:learningsparsifyingtransforms}. See \cite{fessler:20:omf} for a survey of optimization methods for MRI reconstruction and a comparison of the computational challenges for cost functions with synthesis and analysis-based regularizers.
The analysis and synthesis regularizers in \eqref{eq: strict synthesis} and \eqref{eq: analysis opt function} quickly yield infeasibly large operators as the signal size increases. In practice, both approaches are usually implemented with patch-based formulations. For the synthesis approach, the patches typically overlap and there is an averaging effect. Analysis regularizers that have rows corresponding to filters, called the convolutional analysis model, extend very naturally to a global image regularizer. For example, in the lower-level cost function of our running filter learning example \eqref{eq: bilevel for analysis filters}, we can define an analysis regularizer matrix as follows: \begin{equation} \mOmega =
\begin{bmatrix}
\mC_1 \\
\vdots \\
\mC_K
\end{bmatrix}
\in \F^{K \sdim \by \sdim }
\label{eq: stacked convolutional matrix} .\end{equation} \blue{Imposing this convolutional structure on \mOmega helps make learning problems feasible as one only has to learn the \filterdim coefficients of each of the $K$ filters rather than learning the full \mOmega matrix. This structure also ensures translation invariance of the regularizer.} See \cite{chen:2014:insightsanalysisoperator} and \cite{pfister:2019:learningfilterbank} for discussion of the connections between global models and patch-based models for analysis regularizers. The running example in this survey focuses on bilevel learning of \blue{convolutional} analysis regularizers.
\section{Brief History of Analysis Regularizer Learning \label{sec: filter learning history}}
In 2003, \citet{haber:2003:learningregularizationfunctionals} proposed using bilevel methods to learn part of the regularizer in inverse problems. The authors motivate the use of bilevel methods through the task-based nature, noting that \dquotes{the choice of good regularization operators strongly depends on the forward problem.} They consider learning tuning parameters, space-varying weights, and regularization operators (comparable to defining \sparsefcn), all for regularizers based on penalizing the energy in the derivatives of the reconstructed image. Their framework is general enough to handle learning filters. Ref. \citep{haber:2003:learningregularizationfunctionals} was published a few years earlier than the other bilevel methods we consider in this review and was not cited in most other early works; \citep{afkham:2021:learningregularizationparameters} calls it a \dquotes{groundbreaking, but often overlooked publication.}
In 2005, \citet{roth:2005:fieldsexpertsframework} proposed the Field of Experts (FoE) model to learn filters. \blue{Although the FoE is not formulated as a bilevel method,} many papers on bilevel methods for filter learning cite FoE as a starting or comparison point. The FoE model is a translation-invariant analysis operator model, built on convolutional filters. It is motivated by the local operators and presented as a Markov random field model, with the order of the field determined by the filter size.
Under the FoE model, the negative log \footnote{
By taking the log of the probability model
in \citep{roth:2005:fieldsexpertsframework},
the connection between the FoE and
the regularization term in the lower-level
of the running filter learning example
\eqref{eq: bilevel for analysis filters} is more evident. } of the probability of a full image, \vx, is proportional to \begin{align}
\sum_k \beta_k \; \sparsefcn.(\xmath{\vc_k} \conv \vx) \text{ where }
\sparsefcn(z) = \text{log}\left( 1 + \onehalf z^2 \right). \label{eq: FoE} \end{align} This (non-convex) choice of sparsity function $\phi$ stems from the Student-t distribution. Ref. \citep{roth:2005:fieldsexpertsframework} learns the filters and filter-dependent tuning parameters such that the model distribution is as close as possible (defined using Kullback-Leibler divergence) to the training data distribution.
In 2007, \citet{tappen:2007:learninggaussianconditional} proposed a different model based on convolutional filters: the Gaussian Conditional Random Field (GCRF) model. Rather than using a sparsity promoting regularizer, the GCRF uses a quadratic function for \sparsefcn. The authors introduce space-varying weights, \mW, so that the quadratic model does not overly penalize sharp features in the image. The general idea behind $\mW$ is to use the given (noisy) image to guess where edges occur, and correspondingly penalize those areas less to avoid blurring edges. The likelihood for GCRF model is thus (to within a proportionality constant and monotonic function transformations): \begin{align*} \sum_k \normsq{\xmath{\vc_k} \conv \vx - e_k\{\vx\}}_{\mW_k}, \end{align*} where the term $e_k\{\vx\}$ captures the estimated value of the filtered image. For example, \citep{tappen:2007:learninggaussianconditional} used one averaging filter and multiple differencing filters for the \xmath{\vc_k}'s. The corresponding estimated values are \vx for the averaging filter and zero for the differencing filters.
The filters, \xmath{\vc_k}, are pre-determined in the GCRF model; the learned element is how to form the weights as a function of image features. Specifically, each $\mW_k$ is formed as a linear combination of the (absolute) responses to a set of edge-detecting filters, with the linear combination coefficients learned from training data. Rather than maximizing the likelihood of training data as in \citep{roth:2005:fieldsexpertsframework}, \citep{tappen:2007:learninggaussianconditional} learns these coefficients to minimize the (corner-rounded) $l_1$ norm of the error of the predicted image, which is a form of bilevel learning even though not described with that terminology.
Apparently one of the first papers to explicitly propose using bilevel methods to learn filters appeared in 2009, where \textcite{samuel:2009:learningoptimizedmap} considered a bilevel formulation where the upper-level loss was the squared Euclidean norm of training data and the lower-level cost was a denoising task based on filter sparsity equivalent to \eqref{eq: bilevel for analysis filters}. The method builds on the FoE model, using the same \sparsefcn as in \citep{roth:2005:fieldsexpertsframework}, but now learning the filters using a bilevel formulation rather than by maximizing a likelihood.
In 2011, \textcite{peyre:2011:learninganalysissparsity} proposed a similar bilevel method to learn analysis regularizers. The authors generalized the denoising task to use an analysis operator matrix and a wider class of sparsifying functions. Their results concentrate on the convolutional filter case with a corner-rounded 1-norm for \sparsefcn.
Both \citep{samuel:2009:learningoptimizedmap} and \citep{peyre:2011:learninganalysissparsity} focus on introducing the bilevel method for analysis regularizer learning, with denoising or inpainting as illustrations. \cref{chap: ift and unrolled} further discusses the methodology of both papers. Many of the bilevel based papers in this review build on one or both of their efforts. The rest of the review will summarize other bilevel based papers; here, we highlight some of papers in the non-bilevel thread of the literature for context and comparison.
\citet{ophir:2011:sequentialminimaleigenvalues} proposed another approach to learning an analysis operator. The method learns the operator one row at a time by searching for vectors orthogonal to the training signals. Algorithm parameters were chosen empirically without an upper-level loss function as a guide.
Between 2011 \citep{yaghoobi:2011:analysisoperatorlearning} and 2013 \citep{yaghoobi:2013:constrainedovercompleteanalysis}, \setmaxcitenames{10} \citeauthor{yaghoobi:2011:analysisoperatorlearning} \setmaxcitenames{3} were among the first to formally present analysis operator learning as an optimization problem. Their conference paper \citep{yaghoobi:2011:analysisoperatorlearning} considered noiseless training data and proposed learning an analysis operator as \begin{equation}
\argmin_\mOmega \normr{\mOmega \mXtrue}_1 \text{ s.t. } \mOmega \in \S \label{eq: noiseless AOL yaghoobi} \end{equation} for some constrained set \S. Each column of $\mXtrue \in \F^{\sdim \by J}$ contains a training sample. The authors discussed varying options for \S, including a row norm, full rank, and tight frame constrained set.
Without any constraint on \mOmega, the trivial solution to \eqref{eq: noiseless AOL yaghoobi} would be to learn the zero matrix, which is not informative for any problem such as image denoising. \sref{sec: filter constraints} discusses in more detail the need for constraints and the various constraint options proposed for filter learning.
Ref. \citep{yaghoobi:2013:constrainedovercompleteanalysis} extends \eqref{eq: noiseless AOL yaghoobi} to the noisy case where one does not have access to \mXtrue. The proposed cost function is \begin{equation}
\argmin_{\mOmega, \, \mX} \norm{\mOmega \mX}_1 + \frac{\beta}{2} \normsq{\mX - \mY}
\text{ s.t. } \mOmega \in \S, \label{eq: AOL yaghoobi} \end{equation} where each column of \mY contains a noisy data vector. Ref. \citep{yaghoobi:2013:constrainedovercompleteanalysis} minimized \eqref{eq: AOL yaghoobi} by alternating updating \mX, using alternating direction method of multipliers (ADMM), and \mOmega, using a projected subgradient method for various constraint sets \S, especially Parseval tight frames.
\blue{ In the same time-frame, \citet{kunisch:2013:bileveloptimizationapproach} started to analyze the theory behind the bilevel problem, building off the ideas in \citep{samuel:2009:learningoptimizedmap, peyre:2011:learninganalysissparsity}. Among the theoretical analysis, \citep{kunisch:2013:bileveloptimizationapproach} proves the existence of upper-level minimizers when the bilevel problem takes the form of \eqref{eq: bilevel for analysis filters}, \params is the tuning parameters (the $\beta_k$ values), and \sparsefcn corresponds to the squared 2-norm or the 1-norm. When $\sparsefcn(z)=z^2$, there is an analytic solution to the lower-level problem and a corresponding closed-form solution to the gradient of the upper-level problem; \citep{kunisch:2013:bileveloptimizationapproach} uses this fact to discuss qualitative properties of the minimizer. Ref. \citep{kunisch:2013:bileveloptimizationapproach} also proposed an efficient semi-smooth Newton algorithm for finding \paramh (using corner rounding for the 1-norm case) and used this algorithm to make empirical comparisons of multiple sparsifying functions (2-norm, 1-norm, and $p=1/2$-norm) and different pre-defined filter banks. }
Also in 2013, \citet{ravishankar:2013:learningsparsifyingtransforms} made a distinction between the analysis model, where one models $\vy = \vx + \vn$ with $\vz = \mOmega \vx$ being sparse, and the transform model, where $\mOmega \vy = \vz + \vn$ where \vz is sparse. The analysis version models the measurement as being a cosparse signal plus noise; the transform version models the measurement as being approximately cosparse. Another perspective on the distinction is that, if there is no noise, the analysis model constrains \vy to be in the range space of \mOmega, while there is no such constraint on the transform model. The corresponding transform learning problem is \begin{align}
\argmin_{\mOmega} \min_\mZ \normsq{\mOmega \mY - \mZ}_2 + &\regfcn(\mOmega)
\quad \text{ s.t. } \norm{\mZ_i}_0 \leq \alpha \;\forall i, \label{eq: transform learning} \end{align} where $i$ indexes the columns of \mZ. Ref. \citep{ravishankar:2013:learningsparsifyingtransforms} considers only square matrices \mOmega. The regularizer, \regfcn, promotes diversity in the rows of \mOmega to avoid trivial solutions, similar to the set constraint in \eqref{eq: AOL yaghoobi}.
A more recent development is directly modeling the convolutional structure during the learning process. In 2020, \citep{chun:2020:convolutionalanalysisoperator} proposed Convolutional Analysis Operator Learning (CAOL) to learn convolutional filters without patches. The CAOL cost function is \begin{align}
\argmin_{[\vc_1, \ldots, \vc_K]} \sum_{k=1}^K \min_\vz
\onehalf \normsq{\xmath{\vc_k} \conv \vx - \vz}_2 + \beta \norm{\vz}_0
\text{ s.t. } [\vc_1 \ldots \vc_K] \in \S \label{eq: CAOL}. \end{align} Unlike the previous cost functions, which typically require patches, CAOL can easily handle full-sized training images \vx due to the nature of the convolutional operator.
While model-based methods were being developed in the signal processing literature, convolutional neural network (CNN) models were being advanced and trained in the machine learning and computer vision literature \cite{haykin:96:nne} \cite{hwang:97:tpp} \cite{lucas:18:udn}. The filters used in CNN models like U-Nets \cite{ronneberger:15:unc} can be thought of as having analysis roles in the earlier layers, and synthesis roles in the final layers \cite{ye:18:dcf}. See also \cite{wen:20:tlf} for further connections between analysis and transform models within CNN models. CNN training is usually supervised, and the supervised approach of bilevel learning of filters strengthens the relationships between the two approaches. A key distinction is that CNN models are generally feed-forward computations, whereas bilevel methods of the form \eqref{eq: generic bilevel lower-level} have a cost function formulation. See \sref{sec: connections} for further discussion of the parallels between CNNs and bilevel methods.
\section{Summary}
This background section focused on the lower-level problem: image reconstruction with a sparsity-based regularizer. After defining the problem and the need for regularization, \sref{sec: filter learning history} reviewed the history of analysis regularizer learning and included many examples of methods to learn hyperparameters.
Bilevel methods are just one, task-based way to learn such hyperparameters. \sref{sec: filter constraints} further expands on this point, but we can already see benefits of the task-based nature of bilevel methods. Without the bilevel approach, filters are often learned such that they best sparsify training data. These sparsifying filters can then be used in a regularizer for image reconstruction tasks. However, they are learned to \textit{sparsify}, not necessarily to best \textit{reconstruct}. In contrast, the bilevel approach aims to learn filters that best reconstruct images (or whatever other task is desired), even if those filters are not the ones that best sparsify. Although this distinction may seem subtle, \citep{chambolle:2021:learningconsistentdiscretizations} shows that different filters work better for image denoising versus image inpainting.
Having provided some background on the lower-level cost function and motivated bilevel methods, this review now turns to defining the upper-level loss function and surveying methods of hyperparameter optimization.
\chapter{Background: Loss Functions and \texorpdfstring{\\}{} Hyperparameter Optimization} \label{chap: hpo}
Most inverse problems involve at least one hyperparameter. For example, the general reconstruction cost function \eref{eq: general data-fit plus reg} requires choosing the tuning parameter $\beta$ that trades-off the influence of the data-fit and regularization terms. The field of hyperparameter optimization is large and encompasses categorical hyperparameters, such as which optimizer to use; conditional hyperparameters, where certain hyperparameters are relevant only if others take on certain values; and integer or real-valued hyperparameters \citep{feurer:2019:chapterhyperparameteroptimization}. Here, we focus on learning real-valued, continuous hyperparameters.
\begin{figure}
\caption{
Example reconstructed simulated MRI images that
demonstrate the importance of tuning parameters.
(a) The original image, $\xtrue \in \reals^\sdim$, is a SheppLogan phantom
\cite{shepp:74:tfr}
and $N$ is the number of pixels.
(b) A simplistic reconstruction
$\frac{1}{N} \mA'\vy$
of the noisy, undersampled data, \vy.
This image is used as initialization, $\vx^{(0)}$,
for the following reconstructions.
(c-e) Reconstructed images, found by optimizing
$\argmin_\vx \onehalf \normrsq{\mA \vx - \vy}_2 + 10^\beta N \sparsefcn(\mC \vx)$,
where \mC is an operator that takes vertical and horizontal finite differences.
The reconstructed images correspond to
(c) $\beta=-6$, resulting in an image that contains ringing artifacts,
(d) $\beta=-3$, resulting in a visually appealing \xhat,
and
(e) $\beta=1$, resulting in a blurred image.
The demonstration code
and more details about the reconstruction set-up
are available on github
\citep{fessler:2020:mirtdemo01recon}.
}
\label{fig: rmse vs beta}
\end{figure}
A hyperparameter's value can greatly influence the properties of the minimizer and a tuned hyperparameter typically improves over a default setting \citep{feurer:2019:chapterhyperparameteroptimization}. \fref{fig: rmse vs beta} illustrates how changing a tuning parameter can dramatically impact the visual quality of the reconstructed image. If $\beta$ is too low, not enough weight is on the regularization term, and the minimizer is likely to be corrupted by noise in the measurements. If $\beta$ is too high, the regularization term dominates, and the minimizer will not align with the measurements.
Generalizing to an arbitrary learning problem that could have multiple hyperparameters, the goal of hyperparameter optimization is to find the ``best'' set of hyperparameters, $\hat{\params}$, to meet a goal, described by a loss function \lfcn. Specifically, we wish to solve \begin{equation}
\hat{\params} = \argmin_{\params \in \Gamma}
\E{ \lfcn(\params) }, \label{eq: general hyperparameter opt} \end{equation} where $\Gamma$ is the set of all possible hyperparameters and the expectation is taken with respect to the distribution of the input data. If evaluating \lfcn uses the output of another optimization problem, \eg, \xhat, then \eqref{eq: general hyperparameter opt} is a bilevel problem as defined in \eqref{eq: generic bilevel upper-level}.
There are two key tasks in hyperparameter optimization. \blue{ \begin{enumerate}
\item The first is to quantify
how good a hyperparameter is;
this step is equivalent to defining \lfcn in \eqref{eq: general hyperparameter opt}.
\sref{sec: loss function design}
focuses on a high-level discussion of loss functions
in the broader image quality assessment (IQA) literature.
\sref{sec: prev results loss function} builds on this discussion
by reviewing specific loss functions used in bilevel methods.
\item The second step is finding a good hyperparameter,
which is equivalent to designing an optimization algorithm to minimize
\eqref{eq: general hyperparameter opt}.
\sref{sec: hyperparameter search strategies}
introduces common approaches,
all of which have computational requirements that scale at least linearly with the number of hyperparameters.
This scaling quickly becomes infeasible for large \params,
which motivates the focus on gradient-based bilevel methods in the remainder of this review. \end{enumerate} The next two sections address each of these tasks in turn. }
\section{Image Quality Metrics} \label{sec: loss function design}
This section concentrates on the part of the upper-level loss function that compares the reconstructed image, \xhatp, to the true image, \xtrue. As mentioned in \cref{chap: intro}, bilevel methods rarely require additional regularization for \params, but it is simple to add a regularization term to any of the loss functions if useful for a specific application. To discuss only the portion of the loss function that measures image quality, we use the notation $\lfcnargs = \xmath{l}(\xhat, \, \xtrue)$.
Picking a loss function is part of the engineering design process. No single loss function is likely to work in all scenarios; users must decide on the loss function that best fits their system, data, and goals. Consequently, there are a wide variety of loss functions proposed in the literature and some approaches combine multiple loss functions \citep{you:2018:structuresensitivemultiscaledeep,hammernik:2020:machinelearningimage}.
One important decision criteria when selecting a loss function is the end purpose of the image. Much of the IQA literature focuses on metrics for images of natural scenes and is often motivated by applications where human enjoyment is the end-goal \citep{wang:2004:imagequalityassessment,wang:2011:reducednoreferenceimage}. In contrast, in the medical image reconstruction field, image quality is not the end-goal, but rather a means to achieving a correct diagnosis. Thus, the perceptual quality is less important than the information content.
There are two major classes of image quality metrics in the IQA literature, called full-reference and no-reference IQA \footnote{There are also reduced-reference image quality metrics, but we will not consider those here.}. The principles are somewhat analogous to supervised and unsupervised approaches in the machine learning literature. This section discusses some of the most common full-reference and no-reference loss functions; see \citep{zhang:2012:comprehensiveevaluationfull} for a comparison of 11 full-reference IQA metrics and \citep{zhang:2020:blindimagequality} for additional no-reference IQA metrics.
Perhaps surprisingly, the bilevel filter learning literature contains few examples of loss functions other than squared error or slight variants (see \sref{sec: prev results loss function}). While this is likely at least partially due to the computational requirements of bilevel methods (see \cref{chap: ift and unrolled} and \ref{chap: bilevel methods}), exploring additional loss functions is an interesting future direction for bilevel research.
\subsection{Full-Reference IQA} \label{sec: hpo supervised}
Full-reference IQA metrics assume that you have a noiseless image, \xtrue, for comparison. Some of the simplest (and most common) full-reference loss functions are: \begin{itemize}[noitemsep,topsep=0pt]
\item Mean squared error (MSE or $\ell_2$ error):
\[
\xmath{l}_{\mathrm{MSE}}(\xhat,\xtrue) = \frac{1}{\sdim} \normsq{\xhat-\xtrue}_2
\]
\item Mean absolute error (or $\ell_1$ error):
$\xmath{l}_{\mathrm{MAE}}(\xhat,\xtrue) = \frac{1}{\sdim} \norm{\xhat-\xtrue}_1$
\item Signal to Noise Ratio (SNR, commonly expressed in dB): \\
\begin{equation}
\xmath{l}_{\mathrm{SNR}}(\xhat,\xtrue) = 10 \log{\frac{\normsq{\xtrue}_2}{\normsq{\xhat-\xtrue}_2}}
\label{eq: snr definition}
\end{equation}
\item Peak SNR (\ac{PSNR}, in dB): $\xmath{l}_{\mathrm{PSNR}}(\xhat,\xtrue)
= 10 \log{\frac{\sdim \norm{\xtrue}_{\infty}}{\normsq{\xhat-\xtrue}_2}}$. \end{itemize} The Euclidean norm is also frequently used as the data-fit term for reconstruction.
\ac{MSE} (and the related metrics SNR and PSNR) are common in the signal processing field; they are intuitive and easy to use because they are differentiable and operate point-wise. However, these measures do not align well with human perceptions of image quality \cite{mason:2020:comparisonobjectiveimage, zhang:2012:comprehensiveevaluationfull}. For example, scaling an image by 2 leads to the same visual quality but causes 100\% MSE. \fref{fig: rmse for various distortions} shows a clean image and five images with different degradations. All five degraded images have almost equivalent squared errors, but humans judge their qualities as very different.
\begin{figure}
\caption{
Example distortions that yield images with identical
normalized squared error values:
$\norm{\xtrue - \vx}/\norm{\xtrue} = 0.17$.
(a) The original image, \xtrue, is a SheppLogan phantom \cite{shepp:74:tfr}.
The remaining images are displayed with the same colormap
and have the following distortions:
(b) blurred with an averaging filter,
(c) additive, white Gaussian noise,
(d) salt and pepper noise,
and
(e) a constant value added to every pixel.
}
\label{fig: rmse for various distortions}
\end{figure}
Tuning parameters using MSE as the loss function tends to lead to images that are overly-smoothed, sacrificing high frequency information \cite{gholizadehansari:20:dlf,seif:18:ebl}. High frequency details are particularly important for perceptual quality as they correspond to edges in images. Therefore, some authors use the MSE on edge-enhanced versions of images to discourage solutions that blur edges. For example, \citep{ravishankar:2011:mrimagereconstruction} used a \dquotes{high frequency error norm} metric consisting of the MSE of the difference of \xhat and \xtrue after applying a Laplacian of Gaussian (LoG) filter.
Another common full-reference IQA is Structural SIMilarity (\ac{SSIM}) \citep{wang:2004:imagequalityassessment} that attempts to address the issues with \ac{MSE} discussed above. SSIM is defined in terms of the local luminance, contrast, and structure in images. A multiscale extension of \ac{SSIM}, called MS-SSIM, considers these features at multiple resolutions \cite{wang:2003:multiscalestructuralsimilarity}. The method computes the contrast and structure measures of SSIM for downsampled versions of the input images and then defines MS-SSIM as the product of the luminance at the original scale and the contrast and structure measures at each scale. However, SSIM and MS-SSIM may not correlate well with human observer performance on radiological tasks \cite{renieblas:17:ssi}.
Recent works, \eg, \citep{bosse:2018:deepneuralnetworks,zhang:2020:blindimagequality}, consider using (deep) CNN models for IQA. CNN methods are increasingly popular and their use as a model for the human visual system \citep{lindsay:2020:convolutionalneuralnetworks} makes them an attractive tool for assessing images. For example, \citep{bosse:2018:deepneuralnetworks} proposed a CNN with convolutional and pooling layers for feature extraction and fully connected layers for regression. They used VGG \citep{simonyan:2015:verydeepconvolutional}, a frequently-cited CNN design with $3 \by 3$ convolutional kernels, as the basis of the feature extraction portion of their network. Ref. \citep{bosse:2018:deepneuralnetworks} showed that deeper networks with more learnable parameters were able to better predict image quality. However, datasets of images with quality labels remain relatively scarce, making it difficult to train deep networks.
\subsection{No-reference IQA}
No-reference, or unsupervised, IQA metrics attempt to quantify an image's quality without access to a noiseless version of the image. These metrics rely on modeling statistical characteristics of images or noise. Many no-reference IQA metrics assume the noise distribution is known.
The discrepancy principle is a classic example of an IQA metric that uses an assumed noise distribution to characterize the expected relation between the reconstructed image and the noisy data. For additive zero-mean white Gaussian noise with known variance $\sigma^2$, the discrepancy principle uses the fact that the expected MSE in the data space is the noise variance \cite{phillips:62:atf}: \[
\E{\frac{1}{\ydim}\normsq{\mA \xhat(\params) - \vy}_2} = \sigma^2 .\] The discrepancy principle can be used as a stopping criteria in machine learning methods or as a loss function, \eg, \[ \lfcnargs = \left(\frac{1}{\ydim}\normsq{\mA \xhat(\params) - \vy}_2 - \sigma^2 \right)^2 .\] However, images of varying quality can yield the same noise estimate, as seen in \fref{fig: rmse for various distortions}. Related methods have been developed for Poisson noise as well \cite{hebert:92:sbm}.
Paralleling MSE's popularity among supervised loss metrics, Stein's Unbiased Risk Estimator (SURE) \citep{stein:1981:estimationmeanmultivariate} is an unbiased estimate of MSE that does not require noiseless images. Let $\vy = \xtrue + \vn$ denote a signal plus noise measurement where \vn is, as above, Gaussian noise with known variance $\sigma^2$. The SURE estimate of the MSE of a denoised signal, \xhat, is \begin{align}
\frac{1}{\sdim} \normsq{\xhat(\vy) - \vy}_2 - \sigma^2 +
\frac{2 \sigma^2}{\sdim}
\Tr{\nabla_y \xhat(\vy)}
\label{eq: SURE}, \end{align} where we write \xhat as a function of \vy to emphasize the dependence \blue{and \Tr{\cdot} denotes the trace operation.} For large signal dimensions \sdim, such as is common in image reconstruction problems, the law of large numbers suggests SURE is a fairly accurate approximation of the true MSE.
It is often impractical to evaluate the divergence term in \eqref{eq: SURE}, due to computational limitations or not knowing the form of $\xhat(\vy)$. A Monte-Carlo approach to estimating the divergence \cite{ramani:2008:montecarlosureblackbox} uses the following key equation: \begin{align}
\Tr{
\nabla_\vy \xhat(\vy)} =
\lim_{\epsilon \rightarrow 0} \E{\vb' \cdot \frac{\xhat(\vy + \epsilon \vb) - \xhat(\vy)}{\epsilon}}, \label{eq: monte carlo sure} \end{align} where \vb is a independent and identically distributed (i.i.d.) random vector with zero mean, unit variance, and bounded higher order moments. Theoretical and empirical arguments show that a single noise vector can well-approximate the divergence \citep{ramani:2008:montecarlosureblackbox}, so only two calls to the lower-level solver $\xhat(\vy)$ are required. This method treats the lower-level problem like a blackbox, thus allowing one to estimate the divergence of complicated functions, including those that may not be differentiable.
See \citep{soltanayev:2018:trainingdeeplearning,kim:20:uto,zhussip:19:tdl} for examples of applying the Monte-Carlo estimation of SURE to train deep neural networks, and \citep{zhang:2020:bilevelnestedsparse,deledalle:2014:steinunbiasedgradient} for two examples of learning a tuning parameter using a bilevel approach with SURE as the upper-level loss function. For extensions to inverse problems (where $\mA \neq \I)$ and to noise from exponential families, see \citep{eldar:08:rbe,eldar:2009:generalizedsureexponential,giryes:11:tpg}.
While SURE and the discrepancy principle are popular no-reference metrics in the signal processing literature, there are many additional no-reference metrics in the image quality assessment literature. These metrics typically depend on modeling one (or more) of three things \citep{wang:2011:reducednoreferenceimage}: \begin{itemize}[noitemsep,topsep=0pt]
\item image source characteristics,
\item image distortion characteristics, \eg, blocking artifact from JPEG compression, and/or
\item human visual system perceptual characteristics. \end{itemize} As an example of a strategy that can capture both image source and human visual system characteristics, natural scene \footnote{
Natural scenes are those captured by optical cameras
(not created by computer graphics or other artificial processes)
and are not limited to outdoor scenes. } statistics characterize the distribution of various features in natural scenes, typically using some filters \citep{mittal:2013:makingcompletelyblind,wang:2011:reducednoreferenceimage}. If a feature reliably follows a specific statistical pattern in natural images but has a noticeably different distribution in distorted images, one can use that feature to assign quality scores to images. Some IQA metrics attempt to first identify the type of distortion and measure features specific to that distortion, while others use the same features for all images.
In addition to their use in full-reference IQA, CNN models have be trained to perform no-reference IQA \citep{kang:2014:convolutionalneuralnetworks,bosse:2018:deepneuralnetworks}. For example, \citep{kang:2014:convolutionalneuralnetworks} proposes a CNN model that extracts small ($32 \by 32$) patches from images, estimates the quality of each one, and averages the scores over all patches to get a quality score for the entire image. Briefly, their method involves local contrast normalization for each patch, applying (learned) convolutional filters to extract features, maximum and minimum pooling, and fully connected layers with rectified linear units (ReLUs). As with most no-reference IQAs, \citep{kang:2014:convolutionalneuralnetworks} trained their CNN on a dataset of human encoded image quality scores (see \citep{laboratoryforimagevideoengineering::imagevideoquality} for a commonly used collection of publicly available test images with quality scores). Unlike most other IQA approaches, \citep{kang:2014:convolutionalneuralnetworks} used backpropagation to learn all the CNN weights rather than learning a transformation from handcrafted features to quality scores.
Interestingly, some of the no-reference IQA metrics \citep{mittal:2013:makingcompletelyblind,kang:2014:convolutionalneuralnetworks, wang:2011:reducednoreferenceimage} approach the performance of the full-reference IQAs in terms of their ability to match human judgements of image quality. This observation suggests that there is room to improve full-reference IQA metrics and that assessing image quality is a very challenging problem!
\section{Parameter Search Strategies} \label{sec: hyperparameter search strategies}
After selecting a metric to measure how good a hyperparameter is, the next task is devising a strategy to find the best hyperparameter according to that metric. Search strategies fall into three main categories: (i) model-free, \lfcn-only; (ii) model-based, \lfcn-only; and (iii) gradient-based, using both \lfcn and $\nabla \lfcn$. Model-free strategies do not assume any information about about the hyperparameter landscape, whereas model-based strategies use historical \lfcn evaluations to predict the loss function at untested hyperparameter values.
The following sections describe common model-free and model-based hyperparameter search strategies that only use \lfcn. See \cite[Ch.~13 and Ch.~20.6]{dempe:2020:bileveloptimizationadvances} for discussion of additional gradient-free methods for bilevel problems, \eg, population-based evolutionary algorithms, and \citep{larson:2019:derivativefreeoptimizationmethods} for a general discussion of derivative-free optimization methods.
The third class of hyperparameter optimization schemes are approaches based on gradient descent of a bilevel problem. The high-level strategy in bilevel approaches is to calculate the gradient of the upper-level loss function \lfcn with respect to \params and then use any gradient descent method to minimize \params. Although this approach can be computationally challenging, it generalizes well to a large number of hyperparameters. \cref{chap: ift and unrolled} and \cref{chap: bilevel methods} discuss this point further and go into depth on different methods for computing this gradient.
\subsection{Model-free Hyperparameter Optimization}
The most common search strategy is probably an empirical search, where a researcher tries different hyperparameter combinations manually. A punny, but often accurate, term for this manual search is GSD: grad[uate] student descent \citep{gencoglu:2019:harksidedeep}. \citet{bergstra:12:rsf} hypothesized that manual search is common because it provides some insight as the user must evaluate each option, it requires no overhead for implementation, and it can perform reliably in very low dimensional hyperparameter spaces.
Grid search is a more systematic alternative to manual search. When there are only one or two continuous hyperparameters, or the possible set of hyperparameters, $\Params$, is small, a grid search (or exhaustive search) strategy may suffice to find the optimal value, $\paramshat$, to within the grid spacing. However, the complexity of grid search grows exponentially with the number of hyperparameters. Regularizers frequently have many hyperparameters, so one generally requires a more sophisticated search strategy.
One popular approach is random search, which \citep{bergstra:12:rsf} shows is superior to a grid search, especially when some hyperparameters are more important than others. There are also variations on random search, such as using Poisson disk sampling theory to explore the hyperparameter space \citep{muniraju:18:crs}. The simplicity of random search makes it popular, and, even if one uses a more complicated search strategy, random search can provide a useful baseline or an initialization strategy. However, random search, like grid search, suffers from the curse of dimensionality, and is less effective as the hyperparameter space grows.
Another group of model-free blackbox strategies are population-based methods such as evolutionary algorithms. A popular population-based method is the covariance matrix adaption evolutionary strategy (CMA-ES) \citep{beyer:2001:theoryevolutionstrategies}. In short, every iteration, CMA-ES involves sampling a multivariate normal distribution to create a number of \dquotes{offspring} samples. Mimicking natural selection, these offspring are judged according to some fitness function, a parallel to the upper-level loss function. The fittest offspring determine the update to the normal distribution and thus \dquotes{pass on} their good characteristics to the next generation.
\subsection{Model-based Hyperparameter Optimization}
Model-based search strategies assume a model (or prior) for the hyperparameter space and use only loss function evaluations (no gradients). This section discusses two common model-based strategies: Bayesian methods and trust region methods.
Bayesian methods fit previous hyperparameter trials' results to a model to select the hyperparameters that appear most promising to evaluate next \cite{klein:17:fbh}. For example, a common model for the hyperparameters is the Gaussian Process prior. Given a few hyperparameter and cost function points, a Bayesian method involves the following steps. \begin{enumerate}[noitemsep,topsep=0pt]
\item Find the mean and covariance functions for the Gaussian Process.
The mean function will generally interpolate the sampled points.
The covariance function is generally expressed as a kernel function,
often using squared exponential functions \citep{frazier:2018:tutorialbayesianoptimization}.
\item Create an acquisition function.
The acquisition function captures how desirable it is
to sample (\dquotes{acquire}) a hyperparameter setting.
Thus, it should be large (desirable) for hyperparameter values that are predicted
to yield small loss function values
or that have high enough uncertainty that they may yield low losses.
The design of the acquisition function thus trades-off between exploring new areas of the hyperparameter landscape with high uncertainty and a more locally focused exploitation of the current best hyperparameter settings. See \citep{frazier:2018:tutorialbayesianoptimization}
for a discussion of specific acquisition function designs.
\item Maximize the acquisition function
(typically designed to be easy to optimize)
to determine which hyperparameter point to sample next.
\item Evaluate the loss function at the new hyperparameter candidate. \end{enumerate} These steps repeat for a given amount of time or until convergence.
The derivative-free, trust-region method (TRM) \citep{conn:2000:trustregionmethods} is similar to Bayesian optimization in that it involves fitting an easier to optimize function to the loss function of interest, \lfcn, and then minimizing the easier, surrogate function (the \dquotes{model}). The \dquotes{trust-region} in TRM captures how well the model matches the observed \lfcn values and determines the maximum step at every iteration, typically by comparing the actual decrease in \lfcn (based on observed function evaluations) to the predicted decrease (based on the model).
TRM requires only function evaluations, not gradients, to construct and then minimize the model. However, unlike most Bayesian optimization-based approaches, TRM uses a local (often quadratic) model for \lfcn around the current iterate, rather than a surrogate that fits all previous points. In taking a step based on this local information, TRM resembles gradient-based approaches.
Following the methods from \citep{ehrhardt:2021:inexactderivativefreeoptimization}, who assume an additively separable and quadratic upper-level loss function \footnote{
One could generalize the method to
non-quadratic loss functions
by approximating \lfcn
with its second order Taylor expansion. }, \eg, \[
\lfcn(\params) = \frac{1}{\Ntrue} \sum_{\ntrue=1}^\Ntrue \lfcn(\params \, ; \xhat_\ntrue(\params))
= \frac{1}{\Ntrue} \sum_{\ntrue=1}^\Ntrue \parenr{
\underbrace{
\xhat_j(\params) - \xtrue_j
}_{\xmath{r}_j(\params \, ; \xhat_\ntrue(\params)}
}^2 ,\] an outline for a TRM is \begin{enumerate}
\item Create a quadratic model
for the upper-level loss function.
\begin{enumerate}
\item Select a set of upper-level interpolating points and (approximately) evaluate \xmath{r} at each one.
After an initialization,
one can generally reuse samples from previous iterations.
Ref. \citep{ehrhardt:2021:inexactderivativefreeoptimization}
discusses requirements on the interpolation set
to guarantee a good geometry
and conditions for re-setting the interpolation sample.
\item Estimate the gradients of $\xmath{r}_j$
by interpolating a set of \paramsdim samples
(recall $\params \in \F^\paramsdim$)
of the upper-level loss function.
This requires solving a set of \paramsdim linear equations in \paramsdim unknowns.
\item Model the upper-level by replacing $\xmath{r}_j$ with its tangent-plane approximation:
$\xmath{r}_j(\params + \delta) \approx \xmath{r}(\params) + (\tilde{\nabla} \xmath{r}_j(\params))' \delta$,
where $\tilde{\nabla} \xmath{r}_j(\params)$ is the estimated gradient from the previous step.
\end{enumerate}
\item Minimize the model within some trust region to find the next candidate set of upper-level parameters.
By construction, this is a simple convex-constrained quadratic problem.
\item Accept or reject the updated parameters and update the trust region.
If the ratio between the actual reduction and predicted reduction is low, the model may no longer be a good fit,
the update is rejected, and
the trust region shrinks. \end{enumerate}
Recall that evaluating \lfcn is typically expensive in bilevel problems as each upper-level function evaluation involves optimizing the lower-level cost. Thus, even constructing the model for a TRM can be expensive. To mitigate this computational complexity, \citep{ehrhardt:2021:inexactderivativefreeoptimization} incorporated a dynamic accuracy component, with the accuracy for the lower-level cost initially set relatively loose (leading to rough estimates of \lfcn) but increasing with the upper-level iterations (leading to refined estimates of \lfcn as the algorithm nears a stationary point).
A main result from \citep{ehrhardt:2021:inexactderivativefreeoptimization} is a bound on the number of iterations to reach an $\epsilon$-optimal point (defined as $\min_\upperiter \normr{\nabla_\params \lfcn(\iter{\params})} < \epsilon$, where \upperiter indexes the upper-level iterates). The bound derivation assumes (i) \ofcn is differentiable in \vx, (ii) \ofcn is $\mu$-strongly convex, \ie, $\ofcn(\vx) - \frac{\mu}{2}\norm{\vx}^2$ is convex for $\mu > 0$, (iii) the derivative of \ofcn is Lipschitz continuous, and (iv) the first and second derivative of the lower-level cost with respect to \vx exist and are continuous. These requirements are satisfied by the example filter learning problem \eqref{eq: bilevel for analysis filters}, when \mA has full column rank, and more generally when there are certain constraints on the hyperparameters. The iteration bound is a function of the following: \begin{itemize}[noitemsep,topsep=0pt]
\item the tolerance $\epsilon$,
\item the trust region parameters
(parameters that control the increase and decrease in trust region size
based on the actual to predicted reduction,
the starting trust region size,
and
the minimum possible trust region size),
\item the initialization for \params, and
\item the maximum possible error between
the gradient of the upper-level loss function and
the gradient of the model for the upper-level loss
within a trust region
(when the gradient of \lfcn is Lipschitz continuous,
this bound is the corresponding Lipschitz constant). \end{itemize} The number of iterations required to reach such an $\epsilon$-optimal point is \order{\frac{1}{\epsilon^2}} \citep{ehrhardt:2021:inexactderivativefreeoptimization} and the number of required upper-level loss function evaluations depends more than linearly on \paramsdim \citep{roberts:2021:inexactdfobilevel}. The growth with the number of hyperparameters impedes its use in problems with many hyperparameters. However, new techniques such as \citep{cartis:2021:scalablesubspacemethods} may be able to decrease or remove the dependency, \blue{ making TRMs promising alternatives to the gradient-based bilevel methods described in the remainder of this review. }
\section{Summary}
Turning from the discussion of the lower-level problem in \cref{chap: image recon}, this section concentrated on the other two aspects of bilevel problems: the upper-level loss function and the optimization strategy.
The loss function defines what a \dquotes{good} hyperparameter is, typically using a metric of image quality to compare \xhatp to a clean, training image, \xtrue. Variations on squared error are the most common upper-level loss functions. \sref{sec: loss function design} discussed many other full-reference and no-reference options, including ones motivated by human judgements of perceptual quality, from the image quality assessment literature; \sref{sec: prev results loss function} gives examples of bilevel methods that use some of these other loss functions.
The second half of this section concentrated on model-free and model-based hyperparameter search strategies. The grid search, CMA-ES, and trust region methods described above all scale at least linearly with the number of hyperparameters. Similarly, Bayesian optimization is best-suited for small hyperparameter dimensions; \citep{frazier:2018:tutorialbayesianoptimization} suggests it is typically used for problems with 20 or fewer hyperparameters.
The remainder of this review considers gradient-based strategies for hyperparameter optimization. The main benefit of gradient-based methods is that they can scale to the large number of hyperparameters that are commonly used in machine learning applications. Correspondingly, the main drawbacks of a gradient-based method over the methods discussed in this section are the implementation complexity, the per-iteration computational complexity, and the differentiability requirement. \crefs{chap: ift and unrolled}{chap: bilevel methods} discuss multiple options for gradient-based methods.
\chapter{Gradient-Based Bilevel Methodology: \texorpdfstring{\\}{} The Groundwork} \label{chap: ift and unrolled}
When the lower-level optimization problem \eqref{eq: generic bilevel lower-level} has a closed-form solution, \xhat, one can substitute that solution into the upper-level loss function \eqref{eq: generic bilevel upper-level}. In this case, the bilevel problem is equivalent to a single-level problem and one can use classic single-level optimization methods to minimize the upper-level loss. (See \citep{kunisch:2013:bileveloptimizationapproach} for analysis and discussion of some simple bilevel problems with closed-form solutions for \xhat.) This review focuses on the more typical bilevel problems that lack a closed-form solution for \xhat.
Although there are a wide variety of optimization methods for this challenging category of bilevel problems, many methods are built on gradient descent of the upper-level loss. The primary challenge with gradient-based methods is that the gradient of the upper-level function depends on a variable that is itself the solution to an optimization problem involving the hyperparameters of interest. This section describes two common approaches for overcoming this challenge. The first approach uses the fact that the gradient of the lower-level cost function is zero at the minimizer to compute an exact gradient at the exact minimizer. The second approach uses knowledge of the update scheme for the lower-level cost function to calculate the exact gradient for an approximation to the minimizer after a specific number of lower-level optimization steps.
With this (approximation of the) gradient of the lower-level optimization variable with respect to the hyperparameters, one can compute the gradient of the upper-level loss function with respect to the hyperparameters, \params. \cref{chap: bilevel methods} uses the building blocks from this section to explain various bilevel methods based on this gradient.
\section{Set-up}
Recall from \sref{sec: bilevel set-up} that a generic bilevel problem is \begin{align}
\argmin_\params \; &\lfcnargs \text{ where }
\xhat(\params) = \argmin_{\vx} \ofcnargs.
\label{eq:lower-repeat} \end{align} For simplicity, hereafter we focus on the case $\F = \R$. Using the chain rule, the gradient of the upper-level loss function with respect to the hyperparameters is \begin{align}
\uppergrad
&= \dParams{\lfcnargs} + \left( \dParams{\xhatargs} \right) ' \dx{\lfcnargs}
, \label{eq: bilevel first chain rule} \end{align} where on the right hand side $\dParams$ and $\dx$ denote partial derivatives w.r.t. the first and second arguments of $\lfcnparamsvx$, respectively. We typically select the loss function such that it is easy to compute these partials. For example, if \lfcn is the squared error training loss, \ie, $\lfcnargs = \frac{1}{2} \normsq{\xhatargs - \xtrue}_2$, then \begin{equation*}
\dParams{\lfcnargs} = 0
\text{ and }
\dx{\lfcnargs} = \xhatargs - \xtrue. \end{equation*} The following sections survey methods to find the remaining, more challenging piece in \eqref{eq: bilevel first chain rule}: the Jacobian $\dParams{\xhatargs} \in \F^{\sdim \by \paramsdim}$ for a given value of \params.
\section{Minimizer Approach} \label{sec: minimizer approach}
The first approach finds the Jacobian $\dParams{\xhatargs}$ by assuming the gradient of \ofcn at the minimizer is zero. There are two ways to arrive at the final expression: the implicit function theorem (\ac{IFT}) perspective (as in \cite{samuel:2009:learningoptimizedmap, gould:2016:differentiatingparameterizedargmin}) and the Lagrangian/KKT transformation perspective (as in \cite{chen:2014:insightsanalysisoperator, holler:2018:bilevelapproachparameter}). This section presents both perspectives in sequence. The end of the section summarizes the required assumptions and discusses computational complexity and memory requirements.
The first step in both perspectives is to assume we have computed \xhatargs and that the lower-level problem \ref{eq:lower-repeat} is unconstrained (\eg, no non-negativity or box constraints). Therefore, the gradient of \ofcn with respect to \vx and evaluated at \xhat must be zero: \begin{align}
\dx{\ofcnargs}\evalat_{\vx=\xhatargs} =
\dx{\ofcn(\xhat \, ; \params)}
= \vzero \label{eq:dPhi}. \end{align} After this point, the two perspectives diverge.
\subsection{Implicit Function Theorem Perspective} \label{sec: ift approach}
In the IFT perspective, we apply the IFT (\cf. \cite{fessler:96:mav}) to define a function $h$ such that $\xhatargs = \hfunc$. If we could write $h$ explicitly, then the bilevel problem could be converted to an equivalent single-level problem. However, per the \ac{IFT}, we do not need to define $h$, we only state that such an $h$ exists. Combining this definition with \eqref{eq:dPhi} yields \begin{align}
\vzero &= \nabla_{\vx} \ofcnargsh \label{eq:dPhi hfunc} . \end{align} Using the chain rule, we differentiate both sides of \eqref{eq:dPhi hfunc} with respect to \params. The \I in the equation below follows from the chain rule because $\nabla_\params \params =\I$. We then rearrange terms to solve for the desired quantity, noting that $\nabla_\params \xhatargs = \nabla_\params \hfunc$. Thus, evaluating all terms at \xhat leads to the Jacobian expression of interest: \begin{align}
0 =& \nabla_{\vx \vx} \ofcnargsh \dParams{\hfunc} +
\I \cdot \nabla_{\vx \params} \ofcnargsh \nonumber \\
\dParams{\hfunc} =& -[\nabla_{\vx \vx} \ofcnargsh]^{-1} \cdot \nabla_{\vx \params} \ofcnargsh \nonumber \\
\dParams{\xhatargs} =& -[\nabla_{\vx \vx} \ofcn(\xhat; \params)]^{-1} \cdot \nabla_{\vx \params} \ofcn(\xhat; \params) \label{eq: dhdgamma IFT}. \end{align} When \ofcn is strictly convex, the Hessian of \ofcn is positive definite and $\nabla_{\vx \vx} \ofcn(\xhat; \params)$ is invertible.
Substituting \eqref{eq: dhdgamma IFT} into \eqref{eq: bilevel first chain rule} yields the following expression for the gradient of the upper-level loss function with respect to \params: \begin{align*}
\uppergrad
&=
\dParams{\lfcnargs} -
\left(\nabla_{\vx \params} \ofcn(\xhat; \params)\right)'
\Hinv
\dx{\lfcn(\params \, ; \xhat)}
\nonumber .\end{align*} \blue{If there is a closed-form solution to the lower-level problem, one can verify that the \ac{IFT} gradient agrees with the analytic gradient; see \cite{gould:2016:differentiatingparameterizedargmin} for examples. }
\subsection{KKT Conditions \label{sec: minimizer via kkt}}
In the Lagrangian perspective, \eqref{eq:dPhi} is treated as a constraint on the upper-level problem, creating a single-level problem with $\sdim$ equality constraints: \begin{align}
&\argmin_\params \lfcn(\params \, ; \vx) \text{ subject to }
\dx{\ofcnargs} = \mat{0}_\sdim. \label{eq: Lagrange set-up for bilevel} \end{align} \blue{Using the KKT conditions to transform the bilevel problem into a single-level, constrained problem is sometimes called the \dquotes{KKT transformation} of the bilevel problem. This transformation relates bilevel optimization to mathematical programs with equilibrium constraints (MPEC); see \cite[Ch.~12]{dempe:2020:bileveloptimizationadvances} and some authors use approaches from the broader MPEC literature to approach bilevel problems \citep{hintermuller:2015:bileveloptimizationcalibrating}. } The Lagrangian \blue{corresponding to \eqref{eq: Lagrange set-up for bilevel}} is \begin{align}
L(\vx, \params, \vnu)
&= \lfcn(\params \, ; \vx) + \vnu^T \dx{\ofcnargs} \nonumber \end{align} where $\vnu \in \F^\sdim$ is a vector of Lagrange multipliers associated with the $\sdim$ equality constraints in \eqref{eq: Lagrange set-up for bilevel}.
The Lagrange reformulation is generally well-posed because many bilevel problems, such as \eqref{eq: bilevel for analysis filters}, satisfy the linear independence constraint qualification (LICQ) \citep{dempe:2003:annotatedbibliographybilevel,scholtes:2001:howstringentlinear}. The LICQ requires that the matrix of derivatives of the constraint has full row rank \citep{scholtes:2001:howstringentlinear}, \ie, \begin{equation*}
\text{rank}
\paren{
\begin{bmatrix}
\nabla_{\vx \params} \ofcnargs & \nabla_{\vx \vx} \ofcnargs
\end{bmatrix}
}
= \sdim .\end{equation*} Strict convexity of \ofcnargs is therefore a sufficient condition for LICQ to hold. (Note the similarity to the IFT perspective, where strict convexity is sufficient for the Hessian to be invertible.) \blue{Ref. \citep{dempe:2012:bilevelprogrammingspecial} explores more generally how bilevel problems relate to MPECs and when the global and local minimizers of the KKT reformulation are minimizers of the original bilevel problem. }
The first \ac{KKT} condition states that, at the optimal point, the gradient of the Lagrangian with respect to \vx must be \mat{0}. We can use this fact to solve for the optimal Lagrangian multiplier, $\hat{\vnu}$: \begin{align}
\dx{ L(\xhat, \params, \hat{\vnu})}
&= \dx{\lfcn(\params \, ; \xhat)} + \nabla_{\vx \vx} \ofcn(\xhat \,; \params) \hat{\vnu} = \vzero
\nonumber \\
\hat{\vnu} &= \neg \Hinv \dx{\lfcn(\params \, ; \xhat)}. \nonumber \end{align} Substituting the expression for $\hat{\vnu}$ into the gradient of the Lagrangian with respect to \params yields \begin{align}
\nabla_\params L(\xhat, \params, \hat{\vnu}) &=
\dParams{\lfcn(\params \, ; \xhat)} +
\left(
\nabla_{\vx \params} \ofcn(\xhat \, ; \params)
\right)'
\hat{\vnu} \nonumber \\
=&
\dParams{\lfcn(\params \, ; \xhat)} -
\left(\nabla_{\vx \params} \ofcn(\xhat \,; \params) \right)'
\Hinv \dx{\lfcn(\params \, ; \xhat)}
, \nonumber \end{align} which is equivalent to the IFT result.
Ref. \citep{holler:2018:bilevelapproachparameter} generalized the Lagrangian approach to the case where the forward model is defined only implicitly, \eg, as the solution to a differential equation. The authors write the lower-level problem as \begin{equation}
\xhat
= \argmin_{\vx} \min_{\tilde{\vy}}
\normsq{\vy - \tilde{\vy}}_2 + \regfcn(\vx)
\text{ s.t. } e(\tilde{\vy}, \vx) = 0, \label{eq: holler lower level} \end{equation} where the constraint function, $e$, incorporates the implicit system model. For example, when the forward model is linear ($\mA\vx$), taking \mbox{$e(\tilde{\vy}, \vx) = \normsq{\mA\vx - \tilde{\vy}}_2$} shows the equivalence of the approach here to the one in \citep{holler:2018:bilevelapproachparameter}.
\subsection{Summary of Minimizer Approach} \label{sec: summary of minimizer approach}
In summary, the upper-level gradient expression for the minimizer approach (\ie, when one ``exactly'' minimizes the lower-level cost function) is \begin{align}
\uppergrad
&= \dParams{\lfcn(\params \, ; \xhat)} -
\left(\nabla_{\vx \params} \ofcn(\xhat; \params)\right)'
\Hinv
\dx{\lfcn(\params \, ; \xhat)}.
\label{eq: IFT final gradient dldparams} \end{align} Thus, for a given loss function and cost function, calculating the gradient of the upper-level loss function (with respect to \params) requires the following components all evaluated at $\vx = \xhat$: $\dParams{\lfcn(\params \, ; \vx)} \in \F^{\paramsdim}$, $\nabla_{\vx \params} \ofcnargs \in \F^{\sdim \by \paramsdim}$, $\nabla_{\vx \vx} \ofcnargs \in \F^{\sdim \by \sdim}$, and $\nabla_\vx \lfcn(\params \, ; \vx) \in \F^{\sdim}$.
Continuing the specific example of learning filter coefficients and tuning parameters \eqref{eq: bilevel for analysis filters}, the components are: \begin{align}
\nabla_{\vx} \ofcn(\xhat \,; \params) &= \mA' (\mA \vx - \vy)
+ \ebeta{0} \sum_{k=1}^K \ebeta{k} \xmath{\tilde{\vc}_k} \conv \dsparsefcn.(\xmath{\vc_k} \conv \vx; \epsilon) \nonumber
\\
\nabla_{\vx \beta_k} \ofcn(\xhat \,; \params) &=
\ebetazerok \xmath{\tilde{\vc}_k} \conv \dsparsefcn.(\xmath{\vc_k} \conv \xhat)
\nonumber \\
\nabla_{\vx c_{k,s}} \ofcn(\xhat \, ; \params) &=
\ebetazerok \paren{ \dsparsefcn.(\circshift{(\xmath{\vc_k} \conv \xhat)}{\vs})
+ \xmath{\tilde{\vc}_k} \conv \left( \ddsparsefcn.(\xmath{\vc_k} \conv \xhat) \odot \circshift{\xhat}{\neg \vs} \right) }
\nonumber \\
\nabla_{\vx \vx} \ofcn(\xhat \, ; \params) &= \mA'\mA + \ebeta{0} \sum_k \ebeta{k} \mC_k' \diag{\ddsparsefcn.(\xmath{\vc_k} \conv \xhat)} \mC_k \nonumber \\
\dParams{\lfcn(\params \,; \vx)} &= 0
\nonumber \\
\nabla_\vx \lfcn(\params \, ; \xhat) &= \xhatargs - \xtrue.
\label{eq: nablas for filter learning} \end{align} Here, the notation \circshift{\vx}{\vi} means circularly shifting the vector \vx by \vi elements, and $c_{k,\vs}$ denotes the $\vs$th element of the $k$th filter \xmath{\vc_k}, where $\vs$ is a tuple that indexes each dimension of \xmath{\vc_k}. \apref{sec: dh of htilde conv f(h conv x)} gives examples of using the \circshift{\vx}{\vi} notation and derives $\nabla_{\hks} \paren{ \xmath{\tilde{\vc}}_k \conv f.(\xmath{\vc_k} \conv \vx)}$, which is the key step to expressing $\nabla_{\vx c_{k,\vs}} \ofcn(\xhat \, ; \params)$. The other components follow directly from $\nabla_{\vx} \ofcn(\xhat \,; \params)$ using standard gradient tools for matrix expressions \citep{petersen:2012:matrixcookbook}.
The minimizer approach to finding \uppergrad uses the following assumptions: \begin{enumerate}[noitemsep]
\item Both the upper and lower optimization problems have no inequality constraints.
\item \xhat is the minimizer to the lower-level cost function, not an approximation of the minimizer.
This constraint ensures that \eqref{eq:dPhi} holds.
\item The cost function \ofcn is twice-differentiable in \vx and differentiable with respect to \vx and \params.
\item The Hessian of the lower-level cost function, $\nabla_{\vx \vx}\ofcnargs$, is invertible;
this is guaranteed when \ofcn is strictly convex. \end{enumerate}
The first condition technically excludes applications like \ac{CT} imaging, where the image is typically constrained to be non-negative. However, non-negativity constraints are rarely required when good regularizers are used, so the resulting non-constrained image can still be useful in practice \cite{fessler:96:mav}.
The second constraint is often the most challenging since the lower-level problem typically uses an iterative algorithm that runs for a certain number of iterations or until a given convergence criteria is met. As previously noted, if there were a closed-form solution for \xhat, then we would not have needed to use the \ac{IFT} or Lagrangian to find the partial derivative of \xhat with respect to \params. Since one usually does not reach the exact minimizer, the calculated gradient will have some error in it, depending on how close the final iterate is to the true minimizer \xhat. Thus, the practical application of this method is more accurately called Approximate Implicit Differentiation (AID) \citep{ji:2021:bileveloptimizationconvergence,grazzi:2020:iterationcomplexityhypergradient}. \sref{sec: ift unrolled comparison} further discusses gradient accuracy.
The third condition disqualifies sparsity-promoting functions such as the 0-norm and 1-norm as choices for \sparsefcn.
\blue{ Finally, the fourth (strict convexity) condition is easily satisfied in denoising problems where $\mA=\I$ whenever \sparsefcn is convex. Common convex \sparsefcn choices include \eqref{eq: corner rounded 1-norm} and the Fair potential \cite{holland:77:rru}. However, in applications like compressed sensing where $\mA'\mA$ is not positive definite, the strict convexity of \ofcn depends non-trivially on \params. The condition is likely to hold in practice for \dquotes{good} values of \params. Specifically, if \sparsefcn is strictly convex, then the condition will hold for any value of \params such that the null-space of the regularization term is disjoint from the null-space of \mA and the regularization parameters are sufficiently large ($e^{\beta_k}$ cannot approach 0). To interpret this condition, recall that regularization helps compensate for the under-determined nature of \mA (\sref{sec: image recon background}). Values of \params that do not sufficiently \dquotes{fill-in} the null-space of \mA will leave the lower-level cost function under-determined. The task-based nature of the bilevel problem should discourage these \dquotes{bad} values, but this intuition is insufficient to claim that the minimizer approach is well-defined at all iterations. To ensure that the lower-level problem is strongly convex, one could include a term like
$\| \vx \|_2^2$ with a small positive regularization parameter, like is done with elastic-net regularization \cite{zou:05:rav}. }
\subsection{Computational Costs \label{sec: ift complexity} }
The largest cost in computing the gradient of the upper-level loss using \eqref{eq: IFT final gradient dldparams} is often finding (an approximation of) \xhat. However, this cost is difficult to quantify, as the IFT approach is agnostic to the lower-level optimization methodology. To compare the bilevel gradient methods, we will later assume the cost is comparable to the gradient descent calculations used in the unrolled approach (described in \sref{sec: unrolled}). However, this is an over-estimation of the cost, as the IFT approach is not constrained to smooth lower-level updates, and one can use optimization methods with, \eg, warm starts and restarts to reduce this cost.
When the lower-level problem satisfies the assumptions above, and assuming one has already found \xhat, a straight-forward approach to computing the gradient \eqref{eq: IFT final gradient dldparams} would be dominated by the $\order{\sdim^3}$ operations required to compute the Hessian's inverse. For many problems, \sdim is large, and that matrix inversion is infeasible due to computation or memory requirements. Instead, as described in \citep{foo:2007:efficientmultiplehyperparameter}, one can use a conjugate gradient (\ac{CG}) method to compute the matrix-vector product \begin{equation}
\Hinv \dx{\lfcn(\params \,; \xhat)} \label{eq: Hinv step for CG} \end{equation} because the Hessian is symmetric and positive definite (see assumption \#4 in the previous section). For a generic \mA, each CG iteration requires multiplying the Hessian by a vector, which is \order{\sdim^2}.
CG takes \sdim iterations to converge fully (ignoring finite numerical precision), so the final complexity is still \order{\sdim^3} in general. However, the Hessian often has a special structure that simplifies computing the matrix-vector product. Consider the running example of learning filters per \eqref{eq: bilevel for analysis filters}. The Hessian, as given in \eqref{eq: nablas for filter learning}, multiplied with any vector $\vv \in \F^N$ is \begin{align}
\nabla_{\vx \vx} \ofcn(\xhat; \params, \vy) \cdot \vv &= \nonumber \\
\underbrace{\mA' (\mA \vv) }_{2 \sdim^2}
+
&\ebeta{0} \sum_k \ebeta{k}
\underbrace{ \Ck' \cdot }_{\sdim \filterdim}
\overbrace{
\diag{\ddsparsefcn.(\underbrace{\xmath{\vc_k} \conv \xhat}_{\sdim \filterdim})}
\cdot
}^{\sdim}
\underbrace{(\mC_k \vv)}_{\sdim \filterdim}
. \label{eq: Hessian with computational cost} \end{align} The annotations show the multiplications required for each component, where we used the simplifying assumption that the number of measurements matches the number of unknowns ($\ydim=\sdim$).
As written, \eqref{eq: Hessian with computational cost} does not make any assumptions on \mA, so the first term is still computationally expensive. If \mA is the identity matrix (as in denoising), the $\sdim^2$ term could instead be zero cost. If $\mA' \mA$ is circulant, \eg, if \mA is a MRI sampling matrix that can be written in terms of a discrete Fourier transform, then the cost is $\sdim \log{\sdim}$. More generally, the computational cost for one (of \sdim) iterations of CG is \order{\cA \sdim} where $\cA \in [0,\sdim]$ is some constant dependent on the structure of \mA.
For the second addend in \eqref{eq: Hessian with computational cost}, we assume that $\filterdim \ll \sdim$, so direct convolution is most efficient and the matrix-vector product requires \order{\sdim \filterdim} multiplies. When the filters are relatively large, one can use Fourier transforms for the filtering, and the cost is \order{\sdim \text{log}(\sdim)}. The final cost of the Hessian-vector product for \eqref{eq: bilevel for analysis filters} is \order{\cA \sdim + \paramsdim \sdim}. This cost includes a multiplication by $K$ to account for the sum over all filters, which simplifies since $\filterdim K$ is \footnote{ The full parameter dimension includes the filters and tuning parameters, so $\paramsdim = \filterdim (K+1) + 1$. } \order{\paramsdim}.
\blue{ If \sdim is small enough that storing the inverse Hessian is feasible, then one can estimate the Hessian inverse rather than computing it directly. Consider using a quasi-Newton algorithm to find \xhat, which involves estimating the inverse Hessian as a pre-conditioning matrix for the gradient steps. This inverse Hessian estimate can be \dquotes{shared} to efficiently approximate the inverse Hessian-vector product in \eqref{eq: IFT final gradient dldparams} \citep{ramani:2008:montecarlosureblackbox}. Ref.~\citep{ramzi:2021:shinesharinginverse} used this strategy and also incorporated information from the upper-level loss function to improve the estimated inverse Hessian vector product while maintaining the super-linear convergence rate of the quasi-Newton algorithm. }
\section{Translation to a Single-Level \label{sec: translation to a single level}}
Before discussing the other widely used approach to calculating the gradient of the upper-level loss, we summarize a specialized approach for 1-norm regularizers. Like the minimizer approach described above, this approach assumes we have computed an (almost) exact minimizer of the lower-level cost function. It writes the minimizer as an (almost everywhere) differentiable function in terms of that \xhat, then substitutes this expression for the minimizer into the upper-level loss to create a single-level optimization problem that is suitable for one hyperparameter update step.
Ref. \citep{sprechmann:2013:supervisedsparseanalysis} proposed the translation to a single-level approach to solve a bilevel problem with both synthesis and analysis operators. Refs.~\citep{mccann:2020:supervisedlearningsparsitypromoting,ghosh:2021:bilevellearningl1regularizers} more recently presented versions specific to analysis operators. The bilevel problem considered in \citep{mccann:2020:supervisedlearningsparsitypromoting, ghosh:2021:bilevellearningl1regularizers} is: \begin{align}
&\argmin_{\params} \sum_\ntrue \onehalf \normrsq{\xhat_\ntrue(\params) - \xtrue_\ntrue}_2
\nonumber\\
\hat{\vx}_j(\params) = &\argmin_{\vx \in \F^\sdim} \onehalf \normsq{\vx - \vy_j}_2 + \norm{\mOmega_\params \vx}_1,
\label{eq:mccann-lower} \end{align} where $\mOmega_\params \in \F^{\xmath{F} \by \sdim}$ is a matrix constructed based on \params. We write \mOmega without the \params subscript \blue{and $\xhat_j(\params)$ without the $j$ subscript} in the following discussion to simplify notation. As in the minimizer approach, the first step is to compute \xhatp for the current guess of \params, \eg, using ADMM. After optimizing for \xhatp, \citep{mccann:2020:supervisedlearningsparsitypromoting,ghosh:2021:bilevellearningl1regularizers} both used the known sign pattern of the filtered signal, $\mOmega \xhatp$ to rewrite the lower-level problem \eqref{eq:mccann-lower} in a simpler, (almost everywhere) differentiable form. \blue{By rewriting the problem, the translation to a single-level approaches handle the non-smooth 1-norm in \eqref{eq:mccann-lower} directly--they do not require any corner rounding as in the minimizer approach. }
One way to rewrite the lower-level problem is to split the 1-norm into its positive and negative elements, \eg, \[ \norm{\mOmega \xhatp}_1 = \sum_{i \in \cI_+(\params)} [\mOmega \xhatp]_i - \sum_{i \in \cI_-(\params)} [\mOmega \xhatp]_i ,\] where $\cI_+(\params)$ and $\cI_-(\params)$ denote the set of indices where $\mOmega \xhatp$ is positive and negative, respectively. Ref.~\citep{mccann:2020:supervisedlearningsparsitypromoting} used this approach and defined a diagonal sign matrix, $ \mS(\params) = \diag{ \sign{\mOmega \xhatp} } $, having positive and negative diagonal elements at the appropriate indices. \blue{For a single training image,} the lower-level problem \eqref{eq:mccann-lower} is thus equivalent to \begin{align}
\hat{\vx}(\params) =
\argmin_{\vx \in \F^\sdim} \onehalf \normsq{\mA \vx - \vy}_2
+ \beta \mat{1}' \mS(\params) \mOmega \vx ,
\text{ s.t. } [\mOmega \vx]_{\cI_0(\params)} = \mat{0},
\label{eq: mccann rewritten} \end{align} where $\cI_0(\params)$ denotes the set of indices where $[\mOmega \xhat(\params)]_i = 0$. The rewritten problem \eqref{eq: mccann rewritten} it is a quadratic cost function with a linear equality constraint and thus has a closed-form solution. Ref.~\citep{mccann:2020:supervisedlearningsparsitypromoting} states that \xhatp is differentiable everywhere except a set of measure zero when $\mA = \I$ and when \blue{the rows of $\mOmega$ corresponding to $\cI_0(\params)$ are linearly independent}.
\blue{ Another way to rewrite \eqref{eq:mccann-lower} uses the results from \citep{tibshirani:2011:solutionpathgeneralized}. The lower-level problem \eqref{eq:mccann-lower} can be transformed into the dual problem \begin{equation}
\min_{\xmath{\vd} \,\in\, \R^{\xmath{F}}} \onehalf \normsq{\neg \mOmega' \xmath{\vd} + \vy} - \onehalf \normsq{\vy}
\text{ s.t. } \abs{\xmath{d}_i} \leq 1 \;\forall i .\end{equation} where the dual variable \xmath{\vd} is related to the filtered signal by \begin{equation}
\xmath{\vd}_i \in \begin{cases}
\text{sign}([\mOmega \vx]_i) &\text{ if } [\mOmega \xhat]_i \neq 0 \\
[\neg1,1] &\text{ if } [\mOmega \xhat]_i = 0
\end{cases} \end{equation} (compare to \eqref{eq: dual problem 1-norm} and \eqref{eq: tibshirani 15} in \apref{sec: primal dual background}). Ref.~\citep{tibshirani:2011:solutionpathgeneralized} defines boundary indices as the set of indices where the dual variable is at the edges of its allowed range: $\xmath{\mathit{B}} \defeq \{i : \abs{\xmath{\vd}_i}=1\}$. The complement to this set is \mbox{$\xmath{\mathit{\bar{B}}} \defeq \{i : \abs{\xmath{\vd}_i} \neq 1\}$} and contains all coordinates where \xmath{\vd} is in the interior of its allowed range. Let $\mOmega_\xmath{\vd} \in \F^{\abs{\xmath{\mathit{B}}} \by \sdim}$ contain the rows of \mOmega that correspond to \xmath{\mathit{B}} and similarly for $\mOmega_{\xmath{\mathit{\bar{B}}}}$. By taking the gradient of the Lagrangian of the dual formulation and then substituting the dual variable minimizer into \eqref{eq: primal dual minimizer relation}, \citep{tibshirani:2011:solutionpathgeneralized} derives the following closed-form expression for \xhat \begin{align}
\xhat &= (\I - \mOmega_\xmath{\mathit{\bar{B}}}^+ \mOmega_\xmath{\mathit{\bar{B}}})
\, (\vy - \mOmega_\xmath{\mathit{B}} \sign{\mOmega_\xmath{\mathit{B}} \xhat)}
\label{eq: ghosh xhat} ,\end{align} which is a projection onto the null space of $\mOmega_\xmath{\mathit{\bar{B}}}$. Thus, similar to splitting the 1-norm based on the sign of $\mOmega \xhat$, splitting the dual variable into boundary and interior indices yields a rewritten problem with a simpler structure. }
\blue{ Ref. \citep{ghosh:2021:bilevellearningl1regularizers} used \eqref{eq: ghosh xhat} to rewrite the lower-level problem \eqref{eq:mccann-lower} and then used matrix gradient relations to derive a closed-form expression for \dParams{\xhatp}. Unlike \citep{mccann:2020:supervisedlearningsparsitypromoting}, the final upper-level gradient \uppergrad in \citep{ghosh:2021:bilevellearningl1regularizers} does not require that the rows of \mOmega that are orthogonal to \xhatp are linearly independent. }
\blue{ In both \eqref{eq: mccann rewritten} and \eqref{eq: ghosh xhat}, the rewritten problem has the same minimizer as the original problem \eqref{eq:mccann-lower}, but the reformulated problem has a simpler structure. Recall that the rewriting process requires \xhatp, so one cannot use this equivalence to optimize the lower-level problem. However, the closed-form expressions can be differentiated. Because of the discontinuity of the sign function, both methods require the sign pattern of $\mOmega \xhat$ to be constant within a region to compute an accurate gradient \citep{mccann:2020:supervisedlearningsparsitypromoting,ghosh:2021:bilevellearningl1regularizers}. The authors have shown that this condition holds in various empirical settings \citep{ghosh:2022:questionsaboutblorc}. }
In summary, the translation to a single-level approach involves computing \xhat, creating a closed-form expression for \xhat, and then differentiating the closed-form expression to compute the desired \blue{Jacobian}, $\nabla_\params \xhat(\params)$. As in the minimizer approach, $\nabla_\params \xhat(\params)$ is related to the upper-level gradient by the chain rule \eqref{eq: bilevel first chain rule}. In terms of computation, both translation to a single-level approaches require optimizing the lower-level cost sufficiently precisely to ensure the sign pattern converges; \citep{ghosh:2021:bilevellearningl1regularizers} used thousands of iterations of ADMM. \blue{Ref.~\citep{ghosh:2021:bilevellearningl1regularizers} demonstrates that evaluating the closed-form expression for \uppergrad is faster than using automatic differentiation tools that rely on backpropagation.}
\section{Unrolled Approaches \label{sec: unrolled}}
A popular approach to finding $\dParams{\xhatargs}$ is to assume that the lower-level cost function is approximately minimized by applying $T$ iterations of some (sub)differentiable optimization algorithm, where we write the update step at iteration $t \in [1 \ldots T]$ as \begin{equation*}
\vx^{(t)} = \optalgstep (\vx^{(t-1)} \,; \params), \end{equation*} for some mapping $\optalgstep: \F^N \mapsto \F^N$ that should have the fixed-point property $\optalgstep(\xhatp \,; \params) = \xhatp$. For example, GD has \( \optalgstep(\vx \,; \params) = \vx - \sslower \nabla \ofcnargs \) for some step size $\sslower$. We write the update here only in terms of \vx; the idea easily extends to updates in terms of a state vector that allows one to include momentum terms, weights, and other accessory variables in \params \citep{franceschi:2017:forwardreversegradientbased}.
In contrast to the two approaches described above, the ``unrolled'' approach no longer assumes the solution to the lower-level problem is an exact minimizer. Instead, the unrolled approach reformulates the bilevel problem \eqref{eq: generic bilevel lower-level} as \begin{align}
\argmin_\params
&\underbrace{\lfcn \left(\params \, ; \, \vx^{(T)}(\params) \right)}_{
\lfcn(\params)}
\text{ s.t. }
\label{eq: unrolled upper-level} \\
&\vx^{(t)}(\params) = \optalgstep(\vx^{(t-1)} \,; \params)
,\quad \forall t \in [1 \ldots T] \nonumber, \end{align} where $\vx^{(0)}$ is an initialization, \eg, $\mA' \vy$. One can then take the (sub)gradient of a finite number $T$ of iterations of \optalgstep, hoping that $\vx^{(T)}$ approximately minimizes the lower-level function \ofcn.
The chain rule for derivatives is the foundation of the unrolled method. The gradient of interest, \uppergrad, depends on the gradient of the optimization algorithm step with respect to \vx and \params. For readability, define the following matrices for the $t$th unrolled iteration \begin{align}
\mH_{t} &\defeq \franA{t-1} \in \F^{\sdim \by \sdim}
\text{ and }
\mJ_{t} \defeq \franB{t-1} \in \F^{\sdim \by \paramsdim}
\nonumber ,\end{align} for $t \in [1,T]$. We use these letters because, when using gradient descent as the optimization algorithm, $\nabla_\vx \optalgstep(\vx \, ; \params)$ is closely related to the Hessian of \ofcn and $\nabla_\params \optalgstep(\vx \, ; \params)$ is \blue{proportional to} the Jacobian of the gradient \footnote{\blue{
When $\optalgstep(\vx \, ; \params) = \vx - \sslower \nabla_\vx \ofcnargs$,
then
$\nabla_\vx \optalgstep(\vx \, ; \params) = \I - \sslower \nabla_{\vx\vx} \ofcnargs$
and
$\nabla_\params \optalgstep(\vx \, ; \params) = \neg\sslower \nabla_{\vx\params} \ofcnargs$. }}. Thus, when \optalgstep corresponds to GD, an unrolled approach involves computing the same quantities as required by the IFT approach \eqref{eq: IFT final gradient dldparams}.
\renewcommand{\franA}[1] {\xmath{\mH_{#1}}} \renewcommand{\franB}[1] {\xmath{\mJ_{#1}}}
By the chain rule, the gradient of \eqref{eq: unrolled upper-level} is \begin{align}
\uppergrad
=&
\nabla_\params \lfcn(\params \, ; \vx^{(T)}) +
\left( \sum_{t=1}^T \left(\franA{T} \cdots \franA{t+1} \right) \franB{t} \right)'
\finalterm \in \F^{\paramsdim}.
\label{eq: generic lower-level chain rule} \end{align} One can derive this gradient expression using a reverse or forward perspective, with parallels to back-propagation through time and real-time recurrent learning respectively \citep{franceschi:2017:forwardreversegradientbased}. \blue{\apref{sec: foward and backward unrolling} describes the reverse and forward approaches to unrolling.}
\blue{ Most unrolled implementations use the reverse-mode approach (backpropagation) due to its lower computational burden, but unrolling with reverse mode differentiation may have prohibitively high memory requirements if $T$ is large or if the training dataset includes large images \citep{chambolle:2021:learningconsistentdiscretizations}. A strategy to trade-off the memory and computation requirements is checkpointing, which stores \vx every few iterations. Checkpointing is an active research area; see \citep{dauvergne:2006:dataflowequationscheckpointing} for an overview. Another option is to use (some or all) reversible network layers \citep{kellman:2020:memoryefficientlearninglargescale} to trade off the memory and computational requirements. }
\blue{ The following sections overview some design decisions for unrolling and draw some parallels to unrolled methods as used in the (non-bilevel specific) machine learning literature. \sref{sec: connections unrolled} further discusses the relation between bilevel problems and unrolling methods common in the broader literature. }
\subsection{\blue{Number of Iterations}} \label{sec: unrolled number of iterations}
Unlike the minimizer approach, where the goal is to run the lower-level optimization until (close to) convergence so that an optimally condition holds and one can use implicit differentiation to find \uppergrad, most unrolling methods set the number of lower-level iterations $T$ in advance. The set number of lower-level iterations mimics the depth of neural networks and allows a precise estimate of how much computational effort each lower-level optimization takes. The chosen number of iterations is important as, at test time, \dquotes{one cannot deviate from the choice of [number of unrolled iterations] and expect good performance} \citep{gilton:2021:deepequilibriumarchitectures}.
Although it is generally not equal to the gradient of the original bilevel problem \eqref{eq: generic bilevel upper-level}, the unrolled gradient is exact for the reformulated problem \eqref{eq: unrolled upper-level}. Therefore, when $T$ is small enough that the lower-level optimizer is far from convergence, the unrolled method is only loosely tied to the original bilevel optimization problem. To maintain a stronger connection to the bilevel problem while avoiding setting $T$ larger than necessary for convergence, \citep{antil:2020:bileveloptimizationdeep} used a convergence criterion to determine the number of \optalgstep iterations rather than pre-specifying a number of iterations. Unrolling until convergence is also used in deep equilibrium or fixed point networks, see \sref{sec: connection to DEQ}.
A subtle point in unrolling gradient-based methods for the lower-level cost function is that the Lipschitz constant of its gradient is a function of the hyperparameters, so the step size range that ensures convergence cannot be pre-specified. Many unrolled methods use a fixed step size alongside a fixed $T$ and allow the learned parameters to adapt to these set values. An alternative approach is to compute a new step-size as a function of the current parameters, \iter{\params}, every upper-level iteration. For example, from \eqref{eq: lower-level LC}, for a given value \params of the tuning parameters and filter coefficients, a Lipschitz constant of the lower-level gradient for \eqref{eq: bilevel for analysis filters} is \begin{equation}
L = \sigma^2_1(\mA) + \ebeta{0} \Ldsparsefcn \sum_k \ebeta{k} \normsq{\xmath{\vc_k}}_1 \label{eq: bilevel for caol L equation} , \end{equation} where $\Ldsparsefcn$ is a Lipschitz constant for $\dsparsefcn(z)$ (for \eqref{eq: corner rounded 1-norm}, $\Ldsparsefcn=1/\epsilon$). A reasonable step size for the classical gradient descent method would be $1/L$. It is relatively inexpensive to update this $L$ as \params evolves.
The adaptive approach to setting the step size ensures that any theoretical guarantees of the lower-level optimizer hold. This approach may be beneficial when using a convergence criteria for the lower-level optimization algorithm or when running sufficiently many lower-level iterations to essentially converge. However, updating the step-size every upper-level iteration is incompatible with fixing the number of unrolled iterations. To illustrate, consider an upper-level iteration where the tuning parameters increase, leading to a larger $L$ and a smaller step size. In a fixed number of iterations, the smaller step size means the lower-level optimization algorithm will be farther from convergence, and the estimated minimizer, $\xhat(\iter{\params}{+1})$, may be worse (as judged by the upper-level loss function) than $\xhat(\iter{\params})$, even if the updated hyperparameters are better when evaluated with the previous (larger) step-size or more lower-level iterations.
Another approach is to learn the step-size and/or number of iterations. For example, \citep{effland:2020:variationalnetworksoptimal} provides a continuous-time perspective on the unrolling approach and learns the stopping time, which translates to the number of iterations in the discrete approach.
The continuous time perspective on unrolling models the lower-level problem as a differential equation with an initial condition enforcing that \vx at time $0$ is $\vx_0$ \citep{chen:2018:neuralordinarydifferential,effland:2020:variationalnetworksoptimal}. Just as the unrolled approach better approximates the bilevel problem as the number of iterations approaches infinity, the continuous perspective on unrolling approaches the bilevel problem as the stopping time $T \rightarrow \infty$. The discretization of the continuous-time gradient flow corresponds to an unrolled optimization algorithm (or, more generally, to a variational network with shared weights) and back-propagation can be seen as a discretization of the continuous-time adjoint equation \citep{chen:2018:neuralordinarydifferential,effland:2020:variationalnetworksoptimal}. Solving the differentiable adjoint equation does not require saving the forward-pass output at every \dquotes{step,} making the backward pass feasible for large problems such as 3D CT image reconstruction \citep{thies:2022:learnedconebeamct}.
Like many other bilevel methods for filter learning, \citep{effland:2020:variationalnetworksoptimal} uses a regularizer based on the Field of Experts \citep{roth:2005:fieldsexpertsframework} and the standard data-fit term. The lower-level problem in \citep{effland:2020:variationalnetworksoptimal} is \begin{align*}
\text{State equation: }\frac{d \vx(t)}{dt} &= \neg \mA'(\mA \vx(t)-\vy) - \sum_k \Ck' \sparsefcn_k(\Ck \vx(t)) \\
\text{Initial condition: } \vx(0) &= \vx_0 ,\end{align*} where \citep{effland:2020:variationalnetworksoptimal} learns a separate penalty function for each filter. Ref. \citep{effland:2020:variationalnetworksoptimal} found that beyond a certain depth, increasing the number of layers did not significantly decrease the upper-level loss. Further, following intuition, the learned stopping time increased with higher noise levels or blur strengths in the denoising and deblurring problem settings \citep{effland:2020:variationalnetworksoptimal}.
\subsection{\blue{Application to Non-smooth Cost Functions}} \label{sec: unrolling non-smooth functions}
An important distinction between the minimizer approach and the unrolled approach is that the unrolled approach depends on the optimization algorithm. Therefore, in addition to the number of iterations and step size, one must select an optimization algorithm to unroll. The choice is typically driven by parameters such as memory availability and desired run-time, with the one requirement being that \optalgstep be differentiable in both \vx and \params. For certain cost functions, a resulting advantage of the unrolling method is that one can use a smooth \optalgstep to optimize a non-smooth cost function, removing the need for smoothing techniques such as used in \eqref{eq: corner rounded 1-norm}.
Ochs \textit{et al.} \citep{ochs:2016:techniquesgradientbasedbilevel} describe one such smooth update algorithm for a non-smooth cost function. At a high-level, their approach is to: \begin{enumerate}[itemsep=0pt]
\item transform the lower-level cost function
to a primal-dual, saddle-point problem,
using the Legendre-Fenchel conjugate of \sparsefcn
(defined in \apref{sec: primal dual background}),
\item use a forward-backward splitting algorithm to
alternatively update the primal (\vx) and dual (\xmath{\vd}) variables, and
\item replace the Euclidean norm in the proximal operator
in the dual variable update equation
with a Bregman divergence measure. \end{enumerate} If the Bregman divergence measure is chosen carefully, the resulting update is smooth and standard backpropagation tools can compute \uppergrad. This section overviews how the approach in \citep{ochs:2016:techniquesgradientbasedbilevel} applies to \eqref{eq: bilevel for analysis filters}. Ref. \citep{ochs:2016:techniquesgradientbasedbilevel} derives the full backpropagation formula and uses Bregman divergences to unroll non-smooth cost functions in a multi-label segmentation problem, but the approach generalizes to image reconstruction as shown here.
Using the stacked convolutional matrix notation for the learned filters defined in \eqref{eq: stacked convolutional matrix} and selecting \sparsefcn to be the absolute value function \footnote{When using the absolute value, one can absorb the tuning parameters $\beta_k$ into the filter magnitudes, conveniently reducing the dimension of \params. }, the lower-level optimization problem is \begin{align*}
&\argmin_\vx \onehalf \normrsq{\mA \vx - \vy} + \norm{\mOmega \vx}_1 .\end{align*} From \eqref{eq: saddle point 1-norm}, the corresponding saddle-point formulation is \begin{align*}
&\argmin_\vx \min_\xmath{\vd} \onehalf \normrsq{\mA \vx - \vy} - \langle \xmath{\vd}, \mOmega \vx \rangle \text{ s.t. } \abs{\xmath{d}_i} \leq 1 \;\forall i ,\end{align*} where \xmath{\vd} is the dual variable. The minimum cost value and corresponding minimizer, \xhat, of the saddle-point problem are equivalent to those of the original problem because the 1-norm is convex.
To optimize the saddle-point problem, one can alternate \vx and \vz updates. Ref. \citep{ochs:2016:techniquesgradientbasedbilevel} uses the primal-dual algorithm from \citep{chambolle:2016:ergodicconvergencerates} that introduces a proximity function to each update step: \begin{align}
\vx^{(\loweriter+1)} = &\argmin_\vx \onehalf \normrsq{\mA \vx - \vy} - \langle \xmath{\vd}^{(\loweriter)}, \mOmega \vx \rangle + \frac{1}{\xmath{\alpha_\mathrm{x}}} \vone' \xmath{D}.(\vx, \vx^{(\loweriter)}) \nonumber \\
\xmath{\vd}^{(\loweriter+1)} &= \argmin_\xmath{\vd} \frac{1}{\xmath{\alpha_\mathrm{\dual}}} \vone' \xmath{D}.(\xmath{\vd}, \tilde{\xmath{\vd}})
- \langle \xmath{\vd}, \mOmega \tilde{\vx} \rangle
\mathrm{\; s.t. \;} \abs{\xmath{\vd}_i} \leq 1 \; \forall i
\label{eq: primal-dual update step z} ,\end{align} where $\tilde{\vx}$ and $\tilde{\xmath{\vd}}$ are defined in terms of previous iterates, \eg, when including momentum, and $\xmath{\alpha_\mathrm{x}}$ and $\xmath{\alpha_\mathrm{\dual}}$ are step size parameters chosen according to the theory in \citep{chambolle:2016:ergodicconvergencerates}. The \vx update is a smooth, quadratic problem and is straight-forward. However, the standard dual update involves a non-smooth projection; in particular, if the proximal distance function is the standard Euclidean 2-norm, i.e., \( D(d,\tilde{d}) = \frac{1}{2} (d - \tilde{d})^2, \) then the \xmath{\vd} update is the projection \[ \xmath{\vd}^{(\loweriter+1)} = \text{sign}.(\tilde{\xmath{\vd}} + \xmath{\alpha_\mathrm{\dual}} \mOmega \tilde{\vx}) \dottimes \text{min}.(1, \abs{\tilde{\xmath{\vd}} + \xmath{\alpha_\mathrm{\dual}} \mOmega \tilde{\vx}}) ,\] which is non-smooth.
To make the \xmath{\vd} update smooth, \citep{ochs:2016:techniquesgradientbasedbilevel} replaces the standard Euclidean norm in the proximity operator with a Bregman divergence. For the 1-norm regularizer, \citep{ochs:2016:techniquesgradientbasedbilevel} considers the divergence measure \begin{equation}
\xmath{D}(\xmath{d}, \tilde{\xmath{d}}) = \psi(\xmath{d}) - \psi(\tilde{\xmath{d}}) - \nabla \psi(\tilde{\xmath{d}})' (\xmath{d}-\tilde{\xmath{d}}) \end{equation} where $\psi(\xmath{d}) = \onehalf \paren{(\xmath{d}+1)\log{\xmath{d}+1} + (1-\xmath{d})\log{1-\xmath{d}}}$. Similar to standard distance metrics, this Bregman divergence is zero when $\xmath{d}=\tilde{\xmath{d}}$. However, it is not symmetric, \ie, $\xmath{D}(\xmath{d}, \tilde{\xmath{d}}) \neq \xmath{D}(\tilde{\xmath{d}}, \xmath{d})$ in general. Using this definition for \xmath{D}, one can differentiate and solve for the minimizer in the \xmath{\vd} update \eqref{eq: primal-dual update step z} \citep{ochs:2016:techniquesgradientbasedbilevel}. Because all the functions are separable, the update can be done independently for each \xmath{\vd} coordinate: \begin{equation}
\xmath{d}_i^{(\loweriter+1)} = \frac{ e^{2\xmath{\alpha_\mathrm{\dual}}[\mOmega\vx]_i} - \frac{1-\tilde{\xmath{d}}_i}{1+\tilde{\xmath{d}}_i} }{
e^{2\xmath{\alpha_\mathrm{\dual}}[\mOmega\vx]_i} + \frac{1-\tilde{\xmath{d}}_i}{1+\tilde{\xmath{d}}_i}} .\end{equation}
When the step-size $\xmath{\alpha_\mathrm{\dual}}$ approaches infinity,
$\xmath{d}_i^{(\loweriter+1)}$ approaches $\pm 1$ (its extreme values).
When $\xmath{\alpha_\mathrm{\dual}}$ approaches 0, $\xmath{d}_i^{(\loweriter+1)} = \tilde{\xmath{d}}_i$. The updated coordinate is guaranteed to satisfy the constraint $\abs{\xmath{d}_i} \leq 1$ whenever $\tilde{\xmath{d}}_i$ does, so there is no need for a (non-smooth) projection. Although this approach allows for applying the unrolled method to non-smooth cost functions, \citep{ochs:2016:techniquesgradientbasedbilevel} comments that \dquotes{the [equivalent of a] `smoothing parameter' in our approach is the number of iterations of the algorithm that replaces the lower level problem.} \fref{fig: bregman prox function} demonstrates how the number of iterations impacts the effective smoothing for a simple version of the problem where $\mA=\I$ and $\mOmega=\I$.
Ref. \citep{chambolle:2021:learningconsistentdiscretizations} uses the same saddle-point problem as in \citep{ochs:2016:techniquesgradientbasedbilevel} to propose another approach to computing \uppergrad. Instead of unrolling an algorithm and then back-propagating, \citep{chambolle:2021:learningconsistentdiscretizations} uses a sensitivity analysis and introduces additional adjoint variables that allow for simultaneously computing \uppergrad in the same forward iteration as \xhatp, without incurring the large matrix-matrix multiplications costs as in the forward-mode method of computing \eqref{eq: generic lower-level chain rule}. Although the theoretical analysis of the resulting \dquotes{piggy-backing} optimization algorithm is for smooth functions, \citep{chambolle:2021:learningconsistentdiscretizations} found it worked well empirically in non-smooth settings.
\begin{figure}
\caption{
Proximal operators for $\regfcn(x) = \onehalf \abs{x}$
and some smooth relatives.
The black line in both plots is the soft thresholding function,
which is the proximal operator for the absolute value function,
\ie,
$\text{prox}(y) = \argmin_x \onehalf (x-y)^2 + \onehalf \abs{x}$.
(a) As described in \citep{ochs:2016:techniquesgradientbasedbilevel},
the number of iterations of the primal-dual algorithm
with the Bregman proximity function
acts as a smoothing parameter for the proximal operator estimate and
the estimate improves as the number of iterations
increases (from light to dark lines).
(b) Smooth proximal operator for the non-smooth penalty function \eqref{eq: christof prox}
for $p=3/2$, $\beta=0.5$, and
four different values of $\tilde{\beta}$.
The proximal operator is closer to soft thresholding
for smaller values of $\tilde{\beta}$
(darker lines).
}
\label{fig: bregman prox function}
\label{fig: christof prox}
\end{figure}
\citet{christof:2020:gradientbasedsolutionalgorithms} shows another approach to achieving a smooth optimization algorithm for a non-smooth cost function. Ref. \citep{christof:2020:gradientbasedsolutionalgorithms} specifically considers cost functions with penalty functions of the form \begin{equation}
\sparsefcn(z) = \beta \abs{z} + 2 \tilde{\beta} \frac{\abs{z}^p}{p} \text{ for } 1 < p < 2
\label{eq: christof prox} .\end{equation} As a simple demonstration, in the case where there are no convolutional filters and $p=3/2$, the lower-level cost function is the proximal operator \begin{align*}
\mathrm{prox}_{\sparsefcn}(y) &= \argmin_x \onehalf (x-y)^2 + \sparsefcn(x) .\end{align*} Differentiating and solving for the minimizer yields \begin{align*}
\mathrm{prox}_{\sparsefcn}(y)
&=
\begin{cases}
\mathrm{sign}(y) \paren{ \sqrt{\tilde{\beta}^2 + \abs{y} - \beta} - \tilde{\beta} }^2 & \text{ if } \abs{y} > \beta \\
0 & \mathrm{ else, }
\end{cases} \end{align*} which is continuous and differentiable everywhere with respect to $y$ despite the non-differential absolute value function in \sparsefcn! \fref{fig: christof prox} shows this proximal operator alongside the proximal operator when \mbox{$\sparsefcn(z)=\abs{z}$} (soft thresholding). Ref. \citep{christof:2020:gradientbasedsolutionalgorithms} proves that this simple example generalizes to the bilevel problem of learning filters.
\section{Summary \label{sec: ift unrolled comparison}}
This section focused on computing \uppergrad, the gradient of the upper-level loss function with respect to the learnable parameters. \cref{chap: bilevel methods} builds on this foundation to consider optimization methods for bilevel problems. Many of those optimization methods can be used in conjunction with the minimizer, translation to a single-level, or unrolled approaches to compute \uppergrad. Thus, how one selects an approach may depend on the structure of the specific bilevel problem, how closely tied one wishes to be to the original bilevel problem, computational cost, and/or gradient accuracy.
The translation to a single-level approach is tailored to a specific type of bilevel problem. A benefit of the translation approach is the ability to use the 1-norm (without any corner rounding) in the lower-level cost function. However, the corresponding drawback is the (current) lack of generality in the minimizer approach; the closed-form expression derived in \citep{sprechmann:2013:supervisedsparseanalysis,mccann:2020:supervisedlearningsparsitypromoting,ghosh:2021:bilevellearningl1regularizers} is specific to using the 1-norm as \sparsefcn. Expanding this approach to regularizers other than the 1-norm is a possible avenue for future work.
One difference among the methods is whether they depend on the lower-level optimization algorithm; while the unrolled approach depends on the specific optimization algorithm, the minimizer approach and the translation to a single-level approach do not. A resulting downside of unrolling is that one cannot use techniques such as warm starts and non-differentiable restarts, so $\vx^{(T)}$ may be farther from the minimizer than the approximation from a similar number of iterations of a more sophisticated, non-differentiable update method. However, the unrolled method's dependence on \optalgstep is also a benefit, as an unrolled method can be applied to non-smooth cost functions, as long as the resulting update mapping \optalgstep is smooth. \blue{Further, defining \optalgstep and the initial starting point ensures that $\vx^{(T)}$ is unique, avoiding concerns about non-unique minimizers. }
Another advantage of unrolling is that one can run a given number of iterations of the optimization algorithm, without having to reach convergence, and still calculate a valid gradient. Particularly in image reconstruction problems, where finding \xhat exactly can be time intensive, the benefit of a more flexible run-time could outweigh the disadvantages. However, the corresponding downside of unrolling is that the learned hyperparameters are less clearly tied to the original cost function than when one uses the minimizer approach. \sref{sec: connections unrolled} further discusses this point in connection to how unrolling for bilevel methods can differ from (deep) learnable optimization algorithms.
\blue{ One way to connect the minimizer and unrolling strategies is to consider the limit as the number of unrolled iterations approaches infinity. Assuming the optimization algorithm converges, this \dquotes{fixed point} approach is strongly related to the minimizer approach. For instance, \citep{shaban:2019:truncatedbackpropagationbilevel} shows that backpropagating through the last $\tilde{T}$ iterations of a converged unrolled algorithm can be viewed as approximating the matrix inverse in the minimizer gradient equation \eqref{eq: IFT final gradient dldparams} with an order-$\tilde{T}$ Taylor series. \sref{sec: connection to DEQ} further discusses how fixed point networks (or \dquotes{equilibrium networks}) relate the unrolled-to-convergence and minimizer approaches. }
Gradient accuracy and computational cost are, unsurprisingly, trade-offs. \tref{tab: ift and unrolled complexities} summarizes the cost of the minimizer and unrolled approaches, derived in \sref{sec: ift complexity} and \apref{sec: unrolled complexity} respectively, but the total computation will depend on the required gradient accuracy. By accuracy, we mean error from the true bilevel gradient \[
\lVert \underbrace{\nablahat_T \lfcn(\params)}_{\substack{\text{Estimated} \\ \text{ gradient}}}
- \underbrace{\nabla \lfcn(\params)}_{\substack{ \text{True bilevel} \\ \text{gradient}}} \!\!\! \rVert ,\] where $T$ denotes the number of lower-level optimization steps. The unrolled gradient is always accurate for the unrolled mapping, but not for the original bilevel problem. Therefore, unrolling may be more computationally feasible when one cannot run a sufficient number of lower-level optimization steps to reach close enough to a minimizer to assume the gradient in \eqref{eq:dPhi} is approximately zero \citep{kobler:2021:totaldeepvariation}.
\begin{table}[b!]
\input{TablesAndAlgs/tab,iftunrolledcomplexity}
\iffigsatend \tabletag{4.1} \fi
\caption{
Memory and computational complexity of the minimizer approach
\eqref{eq: IFT final gradient dldparams},
reverse-mode unrolled approach \eqref{eq: reverse mode}, and
forward-mode unrolled approach \eqref{eq: forward mode} to computing \dParams{\lfcnargs}.
Computational costs do not include running the optimization algorithm
(typically expensive but often comparable across methods),
computing \finalterm (typically cheap),
or computing \dParams{\lfcn(\params \, ; \vx)} (frequently zero).
Memory requirements do not include
storing a single copy of \vx, \mA, \params, \franA, and \franB{}.
Recall $\vx \in \F^\sdim$, $\params \in \F^\paramsdim$,
and there are $T$ iterations of the lower-level optimization algorithm for the unrolled method. Hessian-vector products (first row) and Hessian-inverse-vector products (middle row) are listed
separately from all other multiplications
(last row)
as the computational cost of Hessian operations
can vary widely;
see discussion in \sref{sec: ift complexity}.
}
\label{tab: ift and unrolled complexities} \end{table}
In all of the approaches considered, the accuracy of the estimated hyperparameter gradient in turn depends on the solution accuracy or number of unrolled iterations of the lower-level cost function. Ref. \citep{mccann:2020:supervisedlearningsparsitypromoting} notes that their translation to a single-level approach failed if they did not optimize the lower-level problem to a sufficient accuracy level. However, \citep{sprechmann:2013:supervisedsparseanalysis,mccann:2020:supervisedlearningsparsitypromoting,ghosh:2021:bilevellearningl1regularizers} did not investigate how the solution accuracy of the lower-level problem impacts the upper-level gradient estimate.
For the minimizer and unrolled approaches, \citep{grazzi:2020:iterationcomplexityhypergradient,ji:2021:bileveloptimizationconvergence} found that the gradient estimate from the minimizer approach converges to the true gradient faster than the unrolled approach (in terms of computation). To state the bounds, \citep{grazzi:2020:iterationcomplexityhypergradient,ji:2021:bileveloptimizationconvergence} assert conditions on the structure of the bilevel problem. They assume that \xhatp is the unique minimizer of the lower-level cost function, the Hessian of the lower-level is invertible, the Hessian and Jacobian of \ofcn are Lipschitz continuous with respect to \vx, the gradients of the upper-level loss are Lipschitz continuous with respect to \vx, the norm of \vx is bounded, and the lower-level cost is strongly convex and Lipschitz smooth for every \params value. \sref{sec: assumptions for double and single loop complexity analysis} discusses similar investigations that use these conditions, how easy or hard they are to satisfy, and how they apply to \eqref{eq: bilevel for analysis filters}.
Ref. \citep{grazzi:2020:iterationcomplexityhypergradient} initializes the lower-level iterates for both the unrolled and minimizer approach with the zero vector, \ie, $\vx^{(0)} = \vzero$. Under their assumptions, \citep{grazzi:2020:iterationcomplexityhypergradient} prove that both the unrolled and minimizer gradients converge linearly in the number of lower-level iterations when the lower-level optimization algorithm and conjugate gradient algorithm for the minimizer approach converge linearly. Although the rate of the approaches is the same, the minimizer approach converges at a faster linear rate and \citep{grazzi:2020:iterationcomplexityhypergradient} generally recommends the minimizer approach, though they found empirically that the unrolled approach may be more reliable when the strong convexity and Lipschitz smooth assumptions on the lower-level cost do not hold.
Ref. \citep{ji:2021:bileveloptimizationconvergence} extended the analysis from \citep{grazzi:2020:iterationcomplexityhypergradient} to consider a warm start initialization for the lower-level optimization algorithm. They similarly find that the minimizer approach has a lower complexity than the unrolled approach. \srefs{sec: double-loop complexity analysis}{sec: single-loop complexity} further discuss complexity results after introducing specific bilevel optimization algorithms.
\chapter{Gradient-Based Bilevel Optimization Methods} \label{chap: bilevel methods}
The previous section discussed different approaches to finding \uppergrad, the gradient of the upper-level loss function with respect to the learnable parameters. Building on those results, we now consider approaches for optimizing the bilevel problem. In particular, this section concentrates on gradient-based algorithms for optimizing the hyperparameters. While there is some overlap with single-level optimization methods, this section focuses on the challenges due to the bilevel structure. Therefore, we do not discuss the lower-level optimization algorithms in detail; for overviews of lower-level optimization, see, \eg, \cite{palomar:11:coi,chambolle:2016:introductioncontinuousoptimization}.
Gradient-based methods for bilevel problems are an alternative to the approaches described in \sref{chap: hpo}, \eg, grid or random search, Bayesian optimization, and trust region methods. By incorporating gradient information, the methods presented in this section can scale to problems having many hyperparameters. In fact, \sref{sec: double and single loop complexity} reviews papers that provide bounds on the number of upper-level gradient descent iterations required to reach a point within some user-defined tolerance of a solution. While the bounds depend on the regularity of the upper-level loss and lower-level cost functions, they do not depend on the number of hyperparameters nor the signal dimension. Although having more hyperparameters will increase computation per iteration, using a gradient descent approach means the number of iterations need not scale with the number of hyperparameters, \paramsdim.
Bilevel gradient methods fall into two broad categories. Most gradient-based approaches to the bilevel problem fall under the first category: double-loop algorithms. These methods involve (i) optimizing the lower-level cost, either to some convergence tolerance if using a minimizer approach or for a certain number of iterations if using an unrolled approach, (ii) calculating \uppergrad, (iii) taking a gradient step in \params, and (iv) iterating. The first step is itself an optimization algorithm and may involve many inner iterations, thus the categorization as a \dquotes{double-loop algorithm.}
The second category, \dquotes{single-loop} algorithms, involve one loop, with each iteration containing one gradient step for both the lower-level optimization variable, \vx, and the upper-level optimization variable, \params. Single-loop algorithms may alternate updates or update the variables simultaneously. \cref{chap: ift and unrolled} used \loweriter to denote the lower-level iteration counter; this section introduces \upperiter as the iteration counter for the upper-level iterations and as the single iteration counter for single-loop algorithms.
\section{Double-Loop Algorithms} \label{sec: double-loop design decisions}
After using one of the approaches in \cref{chap: ift and unrolled} to compute the hyperparameter gradient \uppergrad, typical double-loop algorithms for bilevel problems run some type of gradient descent on the upper-level loss. \aref{alg: hoag} shows an example double-loop algorithm \cite{pedregosa:2016:hyperparameteroptimizationapproximate}. \blue{ Line \ref{alg step HOAG CG} of \aref{alg: hoag} uses the CG method to compute the product of the Hessian inverse with a vector in \eqref{eq: IFT final gradient dldparams}. Thus, \aref{alg: hoag} actually involves three loops. However, the third, CG loop is often left as an implementation detail and we will continue to use the term \dquotes{double-loop} for the overall strategy. There is similarly a third, hidden loop in approaches that use the reverse mode method for backpropogation in the unrolled approaches described in \sref{sec: unrolled}. }
The final iterate of a lower-level optimizer is only an approximation of the lower-level minimizer. However, the minimizer approach to calculating the upper-level gradient \uppergrad from \sref{sec: minimizer approach} assumes $\nabla_\vx \ofcn(\xhat \, ; \params) = \vzero$. Any error stemming from not being at an exact critical point can be magnified in the full calculation \eqref{eq: IFT final gradient dldparams}, and the resulting hyperparameter gradient will be an approximation of the true gradient, as illustrated in \fref{fig: grad accuracy example}. \blue{Thus, how accurately one optimizes the lower-level problem can greatly impact the quality of the learned parameters, \paramh \citep{chen:2013:revisitinglossspecifictraining}.} Alternatively, if one uses the unrolled approach with a set number of iterations \eqref{eq: unrolled upper-level}, the gradient is accurate for that specific number of iterations, but the lower-level optimization sequence may not have converged and the overall method may not accurately approximate the original bilevel problem.
\begin{figure}
\caption{
Error in the upper-level gradient, \uppergrad,
for various convergence thresholds for the lower-level optimizer.
The bilevel problem is \eqref{eq: bilevel for analysis filters}
with a single filter,
$\vc = \begin{bmatrix} c_0 & c_1 \end{bmatrix}$,
$\ebeta{0}=0$, $\ebeta{1}=-5$, and
$\sparsefcn(z) = z^2$ so there is an analytic solution for \uppergrad.
The training data is piece-wise constant 1d signals
and the learnable hyperparameters are the filter coefficients.
(a) Upper-level loss function, $\lfcn(\params)$.
The cost function is low (dark)
where $c_1 \approx c_0$,
corresponding to approximate finite differences.
The star indicates the minimum.
(b-d) Error in the estimated gradient angle
using the minimizer approach \eqref{eq: IFT final gradient dldparams},
defined as the angle between
$\nablahat \lfcn(\params)$ and \uppergrad,
when the lower-level optimization is run until
$\norm{\nabla_\vx \ofcnargs}_2 < \epsilon$.
}
\label{fig: grad accuracy example}
\end{figure}
Due to such inevitable inexactness when computing \uppergrad, one may wonder about the convergence of double-loop algorithms for bilevel problems. Considering the unrolled method of computing \uppergrad, \citep{franceschi:2018:bilevelprogramminghyperparameter} showed that the sequence of hyperparameter values in a double-loop algorithm, \iter{\params}, converges as the number of unrolled iterations increases. To prove this result, \citep{franceschi:2018:bilevelprogramminghyperparameter} assumed the hyperparameters were constrained to a compact set, \lfcnparamsvx and \ofcnargs are jointly continuous, there is a unique solution \xhatp to the lower-level cost for all \params; and \xhatp is bounded for all \params. These conditions are satisfied for problems with strictly convex lower-level cost functions and suitable box constraints on \params. \sref{sec: double-loop complexity analysis} further discusses convergence results for double-loop algorithms.
\begin{algorithm}[t!]
\caption{Hyperparameter optimization with approximate gradient (HOAG) from \cite{pedregosa:2016:hyperparameteroptimizationapproximate}.
As written below, the HOAG algorithm is impractical because
it uses $\xhat(\iter{\params})$
in the convergence criteria;
however, for strongly convex lower-level problems,
the convergence criteria,
$\normr{\xhat(\iter{\params}) - \lliter (\iter{\params}) }$,
is easily upper-bounded.
}
\label{alg: hoag}
\input{TablesAndAlgs/alg,hoag} \end{algorithm}
\citet{pedregosa:2016:hyperparameteroptimizationapproximate} proved a similar result for the minimizer formula \eqref{eq: IFT final gradient dldparams} using CG to compute \eqref{eq: Hinv step for CG}. Specifically, \citep{pedregosa:2016:hyperparameteroptimizationapproximate} showed that the hyperparameter sequence convergences to a stationary point if the sequence of positive tolerances, $\{\iter{\epsilon}, \upperiter=1,2,\ldots\}$ in \aref{alg: hoag}, is summable. The convergence results are for the algorithm version shown in \aref{alg: hoag} that uses a Lipschitz constant of $\lfcn(\params)$, which is generally unknown. Although \citep{pedregosa:2016:hyperparameteroptimizationapproximate} discusses various empirical strategies for setting the step size, the convergence theory does not consider those variations. Thus, the double-loop algorithm \citep{pedregosa:2016:hyperparameteroptimizationapproximate} requires multiple design decisions.
There are four key design decisions for double-loop algorithms: \begin{enumerate}[nolistsep]
\item How accurately should one solve the lower-level problem?
\item What upper-level gradient descent algorithm should one use?
\item
How does one pick the step size for the upper-level descent step?
\item What stopping criteria should one use for the upper-level iterations? \end{enumerate} This section first reviews some (largely heuristic) approaches to these design decisions and presents example bilevel gradient descent methods with no (or few) assumptions beyond those made in \cref{chap: ift and unrolled}. Without any further assumptions, the answers to the questions above are based on heuristics, with few theoretical guarantees but often providing good experimental results. \sref{sec: double-loop complexity analysis} discusses recent methods with stricter assumptions on the bilevel problem and their theory-backed answers to the above questions.
The first step in a double-loop algorithm is to optimize the lower-level cost, for which there are many optimization approaches. The only restriction is computability of the gradient of the upper-level loss \uppergrad, which typically includes a smoothness assumption (see \cref{chap: ift and unrolled} for discussion). Many bilevel methods use a standard optimizer for the lower-level problem, although others propose new variants, \eg, \citep{chen:2021:learnabledescentalgorithm}.
The \textbf{first design decision} (how accurately to solve the lower-level problem) involves a trade-off between computational complexity and accuracy. Example convergence criteria are fairly standard to the optimization literature, \eg, the Euclidean norm of the lower-level gradient \citep{hintermuller:2020:dualizationautomaticdistributed, chen:2014:insightsanalysisoperator} or the normalized change in the estimate \vx \citep{sixou:2020:adaptativeregularizationparameter} being less than some threshold. For example, \citep{chen:2014:insightsanalysisoperator} used a convergence criteria of $\normr{\nabla_\vx \ofcn(\lliter \, ; \params)}_2 \leq 10^{\neg 3}$ (where the image scale is 0-255). As mentioned above, \citep{pedregosa:2016:hyperparameteroptimizationapproximate} uses a sequence of convergence tolerances so that the lower-level cost function is optimized more accurately as the upper-level iterations continue.
\blue{ Ref. \citep{chen:2013:revisitinglossspecifictraining} investigated the importance of lower-level optimization accuracy. The authors use the same training model as in \citep{samuel:2009:learningoptimizedmap}, which is the bilevel extension of the Field of Experts \citep{roth:2005:fieldsexpertsframework}, but varied the convergence criteria for the lower-level problem. When using a convergence tolerance of $\normr{\nabla_\vx \ofcn(\lliter \, ; \params)}_2/\sqrt{\sdim} \leq 10^{\neg5}$, \citep{chen:2013:revisitinglossspecifictraining} found an average improvement of 0.65dB in the PSNR for test images over \citep{samuel:2009:learningoptimizedmap}, who ran their lower-level optimization algorithm for a set number of iterations. Ref. \citep{chen:2013:revisitinglossspecifictraining} also plots the test PSNR and training loss versus the lower-level convergence criteria and shows how test PSNR increases and training loss decreases with increased lower-level solution accuracy for this specific filter learning bilevel problem. }
Many publications do not report a specific threshold or discuss how they chose a convergence criteria or number of lower-level iterations. However, a few note the importance of such decisions. For example, \citep{mccann:2020:supervisedlearningsparsitypromoting} found that their learning method fails if the lower-level optimizer is insufficiently close to the minimizer and \citep{chen:2014:insightsanalysisoperator} stated their results are \dquotes{significantly better} than \citep{samuel:2009:learningoptimizedmap} because they solve the lower-level problem \dquotes{with high[er] accuracy.}
After selecting a level of accuracy, finding (an approximation of) \xhat, and calculating \uppergrad using one of the approaches from \cref{chap: ift and unrolled}, one must make the \textbf{second design decision}: which gradient-based method to use for the upper-level problem. Many bilevel methods suggest a simple gradient-based method such as plain gradient descent (GD) \citep{peyre:2011:learninganalysissparsity}, GD with a line search (see the third design decision), projected GD \citep{antil:2020:bileveloptimizationdeep}, or stochastic GD \citep{mccann:2020:supervisedlearningsparsitypromoting}. These methods update \params based on only the current upper-level gradient; they do not have memory of previous gradients nor require/estimate any second-order information.
Methods that incorporate some second-order information use more memory and computation per iteration, but may converge faster than basic GD methods. For example, Broyden-Fletcher-Goldfarb-Shanno (BFGS) and L-BFGS (the low-memory version of BFGS) \cite{byrd:95:alm} are quasi-Newton algorithms that store and update an approximate Hessian matrix that serves as a preconditioner for the gradient. The $\paramsdim \by \paramsdim$ size of the Hessian grows as the number hyperparameters increases, but quasi-Newton methods like L-BFGS use practical rank-1 updates with storage \order{\paramsdim}. Adam \citep{kingma:2015:adammethodstochastic} is a popular GD method, especially in the machine learning community, that tailors the step size (equivalently the learning rate) for each hyperparameter based on moments of the gradient. Although Adam requires its own parameters, the parameters are relatively easy to set and the default settings often perform adequately. Example bilevel papers using methods with second-order information include those that use BFGS \citep{holler:2018:bilevelapproachparameter}, L-BFGS \citep{chen:2014:insightsanalysisoperator}, Gauss-Newton \citep{fehrenbach:2015:bilevelimagedenoising}, and Adam \citep{chen:2021:learnabledescentalgorithm}.
Many gradient-based methods require selecting a step size parameter, \eg, one must choose a step size \ssupper in classical GD: \begin{equation*}
\params^{(\upperiter+1)} = \params^{(\upperiter)}
- \ssupper \, \uppergradu .\end{equation*} This choice is the \textbf{third design decision}. Bilevel problems are generally non-convex, and typically a Lipschitz constant is unavailable, so line search strategies initially appear appealing. However, any line search strategy that involves attempting multiple values quickly becomes computationally intractable for large-scale problems. The upper-level loss function in bilevel problems is particularly expensive to evaluate because it requires optimizing the lower-level cost! Further, recall that the upper-level loss is typically an expectation over multiple training samples \eqref{eq: generic bilevel upper-level}, so evaluating a single step size involves optimizing the lower-level cost \Ntrue times (or using a stochastic approach and selecting a batch size).
Despite these challenges, a line search strategy may be viable if it rarely requires multiple attempts. For example, the backtracking line search in \cite{hintermuller:2020:dualizationautomaticdistributed} that used the Armijo–Goldstein condition required 57-59 lower-level evaluations (per training example) over 40 upper-level gradient descent steps, so most upper-level steps required only one lower-level evaluation. Other bilevel papers that used backtracking with Armijo-type conditions include \citep{holler:2018:bilevelapproachparameter, sixou:2020:adaptativeregularizationparameter,calatroni:2017:bilevelapproacheslearning}; \citep{lecouat:2020:flexibleframeworkdesigning} used the Barzilai-Borwein method for picking an adaptive step size.
Other approaches to determining the step size are: (i) normalize the gradient by the dimension of the data and pick a fixed step size \citep{mccann:2020:supervisedlearningsparsitypromoting}, (ii) pick a value that is small enough based on experience \citep{peyre:2011:learninganalysissparsity}, or (iii) adapt the step size based on the decrease from the previous iteration \citep{pedregosa:2016:hyperparameteroptimizationapproximate}.
The \textbf{fourth design decision} is the convergence criteria for the upper-level loss. As with the lower-level convergence criteria, few publications include a specific threshold, but most bilevel methods tend to use traditional convergence criteria such as the norm of the hyperparameter gradient falling below some threshold \citep{holler:2018:bilevelapproachparameter}, the norm of the change in parameters falling below some threshold \citep{chen:2014:insightsanalysisoperator}, and/or reaching a maximum iteration count (many papers). One specific example is to terminate when the normalized change in learned parameters, \( \normr{\iter{\params}{+1} - \iter{\params}} / \normr{\iter{\params}}, \) is below 0.01 \citep{sixou:2020:adaptativeregularizationparameter}. The normalized change bound is convenient because it is unitless and thus invariant to scaling of \params.
\begin{figure}
\caption{Example convergence plots for a double-loop bilevel method
when \params includes \vc and \vbeta (solid lines)
and when $\params = \vbeta$ (dotted lines).
(a) Estimated upper-level loss function
evaluated at the current estimate of the lower-level minimizer,
$\vx^{(T)} = \vx^{(T)}(\params^{(\upperiter)})$,
versus upper-level iteration \upperiter.
(b) Lower-level convergence metric, averaged
over all training samples,
versus upper-level iteration.
The estimated lower-level minimizer
remains close to convergence
throughout the double-loop method.
}
\label{fig: cameraman training loss}
\end{figure}
\blue{\fref{fig: cameraman training loss} shows example upper-level convergence plots for a double-loop algorithm for the bilevel problem \eqref{eq: bilevel for analysis filters}. After an initial first run of OGM to get the lower-level initialization $\xhat(\params^{(0)})$ such that $\tfrac{1}{\sqrt{\sdim}}\norm{\dx{\ofcn\left(\xhat(\params^{(0)})\, ; \, \params^{(0)}\right)}}_2 < 10^{\neg 7}$, the lower-level optimizer consisted of 10 iterations of OGM \citep{kim:18:aro}, initialized with the estimate from the previous upper-level iteration. The upper-level optimizer is Adam \citep{kingma:2015:adammethodstochastic} with the default parameters, negating the need for a separate upper-level step-size parameter. We ran 10,000 outer-loop iterations. The final norm of the upper-level gradient, $\tfrac{1}{\sqrt{\paramsdim}} \normr{ \nabla(\params^{(U)}) }$ was 0.08 when learning the filter coefficients and tuning parameters and $5\cdot10^{\neg 4}$ when learning only \vbeta. \fref{fig: cameraman example results} shows the corresponding denoised images and \apref{sec: cameraman training details} further details the experiment settings.}
\section{Single-Loop Algorithms}
Unlike double-loop algorithms, single-loop algorithms take a gradient step in \params without optimizing the lower-level problem each step. Two early bilevel method papers \citep{haber:2003:learningregularizationfunctionals,kunisch:2013:bileveloptimizationapproach} proposed single-loop approaches based on solving the system of equations that arises from the Lagrangian.
The system of equations approach in \citep{haber:2003:learningregularizationfunctionals,kunisch:2013:bileveloptimizationapproach} closely follows the KKT perspective on the minimizer approach in \sref{sec: minimizer via kkt}. Recall that the gradient of the lower-level problem is zero at a minimizer, \xhat, and one can use this equality as a constraint on the upper-level loss function. The corresponding Lagrangian is \begin{align}
L(\vx, \params, \vnu)
&= \lfcn(\params \, ; \vx) + \vnu' \dx{\ofcnargs} \label{eq: lagrangian} ,\end{align} where \vnu is a vector of Lagrange multipliers. For the filter learning example \eqref{eq: bilevel for analysis filters}, the Lagrangian is \begin{align}
L(\vx, \params, \vnu)
&= \onehalf \normrsq{\vx - \xtrue}_2 + \nonumber \\
& \, \, \vnu' \left(
\mA' (\mA \vx - \vy)
+ \ebeta{0} \sum_{k=1}^K \ebeta{k} \xmath{\tilde{\vc}_k} \conv \sparsefcn.(\xmath{\vc_k} \conv \vx; \epsilon)
\right)
\nonumber .\end{align}
As in \sref{sec: minimizer via kkt}, we consider derivatives of the Lagrangian with respect to \vnu, \vx, and \params. Here are the general expressions and the specific equations for the filter learning example \eqref{eq: bilevel for analysis filters} when considering the element of \params corresponding to $\beta_k$: \begin{align*}
\nabla_\vnu L(\vx, \params, \vnu) &= \dx{\ofcnargs} \\
&= \mA' (\mA \vx - \vy)
+ \ebeta{0} \sum_{k=1}^K \ebeta{k} \xmath{\tilde{\vc}_k} \conv \sparsefcn.(\xmath{\vc_k} \conv \vx; \epsilon) \\
\dx{ L(\vx, \params, \vnu)}
&= \dx{\lfcn(\params \, ; \vx)} + \nabla_{\vx \vx} \ofcn(\vx \,; \params) \vnu \\
&= \vx - \xtrue + \mA'\mA \vnu + \ebeta{0} \sum_k \ebeta{k} \mC_k' \diag{\ddsparsefcn.(\xmath{\vc_k} \conv \xhat)} \mC_k \vnu \\
\dParams{L(\vx, \params, \vnu)} &= \dParams{\lfcnparamsvx} + \vnu' \nabla_{\vx \params} \ofcn(\vx \,; \params)
\\
&= \vnu' \left( \ebeta{0} \ebeta{k} \xmath{\tilde{\vc}_k} \conv \dsparsefcn.(\xmath{\vc_k} \conv \xhat) \right)
\text{ when } \params=\beta_k .\end{align*} These expressions are equivalent to the primal, adjoint, and optimality conditions respectively in \citep{kunisch:2013:bileveloptimizationapproach}.
Here the minimizer and single-loop approach diverge. \sref{sec: minimizer via kkt} used the above Lagrangian gradients to solve for $\hat{\vnu}$, substitute $\hat{\vnu}$ into the gradient of the Lagrangian with respect to \params, and thus find the minimizer expression for \uppergrad. The single-loop approach instead considers solving the system of gradient equations directly: \[
\mG(\vx, \params, \vnu)
= \begin{bmatrix}
\nabla_\vnu L(\vx, \params, \vnu) \\
\dx{ L(\vx, \params, \vnu)} \\
\dParams{ L(\vx, \params, \vnu)}
\end{bmatrix} = \vzero .\] For example, \citep{kunisch:2013:bileveloptimizationapproach} proposed a Newton algorithm using the Jacobian of the gradient matrix \mG.
\blue{ Another approach to single-loop algorithms is to replace the \dquotes{while} loop in \aref{alg: hoag} line \ref{alg: HOAG while loop for LL} with a single gradient step in the lower-level optimization variables. Two single-loop algorithms are the two-timescale stochastic approximation (TTSA) method \citep{hong:2020:twotimescaleframeworkbilevel} and the Single Timescale stochAstic BiLevEl optimization (STABLE) method \citep{chen:2021:singletimescalestochasticbilevel}. \aref{alg: ttsa} shows TTSA as an example single-loop algorithm. Both TTSA and STABLE alternate between one gradient step for the lower-level cost and one gradient step for the upper-level problem. }
There are two main challenges in designing such a single loop algorithm for bilevel optimization. Because both TTSA and STABLE use the minimizer approach \eqref{eq: IFT final gradient dldparams} to finding the upper-level gradient, the first challenge is ensuring the current lower-level iterate is close enough to the minimizer to calculate a useful upper-level gradient. TTSA addresses this challenge by taking larger steps for the lower-level problem while STABLE addresses this using a lower-level update that better predicts the next lower-level minimizer, $\xhat(\iter{\params}{+1})$.
The second main challenge is estimating the upper-level gradient, even given stochastic estimates of $\nabla_{\vx \vx} \ofcn$ and $\nabla_{\vx \params} \ofcn$, because the minimizer equation \eqref{eq: IFT final gradient dldparams} is nonlinear. The theoretical results about TTSA are built on the assumption that the upper-level gradient is biased due to this nonlinearity. In contrast, STABLE uses recursion to update estimates of the gradients and thus reduce variance. \sref{sec: single-loop complexity} goes into more detail about both algorithms.
\begin{algorithm}[t!]
\caption{Two-Timescale Stochastic Approximation (TTSA) method from \citep{hong:2020:twotimescaleframeworkbilevel}.
TTSA includes a possible projection of the hyperparameter after each gradient step onto a constraint set,
not shown here.
The tildes denote stochastic approximations for the corresponding expressions.
}
\label{alg: ttsa}
\input{TablesAndAlgs/alg,ttsa} \end{algorithm}
\section{Complexity Analysis} \label{sec: double and single loop complexity}
A series of recent papers established finite-time sample complexity bounds for stochastic bilevel optimization methods based on gradient descent for the upper-level loss and lower-level cost. Ref.s \citep{ghadimi:2018:approximationmethodsbilevel, ji:2021:bileveloptimizationconvergence} use double-loop approaches and \citep{hong:2020:twotimescaleframeworkbilevel, chen:2021:singletimescalestochasticbilevel} use single-loop algorithms. Unlike most of the methods discussed in \sref{sec: double-loop design decisions}, these papers make additional assumptions about the upper and lower-level functions then select the upper and lower-level step sizes to ensure convergence.
In these works, ``finite-time sample complexity'' refers to big-O bounds on a number of iterations that ensures one reaches a minimizer to within some desired tolerance. In contrast to asymptotic convergence analysis, finite-time bounds provide information about the estimated hyperparameters, \iter{\params}, after a finite number of upper-level iterations. These bounds depend on problem-specific quantities, such as Lipschitz constants, but not on the hyperparameter or signal dimensions.
To summarize the results, this section returns to the notation from the introduction where the upper-level loss may be deterministic or stochastic, \eg, the bilevel problem is \begin{align}
\paramh &= \argmin_\params \lfcn(\params) \text{ with }
\lfcn(\params) = \begin{cases}
\lfcn(\params, \xhat(\params)) & \text{deterministic} \\
\E{\lfcn(\params, \xhat(\params))} & \text{ stochastic. }
\end{cases}
\label{eq:bilevel,E} \end{align} The expectation in \eqref{eq:bilevel,E} can have different meanings depending on the setting. When one has $J$ training images with one noise realization per image, one often picks a random subset (``minibatch'') of those $J$ images for each update of \params, corresponding to stochastic gradient descent of the upper-level loss. In this setting, the randomness is a property of the algorithm, not of the upper-level loss, and the expectation reduces to the deterministic case. \sref{sec: bilevel future directions} discusses other possible definitions of the stochastic bilevel formulation.
The complexity results (summarized in \tref{tab: stochastic complexity summaries}) are all in terms of finding \xmath{\params_\epsilon}, defined as an $\epsilon$-optimal solution. In the (atypical) setting where $\lfcn(\params)$ is convex, \xmath{\params_\epsilon} is an $\epsilon$-optimal solution if it satisfies either $\lfcn(\xmath{\params_\epsilon}) - \lfcn(\paramh) \leq \epsilon$ \citep{ghadimi:2018:approximationmethodsbilevel, ji:2021:bileveloptimizationconvergence,hong:2020:twotimescaleframeworkbilevel} or $\normsq{\paramh - \xmath{\params_\epsilon}} \leq \epsilon$ \citep{chen:2021:singletimescalestochasticbilevel}. (These conditions are equivalent if \lfcn is strongly convex in \params, but can differ otherwise.) In the (common) non-convex setting, \xmath{\params_\epsilon} is typically called an $\epsilon$-stationary point if it satisfies $\normsq{\nabla \lfcn(\xmath{\params_\epsilon})} \leq \epsilon$ \citep{ghadimi:2018:approximationmethodsbilevel, ji:2021:bileveloptimizationconvergence, chen:2021:singletimescalestochasticbilevel}. In the stochastic setting, \xmath{\params_\epsilon} must satisfy these conditions in expectation.
\begin{table}[bt]
\input{TablesAndAlgs/tab,stochasticbilevelcomplexities}
\iffigsatend \tabletag{5.3} \fi
\caption{
Finite-time sample complexities
for the stochastic bilevel problem
in the common scenario where \lfcn is non-convex
when using
BA \citep{ghadimi:2018:approximationmethodsbilevel},
stocBiO \citep{ji:2021:bileveloptimizationconvergence},
TTSA \citep{hong:2020:twotimescaleframeworkbilevel},
and
STABLE \citep{chen:2021:singletimescalestochasticbilevel}.
When \lfcn is strongly convex,
the sample complexity of STABLE
is $\order{\frac{1}{\epsilon^1}}$
(for the upper- and lower-level gradients),
which is the same as single level stochastic gradient algorithms.
See cited papers for other complexity
results when \lfcn is strongly convex.
}
\label{tab: stochastic complexity summaries} \end{table}
\blue{ The following sections briefly describe the BA, stocBiO, TTSA, and STABLE algorithms. The literature in this area is quickly evolving; between the writing and editing of this work, new double-loop and single-loop methods appeared with improved complexity results. For example, \citep{yang:2021:provablyfasteralgorithms, khanduri:2021:nearoptimalalgorithmstochastic} concurrently proposed bilevel optimization methods that leverage momentum and variance reduction techniques to reduce the bound on the number of iterations to \ordertil{\frac{1}{\epsilon^{1.5}}} for both upper-level and lower-level gradients. Ref. \citep{yang:2021:provablyfasteralgorithms} achieved this complexity result for both a double-loop method and a single-loop method. }
Whether double-loop or single-loop methods are preferred is an open question. Refs.~ \citep{ji:2021:bileveloptimizationconvergence,yang:2021:provablyfasteralgorithms} find that double-loop methods converge faster (in terms of wall time) than single-loop methods. The authors hypothesize that \uppergrad is sensitive enough to changes in the estimate of the lower-level optimizer that the increased accuracy of the double-loop estimates of \uppergrad is worth the additional lower-level optimization time. Future work should test this hypothesis in different experimental settings and establish guidelines on when to use a double-loop or single-loop algorithm.
\subsection{Assumptions} \label{sec: assumptions for double and single loop complexity analysis}
References \citep{ghadimi:2018:approximationmethodsbilevel, ji:2021:bileveloptimizationconvergence, hong:2020:twotimescaleframeworkbilevel, chen:2021:singletimescalestochasticbilevel} all make similar assumptions about \lfcn and \ofcn to derive theoretical results for their proposed bilevel optimization methods. We first summarize the set of sufficient conditions from \citep{ghadimi:2018:approximationmethodsbilevel}, and later note any additional assumptions used by the other methods. The conditions in \citep{ghadimi:2018:approximationmethodsbilevel} on the upper-level function, \lfcnparamsvx, are: \begin{enumerate}[label=A\lfcn\!\arabic*.,leftmargin=2cm,itemsep=0pt,ref=A\lfcn\!\arabic*]
\item
$\forall \params \in \F^\paramsdim$,
$\nabla_\params \lfcn(\params, \vx)$ and
$\nabla_\vx \lfcn(\params, \vx)$
are Lipschitz continuous with respect to \vx,
with corresponding Lipschitz constants
\xmath{L_{\vx,\nabla_\params \lfcn}} and \xmath{L_{\vx,\nabla_\vx \lfcn}}.
(These constants are independent of \vx and \params.)
\label{BA assumption upper-level 1}
\item The gradient with respect to \vx
is bounded, \ie,
\\
$\norm{\nabla_\vx \lfcn(\params, \vx)} \leq \xmath{C_{\nabla_\vx \lfcn}}
,\ \forall \vx \in \F^\sdim
$.
\label{BA assumption upper-level 2}
\item
$\forall \vx \in \F^\sdim$,
$\nabla_\vx \lfcn(\params, \vx)$ is Lipschitz continuous with respect to \params,
with corresponding Lipschitz constant
\xmath{L_{\params,\nabla_\vx \lfcn}}.
\label{BA assumption upper-level end} \end{enumerate}
The conditions in \citep{ghadimi:2018:approximationmethodsbilevel} on the lower-level function, \ofcnargs, are: \begin{enumerate}[label=A\ofcn\!\!\arabic*.,leftmargin=2cm,itemsep=0pt,ref=A\ofcn\!\!\arabic*]
\item \ofcn is continuously twice differentiable in \params and \vx.
\label{BA assumption lower-level 1}
\item
$\forall \params \in \F^\paramsdim$,
$\nabla_\vx \ofcnargs$ is Lipschitz continuous with respect to \vx
with corresponding constant \xmath{L_{\vx,\nabla_\vx \ofcn}}.
\label{BA assumption lower-level 2}
\item
$\forall \params \in \F^\paramsdim$,
$\ofcnargs$ is strongly convex with respect to \vx, \ie,
$\xmath{\mu_{\vx,\ofcn}} \I \preceq \nabla_\vx^2 \ofcn(\params\, ; \vx)$,
for some $\xmath{\mu_{\vx,\ofcn}} > 0$.
\label{BA assumption lower-level 3}
\item
$\forall \params \in \F^\paramsdim$,
$\nabla_{\vx \vx} \ofcnargs$ and $\nabla_{\params \vx} \ofcnargs$
are Lipschitz continuous with respect to \vx with Lipschitz constants \xmath{L_{\vx,\nabla_{\vx\vx}\ofcn}} and \xmath{L_{\vx,\nabla_{\params\vx}\ofcn}}.
\label{BA assumption lower-level 4}
\item The mixed second gradient of \ofcn is bounded, \ie,
\\
$\norm{\nabla_{\params \vx} \ofcnargs} \leq \xmath{C_{\nabla_{\params\vx}\ofcn}},
\quad \forall \params, \vx$.
\label{BA assumption lower-level 5}
\item
$\forall \vx \in \F^\sdim$,
$\nabla_{\params \vx} \ofcnargs$ and $\nabla_{\vx \vx} \ofcnargs$
are Lipschitz continuous with respect to \params
with Lipschitz constants
$\xmath{L_{\params,\nabla_{\params\vx}\ofcn}}$
and
$\xmath{L_{\params,\nabla_{\vx\vx}\ofcn}}$.
\label{BA assumption lower-level end} \end{enumerate}
In addition to the assumptions above on \lfcn and \ofcn, analyses of optimization algorithms for the stochastic bilevel problem assume that (i) all estimated gradients are unbiased and (ii) the variance of the estimation errors is bounded by \xmath{\sigma^2_{\nabla_\params \lfcn}}, \xmath{\sigma^2_{\nabla_\vx \lfcn}}, \xmath{\sigma^2_{\nabla_{\vx} \ofcn}}, \xmath{\sigma^2_{\nabla_{\params\vx} \ofcn}}, and \xmath{\sigma^2_{\nabla_{\vx\vx} \ofcn}}. The stochastic methods discussed here are all based on the minimizer approach to finding the upper-level gradient. Therefore, the methods use estimates of $\nabla_\params \lfcnparamsvx$, $\nabla_\vx \lfcnparamsvx$, $\nabla_\vx \ofcnargs$, $\nabla_{\params,\vx} \ofcnargs$, and $\nabla_{\vx,\vx}\ofcnargs$. We denote the estimates of these gradient using tildes, \eg, $\nablatil_\params \lfcnparamsvx$. Following \eqref{eq: IFT final gradient dldparams}, an estimate of the upper-level gradient approximation is thus \begin{align}
\uppergradhat
&= \nablatil_\params \lfcn(\params, \vx)
- \parenr{\nablatil_{\vx \params} \ofcn(\vx \, ; \params)}'
\parenr{\nablatil_{\vx \vx} \ofcn(\vx \, ; \params)}^{\neg 1}
\nablatil_\vx \lfcn(\params, \vx). \nonumber \end{align} As an example of the bounded variance assumption, \citep{ghadimi:2018:approximationmethodsbilevel} assumes \begin{equation*}
\E{\normrsq{\nabla_\params \lfcnparamsvx - \nablatil_\params \lfcn(\params\, ; \vx)} }
\leq \xmath{\sigma^2_{\nabla_\params \lfcn}}
\quad \forall \vx, \params . \end{equation*}
To consider how the complexity analysis bounds may apply in practice, \apref{sec: ghadimi bounds applied} examines how assumptions \ref{BA assumption upper-level 1}-\ref{BA assumption upper-level end} and assumptions \ref{BA assumption lower-level 1}-\ref{BA assumption lower-level end} apply to the running filter learning example \eqref{eq: bilevel for analysis filters}. Although a few of the conditions are easily satisfied, most are not. \apref{sec: ghadimi bounds applied} shows that the conditions are met if one invokes box constraints on the variables \vx and \params. Although imposing box constraints requires modifying the algorithms, \eg, by including a projection step, the iterates remain unchanged if the constraints are sufficiently generous. However, such generous box constraints are likely to yield large Lipschitz constants and bounds, leading to overly-conservative predicted convergence rates. Further, any differentiable upper-level loss and lower-level cost function would meet the conditions above with such box constraints. Generalizing the following complexity analysis for looser conditions is an important avenue for future work.
\subsection{Double-loop} \label{sec: double-loop complexity analysis}
\citet{ghadimi:2018:approximationmethodsbilevel} were the first to provide a finite-time analysis of the bilevel problem. The authors proposed and analyzed the Bilevel Approximation (BA) method (see \aref{alg: ba}). BA uses two nested loops. The inner loop minimizes the lower-level cost to some accuracy, determined by the number of lower-level iterations; the more inner iterations, the more accurate the gradient will be, but at the cost of more computation and time. The outer loop is (inexact) projected gradient steps on \lfcn. Ref. \citep{ghadimi:2018:approximationmethodsbilevel} used the minimizer result \eqref{eq: IFT final gradient dldparams} (with the IFT perspective for the derivation) to estimate the upper-level gradient.
\begin{algorithm}[t!] \caption{
Bilevel Approximation (BA) Method from \citep{ghadimi:2018:approximationmethodsbilevel}.
The differences for the AID-BiO and ITD-BiO methods from \citep{ji:2021:bileveloptimizationconvergence} are:
(1) when $\upperiter > 0$,
the BiO methods replace \lref{alg: BA line: lower level init}
with \mbox{$\vx^{(0)} = \vx^{(T_{\upperiter-1})}$},
(2) $T_i$ does not vary with upper-level iteration,
(3) the upper-level gradient calculation in \lref{alg: BA line: hypergradient calc}
can use the minimizer approach \eqref{eq: IFT final gradient dldparams}
or backpropagation \eqref{eq: reverse mode},
and
(4) the hyperparameter update is standard gradient descent,
so \lref{alg: BA line: hyperparameter update} becomes
\mbox{$\params^{(\upperiter+1)} = \params^{(\upperiter)} - \ssupper \vg $}. } \label{alg: ba} \input{TablesAndAlgs/alg,ba} \end{algorithm}
\newcommand{\nowidth}[2] {\makebox[0pt][#1]{#2}} \newcommand{\textnw}[1] {\text{\nowidth{c}{#1}}} \newcommand{\textnwl}[1] {\text{\nowidth{l}{#1}}} \newcommand{\textnwr}[1] {\text{\nowidth{r}{#1}}}
To bound the complexity of BA, \citep{ghadimi:2018:approximationmethodsbilevel} first related the error in the lower-level solution to the error in the upper-level gradient estimate as \begin{align*}
\lVert
\underbrace{\nablahat_\params \lfcn (\params, \vx^{(T)})}_{\textnw{Estimated gradient}} -
\underbrace{\nabla_\params \lfcn (\params, \xhat(\params))}_{\text{True gradient}}
\rVert
\leq \xmath{C_{\textsc{GW}}}
\underbrace{\norm{\vx^{(T)} - \xhat(\params)}}_{\textnw{Error in lower-level}}, \end{align*} where \xmath{C_{\textsc{GW}}} is a constant that depends on many of the bounds defined in the assumptions above \citep{ghadimi:2018:approximationmethodsbilevel}. Combing the above error bound with known gradient descent bounds for the accuracy of the lower-level problem yields bounds on the accuracy of the upper-level gradient. The standard lower-level bounds can vary by the specific algorithm (\citep{ghadimi:2018:approximationmethodsbilevel} uses plain GD), but are in terms of $Q_\ofcn = \frac{\xmath{L_{\vx,\nabla_\vx \ofcn}}}{\xmath{\mu_{\vx,\ofcn}}}$ (the \dquotes{condition number} for the strongly convex lower-level function) and the distance between the initialization and the minimizer.
Ref. \citep{ghadimi:2018:approximationmethodsbilevel} shows that $\xhat(\params)$ is Lipschitz continuous in \params under the above assumptions, which intuitively states that the lower-level minimizer does not change too rapidly with changes in the hyperparameters. Further, $\uppergrad$ is Lipschitz continuous in \params with a Lipschitz constant, \xmath{L_{\params,\nabla_\params \lfcn}}, that depends on many of the constants given above.
The main theorems from \citep{ghadimi:2018:approximationmethodsbilevel} hold when the lower-level GD step size is $\sslower = \frac{2}{\xmath{L_{\vx,\nabla_\vx \ofcn}} + \xmath{\mu_{\vx,\ofcn}}}$ and the upper-level step size satisfies $\ssupper \leq \frac{1}{\xmath{L_{\params,\nabla_\params \lfcn}}}$. Then, the distance between the $\upperiter$th loss function value and the minimum loss function value, $\lfcn(\iter{\params}, \xhat(\iter{\params})) - \lfcn(\paramh, \xhat(\paramh))$, is bounded by a constant that depends on the starting distance from a minimizer (dependent on the initialization of \params and \vx), $Q_\ofcn$, \xmath{C_{\textsc{GW}}}, the number of inner iterations, and the upper-level step size. The bound differs for strongly convex, convex, and possibly non-convex upper-level loss functions. \tref{tab: BA deterministic bilevel complexity convexity scale} summarizes the sample complexity required to reach an $\epsilon$-optimal point in each of these scenarios.
\begin{table}[htbp]
\centering
\input{TablesAndAlgs/tab,BAsamplecomplexities}
\iffigsatend \tabletag{5.1} \fi
\caption{
Sample complexity
to reach an $\epsilon$-optimal solution
of the deterministic bilevel problem
using BA \citep{ghadimi:2018:approximationmethodsbilevel},
for various assumptions on the upper-level loss function.
Usually $\lfcn(\params)$ is non-convex
and that case has the worst-case order results.
The complexities show the total number of partial gradients
of the upper-level loss
(equal to the number of lower-level Hessians needed
for estimating \uppergrad using \eqref{eq: IFT final gradient dldparams})
and the partial gradients of the lower-level.
The convex results use the accelerated BA method,
which uses acceleration techniques similar to Nesterov's method
\cite{nesterov:83:amo}
applied to the upper-level gradient step in \aref{alg: ba}.
}
\label{tab: BA deterministic bilevel complexity convexity scale} \end{table}
\citet{ji:2021:bileveloptimizationconvergence} proposed two methods for Bilevel Optimization that improve on the sample complexities from \citep{ghadimi:2018:approximationmethodsbilevel} for non-convex loss functions under similar assumptions. The first, ITD-BiO (ITerative Differentiation), uses the unrolled method for calculating the upper-level gradient (see \sref{sec: unrolled}). The second, AID-BiO (Approximate Implicit Differentiation), uses the minimizer method with the implicit function theory perspective (see \sref{sec: minimizer approach}). \tref{tab: deterministic bilevel complexity} summarizes the sample complexities \citep{ji:2021:bileveloptimizationconvergence}. Much of the computational advantage of ITD-BiO and AID-BiO is in improving the iteration complexity with respect to the condition number \blue{(not shown in the summary table).}
One of the main computational advantages of the AID-BiO and IFT-BiO methods in \citep{ji:2021:bileveloptimizationconvergence} over the BA algorithm \aref{alg: ba} is a warm restart for the lower-level optimization. Although the hyperparameters change every outer iteration, the change is generally small enough that the stopping point of the previous lower-level descent is a better initialization than the noisy data (recall that \citep{ghadimi:2018:approximationmethodsbilevel} showed the lower-level minimizer is Lipschitz continuous in \params). One can account for this warm restart when using automatic differentiation tools (backpropagation) \citep{ji:2021:bileveloptimizationconvergence}. The caption for \aref{alg: ba} summarizes the other differences between BA and the BiO methods.
\begin{table}[htbp]
\centering
\input{TablesAndAlgs/tab,BAvsBiOsamplecomplexity}
\iffigsatend \tabletag{5.2} \fi
\caption{
A comparison of the
finite-time sample complexity
to reach an $\epsilon$-solution
of the deterministic bilevel problem
when the upper-level loss function is non-convex using
BA \citep{ghadimi:2018:approximationmethodsbilevel},
AID-BiO \citep{ji:2021:bileveloptimizationconvergence},
and ITD-BiO \citep{ji:2021:bileveloptimizationconvergence}.
\ordertil{\cdot} = order omits any $\log \epsilon^{\neg1}$ term.
}
\label{tab: deterministic bilevel complexity} \end{table}
The Bilevel Stochastic Approximation (BSA) method replaces the lower-level update in BA (see \aref{alg: ba}) with standard stochastic gradient descent. The corresponding upper-level step in BSA is a projected gradient step with stochastic estimates of all gradients. Another difference in the stochastic versions of the BA \citep{ghadimi:2018:approximationmethodsbilevel} and BiO \citep{ji:2021:bileveloptimizationconvergence} methods is that they use an inverse matrix theorem (based on the Neumann series) to estimate the Hessian inverse. Ref. \citep{ji:2021:bileveloptimizationconvergence} simplifies the inverse Hessian calculation to replace expensive matrix-matrix multiplications with matrix-vector multiplications. This same strategy makes backpropagation more computationally efficient than the forward mode computation for the unrolled gradient; see \apref{sec: foward and backward unrolling}.
\subsection{Single-Loop} \label{sec: single-loop complexity}
Recently, \citep{hong:2020:twotimescaleframeworkbilevel,chen:2021:singletimescalestochasticbilevel} extended the double-loop analysis of \citep{ghadimi:2018:approximationmethodsbilevel,ji:2021:bileveloptimizationconvergence} to single-loop algorithms that alternate gradient steps in \vx and \params.
\aref{alg: ttsa} summarizes the single-loop algorithm TTSA \citep{hong:2020:twotimescaleframeworkbilevel}. The analysis of TTSA uses the same lower-level cost function assumptions as mentioned above for BSA \citep{ghadimi:2018:approximationmethodsbilevel} and one additional upper-level assumption: that \lfcn is weakly convex with parameter $\mu_\lfcn$, \ie, \[
\lfcn(\params+\vdelta) \geq \lfcn(\params) \langle \nabla \lfcn(\params), \vdelta \rangle + \mu_\lfcn \normsq{\vdelta}
,\quad \forall \params, \vdelta \in \R^\paramsdim. \] TTSA assumes the lower-level gradient estimate is still unbiased and that its variance is now bounded as \[
\E{\normrsq{\nabla_\vx \ofcn(\vx, \params) - \nablatil_\vx \ofcn(\vx, \params) }}
\leq \xmath{\sigma^2_{\nabla_{\vx} \ofcn}} \, (1 + \normsq{\nabla_\vx \ofcn(\vx, \params)}). \] Further, the stochastic upper-level gradient estimate, $\nablatil_\params \lfcn(\iter{\params},\iter{\vx}{+1})$, includes a bias that stems from the nonlinear dependence on the lower-level Hessian. This bias decreases as the batch size increases.
The \dquotes{two-timescale} part of TTSA comes from using different upper and lower step size sequences. The lower-level step size is larger and bounds the tracking error (the distance between \xhat and the \vx iterate) as the hyperparameters change (at the upper-level loss's relatively slower rate). Thus, \citep{hong:2020:twotimescaleframeworkbilevel} chose step-sizes such that \mbox{$\ssupper(\upperiter) /\sslower(\upperiter) \rightarrow 0 $}. Specifically, if \lfcn is strongly convex, then \ssupper is $\order{\upperiter^{\neg1}}$ and \sslower is $\order{\upperiter^{\neg2/3}}$. If \lfcn is convex, then \ssupper is $\order{\upperiter^{\neg3/4}}$ and \sslower is $\order{\upperiter^{\neg1/2}}$.
\citet{chen:2021:singletimescalestochasticbilevel} improved the sample complexity of TTSA. By using a single timescale, their algorithm, STABLE, achieves the \dquotes{same order of sample complexity as the stochastic gradient descent method for the single-level stochastic optimization} \citep{chen:2021:singletimescalestochasticbilevel}. However, the improved sample complexity comes at the cost of additional computation per iteration as STABLE can no longer trade a matrix inversion (of size $\paramsdim \by \paramsdim$) for matrix-vector products, as done in the \citep{ji:2021:bileveloptimizationconvergence}. Ref. \citep{chen:2021:singletimescalestochasticbilevel} therefore recommended STABLE when sampling is more costly than computation or when \paramsdim is relatively small.
The analysis of STABLE uses the same upper-level loss and lower-level cost function assumptions as listed above for BSA. Additionally, STABLE assumes that, \xmath{\forall \vx, \, \nabla_\params \lfcnparamsvx} is Lipschitz continuous in \params. This condition is easily satisfied as many upper-level loss functions do not regularize \params. Further, those that do often use a squared 2-norm, \ie, Tikhonov-style regularization, that has a Lipschitz continuous gradient. Additionally, rather than bounding the gradient norms as in assumptions \ref{BA assumption upper-level 2} and \ref{BA assumption lower-level 5}, \citep{hong:2020:twotimescaleframeworkbilevel} assumes the following moments are bounded: \begin{itemize}[noitemsep,topsep=0pt]
\item the second and fourth moment of $\nabla_\params \lfcnparamsvx$ and $\nabla_\vx \lfcnparamsvx$
and
\item the second moment of $\nabla_{\params \vx} \ofcnargs$ and $\nabla_{\vx \vx} \ofcnargs$, \end{itemize} ensuring that the upper-level gradient is Lipschitz continuous.
Like the previous algorithms discussed, STABLE evaluates the minimizer result \eqref{eq: IFT final gradient dldparams} at non-minimizer lower-level iterates, $\vx^{(T)}(\params^{(\upperiter)})$, to estimate the hyperparameter gradient. However, it differs in how it estimates and uses the gradients. STABLE replaces the upper-level gradient in TTSA \lref{alg: ttsa line: upper-level gradient calc} with \begin{align} \vg = \nabla_\params \iter{\lfcn}
-
\underbrace{\parenr{ \iter{\xmath{\Delta}_{\vx \params}} }'}_{\mathllap{
\text{Prev. }
\nablatil_{\vx \params} \iter{\ofcn} \!\!\!\!\!\!\!\!\!\!\!}
}
\underbrace{ (\iter{\xmath{\Delta}_{\vx \vx}}}
_{\mathrlap{\hspace{-.2cm} \text{Prev. }
\nablatil_{\vx \vx} \iter{\ofcn} }}
)^{\neg 1}
\nabla_\vx \iter{\lfcn}. \label{eq: stable gradient} \end{align} \blue{Taking inspiration from variance reduction techniques for single-level optimization problems, \eg, \citep{nguyen:2017:sarahnovelmethod},} STABLE recursively updates the newly defined matrices as follows: \begin{align*}
\iter{\xmath{\Delta}_{\vx \params}} &=
\mathcal{P}_{ \norm{\xmath{\Delta}} \leq \xmath{C_{\nabla_{\params\vx}\ofcn}} } \left(
(1-\tau_\upperiter)
\underbrace{( \iter{\xmath{\Delta}_{\vx \params}}{-1} - \nablatil_{\vx \params} \iter{\ofcn}{-1})}_{\text{Recursive update}}
+
\underbrace{\nablatil_{\vx \params} \iter{\ofcn}}_{\text{New estimate}}
\right) \\
\iter{\xmath{\Delta}_{\vx \vx}} &=
\mathcal{P}_{\xmath{\Delta} \psd \xmath{\mu_{\vx,\ofcn}} \I } \left(
(1-\tau_\upperiter)
\overbrace{(\iter{\xmath{\Delta}_{\vx \vx}}{-1}-\nablatil_{\vx \vx} \iter{\ofcn}{-1})}
+
\overbrace{\nablatil_{\vx \vx} \iter{\ofcn}}
\right) .\end{align*} In the $\iter{\xmath{\Delta}_{\vx \params}}$ update, the projection onto the set of matrices with a maximum norm helps ensure stability by not allowing the gradient to get too large. The projection in the $\iter{\xmath{\Delta}_{\vx \vx}}$ update is an eigenvalue truncation that ensures positive definiteness of the estimated Hessian in this Newton-based method. After computing the gradient \vg \eqref{eq: stable gradient}, the upper-level update is a standard descent step as in \aref{alg: ttsa} \lref{alg: ttsa line: upper-level step}.
STABLE \citep{chen:2021:singletimescalestochasticbilevel} also uses the recursively estimated gradient matrices in the lower-level cost function descent. It replaces the standard gradient descent step in \aref{alg: ttsa} \lref{alg: ttsa line: lower-level update} with one that uses second order information: \begin{align*}
\iter{\vx}{+1} &= \iter{\vx} -
\underbrace{\sslower(\upperiter) \nablatil_\vx \ofcn(\iter{\vx}; \iter{\params})}_{\text{Standard GD step}}
- \underbrace{(\xmath{\Delta}^{(\upperiter)}_{\vx \vx})^{\neg 1} (\xmath{\Delta}^{(\upperiter)}_{\params \vx})'(\iter{\vx}{+1} - \iter{\vx})}_{\text{New term}}. \end{align*} With these changes, STABLE is able to reduce the iteration complexity relative to TTSA as summarized in \tref{tab: stochastic complexity summaries}.
\section{Summary of Methods}
There are many variations of gradient-based methods for optimizing bilevel problems, especially when one considers that many of the upper-level descent strategies can work with either the minimizer or unrolled approach discussed in \cref{chap: ift and unrolled}. There is no clear single \dquotes{best} algorithm for all applications; each algorithm involves trade-offs.
Building on the minimizer and unrolled methods for finding the upper-level gradient with respect to the hyperparameters, \uppergrad, double-loop algorithms are an intuitive approach. Although optimizing the lower-level problem every time one takes a gradient step in \params is computationally expensive, the lower-level problem is is embarrassingly parallelizable across samples. Specifically, one can optimize the lower-level cost for each training sample independently before averaging the resulting gradients to take an upper-level gradient step. In the typical scenario when training is performed offline, training wall-time can therefore be dramatically reduced by using multiple processors.
Single-loop algorithms remove the need to optimize the lower-level cost function multiple times. The single-loop algorithms that consider a system of equations often accelerate convergence using Newton solvers \citep{kunisch:2013:bileveloptimizationapproach,calatroni:2017:bilevelapproacheslearning}. However, the optimality system grows quickly when there are multiple training images, and may become too computationally expensive as \Ntrue increases \citep{chen:2014:insightsanalysisoperator}. \blue{Another type of single-loop algorithm uses alternating gradient steps in \vx and \params \citep{hong:2020:twotimescaleframeworkbilevel,chen:2021:singletimescalestochasticbilevel}. Although each method has slight variations (such as whether it uses momemtum), these single-loop methods are generally equivalent to considering $T=1$ in the double-loop methods.}
This section organized algorithms based on the number of for-loops; double-loop algorithms have two loops while single-loop algorithms have one \footnote{As noted at the start of the section, this loop counting does not include the loop in CG or in backpropagation.}. However, \blue{there are many other ways in which bilevel optimization methods differ and} not all methods fall cleanly into one group. One such example is the Penalty method \citep{mehra:2021:penaltymethodinversionfree}. The Penalty method forms a single-level, constrained optimization problem, with the constraint that the gradient of the lower-level cost function should be zero, $\nabla_\vx \ofcnargs = \vzero$. (This step is similar to the derivation of the minimizer approach via KKT conditions; see \sref{sec: minimizer via kkt}.) Rather than forming the Lagrangian as in \eqref{eq: lagrangian}, \citep{mehra:2021:penaltymethodinversionfree} penalizes the norm of the gradient, with increasing penalties as the upper iterations increase. Thus, the Penalty cost function \footnote{
This is a simplification;
\citep{mehra:2021:penaltymethodinversionfree}
allows for constraints on \vx and \params. } at iteration \upperiter is \begin{equation*}
p(\params \, , \vx) = \lfcnargs + \iter{\lambda} \normsq{\nabla_\vx \ofcnargs}_2 .\end{equation*} The penalty variable sequence, ${\iter{\lambda}}$, must be positive, non-decreasing, and divergent ($\iter{\lambda} \rightarrow \infty$).
Penalty \citep{mehra:2021:penaltymethodinversionfree} incorporates elements of both double-loop and single-loop algorithms. Similar to the double-loop algorithms, Penalty takes multiple gradient descent steps in the lower-level optimization variable, \vx, before calculating and updating the hyperparameters. However, Penalty forms a single-level optimization problem that could be optimized using techniques such as those used in single-loop algorithms.
\blue{ Another variant on a double-loop bilevel optimization method is to optimize a lower-level surrogate function $\tilde{\ofcn}(\vx \, ; \, \iter{\params})$ instead of optimizing $\ofcn(\vx \, ; \, \iter{\params})$. For example, \citep{hoeltgen:2013:optimalcontrolapproach} replaces \ofcn with its first-order approximation around the current solution point $(\iter{\params}, \, \xhat(\iter{\params}))$. Because this approximation is only reliable in the neighborhood of $(\iter{\params}, \, \xhat(\iter{\params}))$, \citep{hoeltgen:2013:optimalcontrolapproach} adds the proximal term $\lambda \normrsq{\params - \iter{\params}}$ to the upper-level loss function at each outer iteration, where $\lambda$ is a positive tuning parameter.}
The finite-time complexity analyses \citep{ghadimi:2018:approximationmethodsbilevel,ji:2021:bileveloptimizationconvergence,hong:2020:twotimescaleframeworkbilevel,chen:2021:singletimescalestochasticbilevel,yang:2021:provablyfasteralgorithms} justify the use of gradient-based bilevel methods for problems with many hyperparameters, as none of the sample complexity bounds involved the number of hyperparameters. This is in stark contrast with the hyperparameter optimization strategies in \cref{chap: hpo}. However, the per-iteration cost for bilevel methods is still large and increasing with the hyperparameter dimension. Further, the conditions on the lower-level cost function \ref{BA assumption lower-level 1}-\ref{BA assumption lower-level end} seem restrictive and may not be satisfied in practice. Complexity analysis based on more relaxed conditions could be very valuable.
\blue{ Because of the restrictive conditions in the complexity analysis, it is generally infeasible to compute theoretically justified step-sizes and other algorithm parameters in the single-loop and double-loop methods \citep{ghadimi:2018:approximationmethodsbilevel,ji:2021:bileveloptimizationconvergence,hong:2020:twotimescaleframeworkbilevel,chen:2021:singletimescalestochasticbilevel,yang:2021:provablyfasteralgorithms}. Thus, one must often resort to grid searches or use heuristics, such as those discussed in \sref{sec: double-loop design decisions}, to select these algorithm parameters. Ref. \citep{yang:2021:provablyfasteralgorithms} comments on one example of how empirical practice can differ from theory. Although their theory requires that the number of iterates of the Neumann series used to approximate the inverse Hessian matrix grows with the desired solution accuracy, the authors found that using a few iterates was sufficient (and faster) in practice. }
Gradient-based and other hyperparameter optimization methods are active research areas, and the trade-offs continue to evolve. Although it currently seems that gradient-based bilevel methods make sense for problems with many hyperparameters, new methods may overtake or combine with what is presented here. For example, \blue{many} bilevel methods (and convergence analyses thereof) use classical gradient descent for the lower-level optimization algorithm, whereas \citep{kim:2017:convergenceanalysisoptimized} showed that the Optimized Gradient Method (OGM) has better convergence guarantees and is optimal among first-order methods for smooth convex problems \cite{drori:17:tei}. These advances provide opportunities for further acceleration of bilevel methods.
\chapter{Survey of Applications} \label{chap: applications}
Bilevel methods have been used in many image reconstruction applications, including 1D signal denoising \citep{peyre:2011:learninganalysissparsity}, image denoising (see following sections), compressed sensing \citep{chen:2021:learnabledescentalgorithm}, spectral CT image reconstruction \citep{sixou:2020:adaptativeregularizationparameter}, and MRI image reconstruction \citep{chen:2021:learnabledescentalgorithm}. \blue{Bilevel methods are also used for classification problems. For example, \citep[Sec. 6]{nowozin:2011:structuredlearningprediction} shows how the structured support vector machine (SSVM) is a convex surrogate for the bilevel model when the lower-level cost is linear in \params.} This section discusses trends and highlights specific applications to provide concrete examples of bilevel methods for image reconstruction.
Many papers present or analyze bilevel optimization methods for general upper-level loss functions and lower-level cost functions, under some set of assumptions about each level. \crefs{chap: ift and unrolled}{chap: bilevel methods} summarized many of these methods. Although there are cases when the choice of a loss function and/or cost impacts the optimization strategy, many bilevel problems could use any optimization method. Thus, this section concentrates on the specific applications, rather than methodology.
This section is split into a discussion of lower-level cost and upper-level loss functions. (Lower-level cost functions that involve CNNs are discussed separately; see \sref{sec: connections unrolled}.) The conclusion section discusses examples where the loss function is tightly connected to the cost function.
\section{Lower-level Cost Function Design} \label{sec: prev results lower level}
Once a bilevel problem is optimized to find \paramh, the learned parameters are typically deployed in the same lower-level problem as used during training but with new, testing data. Thus, it is the lower-level cost function that specifies the application of the bilevel problem, \eg, CT image reconstruction or image deblurring.
Denoising applications consider the case where the forward model is an identity operator ($\mA=\I$). This case has the simplest possible data-fit term in the cost function and requires the least amount of computation when computing gradients or evaluating \ofcn. Because bilevel methods are generally already computationally expensive, it is unsurprising that many papers focus on denoising, even if only as a starting point towards applying the proposed bilevel method to other applications.
More general image reconstruction problems consider non-identity forward models. Few papers learn parameters for image reconstruction in the fully task-based manner described in \eqref{eq: generic bilevel upper-level}, likely due to the additional computational cost. Some papers, \eg, \cite{kobler:2021:totaldeepvariation,chen:2014:insightsanalysisoperator,chambolle:2021:learningconsistentdiscretizations} consider learning parameters for denoising, and then apply \paramh in a reconstruction problem with the same regularizer but introducing the new \mA to the data-fit term. These \dquotes{crossover experiments} \citep{chambolle:2021:learningconsistentdiscretizations} test the generalizability of the learned parameters, but they sacrifice the specific task-based nature of the bilevel method.
\blue{ Recall from \cref{chap: image recon} that the regularizer (with its learned parameters) can be related to a prior for \vx in a maximum \textit{a posteriori} probability perspective. If this perspective is valid, then the \paramh should generalize to other system matrices. However, the exact connection between the regularizer and the probability distribution is not straight-forward \citep{nikolova:2007:modeldistortionsbayesian} and previous results suggest that \paramh varies with different \mA's \citep{chambolle:2021:learningconsistentdiscretizations,effland:2020:variationalnetworksoptimal}. Further, \mA often is an imperfect model for the true underlying phenomena and \paramh may end up compensating for modeling errors that are specific to a given \mA, and thus may not generalize to other imaging system models. }
Many bilevel methods, especially in image denoising \citep{peyre:2011:learninganalysissparsity,fehrenbach:2015:bilevelimagedenoising,samuel:2009:learningoptimizedmap,kunisch:2013:bileveloptimizationapproach,chen:2014:insightsanalysisoperator}, but also in image reconstruction \citep{holler:2018:bilevelapproachparameter}, use the same or a very similar lower-level cost as the running example in this review. From \sref{sec: bilevel set-up}, the running example cost function is: \begin{equation}
\xhat(\params, \vy) = \argmin_\vx \overbrace{\onehalf \norm{\mA \vx-\vy}^2_2 + \ebeta{0}
\underbrace{\sum_{k=1}^K \ebeta{k} \mat{1}' \sparsefcn(\xmath{\vc_k} \conv \vx; \epsilon)}_{R(\vx \, ; \params)}
}^{\ofcnargs}
\label{eq: lower-level repeat 2} .\end{equation} The learned hyperparameters, \params, include the tuning parameters, $\beta_k$ and/or the filter coefficients, \xmath{\vc_k}. The image reconstruction example in \citep{holler:2018:bilevelapproachparameter} generalized \eqref{eq: lower-level repeat 2} for implicitly defined forward models by using a different data-fit term, as given in \eqref{eq: holler lower level}. Their two example problems involve learning parameters to estimate the diffusion coefficient or forcing function in a second-order elliptic partial differential equation.
Two common variations among applications using \eqref{eq: lower-level repeat 2} are (1) the choice of which tuning parameters to learn and (2) what sparsifying function, \sparsefcn, to use. Some methods \citep{kunisch:2013:bileveloptimizationapproach,fehrenbach:2015:bilevelimagedenoising,holler:2018:bilevelapproachparameter} learn only the tuning parameters; these methods typically use finite differencing filters or discrete cosine transform (DCT) filters (excluding the DC filter) as the \xmath{\vc_k}'s. Other methods learn only filter coefficients \citep{peyre:2011:learninganalysissparsity}. \blue{\fref{fig: cameraman learned filters} shows filters learned from patches of the \dquotes{cameraman} image when $\params = (\vbeta, \vc)$ and shows filter strengths when $\params=\vbeta$. The corresponding bilevel problem is \eqref{eq: bilevel for analysis filters} with \sparsefcn given in \eqref{eq: corner rounded 1-norm}. \fref{fig: cameraman example results} shows the corresponding denoised image and \apref{sec: cameraman training details} describes the experiment settings and additional results.}
\begin{figure}
\caption{The DCT filter bank
and example learned filters
for \eqref{eq: bilevel for analysis filters}
with training data from the \dquotes{cameraman} image.
(a) The 48 non-constant $7\by7$ DCT filters
used to initialize \params.
The dark, top-left square represents the removed DC filter.
(b) The DCT filters multiplied by their respective tuning parameter $\beta_k$ when $\params=\vbeta$.
The range of $e^{\beta_0 + \beta_k}$ is 0.001-1.08.
The learned tuning parameters emphasize the higher-frequency DCT filters.
(c) Learned filters when $\params=(\vbeta, \vc)$
(scaled to have unit-norm for visualization).
}
\label{fig: cameraman learned filters}
\end{figure}
A slight variation on learning the filters is to learn coefficients for a linear combination of filter basis elements \citep{samuel:2009:learningoptimizedmap,chen:2014:insightsanalysisoperator}, \ie, learning $a_{k,i}$ where \[
\xmath{\vc_k} = \sum_i a_{k,i} \vb_i ,\] for some set of basis filter elements, $\vb_i$. One benefit of imposing a filter basis is the ability to ensure the filters lie in a given subspace. For example, \citep{samuel:2009:learningoptimizedmap,chen:2014:insightsanalysisoperator} use the DCT as a basis and remove the constant filter so that all learned filters are guaranteed to have zero-mean.
\begin{figure}
\caption{Example denoising results
for the full \dquotes{cameraman} test image
and two of the training patches.
(a) Noiseless training \dquotes{cameraman} test image.
(b) Noisy image and its SNR.
(c) Denoised image using the learned tuning parameters
that weight the DCT filters as shown in
\fref{fig: cameraman learned filters}b.
(d) Denoised image using the learned filter coefficients and tuning parameters as shown in \fref{fig: cameraman learned filters}c.
For comparison,
the denoised image using BM3D \citep{dabov:2007:imagedenoisingsparse}
has a SNR of 26.87.
See \apref{sec: cameraman training details}
for more details.
}
\label{fig: cameraman example results}
\end{figure}
In terms of sparsifying functions, \citep{peyre:2011:learninganalysissparsity,fehrenbach:2015:bilevelimagedenoising} used the same corner rounded 1-norm as in \eqref{eq: corner rounded 1-norm}, \citep{samuel:2009:learningoptimizedmap} used $\sparsefcn = \log{1+z^2}$ to relate their method to the Field of Experts framework \citep{roth:2005:fieldsexpertsframework}, \citep{holler:2018:bilevelapproachparameter} used a quadratic penalty, and \citep{kunisch:2013:bileveloptimizationapproach,chen:2014:insightsanalysisoperator} both consider multiple \sparsefcn options to examine the impact of non-convexity in \sparsefcn. Ref. \citep{kunisch:2013:bileveloptimizationapproach} compared $p$-norms, $\norm{\xmath{\vc_k} \conv \vx}_p^p$, for $p \in \{\onehalft, 1, 2\}$, where the $p=\onehalft$ and $p=1$ cases are corner-rounded to ensure \sparsefcn is smooth. (The $p=\onehalft$ case is non-convex.) Ref. \citep{chen:2014:insightsanalysisoperator} compared the convex corner-rounded 1-norm in \eqref{eq: corner rounded 1-norm} with two non-convex choices: the log-sum penalty $\log{1+z^2}$, and the Student-t function $\log{10\epsilon + \sqrt{z^2+\epsilon^2}}$.
Both \citep{kunisch:2013:bileveloptimizationapproach,chen:2014:insightsanalysisoperator} found that non-convex penalty functions led to denoised images with better (higher) PSNR. They hypothesize that the improvement is due to the non-convex penalty functions better matching the heavy-tailed distributions in natural images. As further evidence of the importance of non-convexity, \citep{chen:2014:insightsanalysisoperator} found that untrained $7 \by 7$ DCT filters (excluding the constant filter) with learned tuning parameters and a non-convex \sparsefcn outperformed learned filter coefficients with a convex \sparsefcn, despite the increased data adaptability when learning filter coefficients. The trade-off for using non-convex penalty functions is the possibility of local minimizers of the lower-level cost.
\begin{figure}
\caption{Example denoising results
from \citep{chen:2014:insightsanalysisoperator}
comparing filters learned using bilevel methods
to other denoising methods.
(a) The original image \xtrue.
(b) The noisy image \vy.
(c-d) Denoised images using
FoE \citep{roth:2005:fieldsexpertsframework},
BM3D \citep{dabov:2007:imagedenoisingsparse},
and a bilevel approach
using a set-up equivalent to
\eqref{eq: bilevel for analysis filters}
with a non-convex penalty function,
$\sparsefcn(z) = \log{1+z^2}$
\citep{chen:2014:insightsanalysisoperator}.
The PSNR values in dB are given in parenthesis.
\copyright
2014 IEEE. Reprinted, with permission, from \citep{chen:2014:insightsanalysisoperator}.
}
\label{fig: example results}
\end{figure}
\citet{chen:2014:insightsanalysisoperator} also investigated how the number of learned filters and the size of the filters impacted denoising PSNR. They concluded that increasing the number of filters to achieve an over-complete filter set may not be worth the increased computational expense and that increasing the filter size past $11 \by 11$ is unlikely to improve PSNR. Using 48 filters of size $7 \by 7$ and the log-sum penalty function, \citep{chen:2014:insightsanalysisoperator} achieved denoising results on natural images comparable to algorithms such as BM3D \citep{dabov:2007:imagedenoisingsparse}, as seen in \fref{fig: example results}. Although results will vary between applications and training data sets, the results from \citep{chen:2014:insightsanalysisoperator} provide motivation for filter learning and an initial guide for designing bilevel methods.
In addition to variations on the running example for \ofcn \eqref{eq: lower-level repeat 2}, a common regularizer for the lower-level cost is Total Generalized Variation \blue{with order 2} (TGV$^2$) \citep{bredies:2010:totalgeneralizedvariation}. Whereas TV encourages images to be piece-wise constant, TGV$^2$ is a generalization of TV designed for piece-wise linear images.
Another generalization of TV
for piece-wise linear images
is Infimal Convolutional Total Variation (ICTV) \citep{chambolle:1997:imagerecoverytotal}.
Bilevel papers that investigate ICTV include \citep{delosreyes:2017:bilevelparameterlearning,calatroni:2017:bilevelapproacheslearning};
these papers also investigate TGV$^2$.
See \citep{benning:2013:higherordertvmethods}
for a comparison of the two.
TGV cost functions are typically expressed in the continuous domain, at least initially, but then discretized for implementation, \eg, \cite{knoll:11:sot,setzer:11:icr}. One discrete approximation of the TGV$^2$ regularizer is: \begin{align*}
R_{\mathrm{TGV}}(\vx) = \min_{\vz} \ebeta{1} \norm{\xmath{\vc_{\mathrm{TV}}} \conv \vx - \vz}_1 + \ebeta{2} \norm{\partial \vz}_1 ,\end{align*} where \xmath{\vc_{\mathrm{TV}}} is a filter that takes finite differences and $\partial$ is a filter that approximates a symmetrized gradient. In TV, one usually thinks of \vz as a sparse vector; here \vz is a vector whose finite differences are sparse, so \vz is approximately piece-wise constant. Encouraging \vz to be piece-wise constant in turn makes \vx approximately piece-wise linear, since $\xmath{\vc_{\mathrm{TV}}} \conv \vx \approx \vz$ from the first term. Bilevel methods for learning $\beta_1$ and $\beta_2$ for the TGV$^2$ regularizer include \citep{delosreyes:2017:bilevelparameterlearning,calatroni:2017:bilevelapproacheslearning}. An extension to the TGV$^2$ regularizer model is to learn a space-varying tuning parameter~ \citep{hintermuller:2020:dualizationautomaticdistributed}.
As an example of how the regularizer should be chosen based on the application, \citep{hintermuller:2020:dualizationautomaticdistributed} found that standard TV with a learned tuning parameter performed best (in terms of SSIM) for approximately piece-wise constant images while TGV$^2$ with learned tuning parameters performed best for approximately piece-wise linear images.
\section{Upper-Level Loss Function Design} \label{sec: prev results loss function}
From some of the earliest bilevel methods, \eg, \citep{haber:2003:learningregularizationfunctionals,peyre:2011:learninganalysissparsity}, to some of the most recent bilevel methods, \eg, \citep{kobler:2021:totaldeepvariation,antil:2020:bileveloptimizationdeep}, square error or mean squared error (MSE) remains the most common upper-level loss function. In the unsupervised setting, \citep{zhang:2020:bilevelnestedsparse,deledalle:2014:steinunbiasedgradient} used SURE (an estimate of the MSE, see \sref{sec: loss function design}) as the upper-level loss function. Unlike many perceptually motivated image quality measures, MSE is convex in \vx and it is easy to find \dx{\lfcnargs}. However, MSE does not capture perceptual quality nor image utility (see \sref{sec: loss function design}). This section discusses a few bilevel methods that used different loss functions.
Ref. \citep{delosreyes:2017:bilevelparameterlearning} compared a squared error upper-level loss function with a Huber (corner rounded 1-norm) loss function. The corresponding lower-level problem was a denoising problem with a standard 2-norm data-fit term and three different options for a regularizer: TV, TGV$^2$, and ICTV. The authors learned tuning parameters for a natural image dataset using both upper-level loss function options for each of the lower-level regularizers.
Since SNR is equivalent to MSE, the MSE loss will always perform the best according to any SNR-based metric (assuming the bilevel model is well-trained). However, \citep{delosreyes:2017:bilevelparameterlearning} found the tuning parameters learned using the Huber loss yielded denoised images with better qualitative properties and better SSIM, especially at low noise levels. Like MSE, the Huber loss operates point-wise and is easy to differentiate. Thus, the authors conclude that the Huber loss is a good trade-off between tractability and improving on MSE as an image quality measure.
A set of loss functions in \citep{fehrenbach:2015:bilevelimagedenoising, hintermuller:2020:dualizationautomaticdistributed, sixou:2020:adaptativeregularizationparameter} consider the unsupervised or \dquotes{blind} bilevel setting, where one wishes to reconstruct an image without clean samples. Therefore, rather than using an image quality metric that compares a reconstructed image, \xhat, to some true image, \xtrue, these loss function consider the estimated residual, \[
\hat{\vn}
= \hat{\vn}(\params)
= \mA \xhat(\params) - \vy, \] where \params is learned using only noisy data. Unsupervised bilevel methods may be beneficial when there is no clean data and one has more knowledge of noise properties than of expected image content. All three methods \citep{fehrenbach:2015:bilevelimagedenoising, hintermuller:2020:dualizationautomaticdistributed, sixou:2020:adaptativeregularizationparameter} assume the noise variance, $\sigma^2$, is known.
The earliest example \citep{fehrenbach:2015:bilevelimagedenoising}, learned tuning parameters \params such that $\hat{\vn}$ matched the second moment of the assumed Gaussian distribution for the noise. Their lower-level cost is comparable to \eqref{eq: bilevel for analysis filters}, but re-written in terms of \vn and with pre-defined finite differencing or $5 \by 5$ DCT filters, \ie, they learn only the tuning parameters, $\beta_k$. Their upper-level loss encourages the empirical variances of the noise in different frequency bands to match the expected variances: \begin{align*}
\lfcn(\params \, ; \vn(\params)) = \onehalf \sum_i \frac{\left( \normsq{\vf_i \conv \vn}_{2} - \mu_i \right)^2 }{v_i} \\
\mu_i = \E{\normsq{\vf_i \conv \vn}_2} \text{ and } v_i = \text{Var}\left[\normsq{\vf_i \conv \vn}_2 \right], \end{align*} where $\vf_i$ are predetermined filters that select specific frequency components. By using bandpass filters that partition Fourier space, the corresponding means and variances of the second moments of the filtered noise are easily computed, with \begin{align*}
\mu_i = \sdim \sigma^2 \normsq{\vf_i}
\quad \text{ and } \quad
v_i = \sdim \sigma^4 \norm{\vf_i}^4 .\end{align*} Although the experimental results are promising, \citep{fehrenbach:2015:bilevelimagedenoising} does not claim state-of-the-art results since their lower-level denoiser is relatively simple.
As an alternative to the Gaussian-inspired approach in \citep{fehrenbach:2015:bilevelimagedenoising}, \citep{hintermuller:2020:dualizationautomaticdistributed} and \citep{sixou:2020:adaptativeregularizationparameter} use loss functions that penalize noise outside a set \dquotes{noise corridor.} Both methods learn space-varying tuning parameters, and the upper-level loss consists of a data-fit term (that measures noise properties) and a regularizer on \params. The data-fit term in the upper-level loss function in \citep{fehrenbach:2015:bilevelimagedenoising} defines the noise corridor between a maximum variance, $\Bar{\sigma}^2$, and a minimum variance, $\underline{\sigma}^2$: \begin{align}
\bmath{1}' &F.\left(\vw \odot (\vn(\params) \odot \vn(\params))\right) \text{ for } \nonumber \\
&F(n) =
\onehalf \text{max}(n - \Bar{\sigma}^2, 0)^2
+
\onehalf \text{min}(n - \underline{\sigma}^2, 0)^2 \label{eq: noise corridor} ,\end{align} where \vw is a predetermined weighting vector. The noise corridor function, $F(n)$, penalizes any noise outside of the expected range as shown in \fref{fig: noise corridor plot}. Ref. \citep{sixou:2020:adaptativeregularizationparameter} uses the same noise corridor function, but extends the bilevel method for images with Poisson noise; \citep{sixou:2020:adaptativeregularizationparameter} thus estimates the noisy image using the Kullback-Leibler distance. In addition to the noise corridor function as the data-fit component of the upper-level loss function, \citep{hintermuller:2020:dualizationautomaticdistributed,sixou:2020:adaptativeregularizationparameter} include a smoothness-promoting regularizer on \params, which is a spatially varying tuning parameter vector in both methods.
\begin{figure}
\caption{Noise corridor function \eqref{eq: noise corridor}
used as part of the upper-level loss function
for the unsupervised bilevel method in
\citep{hintermuller:2020:dualizationautomaticdistributed}.}
\label{fig: noise corridor plot}
\end{figure}
The task-based nature of bilevel typically makes regularizers or constraints on \params unnecessary (see \sref{sec: filter constraints} for common options for other forms of learning). However, there are two general cases where a regularizer on \params is useful in the upper-level loss function. First, a regularizer can help avoid over-fitting when the amount of training data is insufficient for the number of learnable hyperparameters. This is often the case when learning space-varying parameters that have similar dimensions as the input data, \eg, \citep{haber:2003:learningregularizationfunctionals,delia:2020:bilevelparameteroptimization, hintermuller:2020:dualizationautomaticdistributed, sixou:2020:adaptativeregularizationparameter}. In such cases, the regularization often takes the form of a 2-norm on the learned hyperparameters, $\normsq{\params}_2$.
Second, some problems require application-specific constraints, \blue{\eg, \citep{chambolle:2021:learningconsistentdiscretizations} incorporates constraints in the upper-level loss to ensure that the learned parameters are valid interpolation kernels.} Many other hyperparameter constraints do not require a regularization term, For example, non-negativity constraints on tuning parameters are easily handled by redefining the tuning parameter in terms of an exponential, as in \eqref{eq: bilevel for analysis filters}, and box constraints are common and easy to incorporate with a projection step if using a gradient-based method. Constraints that require sparsity on the learned parameters may benefit from regularization in the upper-level loss function.
An example of an application-specific constraint is found in \citep{ehrhardt:2021:inexactderivativefreeoptimization,sherry:2020:learningsamplingpattern}, which consider MRI reconstruction with a data-fit term and a variational regularizer. Both papers extend the bilevel model in \eqref{eq: bilevel for analysis filters} to include part of the forward model in the learnable parameters, \params. Specifically, \citep{ehrhardt:2021:inexactderivativefreeoptimization,sherry:2020:learningsamplingpattern} learned the sparse sampling matrix for MRI. (Ref.~\citep{sherry:2020:learningsamplingpattern} additionally learns tuning parameters for predetermined filters, whereas \citep{ehrhardt:2021:inexactderivativefreeoptimization} sets the tuning parameters and filters and learns only the sampling matrix.) Here, the forward model is \[
\mA =
\mathrm{diag}\Big( \underbrace{s_1, s_2, \ldots , s_\ydim }_{\vs(\params)} \Big)
\mF ,\] where \mF is the DFT matrix and $s_i$ are learned binary values that specify whether a frequency location should be sampled.
The motivation for learning a sparse sampling matrix comes from the lower-level MRI reconstruction problem; designing more effective sparse sampling patterns in MRI can decrease scan time and thus improve patient experience, decrease cost, and decrease artifacts from patient movement. This goal requires the learned parameters, $s_i$, to be binary, which in turn influences the upper-level loss function design. Thus, \citep{ehrhardt:2021:inexactderivativefreeoptimization,sherry:2020:learningsamplingpattern} include regularization in the upper-level to encourage \vs to be sparse, \eg, \citep{sherry:2020:learningsamplingpattern} uses an upper-level loss with a squared error term and regularizer on \vs: \begin{equation}
\lfcnargs = \normrsq{\xhatp - \xtrue}_2 + \lambda \sum_i \left(s_i + s_i (1-s_i) \right)
\label{eq:binary-s-regularizer} ,\end{equation} where $\lambda$ is a upper-level tuning parameter that one must set manually. (In experiments, they thresholded the learned $s_i$ values to be exactly binary.) An alternative approach is to constrain the number of samples \cite{gozcu:18:lbc}, though that formulation requires other optimization methods.
\section{Conclusion}
This section split the discussion of lower-level cost and upper-level loss functions to discuss trends in both areas. However, when designing a bilevel problem, design decisions can impact both levels. For example, the unsupervised nature of \citep{fehrenbach:2015:bilevelimagedenoising,sixou:2020:adaptativeregularizationparameter} clearly impacted their choice of upper-level loss function to use noise statistics rather than squared error calculated with ground-truth data. Since it can be challenging to learn many good parameters from noisy training data, the unsupervised nature also likely impacted the authors' decision to learn only tuning parameters and set the filters manually. Another example of coupling between lower-level and upper-level design is when one enforces application-specific constraints on the learned parameters, \eg, using a regularizer like \eqref{eq:binary-s-regularizer} in the upper-level loss to promote sparsity of the MRI sampling matrix \citep{ehrhardt:2021:inexactderivativefreeoptimization,sherry:2020:learningsamplingpattern}.
In addition to design decisions influencing both levels, bilevel methods may adopt common techniques for the upper-level loss function and lower-level cost function. For example, a common theme is the tendency to use smooth functions, such as replacing the 1-norm with a corner-rounded 1-norm. This approach requires setting a smoothing parameter, \eg, $\epsilon$ in \eqref{eq: corner rounded 1-norm}, which in turn impacts the Lipschitz constant and optimization speed. More accurate approximations generally lead to larger Lipschitz constants and slower convergence. One approach to trading-off the accuracy of the smoothing with optimization speed is to use a graduated approach and approximate the non-smooth term more and more closely as the optimization progresses \citep{chen:2021:learnabledescentalgorithm}.
The prevalence of smoothing is unsurprising considering that this review focuses on gradient-based bilevel methods. Rare exceptions include \citep{mccann:2020:supervisedlearningsparsitypromoting,ghosh:2021:bilevellearningl1regularizers}, which used the (not corner-rounded) one-norm to define \sparsefcn to learn convolutional filters using the translation to a single level approach described in \sref{sec: translation to a single level}. The impact of smoothing and how accurately one should approximate a non-differentiable point remains an open question.
From an image quality perspective, ideally one would independently design the lower-level cost function and upper-level training loss. The lower-level cost would depend on the imaging physics and would incorporate regularizers that expected to provide excellent image quality when tuned appropriately, and the upper-level loss would use terms that are meaningful for the imaging tasks of interest. As we have seen, in practice one often makes compromises to facilitate optimization and reduce computation time.
\chapter{Connections and Future Directions \label{chap: conclusion} \label{sec: connections}}
This final section connects bilevel methods with related approaches and mentions some additional future directions beyond those already described in previous sections.
Shlezinger \textit{et al.} \citep{shlezinger:2020:modelbaseddeeplearning} recently proposed a framework, summarized in \fref{fig: model-based to learning spectrum}, for categorizing learning-based approaches that combine inferences, or prior knowledge \footnote{
Ref. \citep{shlezinger:2020:modelbaseddeeplearning}
uses the term \dquotes{model-based},
but this review uses \dquotes{inferences}
to differentiate from other definitions
of model-based learning in the literature. }, and deep learning. Inferences can include information about the structure of the forward model, \mA, or about the object \vx being imaged. For example, any known statistical properties of the object of interest could be used to design a regularizer that encourages the minimizer \xhat to be compatible with that prior information. At one extreme, inference-based approaches rely on a relatively small number of handcrafted regularizers with a few, if any, tuning parameters learned from training data. At the other extreme, fully learned approaches assume no information about the application or data and learn all hyperparameters from training data.
\begin{figure}
\caption{
Spectrum of learning to inference-based
methods from \citep{shlezinger:2020:modelbaseddeeplearning}.
}
\label{fig: model-based to learning spectrum}
\end{figure}
Ref. \citep{shlezinger:2020:modelbaseddeeplearning} proposed two general categories for methods that mix elements of inference-based and learning-based methods. The first category, inference-aided networks, includes deep neural networks (DNNs) with architectures based on an inference-based method. For example, in deep unrolling, one starts with a fixed number of iterations of an optimization algorithm derived from a cost function and then learns parameters that may vary between iterations, or ``layers,'' or may be shared across such iterations. \sref{sec: connections unrolled} further discusses unrolling, which is a common inference-aided network design strategy, and the connection to the bilevel unrolling method described in \sref{sec: unrolled}.
The second general category is DNN-aided inference methods \citep{shlezinger:2020:modelbaseddeeplearning}. These methods incorporate a deep learning component into traditional inference-based techniques (typically a cost function in image reconstruction). The learned DNN component(s) can be trained separately for each iteration or end-to-end. Because prior knowledge takes a larger role than in the inference-aided networks, these methods typically require smaller training datasets, with the amount of training data required varying with the number of hyperparameters. \sref{sec: connections plug and play} discusses how bilevel methods compare to Plug-and-Play, which is an example DNN-aided inference model.
While \citep{shlezinger:2020:modelbaseddeeplearning} focused on DNNs due to their highly expressive nature and the abundance of interest in them, the idea of trading off prior knowledge and learning components applies to machine learning more broadly. \sref{sec: connections unrolled} through \ref{sec: connections plug and play} describe how bilevel methods fit into the framework from \citep{shlezinger:2020:modelbaseddeeplearning} and relate bilevel methods to other methods in the framework. Although not covered in the above framework, \sref{sec: connections single-level} also compares bilevel methods to a third general category: \dquotes{single-level} hyperparameter learning methods. Like bilevel methods, single-level methods learn hyperparameters in a supervised manner. However, they generally learn parameters that sparsify the training images, $\{\xtrue_j\}$, and do not use the noisy data, $\{\vy_j\}$. This last comparison demonstrates the benefit of task-based approaches. Of course, there is variety among bilevel methods; this discussion is meant to provide perspective and general relations to increase understanding, rather than to narrow the definition or application of any method.
\section{Connection: Learnable Optimization Algorithms \label{sec: connections unrolled} }
Learning parameters in unrolled optimization algorithms to create an inference-aided network, often called a Learnable Optimization Algorithm (LOA), is a quickly growing area of research \citep{monga:2021:algorithmunrollinginterpretable}. The first such instance was a learned version of the Iterative Shrinkage and Thresholding Algorithm (ISTA), called LISTA \cite{gregor:10:lfa}. Similar to the bilevel unrolling method, a LOA typically starts from a traditional, inference-based optimization algorithm, unrolls multiple iterations, and then learns parameters using end-to-end training.
There are many unrolled methods for image reconstruction \citep{monga:2021:algorithmunrollinginterpretable}. Two examples that explicitly state the bilevel connection are \citep{chen:2021:learnabledescentalgorithm,bian:2020:deepparallelmri}; both set-up a bilevel problem with a DNN as a regularizer and then allow the parameters to vary by iteration, \ie, learning $\lliter{\xmath{\vc_k}}$ where $t$ denotes the lower-level iteration. Ref. \citep{bian:2020:deepparallelmri} motivated the use of an unrolled DNN over more inference-based methods by the lack of an accurate forward model, specifically coil sensitivity maps, for MRI reconstruction. Other examples of unrolled networks are \citep{hammernik:2018:learningvariationalnetwork}, which unrolls the Field of Experts model \citep{roth:2005:fieldsexpertsframework} (see \srefs{sec: filter learning history}{sec: prev results lower level} for how the Field of Experts model has inspired many bilevel methods); \citep{lim:2020:improvedlowcountquantitative}, which unrolls the convolutional analysis operator model \citep{chun:2020:convolutionalanalysisoperator} (see \eqref{eq: CAOL}); and \citep{franceschi:2018:bilevelprogramminghyperparameter}, which discusses the connection to meta-learning.
Unlike the unrolled approach to bilevel learning described in \sref{sec: unrolled}, many LOAs depart from their base cost function and \dquotes{only superficially resemble the steps of optimization algorithms} \citep{chen:2021:learnabledescentalgorithm}. For example, unrolled algorithms may \dquotes{untie} the gradient from the original cost function, \eg, using $\widetilde{\mA}' (\mA \vx - \vy)$, instead of $\mA' (\mA \vx - \vy)$ for the gradient of the common 2-norm data-fit term, where $\tilde{\mA}'$ is learned or otherwise differs from the adjoint of \mA. LOAs that allow the learned parameters to vary every unrolled iteration or learn step size and momentum parameters further depart from a cost function perspective.
In addition to selecting which variables to learn, one must decide how many iterations to unroll for both bilevel unrolled approaches and LOAs. Most methods pick a set number of iterations in advance, perhaps based on previous experience, initial trials, or the available computational resources. Using a set number of iterations yields an algorithm with predictable run times and allows the learned parameters to adapt to the given number of iterations. Further, picking a small number of iterations can act as implicit regularization, comparable to early stopping in machine learning, which may be helpful when the amount of training data is small relative to the number of hyperparameters in the unrolled algorithm \citep{franceschi:2018:bilevelprogramminghyperparameter}.
One can also use a convergence criteria to determine the number of iterations to evaluate, rather than selecting a number in advance \cite{antil:2020:bileveloptimizationdeep}. This convergence-based method more closely follows classic inference-based optimization algorithms. A benefit of running the lower-level optimization algorithm until convergence is that one could switch optimization algorithms between training and testing, especially for strictly convex lower-level cost functions, and still expect the learned parameters to perform similarly. This ability to switch optimization algorithms means one could use faster, but not differentiable, algorithms at test-time, such as accelerated gradient descent methods with adaptive restart \cite{kim:18:aro}. We are unaware of any bilevel methods that have exploited this possibility.
Even within the unrolling methodology, one must make several design decisions. To remain most closely tied to the original optimization algorithm, an unrolled method might fix a large number of iterations or run the optimization algorithm until convergence, use the same parameters every layer, and calculate the step size based on the Lipschitz constant every upper-level iteration \blue{(see discussion in \sref{sec: unrolled number of iterations})}. Like all design decisions, there are trade-offs and the literature shows many successful methods that benefit from the increased generality of designing LOAs that are further removed from their cost function roots \citep{monga:2021:algorithmunrollinginterpretable}. Echoing the ideas from \citep{shlezinger:2020:modelbaseddeeplearning}, the design should be based on the specific application and relative availability, reliability, and importance of prior knowledge and training data.
This survey focuses on unrolled methods that are closely tied to the original bilevel formulation; \citep{monga:2021:algorithmunrollinginterpretable} reviews LOAs more broadly. A benefit of maintaining the connection to the original cost function and optimization algorithm is that, once trained, the lower-level problem in an unrolled bilevel method inherits any theoretical and convergence results from the corresponding optimization method. The corresponding benefit for LOAs is increased flexibility in network architecture.
\blue{ \section{Connection: Equilibrium-based Networks} } \label{sec: connection to DEQ}
Equilibrium-based, or fixed point, networks are related to both LOAs and the minimizer approach from \sref{sec: minimizer approach}. The idea was proposed only recently in \citep{bai:2019:deepequilibriummodels}, but has received much attention. From the unrolled perspective, equilibrium networks consider what happens when the number of unrolled iterations approaches infinity. Alternatively, they can be viewed as a single, implicit layer; as in the minimizer approach, the output is the solution to a nonlinear equation.
We first consider the unrolled perspective. If an algorithm \optalgstep is a contraction, \ie, \[ \norm{\optalgstep(\vx_1 \, ; \, \params) - \optalgstep(\vx_2 \, ;\, \params)} \leq \delta \norm{\vx_1-\vx_2}, \, \forall \vx_1, \vx_2 \in \F^\sdim \] for some parameter $\delta \in [0,1)$, then the sequence of iterates will eventually converge to a fixed-point of \optalgstep. If the optimization algorithm optimizes a cost function with a data-fit and regularization term, then the equilibrium network approach is equivalent to a bilevel method. For a given value of \params, the contraction condition is typically easy to satisfy by selecting an appropriate step-size in algorithms like gradient descent. Ref. \citep{gilton:2021:deepequilibriumarchitectures} provides conditions on deep equilibrium models specific to optimization algorithms based on gradient descent, proximal gradient descent, and ADMM that ensure convergence.
Re-using some of our bilevel notation, let \xhatp denote a fixed-point of an equilibrium network. The derivation for finding $\dParams{\xhatp} \in \F^{\sdim \by \paramsdim}$ follows similar steps to the IFT perspective on the bilevel minimizer approach in \sref{sec: ift approach}. The key difference is that rather than using the first-order optimally condition as in the minimizer approach \eqref{eq:dPhi}, the equilibrium method considers the lower-level minimizer to be a fixed point of an optimization algorithm.
When the goal of the lower level problem is to find a fixed point, the bilevel problem becomes \begin{align}
\argmin_\params
&\underbrace{\lfcn \left(\params \, ; \, \xhatp \right)}_{
\lfcn(\params)}
\text{ s.t. }
\label{eq: fixed point bilevel formulation}
\underbrace{\xhatp = \optalgstep(\xhatp \,; \params)}_{\text{Fixed point equation}} .\end{align} Similar to the IFT perspective, one can differentiate both sides of the fixed point equation using the chain rule \begin{align*}
\dParams{\xhatp} &= \paren{\dx{\optalgstep(\xhatp \,; \params)}} \dParams{\xhatp} + \dParams{\optalgstep(\xhatp \,; \params)} \end{align*} and then rearrange to derive an expression for \dParams{\xhatp} \begin{align}
\dParams{\xhatp} &= \parenr{ \I - \underbrace{\paren{\dx{\optalgstep(\xhatp \,; \params) }} }_{\xmath{\hat{\mJ}}}}^{\neg1} \dParams{\optalgstep(\xhatp \,; \params) }
\label{eq: fixed point dxdparams} .\end{align} The matrix \xmath{\hat{\mJ}} is the Jacobian of the optimization algorithm, evaluated at the fixed point \xhatp.
Substituting \eqref{eq: fixed point dxdparams} into the expression for the upper-level gradient \eqref{eq: bilevel first chain rule} yields \begin{align}
\uppergrad
&= \dParams{\lfcnargs} + \paren{\dParams{\optalgstep(\xhatp \,; \params) }}' (\I-\xmath{\hat{\mJ}})^{\neg1} \dx{\lfcnargs} \label{eq: fixed point uppergrad} .\end{align} If the optimization is standard gradient descent, \ie, $\optalgstep(\vx \,; \params) = \vx - \sslower \nabla_{\vx} \ofcnargs$, then \begin{align*}
\dParams{\optalgstep(\xhatp \,; \params) } &= \neg \sslower \nabla_{\vx \params} \ofcnargs
\text{ and } \\
\dx{\optalgstep(\xhatp \,; \params)} &= \I - \sslower \nabla_{\vx\vx} \ofcnargs .\end{align*} Substituting these expressions into \eqref{eq: fixed point dxdparams} yields the gradient as derived using the IFT perspective in the minimizer approach \eqref{eq: dhdgamma IFT}, showing the close connection between the equilibrium and minimizer approach.
Similar to the minimizer approach, one can use any algorithm to find a fixed point \xhatp of \optalgstep. For example, \citep{bai:2019:deepequilibriummodels} used a quasi-Newton method and \citep{gilton:2021:deepequilibriumarchitectures} used a standard fixed-point accelerated method. One can use any fixed point algorithm to find \xhatp; the algorithm used need not correspond to \optalgstep in \eqref{eq: fixed point bilevel formulation}. For example, \optalgstep could be standard gradient descent, even if one uses a more advanced algorithm to initially compute \xhatp. Another similarity to the minimizer approach is that the learned parameters are optimal at convergence of the lower-level problem, rather than after a fixed number of lower-level iterations. Therefore, the end-user can trade-off accuracy and compute requirements at test time, unlike in unrolled approaches where the number of iterations is pre-decided.
Although the equilibrium model is the limit as the number of unrolled iterations approaches infinity, computing \uppergrad does not require backpropagation nor storing any intermediate matrices. The trade-off is that \eqref{eq: fixed point uppergrad} requires multiplying $(\I - \xmath{\hat{\mJ}})^{\neg1}$ by a vector. The remaining computations in the full upper-level gradient \eqref{eq: fixed point uppergrad} are straightforward. Similar to the required Hessian inverse-vector product in the minimizer approach, one can use an iterative algorithm to approximate the matrix inverse. Ref. \citep{gilton:2021:deepequilibriumarchitectures} notes that the inverse matrix-vector product \begin{equation*}
\vv = (\I - \xmath{\hat{\mJ}})^{\neg1} \dx{\lfcnargs} ,\end{equation*} is a fixed point of the equation \begin{equation*}
\vv = \xmath{\hat{\mJ}} \vv + \dx{\lfcnargs} .\end{equation*} Therefore, one can use any fixed-point solver to compute the matrix-vector product. Another way to decrease the computational cost of the Jacobian product is to use the method from \citep{ramzi:2021:shinesharinginverse}: if a quasi-Newton algorithm is used to estimate the Jacobian for the forward step of computing \xhatp, then one can \dquotes{re-use} this estimated Jacobian to find \uppergrad.
Fixed point networks can also be viewed from the perspective of unrolled methods. Although it is often infeasible to backpropagate through the large number of iterations required to reach a fixed point, backpropagating through the last few iterations yields a valid gradient estimate for \dParams{\xhatp} \citep{shaban:2019:truncatedbackpropagationbilevel}. Ref. \citep{shaban:2019:truncatedbackpropagationbilevel} proves that this \dquotes{truncated backpropagation} approach converges to a stationary point of the upper-level loss when the lower-level cost function is locally strongly convex around \xhatp because the backpropagation gradient error decays exponentially with reverse depth. A similar approach is to use \xhatp at every backpropagation step rather than previous iterates. Ref. \citep{lorraine:2020:optimizingmillionshyperparameters} shows this is equivalent to approximating the matrix inverse in the minimizer approach using a Neumann series.
Recently, \citep{fung:2022:jfbjacobianfreebackpropagation} proposed a Jacobian-free method to find \uppergrad that takes the approach from \citep{shaban:2019:truncatedbackpropagationbilevel} to the extreme case: it considers unrolling a single layer. The approach in~ \citep{fung:2022:jfbjacobianfreebackpropagation} is equivalent to viewing the deep equilibrium network as a single layer network where the initialization is the fixed-point, \ie, using $\xhatp = \optalgstep(\vx^{(0)} \, ; \, \params)$ in the unrolled method with $\vx^{(0)} = \xhatp$. With this new perspective, it is easy to use existing backpropagation tools to compute the derivative through the single layer network. Assuming that the network is Lipschitz, contractive, and differentiable and that the upper-level loss is differentiable, \citep{fung:2022:jfbjacobianfreebackpropagation} shows the Jacobian-free gradient is a descent direction for estimates of \xhatp that are within some error bound of the true fixed point.
Deep equilibrium networks can be fully learned or they can incorporate physics-based models into their network architecture and move into the inference-aided networks category in \fref{fig: model-based to learning spectrum}. For example, \citep{gilton:2021:deepequilibriumarchitectures,heaton:2021:feasibilitybasedfixedpoint} incorporated system matrices into fixed point networks and applied them to MRI and CT image reconstruction problems.
\section{Connection: Plug-and-play Priors} \label{sec: connections plug and play}
The Plug-and-Play (PNP) framework \citep{venkatakrishnan:2013:plugandplaypriorsmodel} is an example of a DNN-aided inference method. It is similar to bilevel methods in its dependence on the forward model. However, unlike bilevel methods, the PNP framework need not be connected to a specific lower-level cost function and it leverages pre-trained denoisers rather than training them for a specific task.
As a brief overview of the PNP framework, consider rewriting the generic data-fit plus regularizer optimization problem \eqref{eq: general data-fit plus reg} with an auxiliary variable: \begin{align}
\xhat = \argmin_{\vx \in \F^\sdim}
\underbrace{\overbrace{\dfcnargs}^{\text{Data-fit}} + \;\;\; \beta
\overbrace{\regfcn(\vz \, ; \params)}^{\text{Regularizer}}}_{\ofcnargs}
\quad \text{ s.t. } \vx = \vz
\label{eq: data-fit plus reg split} .\end{align} Using ADMM \cite{eckstein:92:otd} to solve this constrained optimization problem and rearranging variables yields the following iterative optimization approach for \eqref{eq: data-fit plus reg split}: \begin{align*}
\iter{\vx}{+1} &= \argmin_\vx \dfcnargs + \frac{\lambda}{2}
\normrsq{
\vx - \underbrace{(\iter{\vz}-\iter{\vu})}_{\tilde{\vx}}
}_2
&&= \text{prox}_{\frac{1}{\lambda} \dfcnargs}(\tilde{\vx})
\\
\iter{\vz}{+1} &= \argmin_\vz \beta \regfcn(\vz \, ; \params) + \frac{\lambda}{2}
\normrsq{
\vz - \underbrace{(\iter{\vx}+\iter{\vu})}_{\tilde{\vz}}
}_2
&&= \text{prox}_{\frac{\beta}{\lambda} \regfcn(\vz \, ; \params) }(\tilde{\vz})
\\
\iter{\vu}{+1} &= \iter{\vu} + (\iter{\vx}{+1} - \iter{\vz}{+1}), && \end{align*} where $\lambda$ is an ADMM penalty parameter that effects the convergence rate (but not the limit, for convex problems). The first step is a proximal update for \vx that uses the forward model but does not depend on the regularizer. Conversely, the second step is a proximal update for the split variable \vz that depends on the regularizer, but is agnostic of the forward model. This step acts as a denoiser. The final step is the dual variable update and encourages $\iter{\vx} \approx \iter{\vz}$ as $\upperiter \rightarrow \infty$.
The key insight from \citep{venkatakrishnan:2013:plugandplaypriorsmodel} is that the above update equations separate the forward model and denoiser. Thus, one can substitute, or \dquotes{plug in,} a wide range of denoisers for the \vz update, in place of its proximal update, while keeping the data-fit update independent.
Whereas in the original ADMM approach, the parameter $\lambda$ has no effect on the final image for convex cost functions, in the PNP framework that parameter does affect image quality. Thus, one could also use training data to tune the $\lambda$ in a bilevel manner. Although PNP allows one to substitute a pre-trained denoiser, one could additionally tune the parameters in the denoiser. Ref.~\citep{he:2019:optimizingparameterizedplugandplay} provides one such example of starting from a PNP framework then learning denoising parameters and $\lambda$ that vary by iteration.
A large motivation for the PNP framework is the abundance of advanced denoising methods, including ones that are not associated with an optimization problem such as BM3D \citep{dabov:2007:imagedenoisingsparse}. However, using existing denoisers sacrifices the ability to learn parameters to work well with the specific forward model, as is done in task-based methods. As simple examples of how learned parameters may differ when \mA changes, \cite{chambolle:2021:learningconsistentdiscretizations} found that different filters worked better for image denoising versus image inpainting \blue{and \citep{effland:2020:variationalnetworksoptimal} found that unrolled deblurring methods required more upper-level iterations than unrolled denoising methods. } A more complicated example is using bilevel methods to learn some aspect of \mA alongside some aspect of the regularizer, \eg, \citep{sherry:2020:learningsamplingpattern} learned a sparse sampling matrix and tuning parameter for MRI that are adaptive to the regularization for the image reconstruction problem.
\section{Connection: Single-Level Parameter Learning} \label{sec: hpo filter learning} \label{sec: filter constraints} \label{sec: connections single-level}
\sref{sec: filter learning history} briefly discussed some approaches to learning analysis operators. This section further motivates the task-based bilevel set-up by discussing the filter learning constraints imposed in single-level hyperparameter learning methods.
As summarized in \sref{sec: filter learning history}, the earliest methods for learning analysis regularizers had no constraints on the analysis operators. Those approaches learned filters from training data to make a prior distribution match the observed data distribution. In contrast, more recent approaches to filter learning minimize a cost function that requires either a penalty function or constraint on the operators to ensure filter diversity. For reference, the cost functions mentioned in \sref{sec: filter learning history} were: \begin{align*}
\text{AOL}: & \argmin_{\mOmega,\, \mX}
\norm{\mOmega \mX}_1 + \frac{\beta}{2} \normsq{\mY - \mX}
\text{ s.t. } \mOmega \in \S
, \nonumber \\
\text{TL}: & \argmin_{\mOmega \in \F^{\filterdim \by \filterdim},\, \mX}
\normsq{\mOmega \mY - \mX}_2 + \regfcn(\mOmega)
\text{ s.t. } \norm{\mX_i}_0 \leq \alpha \;\forall i
, \nonumber \\
\text{CAOL}: &\argmin_{[\vc_1, \ldots, \vc_K]} \min_\vz
\sum_{k=1}^K \onehalf \normsq{\xmath{\vc_k} \conv \vx - \vz}_2 + \beta \norm{\vz_k}_0
\text{ s.t. } [\vc_1, \ldots, \vc_K] \in \S, \end{align*} where AOL is analysis operator learning \citep{yaghoobi:2013:constrainedovercompleteanalysis}, TL is transform learning \citep{ravishankar:2013:learningsparsifyingtransforms}, and CAOL is convolutional analysis operator learning \citep{chun:2020:convolutionalanalysisoperator}. In the following discussion of constraint sets, the equivalent filter matrix for CAOL has the convolutional kernels as rows: \[
\mOmega_{\mathrm{CAOL}} =
\begin{bmatrix}
\vc_1' \\
\vdots \\
\vc_K'
\end{bmatrix} .\] While there are many other proposed cost functions in the literature, using different norms or including additional variables, these three examples capture the most common structures for filter learning.
In all the above cost functions, if one removed the constraint or regularizer, then the trivial solution would be to learn zero filters for \mOmega. Furthermore, a simple row norm constraint on \mOmega would be insufficient, as then the minimizer would contain a single filter that is repeated many times. (In contrast, a unit norm constraint typically suffices for dictionary learning.) A row norm constraint plus a full rank constraint is also insufficient because \mOmega can have full rank while being arbitrarily close to the rank-1 case of having a single repeated row.
The choice of constraint set $\S$ is important in single-level learning. Many methods constrain analysis operators to satisfy a tight frame constraint. A matrix $\mA$ is a tight frame if there is a positive constant, $\alpha$, such that \begin{equation*}
\normsq{\mA' \vx}_2 = \sum_{i} \abs{\langle \va_i, \vx \rangle}^2 = \alpha \normsq{\vx}_2,
\; \forall \vx \end{equation*} where $\va_i$ is the $i$th column of \mA. This tight frame condition is equivalent to $\mA \mA' = \alpha \I$ for some positive constant $\alpha$. Most analysis operators are defined with filters in their rows, so a tight frame requirement on the filters appears as the constraint $\mOmega'\mOmega = \alpha \I$.
Under the tight frame constraint for the filters, \mOmega must be square or tall, so the filters are complete or over-complete. However, \citep{yaghoobi:2013:constrainedovercompleteanalysis} found that the frame constraint was insufficient when learning over-complete operators, as the \dquotes{excess} rows past full-rank tended to be all zeros. Therefore, \citep{yaghoobi:2013:constrainedovercompleteanalysis} imposed a uniformly-normalized tight frame constraint: each row of the \mOmega had to have unit norm and the filters had to form a tight frame.
Ref. \citep{hawe:13:aol} similarly constrained \mOmega to have unit-norm rows with the filters forming a frame (though not tight). Such loosening of the tight frame constraint to a frame constraint could lead to the problem of learning almost identical rows, as discussed above. To prevent this issue, \citep{hawe:13:aol} additionally included a penalty that encourages distinct rows: \begin{equation}
- \sum_k \sum_{\tilde{k} < k} \log{1- (\vomega_{\tilde{k}}' \vomega_k)^2} \label{eq: encourage distinct filters via correlation} . \end{equation}
One possible concern with a tight frame constraint is that it requires the filters to span all of $\F^\sdim$, so every spatial frequency can pass through at least one filter. However, most images are not zero-mean and have piece-wise constant regions, so the zero frequency component is not sparse. Ref. \citep{yaghoobi:2013:constrainedovercompleteanalysis} modified the tight-frame constraint to require \mOmega to span some space (\eg, the space orthogonal to the zero frequency term). Likewise, \citep{crockett:2019:incorporatinghandcraftedfilters} extended the CAOL algorithm to include handcrafted filters, such as a zero frequency term, that can then be used or discarded when reconstructing images. In the bilevel literature, \citep{samuel:2009:learningoptimizedmap,chen:2014:insightsanalysisoperator} similarly ensured that learned filters had no zero frequency component by learning coefficients for a linear combination of filter basis vectors, rather than learning the filters directly; see \sref{sec: prev results lower level}.
As an alternative to imposing a strict constraint on the filters, one can penalize \mOmega to encourage filter diversity, as in \eqref{eq: encourage distinct filters via correlation}. Using a penalty has the advantage of being able to learn any size (under- or over-complete) \mOmega and not \textit{requiring} the filters to represent all frequencies. For example, as an alternative to the tight frame constraint, \citep{chun:2020:convolutionalanalysisoperator} proposed a version of CAOL using the following regularizer (to within scaling constants) \begin{align}
\regfcn(\mOmega) = \beta \normsq{\mOmega' \mOmega - \I} \nonumber \end{align} and a unit norm constraint on the filters. Ref.~\citep{pfister:2019:learningfilterbank} included a similar penalty to \eqref{eq: encourage distinct filters via correlation}, but with the inner product being divided by the norm of the filters as the filters were not constrained to unit norm. All such variations on this penalty are to encourage filter diversity.
To ensure a square \mOmega is full rank, while also encouraging it to be well-conditioned, \citep{ravishankar:2013:learningsparsifyingtransforms} used a regularizer that includes a term of the form \begin{equation}
\regfcn(\mOmega) = \neg \beta_1 \log{|\mOmega|} \nonumber
. \end{equation} The log determinant term is known as a log barrier; it forces \mOmega to have full rank because of the asymptote of the log function. Ref.~\citep{pfister:2019:learningfilterbank} includes a similar log barrier regularization term in terms of the eigenvalues of \mOmega to ensure it is left-invertible.
As another example of a filter penalty regularizer, both \citep{ravishankar:2013:learningsparsifyingtransforms} and \citep{pfister:2019:learningfilterbank}, include the following regularization term \begin{equation}
\regfcn(\mOmega) = \beta_2 \norm{\mOmega}_F^2 \nonumber ,\end{equation} rather than constraining the norm of the filters. This Frobenius norm addresses the scale ambiguity in the analysis and transform formulations and ensures the filter coefficients do not grow too large in magnitude.
Yet another approach to encouraging filter diversity is to consider the frequency response of the set of filters. \citet{pfister:2019:learningfilterbank} discuss different constraint options for filter banks based on convolution strides to ensure perfect reconstruction. When the stride is one and one considers circular boundary conditions, the filters can perfectly reconstruct any signal as long as they pass the $\sdim$ discrete Fourier transform frequencies. Tight frames satisfy this constraint, but the constraint is more relaxed than a tight frame constraint.
\cref{chap: applications} discussed some (relatively rare) bilevel problems with penalties on the learned hyperparameters, but, notably, there are no constraints nor penalties on the filters in the bilevel method \eqref{eq: bilevel for analysis filters}! Because of its task-based nature, filters learned via the bilevel method should be those that are best for image reconstruction. Thus, one should not have to worry about redundant filters, zero filters, or filters with excessively large coefficients. This property is one of the key benefits of bilevel methods.
\section{Future Directions} \label{sec: bilevel future directions}
Throughout this review, we mentioned a few areas for future work on bilevel methods. This section highlights some of the avenues that we think are particularly promising.
Advancing upper-level loss function design is identified as future work in many bilevel papers. Despite the abundance of research on image quality metrics (see \sref{sec: loss function design}), most bilevel methods use squared error for the upper-level loss function (see \sref{sec: prev results loss function} for exceptions). Using loss functions that better match the end-application of the images is a clear future direction for bilevel methods that nicely aligns with their task-based nature. For example, in the medical imaging field there is a large literature on objective measures of image quality \cite{barrett:90:oao}, often based on mathematical observers designed to emulate human performance on signal detection tasks, \eg, in situations where a lesion's location is unknown \cite{yendiki:07:aoo}. To our knowledge, there has been little if any work to date on using such mathematical observers to define loss functions for bilevel methods or for training CNN models, though there has been work on CNN-based observers \cite{kopp:18:cam}. Using task-based metrics for bilevel methods and CNN training is a natural direction for future work that could bridge the extensive literature on such metrics with the image reconstruction field.
Unsupervised bilevel problems are exceptions to the trend of using squared error for the upper-level loss function. \sref{sec: prev results loss function} considered a few unsupervised bilevel methods that use noise statistics to estimate the quality of the reconstructed images, \eg, \citep{fehrenbach:2015:bilevelimagedenoising, hintermuller:2020:dualizationautomaticdistributed, sixou:2020:adaptativeregularizationparameter} \citep{zhang:2020:bilevelnestedsparse,deledalle:2014:steinunbiasedgradient}. One extension to the unsupervised setting is the semi-supervised setting, where one might have access to a few clean training samples and additional, noisy training samples.
\blue{ A related opportunity for future work is to use bilevel methods to learn patient-adaptive parameters. The population-based learning approach considered in \eqref{eq: stochastic bilevel upper-level} learns hyperparameters that are best \textit{on average} over the set of training images. In contrast, a patient-adaptive approach tunes hyperparameters for every input image. For example, one could learn filters and initial tuning parameters offline from a training dataset and then adjust the tuning parameters when reconstructing a specific image, \eg, using approaches such as the unsupervised approaches in \sref{sec: prev results loss function}. An alternative approach for adapting hyperparameters at test time is to learn a mapping from the input data to the set of hyperparameters \citep{afkham:2021:learningregularizationparameters,xu:2021:patientspecifichyperparameterlearning}. }
Just as considering more advanced image quality metrics for the upper-level loss function is a promising area for future work, bilevel methods can likely be improved by using more advanced lower-level cost functions. \blue{For example, one could use bilevel methods to learn multi-scale filters, which can increase the receptive field of a regularizer and provide a more natural representation for data that is inherently multiscale \citep{mairal:2008:learningmultiscalesparse,liu:2021:learningmultiscaleconvolutional}.} Perhaps due to the already challenging and non-convex nature of bilevel problems, most methods consider relatively simple convex lower-level cost functions. Papers that examine non-convex regularizers, \eg, \citep{kunisch:2013:bileveloptimizationapproach,chen:2014:insightsanalysisoperator}, conclude that non-convex regularizers lead to more accurate image reconstructions, likely due to better matching the statistics of natural images. This observation aligns with the simple denoising experimental results in \citep{crockett:2021:motivatingbilevelapproaches}, where learned filters with \eqref{eq: corner rounded 1-norm} as the regularizer yielded noisier signals than signals denoised with a hand-crafted filter with the non-convex 0-norm regularizer. In other words, the structure of the regularizer matters in addition to how one learns the filters.
In addition to non-convexity, future bilevel methods could consider non-smooth cost functions. Many bilevel methods require the lower-level cost to be smooth. Exceptions include the translation to a single level approach (\sref{sec: translation to a single level}), which uses the 1-norm as the lower-level regularizer, and unrolled methods, which can be applied to non-smooth cost functions as long as the optimization algorithm has smooth updates (\sref{sec: unrolling non-smooth functions}). The impact of smoothing the cost function on the perceptual quality of the reconstructed image is largely unknown.
Another avenue for future work is based on the fact that \xtrue is really a continuous-space function. A few methods, \eg, \citep{calatroni:2017:bilevelapproacheslearning,delosreyes:2017:bilevelparameterlearning}, develop bilevel methods in continuous-space. However, the majority of methods use discretized forward models without considering the impact of this simplification (as done in this review paper). Future investigations of bilevel methods should strive to avoid the ``inverse crime'' \cite{kaipioa:07:sip} implicit in \eqref{eq: y=Ax+n} where the data is synthesized using the same discretization assumed by the reconstruction method.
\blue{ Future work may also consider how to more closely tie the bilevel method to a statistical modeling framework and leverage progress made in that field. Many bilevel methods for filter learning use the Field of Experts \citep{roth:2005:fieldsexpertsframework} as a starting point. Ref. \citep{roth:2005:fieldsexpertsframework} takes a maximum-likelihood perspective and learns parameters to model the training data distribution. In contrast, bilevel methods such as \eqref{eq: bilevel for analysis filters} have their roots in a maximum \textit{a posteriori} perspective. While this approach is motivated by and aligns with the task-based nature of bilevel methods \citep{samuel:2009:learningoptimizedmap}, it is not clear how well the learned parameters reflect a prior or how to use the learned parameters to generate model uncertainties. Ideas from the Bayesian statistics literature, such as Monte Carlo methods, may be a promising avenue for future research. }
\blue{Related to connecting bilevel methods and statistical processes,} an interesting opportunity for a stochastic bilevel formulation is to add different noise realizations in \eqref{eq: y=Ax+n}, providing an uncountable ensemble of $(\vx,\vy)$ training tuples, where the expectation in \eqref{eq: stochastic bilevel upper-level} is over the distribution of noise realizations. Yet another possibility is to have a truly random set of training images \xtrue drawn from some distribution. For example, \cite{jin:17:dcn} trained a CNN-based CT reconstruction method using an ensemble of images consisting of randomly generated ellipses. Other variations, such as random rotations or warps, have also been used for data augmentation \cite{shorten:19:aso}. One could combine such a random ensemble of images with a random ensemble of noise realizations, in which case the expectation in \eqref{eq: stochastic bilevel upper-level} would be taken over both the image and noise distributions. We are unaware of any bilevel methods for imaging that exploit this full generality. Future literature on stochastic methods should clearly state what expectation is used and may consider exploiting a more general definition of randomness.
\section{Summary of Advantages and Disadvantages}
Like the methods described in \citep{shlezinger:2020:modelbaseddeeplearning}, bilevel methods for computational imaging involve mixing inference-based optimization approaches with learning-based approaches to leverage benefits of both techniques.
Inference-based approaches use prior knowledge, usually in the form of a forward model and an object model, to reconstruct images. Typically the forward model, \mA, is under-determined, so some form of regularization based on the object model is essential. Regularizers always involve some number of adjustable parameters; traditionally inference-based methods select such parameters empirically or using basic image properties like resolution and noise \cite{fessler:96:srp,fessler:96:mav}. The regularization parameters may also be learned from training to maximize SNR \cite{qi:06:pml} or detection task performance \cite{yang:14:rdi} in a bilevel manner (often using a grid or random search due to the relatively small number of learnable parameters). When the forward model and object model are well-known and easy to incorporate in a cost function, inference-based methods can yield accurate reconstructions without the need for large datasets of clean training data.
Learning-based approaches use training datasets to learn a prior. Recently, learning-based approaches have achieved remarkable reconstruction accuracy in practice, largely due to the increased availability in computational resources and larger, more accessible training datasets \citep{wang:16:apo,hammernik:2020:machinelearningimage}. However, many (deep) learning methods lack theoretical guarantees and explainability and finding sufficient training data is still challenging in many applications. Both of these challenges may impede adoption of learning-based methods in clinical practice for some applications, such as medical image reconstruction \citep{sahiner:18:dli}. Some deep learning methods for CT image reconstruction were approved for clinical use in 2019 \cite{fda:19:ge-dlir}; early studies have shown such methods can significantly reduce noise but may also compromise low-contrast spatial resolution \cite{solomon:20:nas}.
Combining inference-based and learning-based approaches allows the integration of learning from training data while using smaller training datasets by incorporating prior knowledge. Such mixed methods often maintain interpretability from the inference-based roots while using learning to provide adaptive regularization. Thus, the benefits of bilevel methods in this review's introduction are generally shared among the methods described in \citep{shlezinger:2020:modelbaseddeeplearning}: theoretical guarantees, competitive performance in terms of reconstruction accuracy, and similar performance to learned networks with a fraction of the free parameters, \eg, \citep{chen:2021:learnabledescentalgorithm, kobler:2021:totaldeepvariation}.
What distinguishes bilevel methods from the other methods in the inference-based to learning-based spectrum in \fref{fig: model-based to learning spectrum}? While one can argue that the conventional CNN and deep learning approach is always bilevel in the sense that the hyperparameters are trained to minimize a loss function, this review considered bilevel methods with the cost function structure \eqref{eq: generic bilevel lower-level}. The regularization term in \eqref{eq: generic bilevel lower-level} could be based on a DNN \citep{chen:2021:learnabledescentalgorithm}, but we followed the bilevel literature that focuses on priors/regularizers, such as in \eqref{eq: bilevel for analysis filters}, maintaining a stronger connection to traditional cost function design.
Another lens for understanding bilevel methods is extending single-level hyperparameter optimization approaches to be task-based, bilevel approaches. Single-level approaches to image reconstruction, such as those using dictionary learning \cite{ravishankar:2011:mrimagereconstruction}, convolutional analysis operator learning \citep{chun:2020:convolutionalanalysisoperator}, and convolutional dictionary learning \citep{garcia-cardona:2018:convolutionaldictionarylearning,chun:18:cdl}, generally aim to learn characteristics of a training dataset, with the idea that these characteristics can then be used in a prior for an image reconstruction task. While such an approach may learn more general information, \citep{crockett:2021:motivatingbilevelapproaches,mccann:2020:supervisedlearningsparsitypromoting} showed that a common single-level optimization strategy resulted in learning a regularizer that was suboptimal for the simple task of signal denoising.
As further evidence of the benefit of task-based learning, \citep{mccann:2020:supervisedlearningsparsitypromoting} found that the lack of constraints in the bilevel filter learning problem is important; the learned filters used the flexibility of the model and were not orthonormal, whereas orthonormality is a constraint often imposed in single-level models (see \sref{sec: filter constraints}). Ref. \citep{kunisch:2013:bileveloptimizationapproach} showed how the task-based nature adapts to training data; total variation based regularization works well for piece-wise constant images but less so for natural images. Beyond adapting to the training dataset, bilevel methods are task-based in terms of adapting to the level of noise; \citep{ehrhardt:2021:inexactderivativefreeoptimization} found the learned tuning parameters for image denoising go to $0$ as the noise goes to 0, since no regularization is needed in the absence of noise for well-determined problems.
A primary disadvantage cited for most bilevel methods is the computational cost compared to single-level hyperparameter optimization methods or other methods with a smaller learning component. In turn, the main driver behind the large computational cost of gradient descent based bilevel optimization methods is that one typically has to optimize the lower-level cost function many times, either to some tolerance or for a certain number of iterations. The computational cost involves a trade-off because how accurately one optimizes the lower-level problem can impact the quality of the learned parameters. For example, \citep{kunisch:2013:bileveloptimizationapproach, chen:2014:insightsanalysisoperator} both claim better denoising accuracy than \citep{samuel:2009:learningoptimizedmap} because they optimize the lower-level problem more accurately. Similarly, \citep{mccann:2020:supervisedlearningsparsitypromoting} notes that learning will fail if the lower-level cost is not optimized to sufficient accuracy.
There are various strategies to decrease the computational cost for bilevel methods. Some are relatively intuitive and applicable to a wide range of problems in machine learning. For example, \citep{mccann:2020:supervisedlearningsparsitypromoting} used larger batch size as the iterations continue, \citep{calatroni:2017:bilevelapproacheslearning} increased the batch size if a gradient step in \params does not sufficiently improve the loss function, and \citep{ehrhardt:2021:inexactderivativefreeoptimization} tightened the accuracy requirement for the gradient estimation over iterations. These strategies all save computation by starting with rougher approximations near the beginning of the optimization method, when \iter{\params} is likely far from \paramh, while using a relatively accurate solution by the end of the algorithm.
Another disadvantage of bilevel methods is that, while the optimization algorithm for the lower-level problem often has theoretical convergence guarantees, and the lower-level cost is often designed to be strictly convex, the full bilevel problem \eqref{eq: generic bilevel upper-level} is usually non-convex, so the quality of the learned hyperparameters can depend on initialization. Thus, in practice, one requires a strategy for initializing \params. For example, for \eqref{eq: bilevel for analysis filters}, one may decide to use a single-level filter learning technique such as the Field of Experts \citep{roth:2005:fieldsexpertsframework} to initialize the hyperparameters. Or, one can use a handcrafted set of filters, such as the DCT filters (or a subset thereof). Other hyperparameters often have similar warm start options. Despite the non-convexity, papers that tested multiple initializations generally found similarly good solutions surprisingly often, \eg, \citep{chen:2014:insightsanalysisoperator,ehrhardt:2021:inexactderivativefreeoptimization,hintermuller:2020:dualizationautomaticdistributed}.
There is no one correct answer for how much a method should use prior information or learning techniques, and it is unlikely that any single approach can be the best for all image reconstruction applications. Like most engineering problems, the trade-off is application-dependent. One should (minimally) consider the amount of training data available, how representative the training data is of the test data, how under-determined the forward model is (\ie, how strong of regularization is needed), how well-known the object model is, the importance of theoretical guarantees and explainability, and the available computational resources at training time and at test time. Bilevel methods show particular promise for applications where training data is limited and/or explainability is highly valued, such as in medical imaging.
\begin{equation}
f^*(\xmath{\vd}) = \sup_{\vx \,\in\, \mathrm{domain}(f)}
\xmath{\vd}'\vx - f(\vx)
\label{eq: definition of conjugate function} ,\end{equation} where $\xmath{\vd} \in \R^N$ is a dual variable. The derivations below use the following two conjugate function relations. \begin{enumerate}
\item When $f(\vx) = \onehalf \normrsq{\vx - \vy}$
for $\vy \in \R^N$,
the conjugate function is
\begin{align*}
f^*(\xmath{\vd}) &= \sup_{\vx \,\in\, \R^N} \xmath{\vd}'\vx - \onehalf \normrsq{\vx - \vy}
.\end{align*}
The maximizer of the quadratic cost function $f^*$ is
\begin{equation}
\xhat = \vy + \xmath{\vd}
\label{eq: primal dual minimizer relation 1}
\end{equation}
and the maximum value simplifies to
\begin{equation}
f^*(\xmath{\vd}) = \onehalf \normsq{\xmath{\vd} + \vy} - \onehalf \normrsq{\vy}
\label{eq: conjugate function for 2-norm}
.\end{equation}
\item When $\sparsefcn(z) = \abs{z}$ is defined on \R,
the conjugate function is
\begin{align*}
\sparsefcn^*(\xmath{d}) &= \sup_{z \,\in\, \R} \xmath{d} z - \abs{z}
.\end{align*} One can verify that the conjugate is
\begin{align}
\sparsefcn^*(\xmath{d}) =
\begin{cases}
0 & \text{ if } \abs{\xmath{d}} \leq 1 \\
\infty & \text{ else }
\end{cases}
\label{eq: 1-norm conjugate}
\end{align}
and the corresponding sets of suprema are
\begin{align}
\argmax_{z \,\in\, \R} \xmath{d} z - \abs{z} =
\begin{cases}
\text{sign}(\xmath{d}) \cdot \infty & \text{ if } \abs{\xmath{d}} > 1 \\
0 & \text{ if } \abs{\xmath{d}} < 1 \\
[0,\infty) & \text{ if } \xmath{d} = 1 \\
(\neg \infty, 0] & \text{ if } \xmath{d} = \neg 1.
\end{cases}
\label{eq: maximizer for 1-norm conjugate}
\end{align}
Generalizing \eqref{eq: 1-norm conjugate} to a vector,
the conjugate function of the 1-norm is
a characteristic function
that is infinity if any element of the input vector is larger than $1$ in absolute value. \end{enumerate} Ref. \citep[p. 50]{borwein:2006:fenchelduality} provides a table with many more conjugate functions.
The biconjugate, denoted $f^{**}$, is the conjugate of $f^*$, \ie, \begin{equation}
f^{**}(\vx) = \sup_{\xmath{\vd} \,\in\, \mathrm{domain}(f^*)} \vx'\xmath{\vd} - f^*(\xmath{\vd})
\label{eq: definition of biconjugate} ,\end{equation} and is the largest convex, lower semi-continuous function below $f$. When $f$ is convex and lower semi-continuous, the biconjugate is equal to the original function, \ie, $f^{**} = f$. One can use the equality of the original function and the biconjugate to derive the saddle point and dual problems when $f$ is convex.
Consider the specific lower-level problem with an analysis-based regularizer \begin{equation}
\argmin_{\vx \,\in\, \R^N} \onehalf \normrsq{\mA \vx - \vy} + \vone'\sparsefcn_.(\mOmega \vx)
\label{eq: example primal problem} ,\end{equation} where \( \mOmega \in \R^{\xmath{F} \by \sdim} .\) When \sparsefcn is convex, the corresponding saddle-point problem is \begin{align*}
&\argmin_{\vx \,\in\, \R^N} \onehalf \normrsq{\mA \vx - \vy}
+ \underbrace{\sup_{\xmath{\vd} \,\in\, \R^{\xmath{F}}} \, \langle \xmath{\vd}, \mOmega \vx \rangle
- \vone' \sparsefcn^*.(\xmath{\vd})}_{\vone'\sparsefcn_.^{**}(\mOmega \vx)} ,\end{align*} where $\langle \cdot, \cdot, \rangle$ is the standard inner product. Under very mild conditions (satisfied for the absolute value function) \citep{chambolle:2016:introductioncontinuousoptimization}, one can swap the minimum and supremum operations and write the \textbf{saddle-point problem} as \begin{equation*}
\sup_{\xmath{\vd} \,\in\, \R^\xmath{F}} \min_{\vx \,\in\, \R^N} \onehalf \normrsq{\mA \vx - \vy}
+ \langle \xmath{\vd}, \mOmega \vx \rangle - \vone' \sparsefcn^*.(\xmath{\vd}) .\end{equation*} Substituting the conjugate of the 1-norm \eqref{eq: 1-norm conjugate}, the saddle-point problem is thus \begin{align}
&\min_{\vx \in \R^\sdim} \min_{\xmath{\vd} \,\in\, \R^\xmath{F}} \onehalf \normrsq{\mA \vx - \vy} - \langle \xmath{\vd}, \mOmega \vx \rangle \text{ s.t. } \abs{\xmath{d}_i} \leq 1 \;\forall i
\label{eq: saddle point 1-norm} .\end{align}
We hereafter assume $\mA=\I$ to derive the dual problem from the saddle-point problem. By grouping terms and re-arranging negative signs, the dual problem can be derived from the saddle point problem. For a general \sparsefcn, the saddle-point problem is equivalent to \begin{align*}
&\max_{\xmath{\vd} \,\in\, \R^\xmath{F}} \neg \vone' \sparsefcn^*.(\xmath{\vd}) +
\paren{ \min_{\vx \in \R^\sdim} \langle \xmath{\vd}, \mOmega \vx \rangle + \onehalf \normrsq{ \vx - \vy} } \\
=& \max_{\xmath{\vd} \,\in\, \R^\xmath{F}} \neg \vone' \sparsefcn^*.(\xmath{\vd}) -
\underbrace{\paren{\max_{\vx \in \R^\sdim} \langle \neg \mOmega' \xmath{\vd}, \vx \rangle - \onehalf \normrsq{ \vx - \vy}}}_{f^*(\neg \mOmega' \xmath{\vd})} ,\end{align*} where the last line follows from properties of inner products. The expression in parenthesis is the conjugate function for the data-fit term, given in \eqref{eq: conjugate function for 2-norm}. Therefore, the dual problem for a general, convex \sparsefcn is \begin{equation*}
\max_{\xmath{\vd} \,\in\, \R^\xmath{F}} \neg \vone' \sparsefcn^*.(\xmath{\vd}) - f^*(\neg \mOmega' \xmath{\vd})
=
\neg \min_{\xmath{\vd} \,\in\, \R^\xmath{F}} \vone' \sparsefcn^*.(\xmath{\vd}) + f^*(\neg \mOmega' \xmath{\vd}) .\end{equation*}
Substituting the conjugates for the data-fit term \eqref{eq: conjugate function for 2-norm} and the conjugate for the 1-norm regularizer \eqref{eq: 1-norm conjugate}, the \textbf{dual problem} for \eqref{eq: example primal problem} with $\sparsefcn(z) = \abs{z}$ becomes \begin{equation}
\min_{\xmath{\vd} \,\in\, \R^\xmath{F}} \onehalf \normsq{\neg \mOmega' \xmath{\vd} + \vy} - \onehalf \normsq{\vy}
\text{ s.t. } \abs{\xmath{d}_i} \leq 1 \;\forall i
\label{eq: dual problem 1-norm} .\end{equation} When we require only the minimizer (not the minimum), an equivalent dual problem is \begin{equation}
\hat{\xmath{\vd}} = \argmin_{\xmath{\vd} \,\in\, \R^\xmath{F}}\onehalf \normsq{\neg \mOmega' \xmath{\vd} + \vy}
\text{ s.t. } \abs{\xmath{d}_i} \leq 1 \;\forall i .\end{equation} This dual problem is a constrained least squares problem and can be solved with a projected gradient descent method, optionally with momentum \cite{kim:18:aro}. From \eqref{eq: primal dual minimizer relation 1}, the primal minimizer can be recovered from the dual minimizer by \begin{equation}
\xhat = \vy - \mOmega' \hat{\xmath{\vd}}
\label{eq: primal dual minimizer relation} .\end{equation} Finally, from \eqref{eq: maximizer for 1-norm conjugate}, the dual variable is related to the filtered signal by \begin{equation}
\xmath{d}_i \in \begin{cases}
1 &\text{ if } [\mOmega \xhat]_i > 0 \\
\neg1 &\text{ if } [\mOmega \xhat]_i < 0 \\
[0,\infty) &\text{ if } [\mOmega \xhat]_i = 1 \\
(\neg\infty,0] &\text{ if } [\mOmega \xhat]_i = \neg1
.\end{cases}
\label{eq: dual variable cases} \end{equation} Ref. \citep{tibshirani:2011:solutionpathgeneralized} provides a more general version of the dual function for non-identity system matrices.
Above, we derived the saddle-point and dual problems using the equality of the biconjugate and the original function for a convex regularizer. The dual problem can also be derived using Lagrangian theory, as shown in \citep{tibshirani:2011:solutionpathgeneralized}. Define an auxiliary (split) variable that is constrained to equal the filtered signal, \ie, $\vz = \mOmega \vx$. Considering the specific case of the 1-norm regularizer, the Lagrangian of the constrained version of \eqref{eq: example primal problem} is \begin{equation*}
\onehalf \normsq{\vx - \vy} + \norm{\vz}_1 + \xmath{\vd}'(\mOmega \vx - \vz) ,\end{equation*} where $\xmath{\vd} \in \R^{\xmath{F}}$ is a vector of Lagrange multipliers and we have omitted the KKT conditions. Minimizing the Lagrangian with respect to \vx and \vz yields the conjugate functions for the data-fit term and 1-norm and thus the dual problem.
Using the Lagrangian perspective to derive the dual problem yields a useful relation between the filtered signal and the dual variable \citep{tibshirani:2011:solutionpathgeneralized}. Because the split variable $\vz$ is constrained to equal $\mOmega \vx$, $[\mOmega \vx]_i > 0$ implies $z_i > 0$. From \eqref{eq: maximizer for 1-norm conjugate}, $z_i$ is only positive and finite when $\xmath{d}_i = 1$. A similar argument holds for $[\mOmega \vx]_i < 0$. Therefore, the dual variable and \xhat are related by \begin{equation}
\xmath{d}_i \in \begin{cases}
\sign{[\mOmega \vx]_i} &\text{ if } [\mOmega \xhat]_i \neq 0 \\
[\neg1,1] &\text{ if } [\mOmega \xhat]_i = 0.
\end{cases}
\label{eq: tibshirani 15} \end{equation} The second case follows from observing that $\xmath{d}_i$ can take any value in its constrained range when $z_i=0$ as the minimum in \eqref{eq: dual problem 1-norm} will be $0$ regardless of $\xmath{d}_i$.
The primal-dual results reviewed in this appendix are referenced in \sref{sec: analysis vs synthesis} to relate analysis and synthesis regularizers, \sref{sec: translation to a single level} to re-write the lower-level minimizer as a differentiable function of itself and \params, and in \sref{sec: unrolling non-smooth functions} to unroll a differentiable algorithm for a non-smooth cost function.
\chapter{Forward and Reverse Approaches to Unrolling} \setcounter{section}{0} \renewcommand{D.\arabic{section}}{B.\arabic{section}} \renewcommand*{\theHsection}{appB.\the\value{section}}
\label{sec: unrolled complexity} \label{sec: foward and backward unrolling}
\blue{This appendix provides background on the forward and backward approaches to the unrolled gradient computation introduced in \sref{sec: unrolled}. From \eqref{eq: generic lower-level chain rule}, the gradient of interest is: \begin{align}
\uppergrad
=&
\nabla_\params \lfcn(\params \, ; \vx^{(T)}) +
\left( \sum_{t=1}^T \left(\franA{T} \cdots \franA{t+1} \right) \franB{t} \right)'
\finalterm \in \F^{\paramsdim}
\label{eq: generic lower-level chain rule repeat} .\end{align} If one uses a gradient descent based algorithm to optimize the lower-level cost function \ofcn, then $\franA{t} = \nabla_\vx \optalgstep(\vx^{(t-1)} \, ; \params) \in \F^{\sdim \by \sdim}$ is closely related to the Hessian of \ofcn and $\franB{t} = \nabla_\params \optalgstep(\vx^{(t-1)} \, ; \params) \in \F^{\sdim \by \paramsdim}$ is proportional to the Jacobian of the gradient.}
To compare the forward and reverse approaches to gradient computation for unrolled methods, we introduce notation for an ordered product of matrices. We indicate the arrangement of the multiplications by the set endpoints, $s \in [ s_1 \leftrightarrow s_2 ]$ with the left endpoint, $s_1$, corresponding to the index for the left-most matrix in the product and the right endpoint, $s_2$, corresponding to the right-most matrix. Thus, for any sequence of square matrices $\{\mA\}_i$: \begin{align*}
\prod_{s \in \left[ t \leftrightarrow T \right]} \mA_s
\defeq
\mA_{t} \mA_{t+1} \cdots \mA_T
=
\left(\mA_T' \mA_{T-1}' \cdots \mA_{t}' \right)'
=
\left( \prod_{s \in \left[ T \leftrightarrow t \right]} \mA_s' \right)'
. \end{align*} The above double arrow notation does not indicate order of operations. In the following notation the arrow direction does not affect the product result (ignoring finite precision effects), but rather signifies the direction (order) of calculation: \begin{align*}
\prod_{s \in \left[ T \leftarrow t \right]} \mA_s
&\defeq \mA_T \left( \mA_{T-1} \cdots \left( \mA_{t+1} \left( \mA_{t} \right) \right) \right)
\\
\prod_{s \in \left[ T \rightarrow t \right]} \mA_s
&\defeq \left( \left( \left( \mA_T \mA_{T-1} \right) \cdots \right) \mA_{t+1} \right) \mA_{t} .\end{align*} We use a similar arrow notation to denote the order that terms are computed for sums; as above, the order is only important for computational considerations and does not affect the final result.
\begin{figure}
\caption{
Reverse mode computation of the unrolled gradient from
\eqref{fig: unrolled reverse-mode}.
The first gradient computation requires
$\vx^{(T)}$,
so all computations occur after
the lower-level optimization algorithm is complete.
The final gradient is
$\uppergrad = \nabla_\params \lfcn(\params \, ; \vx^{(T)}) + \vr$.
}
\label{fig: unrolled reverse-mode}
\end{figure}
Using this notation, the reverse gradient calculation of \eqref{eq: generic lower-level chain rule repeat} is \begin{align}
\nabla_\params \lfcn(\params \, ; \vx^{(T)}) +
\sum_{t \in [T \rightarrow 1]} \franB{t}{}'
\left( \prod_{s \in [(t+1) \leftarrow T]} \franA{s}' \right)
\finalterm. \label{eq: reverse mode} \end{align} This expression requires $\prod_{s \in [(T+1) \leftarrow T]} \franA{s}' = \I$, because \franA{T+1} is not defined. For example, for $T=3$, we have \[
\nabla_\params \lfcn(\params \, ; \vx^{(3)}) +
\underbrace{\franB{3}' (\I) \vg}_{t=3} +
\underbrace{\franB{2}' \paren{\franA{3}'} \vg}_{t=2} +
\underbrace{\franB{1}' \paren{\franA{2}'\franA{3}'}\vg}_{t=1} ,\] where \vg is shorthand for \finalterm here. This version is called reverse as all computations (arrows) begin at the end, $T$.
The primary benefit of the reverse mode comes from the ability to group \finalterm with the right-most \franA{T}, such that all products are matrix-vector products, as seen in \fref{fig: unrolled reverse-mode} Further, one can save the matrix-vector products for use during the next iteration and avoid duplicating the computation. Continuing the example for $T=3$, we have \[
\nabla_\params \lfcn(\params \, ; \vx^{(3)}) +
\underbrace{\franB{3}' (\I) \vg}_{t=1} +
\underbrace{\franB{2}' (\overbrace{\franA{3}' \vg}^{\mDelta})}_{t=2} +
\underbrace{\franB{1}' (\franA{2}' \overbrace{\paren{\franA{3}'\vg}}^{\mDelta} ) }_{t=3} ,\] where one only needs to compute $\mDelta$ once. This ability to rearrange the parenthesis to compute matrix-vector products greatly decreases the computational requirement compared to matrix-matrix products. Excluding the costs of the optimization algorithm steps and forming the \franA{s} and \franB{t} matrices (these costs will be the same in the forward mode computation), reverse mode requires $\order{T}$ Hessian-vector multiplies and $\order{T \sdim \paramsdim}$ additional multiplies. The trade-off is that reverse mode requires storing all $T$ iterates, $\vx^{(t)}$, so that one can compute the corresponding Hessians and Jacobians from them as needed, and thus has a memory complexity $\order{T \sdim}$.
\begin{figure}
\caption{Forward mode computation of the unrolled gradient
from \eqref{eq: forward mode}.
The intermediate computation matrix, \mZ,
is initialized to zero ($\mZ_0 = \vzero$)
then updated every iteration.
The final gradient is
$\uppergrad = \nabla_\params \lfcn(\params \, ; \vx^{(T)}) + \mZ_T' \finalterm$.
}
\label{fig: unrolled forward-mode}
\end{figure}
The forward mode calculation of \eqref{eq: generic lower-level chain rule repeat}, depicted in \fref{fig: unrolled forward-mode}, has all computations (arrows) starting at the earlier iterate: \begin{align}
\nabla_\params \lfcn(\params \, ; \vx^{(T)})
+
\left( \sum_{t\in [1\rightarrow T]}
\left( \prod_{s \in [T \leftarrow (t+1)] } \franA{s} \right) \franB{t} \right)' \finalterm.
\label{eq: forward mode} \end{align} As before, \franA{T+1} is not defined, so we take $\prod_{s \in [T \leftarrow (T+1)] } \franA{s} = \I$. For example, for $T=3$ we have \begin{align*}
\nabla_\params& \lfcn(\params \, ; \vx^{(T)}) +
\paren{
\underbrace{\paren{(\franA{3} \franA{2} ) \franB{1}}' }_{t=1} +
\underbrace{\paren{(\franA{3} ) \franB{2}}' }_{t=2} +
\underbrace{\paren{(\I ) \franB{3}}' }_{t=3}
}
\vg .\end{align*} How the forward mode avoids storing \vx iterates is evident after rearranging the parenthesis to avoid duplicate calculations, as illustrated in \fref{fig: unrolled forward-mode}. Continuing the example for $T=3$, we have \[
\nabla_\params \lfcn(\params \, ; \vx^{(T)}) +
\left[
\underbrace{\franA{3}
\paren{
\overbrace{
\franA{2}
\underbrace{\paren{
\franA{1} \cdot \vzero + \franB{1}
}}_{\mZ_1}
+ \franB{2}
}^{\mZ_2}
}
+ \franB{3}
}_{\mZ_3}
\right]' \vg ,\] where $\mZ_{s} = \franA{s} \mZ_{s-1} + \franB{s} \in \F^{\sdim \by \paramsdim}$ stores the intermediate calculations. The above formula also illustrates why \franA{1} is not needed in \eqref{eq: unrolled upper-level}; \dParams{\vx^{(0)} = \vzero} is the last element from applying the chain rule.
There is no way to rearrange the terms in the forward mode formula to achieve matrix-vector products (while preserving the computation order). Therefore, the computation requirement is much higher at \order{T \paramsdim} Hessian-vector multiplications. The corresponding benefit of the forward mode method is that it does not require storing iterates, thus decreasing (in the common case when $T > \paramsdim$) the memory requirement to \order{\sdim \paramsdim} for storing the intermediate matrix $\mZ_s$ during calculation.
As with the minimizer approach in \sref{sec: minimizer approach}, the computational complexity of the unrolled approach is lower than the generic bound when we consider the specific example of learning convolutional filters according to \eqref{eq: bilevel for analysis filters}. Nevertheless, the general comparison that reverse mode takes more memory but less computation holds true. See \tref{tab: ift and unrolled complexities} for a comparison of the computational and memory complexities.
\chapter{Additional Running Example Results} \setcounter{section}{0} \renewcommand{D.\arabic{section}}{C.\arabic{section}} \renewcommand*{\theHsection}{appC.\the\value{section}}
This appendix derives some results that are relevant to the running example used throughout the survey.
\section{Derivatives for Convolutional Filters} \label{sec: dh of htilde conv f(h conv x)}
{
\newcommand{\xmath{\h_{\vs}}}{\xmath{\h_{\vs}}} \newcommand{\dhs} {\frac{\partial}{\partial \xmath{\h_{\vs}}}} \newcommand{\xmath{i_1,\ldots,i_N}}{\xmath{i_1,\ldots,i_N}} \newcommand{\xmath{\neg i_1,\ldots,\neg i_N}}{\xmath{\neg i_1,\ldots,\neg i_N}} \renewcommand{\vc}{\vc}
This section proves the result \begin{align}
\dhs \paren{ \xmath{\tilde{\vc}}_k \conv f.(\xmath{\vc_k} \conv \vx)}
=
f.(\circshift{\xmath{\vc_k} \conv \vz}{\vs}) + \xmath{\tilde{\vc}_k} \conv \paren{\dot{f}.(\xmath{\vc_k} \conv \vx) \odot \circshift{\vx}{-\vs}} \label{eq: bilevel caol pd htilde conv f(h conv x) pd hs} ,\end{align} when considering $\F=\R$. This equation is key to finding derivatives of the lower-level cost function in \eqref{eq: bilevel for analysis filters} with respect to the filter coefficients.
To simplify notation, we drop the indexing over $k$, so \vc is a single filter and \xmath{\h_{\vs}} denotes the $\vs$th element in the filter for $\vs \in \ints^D$. Here, $\vs$ indexes every dimension of \vc, \eg, for a two-dimensional filter, we could equivalently write $\vs$ as $\langle s_1, s_2 \rangle$. Recall that the notation \xmath{\tilde{\vc}} signifies a reversed version of \vc, as needed for the adjoint of convolution.
Define the notation $\circshift{\vx}{\vi}$ as the vector \vx circularly shifted according to the index $\vi$. Thus, if \vx is 0-indexed and we use circular indexing, \[
\parenr{\circshift{\vx}{\vs}}_\vi = \vx_{\vi-\vs} .\] As two examples, \[
\vx = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_{N-1} \\ x_N \end{bmatrix}
\rightarrow
\circshift{\vx}{\neg 1} = \begin{bmatrix} x_2 \\ x_3 \\ \vdots \\ x_N \\ x_1 \end{bmatrix} ,\] and, in two dimensions, if $\mX \in \F^{M \by N}$ \[
\circshift{\mX}{1,2} =
\begin{bmatrix}
x_{M, N-1} & x_{M, N} & x_{M, 1} & \ldots & x_{M, 3} \\
x_{1, N-1} & x_{1, N} & x_{1,1} & \ldots & x_{1, 3} \\
x_{2, N-1} & x_{2, N} & x_{2,1} & \ldots & x_{2, 3} \\
\vdots & & \ddots & & \vdots \\
x_{M-1,N-1} & x_{M-1,N} & x_{M-1,1} & \ldots & x_{M-1,3}
\end{bmatrix} .\]
This circular shift notation is useful in the derivation and statement of the desired gradient.
Define $\vz = \vc \conv \vx$, where \vc and \vx are both $N$-dimensional. By the definition of convolution, \vz is given by \begin{equation*}
\vz = \sum_{i_1} \cdots \sum_{i_N} c_{\xmath{i_1,\ldots,i_N}} \circshift{\vx}{\xmath{\neg i_1,\ldots,\neg i_N}}
\defeq \sum_{\xmath{i_1,\ldots,i_N}} c_{\xmath{i_1,\ldots,i_N}} \circshift{\vx}{\neg \vi} ,\end{equation*} where, for each sum, the indexing variable $i_n$ iterates over the size of \vc in the $i$th dimension and we simplify the index for circularly shifting vectors, \xmath{i_1,\ldots,i_N}, as simply $\langle \vi \rangle$. This expression shows that the derivative of $\vc \conv \vx$ with respect to the $\vs$th filter coefficient is the $\neg \vs$th coefficient in \vx, \ie, \begin{equation}
\dhs (\vc \conv \vx) = \circshift{\vx}{-\vs} \label{eq: bilevel caol pd z pd hs} .\end{equation}
We can now find the partial derivative of interest: \begin{align*}
\xmath{\tilde{\vc}} \conv f.(\vz) &=
\sum_{\xmath{i_1,\ldots,i_N}} [\xmath{\tilde{\vc}}]_{\xmath{i_1,\ldots,i_N}} \circshift{f.(\vz)}{\neg \vi}
&& \text{ by the convolution formula }
\\
&= \sum_{\xmath{i_1,\ldots,i_N}} [\xmath{\tilde{\vc}}]_{\xmath{i_1,\ldots,i_N}} f.\paren{\circshift{\vz}{\neg \vi}}
&& \text{ since $f$ operates point-wise}
\\
&= \sum_{\xmath{i_1,\ldots,i_N}} \h_{\xmath{\neg i_1,\ldots,\neg i_N}} f.\paren{\circshift{\vz}{\neg \vi}}
&& \text{ by definition of } \xmath{\tilde{\vc}}
\\
&= \sum_{\xmath{i_1,\ldots,i_N}} \h_{\xmath{i_1,\ldots,i_N}} f.\paren{\circshift{\vz}{\vi}}
&& \text{ reverse summation order.} \end{align*} Recall that \vz is a function of \xmath{\h_{\vs}}. Therefore, using the chain rule to take the derivative, \begin{align*}
\dhs &\paren{\xmath{\tilde{\vc}} \conv f.(\vz)} \\
&= f.(\circshift{\vz}{s}) + \sum_{i_1} \cdots \sum_{i_N} \h_{\xmath{i_1,\ldots,i_N}} \dot{f}.(\circshift{\vz}{\xmath{i_1,\ldots,i_N}}) \odot \nabla_{\xmath{\h_{\vs}}} \paren{\circshift{\vz}{\vi}} \\
&= f.(\circshift{\vz}{\vs}) + \sum_{i_1} \cdots \sum_{i_N} [\xmath{\tilde{\vc}}]_{\xmath{\neg i_1,\ldots,\neg i_N}} \dot{f}.(\circshift{\vz}{\xmath{i_1,\ldots,i_N}}) \odot \circshift{\vx}{\vi-\vs} ,\end{align*} where the second equality follows from \eqref{eq: bilevel caol pd z pd hs} and the definition of \xmath{\tilde{\vc}}. Recognizing the convolution formula in the second summand, the expression can be simplified to \[
f.(\circshift{\vz}{\vs}) + \xmath{\tilde{\vc}} \conv \paren{\dot{f}.(\vz) \odot \circshift{\vx}{-\vs}} .\] This proves the claim. Note that the provided formula is for a single element in \vc. One can concatenate the partial derivative result for each value of $\vs$ to get the full Jacobian.
}
\section{Evaluating Assumptions for the Running Example} \label{sec: ghadimi bounds applied}
To better understand the upper-level assumptions \ref{BA assumption upper-level 1}-\ref{BA assumption upper-level end} and lower-level assumptions \ref{BA assumption lower-level 1}-\ref{BA assumption lower-level end} in \sref{sec: assumptions for double and single loop complexity analysis}, this section examines whether the filter learning example \eqref{eq: bilevel for analysis filters} meets each assumption.
\subsection{Upper-level Loss Assumptions}
Recall the upper-level loss function in \eqref{eq: bilevel for analysis filters} is squared error: \begin{align}
\lfcnparamsvx = \onehalf \normr{\vx - \xtrue}^2_2
\label{eq: loss function repeat} ,\end{align} where \lfcn is typically evaluated at $\vx=\xhat(\params)$.
The loss function \eqref{eq: loss function repeat} satisfies \ref{BA assumption upper-level 1}. Because there is no dependence on \params in the upper-level, $\xmath{L_{\vx,\nabla_\params \lfcn}}=0$. The gradient with respect to \vx is $\nabla_\vx \lfcnparamsvx = \vx - \xtrue$, so $\xmath{L_{\vx,\nabla_\vx \lfcn}}=1$.
The norm of the upper-level gradient with respect to \vx, \[ \norm{\nabla_\vx \lfcnparamsvx} = \norm{\vx - \xtrue} ,\] can grow arbitrarily large, so condition \ref{BA assumption upper-level 2} is not met in general. However, in most applications, one can assume an upper bound (possibly quite large) on the elements of \xtrue and impose that bound as a box constraint when computing \xhat. Then the triangle inequality provides a bound on \norm{\vx - \xtrue} for all \vx within the constraint box.
Finally, \ref{BA assumption upper-level end} is met by any loss function, including \eqref{eq: loss function repeat}, that lacks cross terms between \vx and \params. We are unaware of any bilevel method papers using such cross terms.
\subsection{Lower-level Cost Assumptions} \label{sec: strict convexity and good params}
One property used below in many of the bounds for the lower-level cost function is that \begin{equation}
\sigma_1(\Ck) = \norm{\xmath{\vc_k}}_1 \label{eq: sigma1 Ck} ,\end{equation} where $\sigma_1(\cdot)$ is a function that returns the first singular value of its matrix argument. This property follows from Young's inequality and is related to bounded-input bounded-output stability of linear and time invariant systems \citep{unser:05:gss}.
As with the upper-level assumptions considered above, \eqref{eq: bilevel for analysis filters} meets the lower-level assumptions \ref{BA assumption lower-level 1}-\ref{BA assumption lower-level end} if we impose additional constraints on the maximum norm of variables. In addition to bounding the elements in \vx, as we did to ensure \ref{BA assumption upper-level 2}, imposing bounds on $\norm{\xmath{\vc_k}}$ and $\abs{\beta_k}$ is sufficient to meet all the lower-level assumptions. We now examine each condition individually.
Recall from \eqref{eq: bilevel for analysis filters} that the example lower-level cost function is \begin{align}
\xhat(\params) &= \argmin_{\vx \in \F^\sdim} \onehalf \norm{\mA \vx-\vy}^2_2
+ \ebeta{0} \sum_{k=1}^K \ebeta{k} \mat{1}' \sparsefcn.(\xmath{\vc_k} \conv \vx; \epsilon)
\nonumber ,\end{align} where \sparsefcn is a corner-rounded 1-norm \eqref{eq: corner rounded 1-norm}.
As described in \sref{sec: minimizer approach}, the minimizer approach requires \ofcn to be twice differentiable. Thus, \ofcn satisfies \ref{BA assumption lower-level 1}. This condition limits the choices of \sparsefcn to twice differentiable functions.
Considering \ref{BA assumption lower-level 2}, the gradient of \ofcn with respect to \vx is Lipschitz continuous in \vx if the norm of the Hessian, $\norm{\nabla_{\vx\vx} \ofcnargs}_2$, is bounded. Using \eqref{eq: nablas for filter learning} and assuming the Lipschitz constant of the derivative of \sparsefcn is $\Ldsparsefcn$ (for \eqref{eq: corner rounded 1-norm}, $\Ldsparsefcn=\tfrac{1}{\epsilon}$), a Lipschitz constant for $\nabla_\vx \ofcn$ is \begin{align}
\xmath{L_{\vx,\nabla_\vx \ofcn}} &= \sigma_1^2(\mA) + L_{\dot{\sparsefcn}} \ebeta{0} \sum_k \ebeta{k} \sigma_1(\Ck'\Ck) \nonumber \\
&= \sigma_1^2(\mA) + L_{\dot{\sparsefcn}} \ebeta{0} \sum_k \ebeta{k} \norm{\xmath{\vc_k}}_1^2
\label{eq: lower-level LC}
\text{ by \eqref{eq: sigma1 Ck}} .\end{align} The Lipschitz constant \xmath{L_{\vx,\nabla_\vx \ofcn}} depends on the values in \params and therefore does not strictly satisfy \ref{BA assumption lower-level 2}. Here if $\beta_0$, $\beta_k$, and $\vc_k$ have upper bounds, then one can upper bound \xmath{L_{\vx,\nabla_\vx \ofcn}}. All of the bounds below have similar considerations.
To consider the strong convexity condition in \ref{BA assumption lower-level 3}, we consider the Hessian, \begin{equation}
\nabla_{\vx \vx} \ofcn(\vx \, ; \params) =
\underbrace{\mA'\mA}_{\text{From data-\\ fit term}} +
\underbrace{\ebeta{0} \sum_k \ebeta{k} \mC_k' \diag{\ddsparsefcn.(\xmath{\vc_k} \conv \vx)} \mC_k}_{\text{From regularizer}} \label{eq: Hessian repeat} .\end{equation} We assume that $\ddsparsefcn(z) \geq 0 \, \forall z$, as is the case for the corner rounded 1-norm. If $\mA'\mA$ is positive-definite with $\sigma_\sdim(\mA'\mA) > 0$ (this is equivalent to \mA having full column rank), then the Hessian is positive-definite and $\xmath{\mu_{\vx,\ofcn}}=\sigma_\sdim^2(\mA)$ suffices as a strong convexity parameter. In applications like compressed sensing, \mA does not have full column rank. In such cases, $\sigma_\sdim(\mA'\mA) = 0$ and as $e^{\beta_0} \rightarrow 0$ the regularizer term vanishes, so there does not exist any universal $\xmath{\mu_{\vx,\ofcn}} > 0$ for all $\params \in \F^\paramsdim$, so the strong convexity condition \ref{BA assumption lower-level 3} is not satisfied. \blue{However, as discussed in \sref{sec: summary of minimizer approach},} the condition may hold in practice for many values of \params. How to adapt the complexity theory to rigorously address these subtleties is an open question.
The fourth condition, \ref{BA assumption lower-level 4}, is that $\nabla_{\vx \vx} \ofcnargs$ and $\nabla_{\params \vx} \ofcnargs$ are Lipschitz continuous with respect to \vx for all \params. For the first part part, a Lipschitz constant results from bounding the difference in the Hessian evaluated at two points, $\vx^{(1)}$ and $\vx^{(2)}$: \begin{align*}
&\norm{\nabla_{\vx \vx} \ofcn(\vx^{(1)} \, ; \params) - \nabla_{\vx \vx} \ofcn(\vx^{(2)} \, ; \params) }_2 \\
&\quad\quad = \norm{ \ebeta{0} \sum_k \ebeta{k} \mC_k'
\diag{\ddsparsefcn.(\xmath{\vc_k} \conv \vx^{(1)}) - \ddsparsefcn(\xmath{\vc_k} \conv \vx^{(2)})}
\mC_k}_2 .\end{align*} Since every element of \ddsparsefcn is bounded in $(0,L_{\dot{\sparsefcn}})$, the difference between any two evaluations of \ddsparsefcn is at most $L_{\dot{\sparsefcn}}$. Thus \begin{align*}
\norm{\nabla_{\vx \vx} \ofcn(\vx^{(1)} \, ; \params) - \nabla_{\vx \vx} \ofcn(\vx^{(2)} \, ; \params) }_2
& \leq
\ebeta{0} L_{\dot{\sparsefcn}} \sum_k \ebeta{k} \norm{\mC_k'\mC_k}_2 \\
&\leq \ebeta{0} L_{\dot{\sparsefcn}} \sum_k \ebeta{k} \normsq{\xmath{\vc_k}}_1 .\end{align*} The final simplification again uses \eqref{eq: sigma1 Ck}. Thus, \[
\xmath{L_{\vx,\nabla_{\vx\vx}\ofcn}} = \ebeta{0} \Ldsparsefcn \sum_k \ebeta{k} \normsq{\xmath{\vc_k}}_1 .\]
For the second part of \ref{BA assumption lower-level 4}, we must look at the tuning parameters and filter coefficients separately. When considering learning a tuning parameter, $\beta_k$, \begin{align*}
\nabla_{\beta_k \vx} \ofcnargs &=
\ebetazerok \Ck' \dsparsefcn.(\Ck \vx) .\end{align*} To find a Lipschitz constant, consider the Jacobian: \begin{align*}
\nabla_\vx \left( \nabla_{\beta_k \vx} \ofcnargs \right) &=
\ebetazerok \Ck' \diag{\ddsparsefcn.(\Ck \vx)} \Ck .\end{align*} A Lipschitz constant of $\nabla_{\beta_k \vx} \ofcnargs$ is given by the bound on the norm of this matrix (we chose to use the matrix 2-norm, also called the spectral norm). Using similar steps as above to simplify the expression, $L_{\vx,\nabla_{\beta_k \vx}\ofcn} = \ebetazerok \Ldsparsefcn \normsq{\xmath{\vc_k}}_1$.
When considering learning the $\vs$th element of the $k$th filter, \begin{align*}
\nabla_{c_{k,\vs} \vx} \ofcn(\vx \, ; \params) &=
\ebetazerok \left( \dsparsefcn.(\circshift{(\Ck \vx)}{s})
+ \Ck' \left( \ddsparsefcn.(\Ck \vx) \odot \circshift{\vx}{\neg s} \right) \right) \\
&= \ebetazerok \left( \underbrace{\dsparsefcn.(\mR_1 \Ck \vx)}_{\text{Expression 1}}
+ \underbrace{\Ck' \left( \ddsparsefcn.(\Ck \vx) \odot \mR_2 \vx \right)}_{\text{Expressions 2-3}} \right)
\in \F^\sdim ,\end{align*} where $\mR_1$ and $\mR_2$ are rotation matrices that depends on $\vs$ such that $\mR_1 \vx = \circshift{\vx}{\vs}$ and $\mR_2 \vx = \circshift{\vx}{\neg \vs}$. For taking the gradient, it is convenient to note that the last term can be expressed in multiple ways: \[
\ddsparsefcn.(\Ck \vx) \odot \circshift{\vx}{\neg \vs}
=
\underbrace{\diag{\ddsparsefcn.(\Ck \vx)} \mR_2 \vx}_{\mathrm{Expression \, 2}}
=
\underbrace{\diag{\mR_2 \vx} \ddsparsefcn.(\Ck \vx)}_{\mathrm{Expression \, 3}} .\] Using the alternate expressions to perform the chain rule with respect to the \vx term that is not in the $\diag{\cdot}$ statement, the gradient with respect to \vx is: \begin{align*}
\nabla_\vx \left( \nabla_{c_{k,s} \vx} \ofcnargs \right) =
\ebetazerok (
&\underbrace{\Ck' \mR_1' \diag{\ddsparsefcn.(\mR_1 \Ck \vx)}}_{\mathrm{Expression \, 1}} \\
&+ \underbrace{\Ck' \diag{\ddsparsefcn.(\Ck \vx)} \mR_2}_{\mathrm{Expression \, 2}} \\
&+ \underbrace{\Ck' \diag{\dddsparsefcn(\Ck \vx)} \diag{\mR_2 \vx}' \Ck}_{\mathrm{Expression \, 3}}
) .\end{align*} The bound on the spectral norm of the first and second expressions are both $\sigma_1(\Ck) \Ldsparsefcn $ because, for any $\vz \in \F^\sdim$, \[
\normr{\diag{\ddsparsefcn.(\vz)}}_2 \leq
\max_z \abs{\ddsparsefcn(z)}
= \Ldsparsefcn .\] The third expression is bounded by $\sigma_1^2(\Ck) \norm{\vx}_2 \Lddsparsefcn$, which requires a bound on the norm of \vx, similar to \ref{BA assumption upper-level 2}. Summing the three expressions and including the tuning parameters gives the final Lipschitz constant \begin{equation}
L_{\vx,\nabla_{\hks \vx}\ofcn} = \ebetazerok
\sigma_1(\Ck)
( 2\Ldsparsefcn + \sigma_1(\Ck) \Lddsparsefcn \norm{\vx}_2 )
\label{eq: L for hks for ghadimi LL4} .\end{equation}
The fifth assumption, \ref{BA assumption lower-level 5} states that the mixed second gradient of \ofcn is bounded. For the tuning parameters, the mixed second gradient is given in \eqref{eq: nablas for filter learning} as \[
\nabla_{\beta_k \vx} \ofcn(\xhat \,; \params) = \ebeta{0} \ebeta{k} \xmath{\tilde{\vc}_k} \conv \dsparsefcn.(\xmath{\vc_k} \conv \xhat) .\] The bound given in \ref{BA assumption lower-level 5} follows easily by considering that \[
\normr{\diag{\dsparsefcn.(\xmath{\vc_k} \conv \xhat)}}_2 \leq \max_z \abs{\dsparsefcn(z)} = \Lsparsefcn .\] For a filter coefficient, the mixed second gradient is more complicated: \[
\nabla_{c_{k,\vs} \vx} \ofcn(\xhat \, ; \params) =
\ebetazerok \Big(
\underbrace{\dsparsefcn.(\circshift{(\xmath{\vc_k} \conv \xhat)}{\vs})}_{\text{Bounded by $\Lsparsefcn$}}
+ \xmath{\tilde{\vc}_k} \conv \Big( \underbrace{\ddsparsefcn.(\xmath{\vc_k} \conv \xhat)}_{\text{Bounded by $\Ldsparsefcn$}}
\odot \circshift{\xhat}{\neg \vs} \Big) \Big) .\] Assuming that the bounds $\Lsparsefcn$ and $\Ldsparsefcn$ exist (they are $1$ and $\frac{1}{\epsilon}$ respectively for \eqref{eq: corner rounded 1-norm}), a bound on the norm of the mixed gradient is \begin{align*}
\normr{\nabla_{c_{k,\vs} \vx} \ofcn(\xhat \, ; \params)}_2
&\leq
\ebetazerok \paren{\Lsparsefcn + \Ldsparsefcn \norm{\xmath{\vc_k}}_1 \norm{\vx}_2} .\end{align*}
The sixth assumption, \ref{BA assumption lower-level end}, is that $\xmath{L_{\params,\nabla_{\params\vx}\ofcn}}$ and $\xmath{L_{\params,\nabla_{\vx\vx}\ofcn}}$ exist. Lipschitz constants for the tuning parameters are \[
L_{\beta_k, \nabla_{\beta_k \vx} \ofcn} = \ebetazerok \norm{\xmath{\vc_k}}_1 \Lsparsefcn
\text{ and }
L_{\beta_k, \nabla_{\vx \vx} \ofcn} = \ebetazerok \normsq{\xmath{\vc_k}}_1 \Ldsparsefcn .\] Using similar derivations as shown above, corresponding Lipschitz constants for the filter coefficients are \begin{align*}
L_{\hks, \nabla_{\hks \vx} \ofcn} &= \ebetazerok \paren{
\Lsparsefcn + \norm{\vx}_2
\paren{
\Ldsparsefcn +
\Lddsparsefcn \norm{\xmath{\vc_k}}_1 \norm{\vx}_2
}
} \\
L_{\hks, \nabla_{\vx \vx} \ofcn} &= \ebetazerok \paren{
2 \Ldsparsefcn \norm{\xmath{\vc_k}}_1 + \Lddsparsefcn \norm{\xmath{\vc_k}}_1^2 \norm{\vx}_2
} .\end{align*} This is the last lower-level condition in \sref{sec: assumptions for double and single loop complexity analysis} for the single-loop and double-loop bilevel optimization method analysis.
\chapter{Implementation Details} \setcounter{section}{0} \renewcommand{D.\arabic{section}}{D.\arabic{section}} \renewcommand*{\theHsection}{appD.\the\value{section}}
This appendix describes the experimental settings used throughout this review. We first present the common settings; the following sub-sections detail any differences specifically for the results in \fref{fig: vertbars simple bilevel filter example} and for the series of figures using the cameraman image (\fref{fig: cameraman training loss}, \fref{fig: cameraman learned filters}, and \fref{fig: cameraman example results}). The code for all experiments is available on github \citep{crockett:2022:bilevelfilterlearningforimagerecon}.
The experiments consider the denoising problem ($\mA = \I$) and use \eqref{eq: corner rounded 1-norm} as the sparsifying function \sparsefcn with $\epsilon=0.01$. The training data is typically on the scale $[0, \, 1]$ and noisy samples are generated from the clean training data using \eqref{eq: y=Ax+n} with zero-mean Gaussian noise with a standard deviation of $\sigma = 25/255$, following \citep{chen:2014:insightsanalysisoperator}.
The lower-level optimizer is the optimized gradient method (OGM) with gradient-based restart \citep{kim:18:aro}. We calculate the step-size based on the Lipschitz constant of the lower-level gradient using \eqref{eq: lower-level LC} every upper-level iteration. Each experiment sets a maximum number of lower-level iterations, but the lower-level optimization will terminate early if it converges, defined as if $\norm{\nabla_{\vx} \ofcnargs} < 10^{\neg 5}$.
The upper-level optimizer follows the general structure of the double-loop procedure outlined in \aref{alg: ba}. To compute \uppergrad, we use the minimizer formulation \eqref{eq: IFT final gradient dldparams}, with the conjugate gradient (CG) method to compute the Hessian-inverse-vector product \eqref{eq: Hinv step for CG}. As suggested in \citep{ji:2021:bileveloptimizationconvergence}, the initialization for the lower-level optimization is the estimated minimizer from the previous outer loop iteration, $\vx^{(T)}(\iter{\params}{\neg1})$ and the initialization for the CG method is the solution from the previous CG iteration. Following \citep{chen:2021:learnabledescentalgorithm} and other bilevel works, the experiments use Adam with the default parameters \citep{kingma:2015:adammethodstochastic} to determine the size of the upper-level gradient descent; this choice avoids introducing the tuning parameter \ssupper.
The learnable parameters include the filter coefficients and the tuning parameters $\beta_k$ for $k \in [1,K]$. The experiments either use random or DCT filters to initialize \vc. An initial grid search determines the tuning parameter $\beta_0$; $\beta_k$ for $k \in [1,K]$ are initialized as 0 such that $e^{\beta_k} = 1$.
\section{Vertical Bar Training Image} \label{sec: vertbars}
This section describes additional details for \fref{fig: vertbars simple bilevel filter example}. This simple proof of concept used 50 lower-level iterations ($T=50$) and 4,000 upper-level iterations ($U=4,000$). The initial grid search for $\beta_0$ yielded $\neg4.6$.
When $\sparsefcn(z) = \abs{z}$, one can absorb the $k$th filter's magnitude into the tuning parameter $\beta_k$ because $\norm{\xmath{\vc_k} \conv \vx}_1 = \norm{\xmath{\vc_k}}_2 \norm{\frac{1}{\norm{\xmath{\vc_k}}_2} \xmath{\vc_k} \conv \vx}_1$. When using \eqref{eq: corner rounded 1-norm}, this equality no longer holds, but \begin{equation}
e^{\beta_0 + \beta_k} \norm{\xmath{\vc_k}}_2
\label{eq: effective beta} \end{equation} still provides a reasonable approximation for the overall regularization strength for the $k$th filter. From left to right, the approximate regularization strengths of the filters in \fref{fig: vertbars simple bilevel filter example} are 0.77, 0.49, 0.17, and 0.05.
The learned filters reflect that the training data is constant along the columns. Visually, the filters resemble vertical (extended) finite differences. This matches our expectations as a filter that takes vertical finite differences will exactly sparsify the noiseless signal. Further, the maximum sum of the columns of the learned filters is $10^{\neg5}$. In contrast, the sum of the rows of the learned filters varies from $\neg2.6$ to 3.0.
\section{Cameraman Training Image} \label{sec: cameraman training details}
This section describes the experimental settings for \fref{fig: cameraman training loss}, \fref{fig: cameraman example results}, and \fref{fig: cameraman learned filters}.
To reduce computation, we selected three $50 \by 50$ patches from the \dquotes{cameraman} image in \fref{fig: cameraman example results} to use as the training data. We hand selected the training patches to contain structure. \fref{fig: cameraman training patches} shows the training image patches.
We set the lower-level initialization $\xhat(\params^{(0)})$ by optimizing the lower-level cost function until the norm of the gradient fell below a threshold for each training patch, \ie, until $\tfrac{1}{\sqrt{\sdim}}\norm{\dx{\ofcn\left(\xhat_j(\params^{(0)})\, ; \, \params^{(0)}\right)}}_2 < 10^{\neg 7}$ for $j \in [1,J]$. The lower-level optimizer consisted of 10 iterations of OGM \citep{kim:18:aro}.
As shown in \fref{fig: cameraman learned filters}, the initial filters are the 48 non-constant DCT filters of size $7 \by 7$. The initial grid search for $\beta_0$ yielded $\neg4$. In summary, the settings are $\Ntrue = 3$, $\sdim = 50 \cdot 50$, $\filterdim = 7 \cdot 7$, $K = 48$, $\paramsdim = 48 (49 + 1) = 2400$, $\beta_0=\neg4$, $T=10$, and $U=10,000$.
\fref{fig: cameraman learned filters} shows the learned filters. To visualize the filters when \params includes \vc, \fref{fig: cameraman learned filters}c scales each learned filter $\hat{\vc}_k$ to have unit norm. \fref{fig: learned filters with effective beta} shows the learned filters with the effective regularization strength printed above each filter.
\begin{figure}
\caption{Patches from the cameraman test images used as the training dataset.}
\label{fig: cameraman training patches}
\end{figure}
\begin{figure}\label{fig: learned filters with effective beta}
\end{figure}
\backmatter
\printbibliography
\end{document} | arXiv |
\begin{definition}[Definition:Northern Hemisphere]
The '''northern hemisphere''' of Earth is the hemisphere between the equator and the North Pole.
Points in the northern hemisphere have latitude between $0 \degrees \, \mathrm N$ (the equator itself) and $90 \degrees \, \mathrm N$ (the North Pole).
\end{definition} | ProofWiki |
Uncertainty theory
Uncertainty theory is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms.
Mathematical measures of the likelihood of an event being true include probability theory, capacity, fuzzy logic, possibility, and credibility, as well as uncertainty.
Four axioms
Axiom 1. (Normality Axiom) ${\mathcal {M}}\{\Gamma \}=1{\text{ for the universal set }}\Gamma $.
Axiom 2. (Self-Duality Axiom) ${\mathcal {M}}\{\Lambda \}+{\mathcal {M}}\{\Lambda ^{c}\}=1{\text{ for any event }}\Lambda $.
Axiom 3. (Countable Subadditivity Axiom) For every countable sequence of events $\Lambda _{1},\Lambda _{2},\ldots $, we have
${\mathcal {M}}\left\{\bigcup _{i=1}^{\infty }\Lambda _{i}\right\}\leq \sum _{i=1}^{\infty }{\mathcal {M}}\{\Lambda _{i}\}$.
Axiom 4. (Product Measure Axiom) Let $(\Gamma _{k},{\mathcal {L}}_{k},{\mathcal {M}}_{k})$ be uncertainty spaces for $k=1,2,\ldots ,n$. Then the product uncertain measure ${\mathcal {M}}$ is an uncertain measure on the product σ-algebra satisfying
${\mathcal {M}}\left\{\prod _{i=1}^{n}\Lambda _{i}\right\}={\underset {1\leq i\leq n}{\operatorname {min} }}{\mathcal {M}}_{i}\{\Lambda _{i}\}$.
Principle. (Maximum Uncertainty Principle) For any event, if there are multiple reasonable values that an uncertain measure may take, then the value as close to 0.5 as possible is assigned to the event.
Uncertain variables
An uncertain variable is a measurable function ξ from an uncertainty space $(\Gamma ,L,M)$ to the set of real numbers, i.e., for any Borel set B of real numbers, the set $\{\xi \in B\}=\{\gamma \in \Gamma \mid \xi (\gamma )\in B\}$ is an event.
Uncertainty distribution
Uncertainty distribution is inducted to describe uncertain variables.
Definition: The uncertainty distribution $\Phi (x):R\rightarrow [0,1]$ of an uncertain variable ξ is defined by $\Phi (x)=M\{\xi \leq x\}$.
Theorem (Peng and Iwamura, Sufficient and Necessary Condition for Uncertainty Distribution): A function $\Phi (x):R\rightarrow [0,1]$ is an uncertain distribution if and only if it is an increasing function except $\Phi (x)\equiv 0$ and $\Phi (x)\equiv 1$.
Independence
Definition: The uncertain variables $\xi _{1},\xi _{2},\ldots ,\xi _{m}$ are said to be independent if
$M\{\cap _{i=1}^{m}(\xi \in B_{i})\}={\mbox{min}}_{1\leq i\leq m}M\{\xi _{i}\in B_{i}\}$
for any Borel sets $B_{1},B_{2},\ldots ,B_{m}$ of real numbers.
Theorem 1: The uncertain variables $\xi _{1},\xi _{2},\ldots ,\xi _{m}$ are independent if
$M\{\cup _{i=1}^{m}(\xi \in B_{i})\}={\mbox{max}}_{1\leq i\leq m}M\{\xi _{i}\in B_{i}\}$
for any Borel sets $B_{1},B_{2},\ldots ,B_{m}$ of real numbers.
Theorem 2: Let $\xi _{1},\xi _{2},\ldots ,\xi _{m}$ be independent uncertain variables, and $f_{1},f_{2},\ldots ,f_{m}$ measurable functions. Then $f_{1}(\xi _{1}),f_{2}(\xi _{2}),\ldots ,f_{m}(\xi _{m})$ are independent uncertain variables.
Theorem 3: Let $\Phi _{i}$ be uncertainty distributions of independent uncertain variables $\xi _{i},\quad i=1,2,\ldots ,m$ respectively, and $\Phi $ the joint uncertainty distribution of uncertain vector $(\xi _{1},\xi _{2},\ldots ,\xi _{m})$. If $\xi _{1},\xi _{2},\ldots ,\xi _{m}$ are independent, then we have
$\Phi (x_{1},x_{2},\ldots ,x_{m})={\mbox{min}}_{1\leq i\leq m}\Phi _{i}(x_{i})$
for any real numbers $x_{1},x_{2},\ldots ,x_{m}$.
Operational law
Theorem: Let $\xi _{1},\xi _{2},\ldots ,\xi _{m}$ be independent uncertain variables, and $f:R^{n}\rightarrow R$ a measurable function. Then $\xi =f(\xi _{1},\xi _{2},\ldots ,\xi _{m})$ is an uncertain variable such that
${\mathcal {M}}\{\xi \in B\}={\begin{cases}{\underset {f(B_{1},B_{2},\cdots ,B_{n})\subset B}{\sup }}\;{\underset {1\leq k\leq n}{\min }}{\mathcal {M}}_{k}\{\xi _{k}\in B_{k}\},&{\text{if }}{\underset {f(B_{1},B_{2},\cdots ,B_{n})\subset B}{\sup }}\;{\underset {1\leq k\leq n}{\min }}{\mathcal {M}}_{k}\{\xi _{k}\in B_{k}\}>0.5\\1-{\underset {f(B_{1},B_{2},\cdots ,B_{n})\subset B^{c}}{\sup }}\;{\underset {1\leq k\leq n}{\min }}{\mathcal {M}}_{k}\{\xi _{k}\in B_{k}\},&{\text{if }}{\underset {f(B_{1},B_{2},\cdots ,B_{n})\subset B^{c}}{\sup }}\;{\underset {1\leq k\leq n}{\min }}{\mathcal {M}}_{k}\{\xi _{k}\in B_{k}\}>0.5\\0.5,&{\text{otherwise}}\end{cases}}$
where $B,B_{1},B_{2},\ldots ,B_{m}$ are Borel sets, and $f(B_{1},B_{2},\ldots ,B_{m})\subset B$ means $f(x_{1},x_{2},\ldots ,x_{m})\in B$ for any$x_{1}\in B_{1},x_{2}\in B_{2},\ldots ,x_{m}\in B_{m}$.
Expected Value
Definition: Let $\xi $ be an uncertain variable. Then the expected value of $\xi $ is defined by
$E[\xi ]=\int _{0}^{+\infty }M\{\xi \geq r\}dr-\int _{-\infty }^{0}M\{\xi \leq r\}dr$
provided that at least one of the two integrals is finite.
Theorem 1: Let $\xi $ be an uncertain variable with uncertainty distribution $\Phi $. If the expected value exists, then
$E[\xi ]=\int _{0}^{+\infty }(1-\Phi (x))dx-\int _{-\infty }^{0}\Phi (x)dx.$
Theorem 2: Let $\xi $ be an uncertain variable with regular uncertainty distribution $\Phi $. If the expected value exists, then
$E[\xi ]=\int _{0}^{1}\Phi ^{-1}(\alpha )d\alpha .$
Theorem 3: Let $\xi $ and $\eta $ be independent uncertain variables with finite expected values. Then for any real numbers $a$ and $b$, we have
$E[a\xi +b\eta ]=aE[\xi ]+b[\eta ].$
Variance
Definition: Let $\xi $ be an uncertain variable with finite expected value $e$. Then the variance of $\xi $ is defined by
$V[\xi ]=E[(\xi -e)^{2}].$
Theorem: If $\xi $ be an uncertain variable with finite expected value, $a$ and $b$ are real numbers, then
$V[a\xi +b]=a^{2}V[\xi ].$
Critical value
Definition: Let $\xi $ be an uncertain variable, and $\alpha \in (0,1]$. Then
$\xi _{sup}(\alpha )=\sup\{r\mid M\{\xi \geq r\}\geq \alpha \}$
is called the α-optimistic value to $\xi $, and
$\xi _{inf}(\alpha )=\inf\{r\mid M\{\xi \leq r\}\geq \alpha \}$
is called the α-pessimistic value to $\xi $.
Theorem 1: Let $\xi $ be an uncertain variable with regular uncertainty distribution $\Phi $. Then its α-optimistic value and α-pessimistic value are
$\xi _{sup}(\alpha )=\Phi ^{-1}(1-\alpha )$,
$\xi _{inf}(\alpha )=\Phi ^{-1}(\alpha )$.
Theorem 2: Let $\xi $ be an uncertain variable, and $\alpha \in (0,1]$. Then we have
• if $\alpha >0.5$, then $\xi _{inf}(\alpha )\geq \xi _{sup}(\alpha )$;
• if $\alpha \leq 0.5$, then $\xi _{inf}(\alpha )\leq \xi _{sup}(\alpha )$.
Theorem 3: Suppose that $\xi $ and $\eta $ are independent uncertain variables, and $\alpha \in (0,1]$. Then we have
$(\xi +\eta )_{sup}(\alpha )=\xi _{sup}(\alpha )+\eta _{sup}{\alpha }$,
$(\xi +\eta )_{inf}(\alpha )=\xi _{inf}(\alpha )+\eta _{inf}{\alpha }$,
$(\xi \vee \eta )_{sup}(\alpha )=\xi _{sup}(\alpha )\vee \eta _{sup}{\alpha }$,
$(\xi \vee \eta )_{inf}(\alpha )=\xi _{inf}(\alpha )\vee \eta _{inf}{\alpha }$,
$(\xi \wedge \eta )_{sup}(\alpha )=\xi _{sup}(\alpha )\wedge \eta _{sup}{\alpha }$,
$(\xi \wedge \eta )_{inf}(\alpha )=\xi _{inf}(\alpha )\wedge \eta _{inf}{\alpha }$.
Entropy
Definition: Let $\xi $ be an uncertain variable with uncertainty distribution $\Phi $. Then its entropy is defined by
$H[\xi ]=\int _{-\infty }^{+\infty }S(\Phi (x))dx$
where $S(x)=-t\ln(t)-(1-t)\ln(1-t)$.
Theorem 1(Dai and Chen): Let $\xi $ be an uncertain variable with regular uncertainty distribution $\Phi $. Then
$H[\xi ]=\int _{0}^{1}\Phi ^{-1}(\alpha )\ln {\frac {\alpha }{1-\alpha }}d\alpha .$
Theorem 2: Let $\xi $ and $\eta $ be independent uncertain variables. Then for any real numbers $a$ and $b$, we have
$H[a\xi +b\eta ]=|a|E[\xi ]+|b|E[\eta ].$
Theorem 3: Let $\xi $ be an uncertain variable whose uncertainty distribution is arbitrary but the expected value $e$ and variance $\sigma ^{2}$. Then
$H[\xi ]\leq {\frac {\pi \sigma }{\sqrt {3}}}.$
Inequalities
Theorem 1(Liu, Markov Inequality): Let $\xi $ be an uncertain variable. Then for any given numbers $t>0$ and $p>0$, we have
$M\{|\xi |\geq t\}\leq {\frac {E[|\xi |^{p}]}{t^{p}}}.$
Theorem 2 (Liu, Chebyshev Inequality) Let $\xi $ be an uncertain variable whose variance $V[\xi ]$ exists. Then for any given number $t>0$, we have
$M\{|\xi -E[\xi ]|\geq t\}\leq {\frac {V[\xi ]}{t^{2}}}.$
Theorem 3 (Liu, Holder's Inequality) Let $p$ and $q$ be positive numbers with $1/p+1/q=1$, and let $\xi $ and $\eta $ be independent uncertain variables with $E[|\xi |^{p}]<\infty $ and $E[|\eta |^{q}]<\infty $. Then we have
$E[|\xi \eta |]\leq {\sqrt[{p}]{E[|\xi |^{p}]}}{\sqrt[{p}]{E[\eta |^{p}]}}.$
Theorem 4:(Liu [127], Minkowski Inequality) Let $p$ be a real number with $p\leq 1$, and let $\xi $ and $\eta $ be independent uncertain variables with $E[|\xi |^{p}]<\infty $ and $E[|\eta |^{q}]<\infty $. Then we have
${\sqrt[{p}]{E[|\xi +\eta |^{p}]}}\leq {\sqrt[{p}]{E[|\xi |^{p}]}}+{\sqrt[{p}]{E[\eta |^{p}]}}.$
Convergence concept
Definition 1: Suppose that $\xi ,\xi _{1},\xi _{2},\ldots $ are uncertain variables defined on the uncertainty space $(\Gamma ,L,M)$. The sequence $\{\xi _{i}\}$ is said to be convergent a.s. to $\xi $ if there exists an event $\Lambda $ with $M\{\Lambda \}=1$ such that
$\lim _{i\to \infty }|\xi _{i}(\gamma )-\xi (\gamma )|=0$
for every $\gamma \in \Lambda $. In that case we write $\xi _{i}\to \xi $,a.s.
Definition 2: Suppose that $\xi ,\xi _{1},\xi _{2},\ldots $ are uncertain variables. We say that the sequence $\{\xi _{i}\}$ converges in measure to $\xi $ if
$\lim _{i\to \infty }M\{|\xi _{i}-\xi |\leq \varepsilon \}=0$
for every $\varepsilon >0$.
Definition 3: Suppose that $\xi ,\xi _{1},\xi _{2},\ldots $ are uncertain variables with finite expected values. We say that the sequence $\{\xi _{i}\}$ converges in mean to $\xi $ if
$\lim _{i\to \infty }E[|\xi _{i}-\xi |]=0$.
Definition 4: Suppose that $\Phi ,\phi _{1},\Phi _{2},\ldots $ are uncertainty distributions of uncertain variables $\xi ,\xi _{1},\xi _{2},\ldots $, respectively. We say that the sequence $\{\xi _{i}\}$ converges in distribution to $\xi $ if $\Phi _{i}\rightarrow \Phi $ at any continuity point of $\Phi $.
Theorem 1: Convergence in Mean $\Rightarrow $ Convergence in Measure $\Rightarrow $ Convergence in Distribution. However, Convergence in Mean $\nLeftrightarrow $ Convergence Almost Surely $\nLeftrightarrow $ Convergence in Distribution.
Conditional uncertainty
Definition 1: Let $(\Gamma ,L,M)$ be an uncertainty space, and $A,B\in L$. Then the conditional uncertain measure of A given B is defined by
${\mathcal {M}}\{A\vert B\}={\begin{cases}\displaystyle {\frac {{\mathcal {M}}\{A\cap B\}}{{\mathcal {M}}\{B\}}},&\displaystyle {\text{if }}{\frac {{\mathcal {M}}\{A\cap B\}}{{\mathcal {M}}\{B\}}}<0.5\\\displaystyle 1-{\frac {{\mathcal {M}}\{A^{c}\cap B\}}{{\mathcal {M}}\{B\}}},&\displaystyle {\text{if }}{\frac {{\mathcal {M}}\{A^{c}\cap B\}}{{\mathcal {M}}\{B\}}}<0.5\\0.5,&{\text{otherwise}}\end{cases}}$
${\text{provided that }}{\mathcal {M}}\{B\}>0$
Theorem 1: Let $(\Gamma ,L,M)$ be an uncertainty space, and B an event with $M\{B\}>0$. Then M{·|B} defined by Definition 1 is an uncertain measure, and $(\Gamma ,L,M\{{\mbox{·}}|B\})$is an uncertainty space.
Definition 2: Let $\xi $ be an uncertain variable on $(\Gamma ,L,M)$. A conditional uncertain variable of $\xi $ given B is a measurable function $\xi |_{B}$ from the conditional uncertainty space $(\Gamma ,L,M\{{\mbox{·}}|_{B}\})$ to the set of real numbers such that
$\xi |_{B}(\gamma )=\xi (\gamma ),\forall \gamma \in \Gamma $.
Definition 3: The conditional uncertainty distribution $\Phi \rightarrow [0,1]$ of an uncertain variable $\xi $ given B is defined by
$\Phi (x|B)=M\{\xi \leq x|B\}$
provided that $M\{B\}>0$.
Theorem 2: Let $\xi $ be an uncertain variable with regular uncertainty distribution $\Phi (x)$, and $t$ a real number with $\Phi (t)<1$. Then the conditional uncertainty distribution of $\xi $ given $\xi >t$ is
$\Phi (x\vert (t,+\infty ))={\begin{cases}0,&{\text{if }}\Phi (x)\leq \Phi (t)\\\displaystyle {\frac {\Phi (x)}{1-\Phi (t)}}\land 0.5,&{\text{if }}\Phi (t)<\Phi (x)\leq (1+\Phi (t))/2\\\displaystyle {\frac {\Phi (x)-\Phi (t)}{1-\Phi (t)}},&{\text{if }}(1+\Phi (t))/2\leq \Phi (x)\end{cases}}$
Theorem 3: Let $\xi $ be an uncertain variable with regular uncertainty distribution $\Phi (x)$, and $t$ a real number with $\Phi (t)>0$. Then the conditional uncertainty distribution of $\xi $ given $\xi \leq t$ is
$\Phi (x\vert (-\infty ,t])={\begin{cases}\displaystyle {\frac {\Phi (x)}{\Phi (t)}},&{\text{if }}\Phi (x)\leq \Phi (t)/2\\\displaystyle {\frac {\Phi (x)+\Phi (t)-1}{\Phi (t)}}\lor 0.5,&{\text{if }}\Phi (t)/2\leq \Phi (x)<\Phi (t)\\1,&{\text{if }}\Phi (t)\leq \Phi (x)\end{cases}}$
Definition 4: Let $\xi $ be an uncertain variable. Then the conditional expected value of $\xi $ given B is defined by
$E[\xi |B]=\int _{0}^{+\infty }M\{\xi \geq r|B\}dr-\int _{-\infty }^{0}M\{\xi \leq r|B\}dr$
provided that at least one of the two integrals is finite.
References
Wikimedia Commons has media related to Uncertainty Theory.
Sources
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• Cuilian You, Some Convergence Theorems of Uncertain Sequences, Mathematical and Computer Modelling, Vol.49, Nos.3-4, 482-487, 2009.
• Yuhan Liu, How to Generate Uncertain Measures, Proceedings of Tenth National Youth Conference on Information and Management Sciences, August 3–7, 2008, Luoyang, pp. 23–26.
• Baoding Liu, Uncertainty Theory, 4th ed., Springer-Verlag, Berlin, 2009
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• Yi Peng, U-Curve and U-Coefficient in Uncertain Environment, Proceedings of the Eighth International Conference on Information and Management Sciences, Kunming, China, July 20–28, 2009, pp. 815–820.
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| Wikipedia |
\begin{document}
\settowidth{\mywidth}{ab} \title{The Gradient Flow of Infinity-Harmonic Potentials} \author{Erik Lindgren, Peter Lindqvist}
\date{\today} \maketitle
{\small \textsc{Abstract:} \textsf{We study the streamlines of $\infty$-harmonic functions in planar convex rings. We include convex polygons. The points where streamlines can meet are characterized: they lie on certain curves.
The gradient has constant norm along streamlines outside the set of meeting points, the \emph{infinity-ridge}. }}
\tableofcontents
{\small \textsf{AMS Classification 2010}: 49N60, 35J15, 35J60, 35J65, 35J70.}
{\small \textsf{Keywords}: Infinity-Laplace Equation, streamlines, convex rings, infinity-potential function} \section{Introduction} The $\infty$-Laplace Equation $$
\Delta_{\infty}u\,\equiv\,\sum_{i,j}\frac{\partial u}{\partial x_i} \frac{\partial u}{\partial x_j}\frac{\partial^2 u}{\partial x_i \partial x_j}\,=\,0
$$ was introduced by G. Aronsson in 1967 (cf. \cite{A1}) to produce optimal Lip\-schitz extensions of boundary values. It has been extensively studied. Some of the highlights are \begin{itemize} \item Viscosity solutions for $\Delta_{\infty}$, \cite{BDM} \item Uniqueness, \cite{J} \item Differentiability, \cite{S}, \cite{ESa} and \cite{ES} \item Tug-of-War (connection with stochastic game theory), \cite{PSW} \end{itemize} We are interested in the two-dimensional equation $$
\Bigl(\frac{\partial u}{\partial x_1}\Bigr)^{\!2}\frac{\partial ^2u}{\partial x_1^2}\,+\,2\, \frac{\partial u}{\partial x_1}\frac{\partial u}{\partial x_2}\frac{\partial ^2u}{\partial x_1 \partial x_2}\,+\, \Bigl(\frac{\partial u}{\partial x_2}\Bigr)^{\!2}\frac{\partial ^2u}{\partial x_2^2}\,=\,0
$$
in so-called convex ring domains $G = \Omega\setminus K$. Here $\Omega$ is a bounded convex domain in $\mathbb{R}^2$ and $K\Subset \Omega$ is a closed convex set.
We continue our investigation in \cite{LL} of the $\infty$-potential $u_\infty$, which is the unique solution in $C(\overline G)$ of the boundary value problem $$
\begin{cases} \Delta_{\infty}u\,=\,0\qquad\text{in}\qquad G\\ \phantom{ \Delta_{\infty}}
u\,=\,0\qquad\text{on}\qquad \partial \Omega\\\phantom {\Delta_{\infty}}
u\,=\,1 \qquad\text{on}\qquad \partial K.
\end{cases}
$$ In \cite{LL} we proved that the \emph{ascending} streamlines, the solutions $\boldsymbol{\alpha}= (\alpha_1,\alpha_2)$ of
$$
\frac{d\boldsymbol{\alpha}(t)}{dt} = +\nabla u_\infty (\boldsymbol{\alpha}(t)), \quad 0\leq t < T_{\boldsymbol{\alpha}}
$$
with given initial point $\boldsymbol{\alpha}(0)\in \overline \Omega\setminus K$, are unique and terminate at $\partial K$. (The descending ones are not!) Streamlines may meet and then continue along a common arc. Uniqueness prevents crossing streamlines.
Along a streamline one would expect that the speed $|\nabla u_\infty (\boldsymbol{\alpha})|$ is constant. Indeed,
$$
\frac{d}{dt} |\nabla u_\infty (\boldsymbol{\alpha}(t))|^2 = 2\,\Delta_\infty u_\infty (\boldsymbol{\alpha} (t)) = 0,
$$
but the calculation requires second derivatives. The main difficulty is the lack of second derivatives. Although, the second derivatives are known to exist almost everywhere with respect to the Lebesgue area, see \cite{KZZ} for this new result, this is of little use since the area of a streamline is zero. In \cite{LL} it was shown that the above calculation fails: for most streamlines the speed is not constant the whole way up to $\partial K$. (We shall see that the speed is constant from the initial point till the streamline meets another streamline.)
We use the approximation with the (unique) solution of the $p$-Laplace equation
$$
\Delta_p u = \ddiv (|\nabla u|^{p-2}\nabla u) = 0, \qquad p > 2.
$$
in $G$ with the same boundary values as $u_{\infty}$.
We shall use several facts about these $p$-harmonic functions due to J. Lewis, cf. \cite{L}. It is decisive that the \emph{level curves} $\{u_p(x)=c\}$ are convex and that $\Delta u_p \leq 0$. See Section \ref{sec:prel} for more details.
We also need the facts that (i) $\nabla u_p\to \nabla u_\infty$ in $L^2_\text{loc}$ and (ii) the family $\{|\nabla u_p|\}$ is locally equicontinuous. (Notice that we wrote $|\nabla u_p|$, not $\nabla u_p$.) We extract a proof of this from the recent pathbreaking work by H. Koch, Y. R-Y. Zhang and Y. Zhou in \cite{KZZ}, complementing their results by applying a simple device, due to Lebesgue in \cite{Leb}, to the norm $|\nabla u_p|$ of the quasiregular mapping
$$
\frac{\partial u_p}{\partial x_1}- i\frac{\partial u_p}{\partial x_2}, \quad i^2=-1.
$$The quasiregularity was obtained by B. Bojarski and T. Iwaniec in \cite{BI}.
We prove the following basic result in Section \ref{sec:eqcont}.
\begin{thm}[Non-decreasing speed] \label{thm:speed} Let $\boldsymbol{\alpha}_\infty = \boldsymbol{\alpha}_\infty(t)$, $0\leq t\leq T$, be a streamline of $u_\infty$, i.e.,
$$
\frac{d\boldsymbol{\alpha}_\infty(t)}{dt} = \nabla u_\infty (\boldsymbol{\alpha}_\infty(t)), \quad 0\leq t< T,
$$
and $\boldsymbol{\alpha}_\infty(0)\in \partial \Omega$, $\boldsymbol{\alpha}_\infty(T)\in \partial K$. Then the function $u_\infty(\boldsymbol{\alpha}_\infty(t))$ is convex when $0\leq t\leq T$. In particular, the speed $|\nabla u_\infty (\boldsymbol{\alpha}_\infty(t))|$, is a non-decreasing function of $t$.
\end{thm}
Combining this with a result in the opposite direction (cf. Lemma 12 in \cite{LL}), we can control the meeting points so that these lie on a few specific streamlines, here called attracting streamlines.
\paragraph{Polygons.} To avoid a complicated description, we begin with a convex polygon as $\Omega$ with $N$ vertices $P_1,P_2,\ldots, P_N$ (set $P_{N+1}=P_1$ for convenience). With $P_k = \boldsymbol{\gamma}_k(0)$ as initial point there is a unique streamline
$$
\boldsymbol{\gamma}_k = \boldsymbol{\gamma}_k(t), \quad 0\leq t \leq T_k,
$$
with terminal point $\boldsymbol{\gamma}_k(T_k)$ on $\partial K$. The $$\text{\emph{attracting streamlines} are}\qquad \boldsymbol{\gamma}_1, \boldsymbol{\gamma}_2, \ldots, \boldsymbol{\gamma}_N.$$
Occasionally, some of them meet and then share a common arc up to $\partial K$. The collection of all the points on the attracting streamlines is called the $\infty$-\emph{ridge} and is denoted by $\Gamma$, i.e.,
$$
\Gamma = \bigcup_{k=1}^N \{\boldsymbol{\gamma}_k(t): \,\, 0\leq t\leq T_k\}.
$$ It seems to play a similar role for the $\infty$-Laplace Equation as the (ordinary) ridge does for the Eikonal Equation.
Before meeting any other streamline, a streamline $\boldsymbol{\alpha}$ either meets an attracting streamline or hits the upper boundary $\partial K$. We formulate this as a theorem, proved in Section \ref{sec:poly}.
\begin{thm}\label{thm:mainpoly} The speed $|\nabla u_\infty(\boldsymbol{\alpha}(t))|$ is constant along the streamline $\boldsymbol{\alpha}$ from the initial point on $\partial \Omega$ until it meets one of the attracting streamlines $\boldsymbol{\gamma}_k$, after which the speed is non-decreasing. It cannot meet any other streamline before it meets an attracting one. \end{thm}
Thus there are no meeting points in $G\setminus \Gamma$, i.e., they all lie on the attracting streamlines $\boldsymbol{\gamma}_1, \boldsymbol{\gamma}_2, \ldots, \boldsymbol{\gamma}_N$. In other words, there is no branching outside the $\infty$-ridge $\Gamma$.
\paragraph{General Domains.} The polygon has a piecewise smooth boundary and at the vertices $|\nabla u_\infty(P_k)|= 0$. Thus the attracting streamlines start at the points of minimal speed. Similar results hold when $\Omega$ is no longer a polygon, but now we have to assume that the following holds:
\noindent\textbf{Assumptions:} \begin{enumerate} \item\emph{$\nabla u_\infty$ is continuous in $\overline \Omega\setminus K$, in particular along $\partial \Omega$}.\footnote{For example, if $\partial \Omega$ is piecewise $C^2$, then the gradient is continuous in $\overline \Omega\setminus K$, see Section \ref{sec:prel}.}
\item \emph{On $\partial \Omega$, the continuous function $|\nabla u_\infty|$ has a finite number of local minimum points, say $P_1,P_2,\ldots, P_N$, and a finite number of local maximum points.} \end{enumerate}
Again, the streamlines with the initial points $P_k$ are called \emph{attracting streamlines}:
$$
\boldsymbol{\gamma}_k = \boldsymbol{\gamma}_k(t), \quad 0\leq t\leq T_k ; \quad \boldsymbol{\gamma}_k(0)=P_k.
$$
The $\infty$-\emph{ridge} is again
$$
\Gamma = \bigcup_{k=1}^N \{\boldsymbol{\gamma}_k(t): \,\, 0\leq t\leq T_k\}.
$$
Theorem \ref{thm:mainpoly} holds also in this setting. As a consequence, streamlines cannot meet, except on $\Gamma$. The theorem below is proved in Section \ref{sec:general}.
\begin{thm}\label{thm:general} The speed $|\nabla u_\infty(\boldsymbol{\alpha}(t))|$ is constant along a streamline $\boldsymbol{\alpha}$ from the initial point on $\partial \Omega$ until it meets one of the attracting streamlines $\boldsymbol{\gamma}_k$. It cannot meet any other streamline before it meets an attracting one.
\end{thm}
The situation when $|\nabla u_{\infty}|$ is constant on some arc on $\partial \Omega$ can happen even for a rectangle, but does not cause extra complications.
\begin{prop} If the speed $|\nabla u_{\infty}|$ is constant along a boundary arc $\overline{ab}$, then the streamlines with initial points on the arc are non-intersecting segments of straight lines. They meet no other streamlines in $G$, except possibly when the initial point is $a$ or $b$.
\end{prop}
This follows from Lemma \ref{lem:constant} and Lemma \ref{lem:trilem}. It allows us to relax assumption 2 to include boundary arcs with constant local maximum speed:
\begin{itemize}
\item[2*.]
\emph{The local maxima and minima of $|\nabla u_{\infty}|$ on $\partial \Omega$ are attained along at most finitely many closed subarcs, which may degenerate to points.}
\end{itemize}
The definition of the attracting streamlines must be amended if the speed attains a local minimum along a boundary arc $\overline{ab}$: it contributes with \emph{two} attracting streamlines, namely the ones with initial points at $a$ and $b$.
\begin{rem} The behavior of the streamlines suggests that the $\infty$-potential is smooth outside the $\infty$-ridge $\Gamma$.
\end{rem}
\paragraph{Examples.}
We mention some examples. \begin{figure}
\caption{The streamlines of $u_\infty$ when $\Omega$ is the square in Example \ref{ex:square}.}
\label{fig:square}
\end{figure}
\begin{ex}\label{ex:square}
Let $\Omega$ be the square $$ -1<x_1<1, \quad -1<x_2<1, \quad $$ and $K$ the origin. The attracting streamlines are the four half-diagonals, constituting the $\infty$-ridge $$
\Gamma = \{(x_1,x_2): \quad x_1=\pm x_2, |x_1|\leq 1, |x_2|\leq 1\}. $$ All streamlines meet at a diagonal, except the four segments along the coordinate axes. See Figure \ref{fig:square}. \end{ex}
\begin{ex}\label{ex:modsq} Let $K$ be the origin and $\Omega$ the square in Example \ref{ex:square} which is truncated in the following symmetric way: in the south west corner we have removed the triangle with corners $(-1,-1), (-1+\delta,-1)$ and $(-1,-1+\delta)$, for some small $\delta$. See Figure \ref{fig:modsq}. We only describe the behavior in the south west quarter of $\Omega$.
The attracting streamlines are those starting in $(-1+\delta,-1)$ and $(-1,-1+\delta)$ (in blue). The only streamlines that do not meet any other before reaching origin, are the medians (in red). Any other streamline will meet one of the attracting streamlines. The streamline starting in the middle of $(-1+\delta,-1)$ and $(-1,-1+\delta)$ (in red) will be a straight line to the origin and will be joined by the attracting streamlines from both sides before terminating at the origin.
\end{ex}
\begin{figure}
\caption{The truncated square in Example \ref{ex:modsq} and some possible streamlines.}
\label{fig:modsq}
\end{figure}
\section{Preliminaries} \label{sec:prel} $\Omega$ is a bounded convex domain in $\mathbb{R}^2$ and $K\Subset \Omega$ is a compact and convex set, which may reduce to a point. We study the equation in the convex ring $G=\Omega\setminus K$. We assume the following \emph{normalization}:
$$\boxed{\dist(\partial \Omega,K)=1.}$$
The boundary value problem $$
\begin{cases} \Delta_{\infty}u\,=\,0\qquad\text{in}\qquad G,\\ \phantom{ \Delta_{\infty}}
u\,=\,0\qquad\text{on}\qquad \partial \Omega,\\\phantom {\Delta_{\infty}}
u\,=\,1 \qquad\text{on}\qquad \partial K,
\end{cases}
$$ has a unique solution $u_\infty \in C(\overline G)$ in general. By \cite{ESa}, $\nabla u_\infty$ is locally H\"older continuous in $G$. We will assume that also $\nabla u_\infty\in C(\overline\Omega\setminus K)$. This is fulfilled if for instance $\partial \Omega$ has a piecewise $C^2$ regular boundary. See Lemma 2 and Theorem 2 in \cite{HL}, Theorem 7.1 in \cite{MPS} and Theorem 1 in \cite{WY}.
In \cite{LL} it was established that, for a given initial point $\xi_0\in \partial\Omega$, the gradient flow $$ \begin{cases} \displaystyle\frac{d \boldsymbol{\alpha} (t)}{d t}& =\, +\nabla u_\infty(\boldsymbol{\alpha} (t)),\quad 0\leq t< T, \\ \boldsymbol{\alpha}(0)&=\,\xi_0, \end{cases} $$
has a unique solution $\boldsymbol{\alpha}=\boldsymbol{\alpha}(t)$, which terminates at some point $\boldsymbol{\alpha}(T)$ on $\partial K$. (Some caution is required if $|\nabla u_\infty(\xi_0)| = 0$.) We say that $\boldsymbol{\alpha}$ is a \emph{streamline}. Although unique, two streamlines may meet, join, and continue along a common arc.
We shall employ the $p$-harmonic approximation $$
\begin{cases} \Delta_{p}u_p\,=\,0\qquad\text{in}\qquad G,\\ \phantom{ \Delta_{p}}
u_p\,=\,0\qquad\text{on}\qquad \partial \Omega,\\\phantom {\Delta_{p}}
u_p\,=\,1 \qquad\text{on}\qquad \partial K,
\end{cases}
$$
for $\mathbf{p>2}$. It is known that $u_p\in C(\overline G)$ and it takes the correct values (in the classical sense) at each boundary point. We shall need the following results from \cite{L} (see also \cite{Ja}):
\begin{enumerate}
\item The level curves $\{u_p=c\}$ are convex, if $0\leq c\leq 1$,
\item $u_p\nearrow u_\infty$ uniformly in $\overline G$,
\item $|\nabla u_p|\neq 0$ in $G$,
\item $u_p$ is real analytic in $G$,
\item $\Delta u_p\leq 0$.
\end{enumerate}
The streamlines of $u_p$ do not meet in $G$. This is due to the regularity of $u_p$ and the Picard-Lindel\"of theorem. Properties 1), 3), and 5) are preserved at the limit $p=\infty$. Especially, $\nabla u_{\infty} \neq 0$ in $G$.
We keep the \emph{normalization} $\dist (\partial\Omega, K) = 1$. Then $|\nabla u_\infty|\leq 1$, but we also need a uniform bound for $|\nabla u_p|$.
The bound \begin{equation} \label{eq:grad1}
|\nabla u_p|\leq 1 \quad \text{on }\partial \Omega. \end{equation}
follows by comparison with the distance function
$$
\delta(x)=\dist(x,\partial \Omega).
$$
In a convex domain, $\delta$ is a supersolution of the $p$-Laplace equation. Since
$$
0\leq u_p(x)\leq \delta(x)\quad \text{on } \partial G,
$$
the same inequality also holds in $G$. In general, $|\nabla u_p|$ is unbounded (but $|\nabla u_\infty|\leq 1$), so we have to consider a subdomain, say $\{u_p < c\}$.
\begin{lemma}\label{lem:gradbound} The uniform bound
\begin{equation}
\label{eq:gradboundlem}
|\nabla u_p(x)|\leq \Bigl(\frac{1}{1-c}\Bigr)^\frac{1}{p-2}
\end{equation}
holds when $u_p(x)\leq c$, $0<c<1$.
\end{lemma}
\begin{proof}
Let $\Upsilon_p(c)$ denote the level curve $\{u_p=c\}$ and
$$
\delta_p(x) = \dist (x,\Upsilon_p(c)).
$$
Since $|\nabla u_p|$ obeys the maximum principle and $|\nabla u_p|\leq 1$ on $\partial \Omega$ by \eqref{eq:grad1}, it is enough to control $|\nabla u_p|$ on $\Upsilon_p(c)$. We see that
\begin{equation}
\label{eq:grad2}
c\leq u_p(x)\leq c+(1-c)\frac{\delta_p(x)}{\dist(\Upsilon_p(c), \partial K)}
\end{equation}
on $\Upsilon_p(c)$ and on $\partial K$, i.e., on the boundary of $\{1 >u_p>c\}$. Again, the majorant is a supersolution to the $p$-Laplace equation, and hence \eqref{eq:grad2} holds in $\{1 >u_p>c\}$ by the comparison principle. It follows that
\begin{equation}
\label{eq:gradbound}
|\nabla u_p(x)|\leq \frac{1-c}{\dist(\Upsilon_p(c),\partial K)},
\end{equation}
on\footnote{Since $u_p\nearrow u_\infty$, $\dist(\Upsilon_p(c),\partial K)$ increases with $p$. Thus we get an upper bound independent of $p$. This is sufficient for our purpose.} $\Upsilon_p(c)$.
To get the explicit upper bound in \eqref{eq:gradboundlem}, we assume that $x_0\in \partial K$ is a point at which the distance $\dist(\Upsilon_p(c),\partial K)$ is attained. Let $R$ be the radius of the largest ball $B_R(x_0)\subset \Omega$. Then
$$
u_p(x)\geq 1-\left(\frac{|x-x_0|}{R}\right)^\frac{p-2}{p-1}\quad \text{in }B_R(x_0)\setminus K
$$
by comparison. Here the minorant is $p$-harmonic in $B_R(x_0)\setminus \{x_0\}$. Now
$$
1-\left(\frac{|x-x_0|}{R}\right)^\frac{p-2}{p-1} = c \iff |x-x_0|=R(1-c)^{1+\frac{1}{p-2}} = r_c
$$
and clearly $\dist(\Upsilon_p(c),\partial K)\geq r_c$. We have by \eqref{eq:gradbound}
$$
|\nabla u_p(x)|\leq \frac{1}{R(1-c)^\frac{1}{p-2}}.
$$
To conclude, use $R\geq \dist(\partial\Omega, \partial K)=1$.
\end{proof}
\section{Equicontinuity of $|\nabla u_p|$}\label{sec:eqcont}
We shall prove that $$
\lim_{p\to \infty}{|\nabla u_p|} = |\nabla u_\infty| $$ locally \emph{uniformly} in $G$. From \cite{KZZ} we can extract the following important properties: If $D\Subset G$, then \[\tag{\textbf{I}}
\iint_D |\nabla u_p-\nabla u_\infty|^2 \, dx_1 dx_2\to 0,\quad \text{as }p\to \infty,\]
\begin{equation}\tag{\textbf{J}}
\iint_D |\nabla (|\nabla u_p|^2)|^2 \, dx_1 dx_2\leq M_D<\infty,\end{equation}
for all (large) $p$.
The constant $M_D$ depends on $\|\nabla u_p\|_{L^\infty(E)}$, where $D\Subset E\Subset G$, and $\dist (D, \partial G)$, but not on $p$.
In \cite{KZZ} the estimates were derived for solutions $u^\varepsilon$ of the auxiliary equation $$ \Delta_\infty u^\varepsilon + \varepsilon \Delta u^\varepsilon = 0 $$ while we use $\Delta_pu_p = 0$ written as $$
\Delta_\infty u_p +\frac{1}{p-2}\,|\nabla u_p|^2\Delta u_p = 0. $$ The advantage of our approach is that the inequality $\Delta u_p\leq 0$ is available in convex domains for $p\geq 2$.
The conversion from $u^\varepsilon$ to $u_p$ requires only obvious changes. Formally, the factor $\varepsilon$ in front of an integral in \cite{KZZ} should be moved in under the integral sign and then replaced by $|\nabla u_p|^2/(p-2)$, upon which every $u^\varepsilon$ be replaced by $u_p$. This procedure is explained in our Appendix.
In order to prove that the family $\{|\nabla u_p|\}$ is locally equicontinuous, we shall use a device due to Lebesgue in \cite{Leb}. A function $f\in C(\overline B_R)\cap W^{1,2}(B_R)$ is monotone (in the sense of Lebesgue) if $$ \osc_{\partial B_r}f = \osc_{\overline B_r} f, \quad 0<r<R, $$ where $B_r$ are concentric discs. For such a function \begin{equation}\label{eq:lebosc}
\Bigl(\osc_{B_r} f\Bigr)^2\ln \frac{R}{r}\leq \pi \iint_{B_R} |\nabla f|^2 \, dx_1 dx_2. \end{equation}
The proof is merely an integration in polar coordinates, cf. \cite{Leb}. We shall apply this oscillation lemma on the function $f = |\nabla u_p|^2$. It was shown by Bojarski and Iwaniec in \cite{BI} that the mapping
$$
\frac{\partial u_p}{\partial x_1}- i\,\frac{\partial u_p}{\partial x_2}, \quad i^2=-1,
$$
is quasiregular. That property implies that its norm $|\nabla u_p|$ satisfies the maximum principle, and, where $|\nabla u_p|\neq 0$, also the minimum principle. Thus $|\nabla u_p|$ is monotone. So is $|\nabla u_p|^2$. From \eqref{eq:lebosc} we obtain
$$
\Bigl(\osc_{B_r}\{|\nabla u_p|^2\}\Bigr)^2\ln \frac{R}{r}\leq \pi \iint_{B_R} |\nabla (|\nabla u_p|^2)|^2 \, dx_1 dx_2.
$$
The uniform bound in (2) and a standard covering argument for compact sets yields the following result.
\begin{thm} \label{thm:eqcont}(Equicontinuity) Let $D\Subset G$. Given $\varepsilon>0$, there is $\delta = \delta (\varepsilon,D)$ such that the inequality
$$
\Big||\nabla u_p(x)|-|\nabla u_p(y)|\Big|<\varepsilon \quad \text{when $|x-y|<\delta$}, \quad x,y\in D,
$$
holds simultaneously for all $p > 2$.
\end{thm}
Since $\nabla u _p\to \nabla u_\infty$ in $L^2_\text{loc}(G)$ we can use Ascoli's theorem to conclude that
$$
\lim_{p\to \infty}|\nabla u_p| = |\nabla u_\infty|
$$
\emph{locally uniformly}. (More accurately, we have to extract a subsequence in Ascoli's theorem, but since the limit $|\nabla u_\infty|$ is unique, this precaution is not called for here.)
\textbf{Caution:} The more demanding convergence $\nabla u_p\to \nabla u_\infty$ holds a.e., but perhaps \emph{not} locally uniformly.
Let us finally mention that the \emph{uniform} convergence is not global. For example, in the ring $0<|x|<1$ we have
$$
u_p(x)=1-|x|^\frac{p-2}{p-1}, \quad u_\infty = 1-|x|.
$$
Now $|\nabla u_p|$ is not even bounded near $x=0$. Thus the convergence cannot be uniform in the whole ring.
\section{Convergence of the Streamlines}
In this section, we study the convergence of the streamlines and prove Theorem \ref{thm:speed}.
It is plain that the level curves $\{u_p=c\}$ converge to the level curves $\{u_\infty = c\}$. However, the convergence of the streamlines requires a more sophisticated proof. (The problem is the identification of the limit as an $\infty$-streamline.)
Suppose that we have the streamlines $\boldsymbol{\alpha}_p$ and $\boldsymbol{\alpha}_\infty$ having the same initial point $\boldsymbol{\alpha}_p(0)=\boldsymbol{\alpha}_\infty(0)=x_0$. Now
$$
\frac{d\boldsymbol{\alpha}_p(t)}{dt} = \nabla u_p (\boldsymbol{\alpha}_p(t)), \quad \frac{d\boldsymbol{\alpha}_\infty(t)}{dt} = \nabla u_\infty (\boldsymbol{\alpha}_\infty(t))
$$
when $0 < t < T_p$, where $u_p(\boldsymbol{\alpha}_p(T_p))= 1$. Thus
$$
\boldsymbol{\alpha}_p(t_2)-\boldsymbol{\alpha}_p(t_1) = \int_{t_1}^{t_2} \nabla u_p(\boldsymbol{\alpha}_p(t)) dt. $$ Using the bound $$
|\nabla u_p|\leq \Bigl(\frac{1}{1-c}\Bigr)^{\frac{1}{p-2}} , \quad \text{when }u_p\leq c, $$ in Lemma \ref{lem:gradbound} we see that \begin{equation} \label{eq:alpha3}
| \boldsymbol{\alpha}_p(t_2)-\boldsymbol{\alpha}_p(t_1)|\leq \Bigl(\frac{1}{1-c}\Bigr)^\frac{1}{p-2}|t_2-t_1| \end{equation} as long as the curves are below the level $u_p=c$, i.e., $u_p(\boldsymbol{\alpha}(t_2))\leq c$. In particular, the bound is valid in the domain $\{u_\infty<c\}$, where $c<1$. Thus, the family of curves is locally equicontinuous. By Ascoli's theorem we can extract a sequence $p_j\to \infty$ such that $$ \boldsymbol{\alpha}_{p_j}(t)\to \boldsymbol{\alpha}(t) $$ uniformly in every domain $\{u_\infty<c\}$. Here $\boldsymbol{\alpha}(t)$ is some curve with initial point $\boldsymbol{\alpha}(0)=x_0$.
The endpoint of $\boldsymbol{\alpha}$ is on $\partial K$. Indeed, let $t_p=t_p(c)$ denote the parameter value at which $u_p(\boldsymbol{\alpha}_p(t_p))=c$. Take any convergent sequence, say $t_p\to t^*$. Then $$ c=\lim_{p\to\infty} u_p(\boldsymbol{\alpha}_p(t_p)) = u_\infty(\boldsymbol{\alpha}(t^*)). $$ Thus $t^* = t_\infty(c)$. Then $t_p(c)\to t_\infty(c)$ for all $c$.
By \eqref{eq:alpha3} $$
| \boldsymbol{\alpha}(t_2)-\boldsymbol{\alpha}(t_1)|\leq |t_2-t_1|. $$ Rademacher's theorem for Lipschitz continuous functions implies that $\boldsymbol{\alpha}(t)$ is differentiable at a.e. $t$.
We claim that $\boldsymbol{\alpha} = \boldsymbol{\alpha}_\infty$. Since they start at the same point, the uniqueness of $\infty$-streamlines shows that it is enough to verify $$ \frac{d\boldsymbol{\alpha}(t)}{dt} = \nabla u_\infty (\boldsymbol{\alpha}(t)). $$ To this end, we shall employ the convex functions $F_p(t)=u_p(\boldsymbol{\alpha}_p(t))$. Indeed, $$
\frac{dF_p(t)}{dt} = \Big\langle \nabla u_p(\boldsymbol{\alpha}_p(t)), \frac{d\boldsymbol{\alpha}_p(t)}{dt}\Big\rangle = |\nabla u_p(\boldsymbol{\alpha}_p(t))|^2 $$ and $$
\frac{d^2F_p(t)}{dt^2} = 2\,\Delta_\infty u_p(\boldsymbol{\alpha}_p(t)) = -\frac{2}{p-2}\,\Delta u_p(\boldsymbol{\alpha}_p(t))\,|\nabla u_p(\boldsymbol{\alpha}_p(t))|^2. $$ By Lewis's theorem, $\Delta u_p\leq 0$ in convex ring domains, if $p\geq 2$. Thus, $$ \frac{d^2F_p(t)}{dt^2}\geq 0 $$ and so \emph{the function $F_p(t)$ is convex}. The convergence $$ F_{p}(t) = u_{p}(\boldsymbol{\alpha}_{p}(t))\to u_\infty(\boldsymbol{\alpha}(t)) = F(t) $$ is at least locally uniform, when $p$ takes the values $p_1, p_2, p_3,\ldots$ extracted above. Also the limit $F(t)$ is convex, of course.
We have the locally uniform convergence $$
|\nabla u_{p}(\boldsymbol{\alpha}_p(t))|^2\to |\nabla u_\infty(\boldsymbol{\alpha}(t))|^2, $$ which follows from Theorem \ref{thm:eqcont} by writing $$
|\nabla u_p(\boldsymbol{\alpha}_p(t))|-|\nabla u_\infty(\boldsymbol{\alpha}(t))| = |\nabla u_p(\boldsymbol{\alpha}_p(t))|-|\nabla u_p(\boldsymbol{\alpha}(t))|+|\nabla u_p(\boldsymbol{\alpha}(t))|-|\nabla u_\infty(\boldsymbol{\alpha}(t))|. $$ Thus, $$
\frac{dF_p(t)}{dt}=|\nabla u_{p}(\boldsymbol{\alpha}_p(t))|^2\,\to\, |\nabla u_\infty(\boldsymbol{\alpha}(t))|^2. $$
It follows that\footnote{$\int |\nabla u_\infty(\boldsymbol{\alpha}(t))|^2\phi(t) dt\leftarrow \int F_p'(t)\phi(t) dt = -\int F_p(t)\phi'(t) dt \to -\int F(t) \phi'(t) dt$} $F'(t) = |\nabla u_\infty(\boldsymbol{\alpha}(t))|^2$ for a.e. $t$. We also have by the chain rule $$ \frac{dF(t)}{dt} = \Big\langle\nabla u_\infty(\boldsymbol{\alpha}(t)), \frac{d\boldsymbol{\alpha}}{dt}\Big\rangle $$ a.e., since $\frac{d\boldsymbol{\alpha}}{dt}$ exists for a.e. $t$.
We have arrived at the identity $$
|\nabla u_\infty(\boldsymbol{\alpha}(t))|^2 = \Big\langle\nabla u_\infty(\boldsymbol{\alpha}(t)), \frac{d\boldsymbol{\alpha}}{dt}\Big\rangle $$ valid for a.e. $t$. From $$
\boldsymbol{\alpha}_p(t_2)-\boldsymbol{\alpha}_p(t_1) \leq \int_{t_1}^{t_2} |\nabla u_p(\boldsymbol{\alpha}_p(t)) |dt, $$ we get $$
\boldsymbol{\alpha}(t_2)-\boldsymbol{\alpha}(t_1) \leq \int_{t_1}^{t_2} |\nabla u_\infty(\boldsymbol{\alpha}(t)) |dt, $$
and, hence for a.e. $t$
$$
\Big\vert\frac{d\boldsymbol{\alpha}(t)}{dt}\Big\vert \leq |\nabla u_\infty(\boldsymbol{\alpha}(t)) |.
$$
We conclude that in the Cauchy-Schwarz inequality
$$
|\nabla u_\infty(\boldsymbol{\alpha}(t)) |^2 = \Big\langle\nabla u_\infty(\boldsymbol{\alpha}(t)), \frac{d\boldsymbol{\alpha}}{dt}\Big\rangle \leq |\nabla u_\infty(\boldsymbol{\alpha}(t))|\Big\vert \frac{d\boldsymbol{\alpha}}{dt}\Big\vert\leq |\nabla u_\infty(\boldsymbol{\alpha}(t)) |^2
$$
we have equality. It follows that
$$
\frac{d\boldsymbol{\alpha}}{dt}= \nabla u_\infty(\boldsymbol{\alpha}(t))
$$
for a.e. $t$. In fact, it holds everywhere because now the identity
$$
\boldsymbol{\alpha}(t_2)-\boldsymbol{\alpha}(t_1) = \int_{t_1}^{t_2} \nabla u_\infty(\boldsymbol{\alpha}(t)) dt
$$
can be differentiated. This concludes our proof of the fact $\boldsymbol{\alpha} = \boldsymbol{\alpha}_\infty$.
We see that the tangent $\frac{d\boldsymbol{\alpha}}{dt}$ is continuous. The proof reveals that the convex functions $F_p\to F$ uniformly and hence $F$ is convex as well. Therefore, its derivative
$$
F'(t)= |\nabla u_\infty(\boldsymbol{\alpha}(t))|^2
$$ is non-decreasing. In other words, $|\nabla u_\infty|^2$ is non-decreasing along the limit streamline.
This proves Theorem \ref{thm:speed}.
\section{Quadrilaterals and Triangles}
Curved quadrilaterals and triangles, bounded by arcs of streamlines and level curves, are useful building blocks. It is tentatively understood that at least the interior of the figures are comprised in $G$; the level arcs can be on $\partial \Omega$ and, occasionally, on $\partial K$.
Recall that the $\infty$-streamline
$$
\boldsymbol{\alpha}(t), \quad 0\leq t\leq T,
$$
with initial point $\boldsymbol{\alpha}(0)=a\in \partial\Omega$ is unique and terminates at $\boldsymbol{\alpha}(T)$ on $\partial K$. On its way, it may (and usually does) meet other streamlines and has common parts with them. By Theorem \ref{thm:speed}, the speed
$$
\left|\frac{d\boldsymbol{\alpha}(t)}{d t}\right| = |\nabla u_\infty (\boldsymbol{\alpha}(t))|
$$
is non-decreasing. Thus we have the bound\footnote{$$|\nabla u_{\infty}(\boldsymbol{\alpha}(T))| = \lim_{t\to T-} |\nabla u_{\infty}(\boldsymbol{\alpha}(t))|$$}
$$
|\nabla u_\infty (\boldsymbol{\alpha}(t_1))|\leq |\nabla u_\infty (\boldsymbol{\alpha}(t_2))|, \quad 0\leq t_1\leq t_2\leq T.
$$
Sometimes the result below (cf. Lemma 12 in \cite{LL}), valid for curved quadrilaterals and triangles, provides us with the reverse inequality, so that we may even conclude that the speed is constant along suitable arcs of streamlines. \begin{figure}
\caption{The quadrilateral $abb'a'$.}
\label{fig:quadbasic}
\end{figure}
\begin{lemma} \label{lem:ll} Suppose that the streamlines $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ together with the level curves $\boldsymbol{\sigma}$ (lower level) and $\boldsymbol{\omega}$ (upper level) form a quadrilateral with vertices $a,b,b'$ and $a'$. If $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ do not meet before reaching $\omega$, then
$$
\max_{\overline{a'b'}} |\nabla u_\infty(\boldsymbol{\omega})|\leq \max_{\overline{ab}} |\nabla u_\infty(\boldsymbol{\sigma})|,
$$
i.e., the maximal speed on the upper level is the smaller one.
\end{lemma}
Suppose now that $\xi\in \overline{ab}$ is a point on the lower level curve $\boldsymbol{\sigma}$ at which
$$
|\nabla u_\infty(\xi)|= \max_{\overline{ab}} |\nabla u_\infty(\boldsymbol{\sigma})| = M.
$$
Let $\boldsymbol{\mu}$ be the streamline that passes through $\xi$. It intersects $\boldsymbol{\omega}$ at some point $\eta\in \overline{a'b'}$ (it may have joined $\boldsymbol{\alpha}$ or $\boldsymbol{\beta}$ before reaching $\eta$). See Figure \ref{fig:quadbasic}. The following result holds:
\begin{lemma}\label{lem:quad} We have $$
|\nabla u_\infty(\boldsymbol{\mu})| = M \quad \text{on } \overline{\xi\eta}. $$ Moreover, $$
\max_{\overline{a'b'}} |\nabla u_\infty(\boldsymbol{\omega})|= \max_{\overline{ab}} |\nabla u_\infty(\boldsymbol{\sigma})|. $$
\end{lemma}
\begin{proof} By Lemma \ref{lem:ll}
$$
|\nabla u_\infty(\xi)|\geq \max_{\overline{a'b'}} |\nabla u_\infty(\boldsymbol{\omega})|\geq |\nabla u_\infty(\eta)|
$$
and the monotonicity of the speed implies
$$
|\nabla u_\infty(\xi)|\leq |\nabla u_\infty(\boldsymbol{\mu}(t))|\leq |\nabla u_\infty(\eta)|
$$
along the arc $\overline{\xi\eta}$ of $\boldsymbol{\mu}$. Thus we have equality.
\end{proof}
We can also formulate a similar result for curved triangles. Suppose that the streamlines $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ together with the level curve $\boldsymbol{\sigma}$ form a curved triangle with vertices $a,b$ and $c$. Assume again that $\xi\in \overline{ab}$ is a point at which
$$
|\nabla u_\infty(\xi)|= \max_{\overline{ab}} |\nabla u_\infty(\boldsymbol{\sigma})| = M.
$$
Let $\boldsymbol{\mu}$ be the streamline that passes through $\xi$. It passes through $c$ (but may have joined $\boldsymbol{\alpha}$ or $\boldsymbol{\beta}$ before reaching $c$). The following result holds:
\begin{cor}\label{cor:tri} For the triangle $a\,b\,c$ we have $$
|\nabla u_\infty(\boldsymbol{\mu})| = M \quad \text{on } \overline{\xi c}. $$ Moreover, $$
|\nabla u_\infty(c)|= \underset{\overline{ab}}{\max} |\nabla u_\infty(\boldsymbol{\sigma})|. $$ \end{cor} \begin{proof} Take $\boldsymbol{\omega}_i$ to be a sequence of level curves approaching $c$ from below. Then apply Lemma \ref{lem:quad} on the quadrilateral formed by $\boldsymbol{\sigma}, \boldsymbol{\omega}_i, \boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ and let $i\to \infty$. \end{proof}
\
\paragraph{The Quadrilateral Rule.}
We provide a practical rule for preventing meeting points. We keep the same notation.
\begin{prop}[Quadrilateral Rule]\label{prop:quad} If $|\nabla u(\boldsymbol{\sigma}(t))|$ is strictly monotone on the arcs $\overline{a\xi}$ and $\overline{\xi b}$ of the level curve $\boldsymbol{\sigma}$ (one of them may reduce to a point), then no streamlines can meet inside the quadrilateral. A streamline with initial point on the arc $\overline{ab}$ (but not $a$ or $b$) has constant speed $|\nabla u_\infty|$ till it meets $\boldsymbol{\alpha}, \boldsymbol{\beta}$ or reaches $\boldsymbol{\omega}$. \end{prop}
\begin{proof} Let $\boldsymbol{\lambda}= \boldsymbol{\lambda}(t)$ be a streamline passing through the point $x\in \overline{\xi b}$, $x\neq \xi$ on the level curve $\boldsymbol{\sigma}$.
\begin{figure}
\caption{Case 1: impossible}
\end{figure}
We have three cases: 1) If $\boldsymbol{\lambda}$ meets $\boldsymbol{\mu}$ at the point $y$, then Lemma \ref{lem:quad} applied on the quadrilateral $x b b'\eta y x$ (or Corollary \ref{cor:tri} if $\boldsymbol{\mu}$ meets $\boldsymbol{\beta}$, so that we have a triangle) implies $$
M=|\nabla u_\infty(\boldsymbol{\lambda})| $$ on the whole arc $\overline{x\eta}$ of $\boldsymbol{\lambda}$ (or until $\boldsymbol{\mu}$ reaches $\boldsymbol{\beta}$). But then $$
|\nabla u_\infty(\xi)| = |\nabla u_\infty(x)|, $$
which contradicts the \emph{strict} monotonicity of $|\nabla u(\boldsymbol{\sigma}(t))|$.
\begin{figure}
\caption{Case 2: possible}
\end{figure}
2) If $\boldsymbol{\lambda}$ meets $\boldsymbol{\beta}$ at $y\in \overline{bb'}$, then Corollary \ref{cor:tri} applied on the triangle $xby$ yields $$
|\nabla u_\infty(\boldsymbol{\lambda})| = \text{constant} $$ on the arc $\overline{xy}$.
\begin{figure}
\caption{Case 3: possible}
\end{figure}
3) If $\boldsymbol{\lambda}$ passes through a point $y\in \overline{\eta b'}$ on the upper level $\boldsymbol{\omega}$, $y\neq \eta$, $y\neq b'$, then Lemma \ref{lem:quad} applied on the quadrilateral $xbb'y$ (or Corollary \ref{cor:tri} in case of a curved triangle) yields $$
|\nabla u_\infty(\boldsymbol{\lambda})| = \text{constant} $$ on the arc $\overline{xy}$.
Finally, if $x$ is chosen from the left level arc $\overline{a\xi}$, the proof consists of three similar cases again. Thus we have established that $\boldsymbol{\lambda}$ has constant speed till it first meets $\boldsymbol{\alpha}, \boldsymbol{\beta}$, or hits $\boldsymbol{\omega}$.
It remains to show that no two streamlines can meet in the quadrilateral. A streamline $\boldsymbol{\lambda}$ passing through the point $x$ at the level curve $\boldsymbol{\sigma}$ has constant speed $$
|\nabla u_{\infty}(x)| = |\nabla u_{\infty}(\boldsymbol{\lambda})| $$
till $\boldsymbol{\lambda}$ meets $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$ or hits $\boldsymbol{\omega}$. But two meetings streamlines must have the same speed, which requires that they pass through $\boldsymbol{\sigma}$ at two points with the same speed $|\nabla u_{\infty}|$. By the \emph{strict} monotonicity of $|\nabla u_{\infty}(\boldsymbol{\sigma})|$, this would require that the points are on different arcs $\overline{a\xi}$ and $\overline{\xi b}$. This is impossible, since no streamlines meet $\boldsymbol{\mu}$. \end{proof}
The Quadrilateral Rule remains true if the monotonicity of $|\nabla u_{\infty}(\boldsymbol{\sigma})|$ is not supposed to be strict. If $|\nabla u_{\infty}(\boldsymbol{\sigma})|$ is constant on some subarc $\overline{cd}$, then the streamlines with initial points on $\overline{cd}$ are non-intersecting straight lines. To see this, we again consider the quadrilateral $a\,b\,b'\,a'$ bounded by $\boldsymbol{\alpha},\boldsymbol{\beta},\boldsymbol{\sigma},\boldsymbol{\omega}.$
\begin{lemma}\label{lem:constant}
Assume that $|\nabla u_\infty (\boldsymbol{\sigma})|$ is constant on the arc $\overline{ab}$. Then no streamlines can meet inside the quadrilateral. Moreover, $|\nabla u_\infty|$ is constant in the quadrilateral and all streamlines are straight lines. \end{lemma}
\begin{proof} By Lemma \ref{lem:quad}, $|\nabla u_{\infty} (\boldsymbol{\omega})|$ is constant on the upper arc $\overline{a'b'}$ .
In particular, $|\nabla u_{\infty}|$ must be constant along $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$. Then $|\nabla u_{\infty}|$ must be constant along any arc of a streamline passing through the quadrilateral. Every point inside the quadrilateral lies on such a streamline. Therefore $|\nabla u_{\infty}|$ is constant in the quadrilateral, which means that it solves the \emph{Eikonal Equation}. Since $u_{\infty}$ is of class $C^1$, we can apply the next proposition
to conclude that all streamlines are non-intersecting straight lines. \end{proof}
\begin{prop} [Eikonal Equation]\label{prop:eikonal} Suppose that $v \in C^1(D)$ is a solution of the Eikonal Equation $|\nabla v| = C$ in the domain $D$, where $C$ denotes a constant. Then the streamlines of $v$ are non-intersecting segments of straight lines. \end{prop}
\begin{proof} A very appeling direct proof is given in Lemma 1 in \cite{A2}. \end{proof}
For the next result we abandon the \emph{strict} monotonicity in Proposition \ref{prop:quad}.
\begin{cor}[Quadrilateral Rule] \label{cor:constant} Assume
that $|\nabla u_\infty (\boldsymbol{\sigma})|$ is monotone on the arc $\overline{ab}$. Then no streamlines can meet inside the quadrilateral. A streamline with initial point on the arc $\overline{ab}$ (but not $a$ or $b$) has constant speed till it meets $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$ or reaches $\boldsymbol{\omega}$. \end{cor}
\begin{proof} Assume that $|\nabla u_{\infty} (\boldsymbol{\sigma})|$ is non-decreasing. Consider the subarc $\overline{x^1x^2}$ on $\boldsymbol{\sigma}$ so that $|\nabla u_{\infty}(x^1)| \leq |\nabla u_{\infty}(x^2)|$, where $x_1 < x_2$. Let $\boldsymbol{\alpha}^j$ be the streamline passing through $x^j$. We claim that $\boldsymbol{\alpha}^1$ does not meet $\boldsymbol{\alpha}^2$ inside the quadrilateral. Indeed, suppose they meet at a point $c$ at the level line $\widetilde \boldsymbol{\omega}$ before reaching $\boldsymbol{\omega}$, where $\widetilde \boldsymbol{\omega}$ intersects $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ at $a''$ and $b''$ respectively. Then Lemma \ref{lem:quad} applied to the quadrilaterals $a\,x^1\,c\,a''$ and $a\,x^2\,c\,a''$ exhibit that the speeds
$$|\nabla u_{\infty}(\boldsymbol{\alpha}^1(t))| = |\nabla u_{\infty}(\boldsymbol{\alpha}^2(t))| = |\nabla u_{\infty}(c)|$$ are constant along the arcs. Again we see that the Eikonal Equation is valid in the triangle
$x^1\,x^2\,c$. At the point $c$ this leads to a contradiction with Proposition \ref{prop:eikonal}. (Thus the eventual point $c$ must lie on $\boldsymbol{\omega}$ and on $\partial K$.)
\end{proof}
\paragraph{The Triangular Rule.} The above results may be formulated for a curved triangle as in Figure \ref{fig:tribasic} (seen as a degenerate quadrilateral). Again, suppose that the streamlines $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ together with the level curve $\boldsymbol{\sigma}$ form a curved triangle with vertices $a,b$ and $c$; $c$ is the meeting point of $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$. Assume that $\xi\in \overline{ab}$ is a point at which
$$
|\nabla u_\infty(\xi)|= \max_{\overline{ab}} |\nabla u_\infty(\boldsymbol{\sigma})| = M.
$$
Let $\boldsymbol{\mu}$ be the streamline that passes through $\xi$. It passes through $c$ (but may have joined $\boldsymbol{\alpha}$ or $\boldsymbol{\beta}$ before reaching $c$).
\begin{figure}
\caption{The curved triangle $abc$.}
\label{fig:tribasic}
\end{figure}
By simply using the results for quadrilaterals, we may deduce the following.
\begin{cor}\label{cor:tricor} If $|\nabla u(\boldsymbol{\sigma}(t))|$ is strictly monotone on the arcs $\overline{a\xi}$ and $\overline{\xi b}$ of the level curve $\boldsymbol{\sigma}$ (one of them may reduce to a point), then no streamlines can meet inside the triangle. A streamline with initial point on the arc $\overline{ab}$ (but not $a$ or $b$) has constant speed $|\nabla u_\infty|$ till it meets $\boldsymbol{\alpha}$ or $ \boldsymbol{\beta}$. \end{cor}
\begin{proof} If two streamlines meet at a point in the triangle we may construct a quadrilateral containing that point by letting $\boldsymbol{\omega}$ be a level curve above $c$. Then Proposition \ref{prop:quad} yields a contradiction. \end{proof}
\begin{lemma}\label{lem:trilem} $|\nabla u_\infty (\boldsymbol{\sigma})|$ cannot be constant on a subarc of $\overline{ab}$, except if $c \in \partial K.$ \end{lemma}
\begin{proof} We can again construct a triangle in which the Eikonal Equation is valid. This yields a contradiction, unless we allow a corner to be outside $G$. \end{proof}
Vi can again abandon the \emph{strict} monotonicity.
\begin{cor}[Triangular Rule] \label{cor:triconstant}
Suppose that $|\nabla u_\infty (\boldsymbol{\sigma})|$ is monotone on the arc $\overline{ab}$ of the level curve $\boldsymbol{\sigma}$. Then no streamlines can meet inside the triangle. A streamline with initial point on the arc $\overline{ab}$ has constant speed till it meets $\boldsymbol{\alpha}$ or $\boldsymbol{\beta}$. \end{cor}
\begin{proof} Reason as in the proof of Corollary \ref{cor:tricor} and apply Corollary \ref{cor:constant}. \end{proof}
\section{Polygons}\label{sec:poly}
Let $\Omega$ be a convex polygon with $N$ vertices $P_1,P_2,\ldots, P_N$ and set $P_{N+1}=P_1$. The gradient $\nabla u_\infty$ is continuous up to the boundary $\partial \Omega$ and especially at the vertices,
$$|\nabla u_\infty(P_j)|=0, \quad j=1,2,\ldots,N.$$
From each vertex $P_j$, there is a unique streamline $\boldsymbol{\gamma}_j$ that terminates on $K$. They are the attracting streamlines.
Let $M_j$ denote a point on the edge $\overline{P_jP_{j+1}}$ at which $|\nabla u_\infty|$ attains its maximum, i.e, $$
|\nabla u_\infty(M_j)|=\max_{\overline{P_jP_{j+1}}} |\nabla u_\infty|. $$ The point divides the edge $\overline{P_jP_{j+1}}$ into two line segments $\overline{P_jM_j}$ and $\overline{M_jP_{j+1}}$. Denote by $\boldsymbol{\mu}_j$ the streamline starting at the point $M_j$.
\begin{lemma}\label{lem:polymon} The normal derivative $$
\frac{\partial u_\infty}{\partial n} = |\nabla u_\infty| $$ is monotone along the half-edges $\overline{P_jM_j}$ and $\overline{M_jP_{j+1}}$ for $j=1,2,\ldots,N$. \end{lemma} \begin{proof}
We arrange it so that the polygon is in the upper half-plane $x_2>0$ and the edge in question is on the $x_1$-axis, say the edge is
$$a\leq x_1\leq b, \quad x_2=0.$$
The convex level curves
$$
\{u_\infty=c\}
$$
approach the $x_1$-axis as $c\to 0$. The shortest distance from the level curve to the edge is attained at some point, say $(x_1(c), x_2(c))$. Choose a sequence $c_j\to 0$ so that $x_1(c_j)\to \xi$ and $x_2(c_j)\to 0$, where $(\xi,0)$ is some point, $a\leq\xi\leq b$ (in fact, $a < \xi < b$). If $\xi >a$, let $a<\xi_1<\xi_2<\xi$ and keep $j$ so large that $\xi_2<x_1(c_j)$. The vertical lines $x_1=\xi_1$ and $x_1=\xi_2$ intersect the level curve $\{u_\infty = c\}$ at the points $(\xi_1,h_1^j)$ and $(\xi_2, h_2^j)$, i.e.
$$
u_\infty(\xi_1,h_1^j)= u_\infty(\xi_2,h_2^j)=c_j.
$$
The convexity of the level curve implies that $h_1^j\geq h_2^j$. (The chord between $(\xi_1,h_1^j)$ and $(x_1(c_j), x_2(c_j))$ must lie inside the set $\{u_\infty\geq c\}$.)
It follows that the difference quotients in the normal direction satisfy
$$
\frac{u_\infty(\xi_1,h_1^j)-u_\infty(\xi_1,0)}{h_1^j}\,\leq \, \frac{u_\infty(\xi_2,h_2^j)-u_\infty(\xi_2,0)}{h_2^j},
$$
since both numerators are $= c_j-0$. As $c_j\to 0$, also $h_1^j\to 0$ and $h_2^j\to 0$. By passing to the limit we obtain $$
|\nabla u_\infty(\xi_1,0)|\,\leq\, |\nabla u_\infty(\xi_2,0)|,\quad \xi_1<\xi_2<\xi $$ as desired.
If $a<\xi<b$ we also obtain the reverse inequality for all $\xi<\xi_1<\xi_2< b$ so that we may conclude the desired result again. It also follows that $(\xi,0)$ is the $M_j$ point of this edge. This excludes that $\xi = a$ or $\xi = b$. \end{proof}
We are now ready to prove our main theorem for polygons.
\begin{proof}[Proof of Theorem \ref{thm:mainpoly}.] Consider the region bounded by $\overline{P_jP_{j+1}}, \boldsymbol{\gamma}_j, \boldsymbol{\gamma}_{j+1}$ and, if $\boldsymbol{\gamma}_j$ does not meet $ \boldsymbol{\gamma}_{j+1}$ also $\partial K$. This can be either a curved triangle (meeting attracting streamlines) or a quadrilateral (the attracting streamlines do not meet). By Lemma \ref{lem:polymon}, $|\nabla u_\infty|$ is monotone along $\overline{P_jM_{j}}$ and $\overline{M_jP_{j+1}}$. Therefore, Corollary \ref{cor:constant} (in the case of a quadrilateral) and Corollary \ref{cor:triconstant} (in the case of a curved triangle) imply that no streamlines can meet (on either side of $\mu_j$)
and that they have constant speed until they meet $\boldsymbol{\gamma}_j$ or $\boldsymbol{\gamma}_{j+1}$, or hit $\partial K.$
\end{proof}
\section{General Domains} \label{sec:general} In this section we assume that $\nabla u_\infty$ is continuous in $\overline \Omega\setminus K$
and that $|\nabla u_\infty|$ has a finite number of local minimum points and maximum points. Denote by $P_1, \ldots, P_N$ (with $P_{N+1}=P_1$ as before) the minimum points. From each $P_j$, there is a unique streamline $\boldsymbol{\gamma}_j$ that terminates in $K$. These streamlines divide $G$ into triangles with corners $P_k, P_k$ and $Q_k$ if $\boldsymbol{\gamma}_k$ and $\boldsymbol{\gamma}_{k+1}$ meet at $Q_k$, and quadrilateras with corners $P_k, P_{k+1}, S_{k+1}$ and $S_{k}$ if $\boldsymbol{\gamma}_k$ and $\boldsymbol{\gamma}_{k+1}$ do not meet but they reach $K$ at the points $S_k$ and $S_{k+1}$. Recall the $\infty$-ridge,
$$
\Gamma = \bigcup_{k=1}^N \{\boldsymbol{\gamma}_k(t), \quad 0\leq T\leq T_k\}.
$$
We give the proof of Theorem \ref{thm:general}.
\begin{proof}[Proof of Theorem \ref{thm:general}.] Consider the region bounded by $\overline{P_jP_{j+1}}, \boldsymbol{\gamma}_j, \boldsymbol{\gamma}_{j+1}$ and perhaps $\partial K$. This can be either a curved triangle or quadrilateral. By construction, $|\nabla u_\infty|$ is monotone along $\overline{P_jM_{j}}$ and $\overline{M_jP_{j+1}}$. Therefore, Corollary \ref{cor:constant} in the case of a quadrilateral and Corollary \ref{cor:triconstant} in the case of a curved triangle imply that no streamlines can meet (on either side of $\mu_j$) and that they are constant until they meet $\boldsymbol{\gamma}_j$ or $\boldsymbol{\gamma}_{j+1}$ or reach $\partial K$.
\end{proof}
\section{Appendix: Estimates of Derivatives of $|\nabla u_p|$}
The fundamental properties
\[\tag{\textbf{I}}\label{eq:eqI}\iint_D |\nabla u_p-\nabla u_\infty|^2 \, dx_1 dx_2\to 0,\quad \text{as }p\to \infty,\]
\begin{equation}\tag{\textbf{J}}\label{eq:eqJ}\iint_D |\nabla (|\nabla u_p|^2)|^2 \, dx_1 dx_2\leq M_D<\infty,\end{equation} for all (large) $p$
used in Section \ref{sec:eqcont} follow directly from \cite{KZZ}, where the corresponding estimates are ingeniously derived for the solution $u^\varepsilon$ of $$ \Delta_\infty u^\varepsilon +\varepsilon\Delta u^\varepsilon = 0. $$ To transcribe the work to the solution $u_p$ of the $p$-Laplace equation $$
\Delta_\infty u_p +\frac{1}{p-2}|\nabla u_p|^2\Delta u_p = 0
$$
one has to replace the constant factor $\varepsilon$ by the \emph{function} $|\nabla u_p|^2/(p-2)$ \emph{under} the integral sign. Below we give just a synopsis of the procedure, referring to the numbering of formulas and theorems in \cite{KZZ}. (The reader is supposed to have access to \cite{KZZ}.)
Formula (2.5) in \cite{KZZ} becomes
$$
-\det(D^2 u_p) = |\nabla |\nabla u_p||^2+\frac{1}{p-2}(\Delta u_p)^2.
$$
Formula (2.7) becomes
$$
I_p(\phi)=\iint_U |\nabla |\nabla u_p||^2 \phi\, dx_1 dx_2 +\frac{1}{p-2}\iint_U (\Delta u_p)^2\phi \,dx_1 dx_2
$$
and (2.8)
$$
I_p(\phi)=\frac12\iint_U \Bigl(\Delta u_p\langle\nabla u_p, \nabla \phi\rangle- \sum_{i,j=1}^2\frac{\partial^2 u_p}{\partial x_i\partial x_j}\frac{\partial u_p}{\partial x_j}\frac{\partial \phi}{\partial x_i} \Bigr) \, dx_1 dx_2 .
$$
Lemma 5.1 is needed only for $\alpha = 2$ (and since $|\nabla u_p|\neq 0$ we can put $\kappa = 0$ in the proof). It becomes \[ \begin{split}
& \iint_U |\nabla |\nabla u_p|^2|^2\xi^2 \, dx_1 dx_2 +\frac{1}{p-2}\iint_U |\nabla u_p|^2(\Delta u_p)^2\xi^2 \,dx_1 dx_2\\
&\leq C(2)\iint_U|\nabla u_p|^4\left(|\nabla \xi|^2+|\xi||D^2\xi|\right) \, dx_1 dx_2. \end{split} \]
This yields Lemma 2.6 and the desired property \eqref{eq:eqJ}, since $|\nabla u_p|$ is locally bounded by Lemma \ref{lem:gradbound}.
Lemma 5.2 is valid with no changes (replace $u^\varepsilon$ with $u_p$), but the proof uses Lemma 5.1 as above. Then Lemma 5.2 implies the flatness estimate in Lemma 2.7: \[ \begin{split}
&\Xint\longminus_{B_r(x)}\left(|\nabla u_p|^2-\langle \nabla P, \nabla u_p\rangle\right)^2 \, dx_1 dx_2\leq C\left(\Xint\longminus_{B_{2r}(x)} |\nabla u_p|^4\, dx_1 dx_2\right)^\frac12 \\
&\times \left(\Xint\longminus_{B_{2r}(x)}\left(\frac{|u_p-P|^2}{r^2}\left(|\nabla P|+|\nabla u_p|\right)^2+\frac{|u_p-P|^4}{r^4}\right) dx_1 dx_2 \right)^\frac12 \end{split} \]
valid for any linear function $P$. This estimate is needed for the proof of Theorem 1.4, when one has to identify the limit of $|\nabla u_p|^2$ in $L^2_\text{loc}$ as $|\nabla u_\infty|^2$. Theorem 1.4 contains our desired property \eqref{eq:eqI}.
\paragraph{Acknowledgments:} Erik Lindgren was supported by the Swedish Research Council, 2017-03736. Peter Lindqvist was supported by The Norwegian Research Council, grant no. 250070 (WaNP).
\noindent {\textsf{Erik Lindgren\\ Department of Mathematics\\ Uppsala University\\ Box 480\\ 751 06 Uppsala, Sweden} \\ \textsf{e-mail}: [email protected]\\
\noindent \textsf{Peter Lindqvist\\ Department of
Mathematical Sciences\\ Norwegian University of Science and
Technology\\ N--7491, Trondheim, Norway}\\ \textsf{e-mail}: [email protected]
\end{document} | arXiv |
Local hidden-variable theory
In the interpretation of quantum mechanics, a local hidden-variable theory is a hidden-variable theory that satisfies the condition of being consistent with local realism. This definition restricts all types of those theories that attempt to account for the probabilistic features of quantum mechanics via the mechanism of underlying inaccessible variables with the additional requirement that distant events be independent, ruling out instantaneous (that is, faster-than-light) interactions between separate events.
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The mathematical implications of a local hidden-variable theory in regard to the phenomenon of quantum entanglement were explored by physicist John Stewart Bell, who in 1964 proved that broad classes of local hidden-variable theories cannot reproduce the correlations between measurement outcomes that quantum mechanics predicts. The most notable exception is superdeterminism. Superdeterministic hidden-variable theories can be local and yet be compatible with observations.
Local hidden variables and the Bell tests
Bell's theorem starts with the implication of the principle of local realism, that separated measurement processes are independent. Based on this premise, the probability of a coincidence between separated measurements of particles with correlated (e.g. identical or opposite) orientation properties can be written:
$P(a,b)=\int d\lambda \cdot \rho (\lambda )\cdot p_{A}(a,\lambda )\cdot p_{B}(b,\lambda ),$
(1)
where $p_{A}(a,\lambda )$ is the probability of detection of particle $A$ with hidden variable $\lambda $ by detector $A$, set in direction $a$, and similarly $p_{B}(b,\lambda )$ is the probability at detector $B$, set in direction $b$, for particle $B$, sharing the same value of $\lambda $. The source is assumed to produce particles in the state $\lambda $ with probability $\rho (\lambda )$.
Using (1), various Bell inequalities can be derived, which provide limits on the possible behaviour of local hidden-variable models.
When John Stewart Bell originally derived his inequality, it was in relation to pairs of entangled spin-1/2 particles, every one of those emitted being detected. Bell showed that when detectors are rotated with respect to each other, local realist models must yield a correlation curve that is bounded by a straight line between maxima (detectors aligned), whereas the quantum correlation curve is a cosine relationship. The first Bell tests were performed not with spin-1/2 particles, but with photons, which have spin 1. A classical local hidden-variable prediction for photons, based on Maxwell's equations, yields a cosine curve, but of reduced amplitude, such that the curve still lies within the straight-line limits specified in the original Bell inequality.
Bell's theorem assumes that measurement settings are completely independent, and not in principle determined by the universe at large. If this assumption, called statistical independence, were to be incorrect, as proposed in superdeterminism, conclusions drawn from Bell's theorem may be invalidated. The theorem also relies on very efficient and space-like separated measurements. Such flaws are generally called loopholes. Except for the statistical independence loophole, a loophole-free experimental verification of a Bell inequality violation was performed in 2015.[1]
Bell tests with no "non-detections"
Consider, for example, David Bohm's thought experiment, in which a molecule breaks into two atoms with opposite spins.[2] Assume that this spin can be represented by a real vector, pointing in any direction. It will be the "hidden variable" in our model. Taking it to be a unit vector, all possible values of the hidden variable are represented by all points on the surface of a unit sphere.
Suppose that the spin is to be measured in the direction a. Then the natural assumption, given that all atoms are detected, is that all atoms the projection of whose spin in the direction a is positive will be detected as spin-up (coded as +1), while all whose projection is negative will be detected as spin-down (coded as −1). The surface of the sphere will be divided into two regions, one for +1, one for −1, separated by a great circle in the plane perpendicular to a. Assuming for convenience that a is horizontal, corresponding to the angle a with respect to some suitable reference direction, the dividing circle will be in a vertical plane. So far we have modelled side A of our experiment.
Now to model side B. Assume that b too is horizontal, corresponding to the angle b. There will be second great circle drawn on the same sphere, to one side of which we have +1, the other −1 for particle B. The circle will be again in a vertical plane.
The two circles divide the surface of the sphere into four regions. The type of "coincidence" (++, −−, +− or −+) observed for any given pair of particles is determined by the region within which their hidden variable falls. Assuming the source to be "rotationally invariant" (to produce all possible states λ with equal probability), the probability of a given type of coincidence will clearly be proportional to the corresponding area, and these areas will vary linearly with the angle between a and b. (To see this, think of an orange and its segments. The area of peel corresponding to a number n of segments is roughly proportional to n. More accurately, it is proportional to the angle subtended at the centre.)
The formula (1) above has not been used explicitly – it is hardly relevant when, as here, the situation is fully deterministic. The problem could be reformulated in terms of the functions in the formula, with ρ constant and the probability functions step functions. The principle behind (1) has in fact been used, but purely intuitively.
Thus the local hidden-variable prediction for the probability of coincidence is proportional to the angle (b − a) between the detector settings. The quantum correlation is defined to be the expectation value of the sum of the individual outcomes, and this is
$E=P_{++}+P_{--}-P_{+-}-P_{-+},$
(2)
where P++ is the probability of a "+" outcome on both sides, P+− that of a "+" on side A, a "−" on side B, etc.
Since each individual term varies linearly with the difference (b − a), so does their sum.
The result is shown in the figure.
Optical Bell tests
In almost all real applications of Bell's inequalities, the particles used have been photons. It is not necessarily assumed that the photons are particle-like. They may be just short pulses of classical light.[3] It is not assumed that every single one is detected. Instead the hidden variable set at the source is taken to determine only the probability of a given outcome, the actual individual outcomes being partly determined by other hidden variables local to the analyser and detector. It is assumed that these other hidden variables are independent on the two sides of the experiment.[4][5]
In this stochastic model, in contrast to the above deterministic case, we do need equation (1) to find the local-realist prediction for coincidences. It is necessary first to make some assumption regarding the functions $p_{A}(a,\lambda )$ and $p_{B}(a,\lambda )$, the usual one being that these are both cosine squares, in line with Malus' law. Assuming the hidden variable to be polarisation direction (parallel on the two sides in real applications, not orthogonal), equation (1) becomes
$P(a,b)=\int d\lambda \cdot \rho (\lambda )\cdot \cos ^{2}(a-\lambda )\cdot \cos ^{2}(b-\lambda )={\frac {1}{8}}+{\frac {\cos ^{2}\phi }{4}},$
(3)
where $\phi =b-a$.
The predicted quantum correlation can be derived from this and is shown in the figure.
In optical tests, incidentally, it is not certain that the quantum correlation is well-defined. Under a classical model of light, a single photon can go partly into the "+" channel, partly into the "−" one, resulting in the possibility of simultaneous detections in both. Though experiments such as by Grangier et al. have shown that this probability is very low,[6] it is not logical to assume that it is actually zero. The definition of quantum correlation is adapted to the idea that outcomes will always be +1, −1 or 0. There is no obvious way of including any other possibility, which is one of the reasons why Clauser and Horne's 1974 Bell test, using single-channel polarisers, should be used instead of the CHSH Bell test. The CH74 inequality concerns just probabilities of detection, not quantum correlations.
Quantum states with a local hidden-variable model
For separable states of two particles, there is a simple hidden-variable model for any measurements on the two parties. Surprisingly, there are also entangled states for which all von Neumann measurements can be described by a hidden-variable model.[7] Such states are entangled, but do not violate any Bell inequality. The so-called Werner states are a single-parameter family of states that are invariant under any transformation of the type $U\otimes U,$ where $U$ is a unitary matrix. For two qubits, they are noisy singlets given as
$\varrho =p\vert \psi ^{-}\rangle \langle \psi ^{-}\vert +(1-p){\frac {\mathbb {I} }{4}},$
(4)
where the singlet is defined as $\vert \psi ^{-}\rangle ={\tfrac {1}{\sqrt {2}}}\left(\vert 01\rangle -\vert 10\rangle \right)$.
R. F. Werner showed that such states allow for a hidden-variable model for $p\leq 1/2$, while they are entangled if $p>1/3$. The bound for hidden-variable models could be improved until $p=2/3$.[8] Hidden-variable models have been constructed for Werner states even if POVM measurements are allowed, not only von Neumann measurements.[9] Hidden variable models were also constructed to noisy maximally entangled states, and even extended to arbitrary pure states mixed with white noise.[10] Beside bipartite systems, there are also results for the multipartite case. A hidden-variable model for any von Neumann measurements at the parties has been presented for a three-qubit quantum state.[11]
Generalizations of the models
By varying the assumed probability and density functions in equation (1), we can arrive at a considerable variety of local-realist predictions.
Time effects
Previously some new hypotheses were conjectured concerning the role of time in constructing hidden-variables theory. One approach was suggested by K. Hess and W. Philipp and relies upon possible consequences of time dependencies of hidden variables; this hypothesis has been criticized by R. D. Gill, G. Weihs, A. Zeilinger and M. Żukowski, as well as D. M. Appleby.[12][13][14]
Optical models deviating from Malus's law
If we make realistic (wave-based) assumptions regarding the behaviour of light on encountering polarisers and photodetectors, we find that we are not compelled to accept that the probability of detection will reflect Malus' law exactly.
We might perhaps suppose the polarisers to be perfect, with output intensity of polariser A proportional to cos2(a − λ), but reject the quantum-mechanical assumption that the function relating this intensity to the probability of detection is a straight line through the origin. Real detectors, after all, have "dark counts" that are there even when the input intensity is zero, and become saturated when the intensity is very high. It is not possible for them to produce outputs in exact proportion to input intensity for all intensities.
By varying our assumptions, it seems possible that the realist prediction could approach the quantum-mechanical one within the limits of experimental error,[15] though clearly a compromise must be reached. We have to match both the behaviour of the individual light beam on passage through a polariser and the observed coincidence curves. The former would be expected to follow Malus' law fairly closely, though experimental evidence here is not so easy to obtain. We are interested in the behaviour of very weak light and the law may be slightly different from that of stronger light.
See also
• EPR paradox
• Bohr–Einstein debates
References
1. Hensen, B.; Bernien, H.; Dréau, A. E.; Reiserer, A.; Kalb, N.; Blok, M. S.; Ruitenberg, J.; Vermeulen, R. F. L.; Schouten, R. N.; Abellán, C.; Amaya, W.; Pruneri, V.; Mitchell, M. W.; Markham, M.; Twitchen, D. J.; Elkouss, D.; Wehner, S.; Taminiau, T. H.; Hanson, R. (2015). "Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres". Nature. 526 (7575): 682–686. doi:10.1038/nature15759. PMID 26503041. S2CID 205246446.
2. Bohm, David (1951). Quantum Theory. Prentice-Hall.
3. Clauser, J. F.; Shimony, A. (1978-12-01). "Bell's theorem. Experimental tests and implications". Reports on Progress in Physics. 41 (12): 1881–1927. doi:10.1088/0034-4885/41/12/002. ISSN 0034-4885. S2CID 250885175.
4. Clauser, John F.; Horne, Michael A. (1974-07-15). "Experimental consequences of objective local theories". Physical Review D. 10 (2): 526–535. doi:10.1103/PhysRevD.10.526. ISSN 0556-2821.
5. Bell, J. S. (2004). Speakable and unspeakable in quantum mechanics : collected papers on quantum philosophy. Cambridge: Cambridge University Press. pp. 29–39. ISBN 0-521-81862-1. OCLC 52947235.
6. Grangier, P; Roger, G; Aspect, A (1986-02-15). "Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences". Europhysics Letters (EPL). 1 (4): 173–179. doi:10.1209/0295-5075/1/4/004. ISSN 0295-5075. S2CID 250837011.
7. R. F. Werner (1989). "Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model". Physical Review A. 40 (8): 4277–4281. Bibcode:1989PhRvA..40.4277W. doi:10.1103/PhysRevA.40.4277. PMID 9902666.
8. A. Acín; N. Gisin; B. Toner (2006). "Grothendieck's constant and local models for noisy entangled quantum states". Physical Review A. 73 (6): 062105. arXiv:quant-ph/0606138. Bibcode:2006PhRvA..73f2105A. doi:10.1103/PhysRevA.73.062105. S2CID 2588399.
9. J. Barrett (2002). "Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality". Physical Review A. 65 (4): 042302. arXiv:quant-ph/0107045. Bibcode:2002PhRvA..65d2302B. doi:10.1103/PhysRevA.65.042302. S2CID 119390251.
10. Almeida, Mafalda L.; Pironio, Stefano; Barrett, Jonathan; Tóth, Géza; Acín, Antonio (23 July 2007). "Noise Robustness of the Nonlocality of Entangled Quantum States". Physical Review Letters. 99 (4): 040403. arXiv:quant-ph/0703018. doi:10.1103/PhysRevLett.99.040403. PMID 17678341. S2CID 7102567.
11. G. Tóth; A. Acín (2006). "Genuine tripartite entangled states with a local hidden-variable model". Physical Review A. 74 (3): 030306. arXiv:quant-ph/0512088. Bibcode:2006PhRvA..74c0306T. doi:10.1103/PhysRevA.74.030306. S2CID 4792051.
12. Hess, K; Philipp, W (March 2002). "Exclusion of time in the theorem of Bell". Europhysics Letters (EPL). 57 (6): 775–781. doi:10.1209/epl/i2002-00578-y. ISSN 0295-5075. S2CID 250792546.
13. Gill, R. D.; Weihs, G.; Zeilinger, A.; Zukowski, M. (2002-11-12). "No time loophole in Bell's theorem: The Hess-Philipp model is nonlocal". Proceedings of the National Academy of Sciences. 99 (23): 14632–14635. arXiv:quant-ph/0208187. doi:10.1073/pnas.182536499. ISSN 0027-8424. PMC 137470. PMID 12411576.
14. Appleby, D. M. (2003). "The Hess-Philipp Model is Non-Local". International Journal of Quantum Information. 1 (1): 29–36. arXiv:quant-ph/0210145. Bibcode:2002quant.ph.10145A. doi:10.1142/S021974990300005X.
15. Marshall, T. W.; Santos, E.; Selleri, F. (October 1983). "Local realism has not been refuted by atomic cascade experiments". Physics Letters A. 98 (1–2): 5–9. doi:10.1016/0375-9601(83)90531-5.
Further reading
• Shadbolt, P. J.; Verde, M. R.; Peruzzo, A.; Politi, A.; Laing, A.; Lobino, M.; Matthews, J. C. F.; Thompson, M. G.; O'Brien, J. L. (January 2012). "Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit". Nature Photonics. 6 (1): 45–49. arXiv:1108.3309. doi:10.1038/nphoton.2011.283. hdl:10072/53103. ISSN 1749-4885. S2CID 56206588. Figure 5 highlights experimental data points inexplicable by local hidden variable theory.
| Wikipedia |
Eric Jang
Technology, A.I., Careers
Uncertainty: a Tutorial
A PDF version of this post can be found here.
Chinese translation by Xiaoyi Yin
Notions of uncertainty are tossed around in conversations around AI safety, risk management, portfolio optimization, scientific measurement, and insurance. Here are a few examples of colloquial use:
"We want machine learning models to know what they don't know.''
"An AI responsible for diagnosing patients and prescribing treatments should tell us how confident it is about its recommendations.''
"Significant figures in scientific calculations represent uncertainty in measurements.''
"We want autonomous agents to explore areas where they are uncertain (about rewards or predictions) so that they may discover sparse rewards.''
"In portfolio optimization, we want to maximize returns while limiting risk.''
"US equity markets finished disappointingly in 2018 due to increased geopolitical uncertainty.''
What exactly then, is uncertainty?
Uncertainty measures reflect the amount of dispersion of a random variable. In other words, it is a scalar measure of how "random" a random variable is. In finance, it is often referred to as risk.
There is no single formula for uncertainty because there are many different ways to measure dispersion: standard deviation, variance, value-at-risk (VaR), and entropy are all appropriate measures. However, it's important to keep in mind that a single scalar number cannot paint a full picture of "randomness'', as that would require communicating the entire random variable itself!
Nonetheless, it is helpful to collapse randomness down to a single number for the purposes of optimization and comparison. The important thing to remember is that "more uncertainty'' is usually regarded as "less good'' (except in simulated RL experiments).
Types of Uncertainty
Statistical machine learning concerns itself with the estimation of models $p(\theta|\mathcal{D})$, which in turn estimate unknown random variables $p(y|x)$. Multiple forms of uncertainty come into play here. Some notions of uncertainty describe inherent randomness that we should expect (e.g. outcome of a coin flip) while others describe our lack of confidence about our best guess of the model parameters.
To make things more concrete, let's consider a recurrent neural network (RNN) that predicts the amount of rainfall today from a sequence of daily barometer readings. A barometer measures atmospheric pressure, which often drops when its about to rain. Here's a diagram summarizing the rainfall prediction model along with different kinds of uncertainty.
Uncertainty can be understood from a simple machine learning model that attempts to predict daily rainfall from a sequence of barometer readings. Aleatoric uncertainty is irreducible randomness that arises from the data collection process. Epistemic uncertainty reflects confidence that our model is making the correct predictions. Finally, out-of-distribution errors arise when the model sees an input that differs from its training data (e.g. temperature of the sun, other anomalies).
Aleatoric Uncertainty
Aleatoric Uncertainty draws its name from the Latin root aleatorius, which means the incorporation of chance into the process of creation. It describes randomness arising from the data generating process itself; noise that cannot be eliminated by simply drawing more data. It is the coin flip whose outcome you cannot know.
In our rainfall prediction analogy, aleatoric noise arises from imprecision of the barometer. There are also important variables that the data collection setup does not observe: How much rainfall was there yesterday? Are we measuring barometric pressure in the present day, or the last ice age? These unknowns are inherent to our data collection setup, so collecting more data from that system does not absolve us of this uncertainty.
Aleatoric uncertainty propagates from the inputs to the model predictions. Consider a simple model $y = 5x$, which takes in normally-distributed input $x \sim \mathcal{N}(0,1)$. In this case, $y \sim \mathcal{N}(0, 5)$, so the aleatoric uncertainty of the predictive distribution can be described by $\sigma=5$. Of course, predictive aleatoric uncertainty is more challenging to estimate when the random structure of the input data $x$ is not known.
One might think that because aleatoric uncertainty is irreducible, one cannot do anything about it and so we should just ignore it. No! One thing to watch out for when training models is to choose an output representation capable of representing aleatoric uncertainty correctly. A standard LSTM does not emit probability distributions, so attempting to learn the outcome of a coin flip would just converge to the mean. In contrast, models for language generation emit a sequence of categorical distributions (words or characters), which can capture the inherent ambiguity in sentence completion tasks.
Epistemic Uncertainty
"Good models are all alike; every bad model is wrong in its own way."
Epistemic Uncertainty is derived from the Greek root epistēmē, which pertains to knowledge about knowledge. It measures our ignorance of the correct prediction arising from our ignorance of the correct model parameters.
Below is a plot of a Gaussian Process Regression model on some toy 1-dimensional dataset. The confidence intervals reflect epistemic uncertainty; the uncertainty is zero for training data (red points), and as we get farther away from training points, the model ought to assign higher standard deviations to the predictive distribution. Unlike aleatoric uncertainty, epistemic uncertainty can be reduced by gathering more data and "ironing out" the regions of inputs where the model lacks knowledge.
1-D Gaussian Process Regression Model showcasing epistemic uncertainty for inputs outside its training set.
There is a rich line of inquiry connecting Deep Learning to Gaussian Processes. The hope is that we can extend the uncertainty-awareness properties of GPs with the representational power of neural networks. Unfortunately, GPs are challenging to scale to the uniform stochastic minibatch setting for large datasets, and they have fallen out of favor among those working on large models and datasets.
If one wants maximum flexibility in choosing their model family, a good alternative to estimating uncertainty is to use ensembles, which is just a fancy way of saying "multiple independently learned models''. While GP models analytically define the predictive distribution, ensembles can be used to compute the empirical distribution of predictions.
Any individual model will make some errors due to randomized biases that occur during the training process. Ensembling is powerful because other models in the ensembles tend to expose the idiosyncratic failures of a single model while agreeing with the correctly inferred predictions.
How do we sample models randomly to construct an ensemble? In Ensembling with bootstrap aggregation, we start with a training dataset of size $N$ and sample $M$ datasets of size $N$ from the original training set (with replacement, so each dataset does not span the entire dataset). The $M$ models are trained on their respective datasets and their resulting predictions collectively form an empirical predictive distribution.
If training multiple models is too expensive, it is also possible to use Dropout training to approximate a model ensemble. However, introducing dropout involves an extra hyperparameter and can compromise single model performance (often unacceptable for real world applications where calibrated uncertainty estimation is secondary to accuracy).
Therefore, if one has access to plentiful computing resources (as one does at Google), it is often easier to just re-train multiple copies of a model. This also yields the benefits of ensembling without hurting performance. This is the approach taken by the Deep Ensembles paper. The authors of this paper also mention that the random training dynamics induced by differing weight initializations was sufficient to introduce a diverse set of models without having to resort to reducing the training set diversity via bootstrap aggregation. From a practical engineering standpoint, it's smart to bet on risk estimation methods that do not get in the way of the model's performance or whatever other ideas the researcher wants to try.
Out-of-Distribution Uncertainty
For our rainfall predictor, what if instead of feeding in the sequence of barometer readings, we fed in the temperature of the sun? Or a sequence of all zeros? Or barometer readings from a sensor that reports in different units? The RNN will happily compute away and give us a prediction, but the result will likely be meaningless.
The model is totally unqualified to make predictions on data generated via a different procedure than the one used to create the training set. This is a failure mode that is often overlooked in benchmark-driven ML research, because we typically assume that the training, validation, and test sets consist entirely of clean i.i.d data.
Determining whether inputs are "valid'' is a serious problem for deploying ML in the wild, and is known as the Out of Distribution (OoD) problem. OoD is also synonymous with model misspecification error and anomaly detection.
Besides its obvious importance for hardening ML systems, anomaly detection models are an intrinsically useful technology. For instance, we might want to build a system that monitors a healthy patient's vitals and alerts us when something goes wrong without necessarily having seen that pattern of pathology before. Or we might be managing the "health" of a datacenter and want to know whenever unusual activity occurs (disks filling up, security breaches, hardware failures, etc.)
Since OoD inputs only occur at test-time, we should not presume to know the distribution of anomalies the model encounters. This is what makes OoD detection tricky - we have to harden a model against inputs it never sees during training! This is exactly the standard attack scenario described in Adversarial Machine Learning.
There are two ways to handle OoD inputs for a machine learning model: 1) catch the bad inputs before we even put them through the model 2) let the "weirdness'' of model predictions imply to us that the input was probably malformed.
In the first approach, we assume nothing about the downstream ML task, and simply consider the problem of whether an input is in the training distribution or not. This is exactly what discriminators in Generative Adversarial Networks (GANs) are supposed to do. However, a single discriminator is not completely robust because it is only good for discriminating between the true data distribution and whatever the generator's distribution is; it can give arbitrary predictions for an input that lies in neither distribution.
Instead of a discriminator, we could build a density model of the in-distribution data, such as a kernel density estimator or fitting a Normalizing Flow to the data. Hyunsun Choi and I investigated this in our recent paper on using modern generative models to do OoD detection.
The second approach to OoD detection involves using the predictive (epistemic) uncertainty of the task model to tell us when inputs are OoD. Ideally, malformed inputs to a model ought to generate "weird'' predictive distribution $p(y|x)$. For instance, Hendrycks and Gimpel showed that the maximum softmax probability (the predicted class) for OoD inputs tends to be lower than that of in-distribution inputs. Here, uncertainty is inversely proportional to the "confidence'' as modeled by the max sofmax probability. Models like Gaussian Processes give us these uncertainty estimates by construction, or we could compute epistemic uncertainty via Deep Ensembles.
In reinforcement learning, OoD inputs are actually assumed to be a good thing, because it represents inputs from the world that the agent does not know how to handle yet. Encouraging the policy to find its own OoD inputs implements "intrinsic curiosity'' to explore regions the model predicts poorly in. This is all well and good, but I do wonder what would happen if such curiousity-driven agents are deployed in real world settings where sensors break easily and other experimental anomalies happen. How does a robot distinguish between "unseen states" (good) and "sensors breaking" (bad)? Might that result in agents that learn to interfere with their sensory mechanisms to generate maximum novelty?
Who Will Watch the Watchdogs?
As mentioned in the previous section, one way to defend ourselves against OoD inputs is to set up a likelihood model that "watchdogs" the inputs to a model. I prefer this approach because it de-couples the problem of OoD inputs from epistemic and aleatoric uncertainty in the task model. It makes things easy to analyze from an engineering standpoint.
But we should not forget that the likelihood model is also a function approximator, possibly with its own OoD errors! We show in our recent work on Generative Ensembles (and also showed in concurrent work by DeepMind), that under a CIFAR likelihood model, natural images from SVHN can actually be more likely than the in-distribution CIFAR images themselves!
Likelihood estimation involves a function approximator that can itself be susceptible to OoD inputs. A likelihood model of CIFAR assigns higher probabilities to SVHN images than CIFAR test images!
However, all is not lost! It turns out that the epistemic uncertainty of likelihood models is an excellent OoD detector for the likelihood model itself. By bridging epistemic uncertainty estimation with density estimation, we can use ensembles of likelihood models to protect machine learning models against OoD inputs in a model-agnostic way.
Calibration: the Next Big Thing?
A word of warning: just because a model is able to spit out a confidence interval for a prediction doesn't mean that the confidence interval actually reflects the actual probabilities of outcomes in reality!
Confidence intervals (e.g. $2\sigma$) implicitly assume that your predictive distribution is Gaussian-distributed, but if the distribution you're trying to predict is multi-modal or heavy-tailed, then your model will not be well calibrated!
Suppose our rainfall RNN tells us that there will be $\mathcal{N}(4, 1)$ inches of rain today. If our model is calibrated, then if we were to repeat this experiment over and over again under identical conditions (possibly re-training the model each time), we really would observe empirical rainfall to be distributed exactly $\mathcal{N}(4, 1)$.
Machine Learning models developed by academia today mostly optimize for test accuracy or some fitness function. Researchers are not performing model selection by deploying the model in repeated identical experiments and measuring calibration error, so unsurprisingly, our models tend to be poorly calibrated.
Going forward, if we are to trust ML systems deployed in the real world (robotics, healthcare, etc.), I think a much more powerful way to "prove our models understand the world correctly'' is to test them for statistical calibration. Good calibration also implies good accuracy, so it would be a strictly higher bar to optimize against.
Should Uncertainty be Scalar?
As useful as they are, scalar uncertainty measures will never be as informative as the random variables they describe. I find methods like particle filtering and Distributional Reinforcement Learning very cool because they are algorithms that operate on entire distributions, freeing us from resorting to simple normal distributions to keep track of uncertainty. Instead of shaping ML-based decision making with a single scalar of "uncertainty", we can now query the full structure of distributions when deciding what to do.
The Implicit Quantile Networks paper (Dabney et al.) has a very nice discussion on how to construct "risk-sensitive agents'' from a return distribution. In some environments, one might favor an opportunitistic policy that prefers to explore the unknown, while in other environments unknown things may be unsafe and should be avoided. The choice of risk measure essentially determines how to map the distribution of returns to a scalar quantity that can be optimized against. All risk measures can be computed from the distribution, so predicting full distributions enables us to combine multiple definitions of risk easily. Furthermore, supporting flexible predictive distributions seems like a good way to improve model calibration.
Performance of various risk measures on Atari games as reported by the IQN paper.
Risk measures are a deeply important research topic to financial asset managers. The vanilla Markowitz portfolio objective minimizes a weighted variance of portfolio returns $\frac{1}{2}\lambda w^T \Sigma w$. However, variance is an unintuitive choice of "risk'' in financial contexts: most investors don't mind returns exceeding expectations, but rather wish to minimize the probability of small or negative returns. For this reason, risk measures like Value-at-Risk, Shortfall Probability, and Target Semivariance, which only pay attention to the likelihood of "bad'' outcomes, are more useful objectives to optimize.
Unfortunately, they are also more difficult to work with analytically. My hope is that research into distributional RL, Monte Carlo methods, and flexible generative models will allow us to build differentiable relaxations of risk measures that can play nicely with portfolio optimizers. If you work in finance, I highly recommend reading the IQN paper's "Risks in Reinforcement Learning" section.
Here's a recap of the main points of this post:
Uncertainty/risk measures are scalar measures of "randomness''. Collapsing a random variable to a single number is done for optimization and mathematical convenience.
Predictive uncertainty can be decomposed into aleatoric (irreducible noise arising from data collection process), epistemic (ignorance about true model), and out-of-distribution (at test time, inputs may be malformed).
Epistemic uncertainty can be mitigated by softmax prediction thresholding or ensembling.
Instead of propagating OoD uncertainty to predictions, we can use a task-agnostic filtering mechanism that safeguards against "malformed inputs''.
Density models are a good choice for filtering inputs at test time. However, it's important to recognize that density models are merely approximations of the true density function, and are themselves susceptible to out-of-distribution inputs.
Self-plug:Generative Ensembles reduce epistemic uncertainty of likelihood models so they can be used to detect OoD inputs.
Calibration is important and underappreciated in research models.
Some algorithms (Distributional RL) extend ML algorithms to models that emit flexible distributions, which provides more information than a single risk measure.
I especially recommend Chapter 3 ("Risk Measurement") of Modern Investment Management by Litterman et al. to learn about risk in a concrete way.
http://uqpm2017.usacm.org/sites/default/files/DStarcuzzi_UQConf.pdf
http://mlg.eng.cam.ac.uk/yarin/blog_2248.html
Posted by Eric at 10:14 AM
Labels: AI, Finance, Machine Learning, Statistics
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Find $a,b,c$ so that $\begin{bmatrix} 0 & 1& 0 \\ 0 & 0 & 1\\ a & b & c \end{bmatrix} $ has the characteristic polynomial $-\lambda^3+4\lambda^2+5\lambda+6=0$
Linear Algebra Matrices Algebra
Mathtutor
You should offer some more for this question.
Lets compute the characteristic equation
\[0=\det \begin{bmatrix} -\lambda & 1& 0 \\ 0& -\lambda & 1 \\ a & b & c-\lambda \end{bmatrix} \]
\[=-\lambda \det \begin{bmatrix} -\lambda & 1 \\ b & c-\lambda \end{bmatrix}-1 \det \begin{bmatrix} 0 & 1 \\ a & c-\lambda \end{bmatrix} \]
\[=-\lambda [\lambda (\lambda-c)-b]-[-a]=-\lambda^3+c\lambda^2+b\lambda+a\]
\[=-\lambda^3+4\lambda^2+5\lambda+6.\]
Hence
\[a=6, b=5, c=4.\]
Solve $abc=2(a-2)(b-2)(c-2)$ where $a,b $ and $c$ are integers
Algebra Word Problem 3
Representation theory question
Prove that: |x| + |y| ≤ |x + y| + |x − y|.
Can enough pizza dough be made to cover the surface of the earth?
Five times the larger of two consecutive odd integers is equal to one more than eight times the smaller. Find the integers.
Need Upper Bound of an Integral
Linear Algebra - matrices and vectors | CommonCrawl |
thai new song
Of course, there will be the curiosity of what the song tries to deliver, you can still search for the lyrics online. 2021 © Copyright Ivan Cheam. This post also great for new listeners who are about to discover Thai songs but unsure where to start. 8. "We are going to eat a dead body of Dhammapala, who will fail to answer three riddles? "Pohn Pee Mai" (พรปีใหม่) literally translated as New Year Blessing, was composed by His Majesty King Bhumibol the Great, and the lyrics were written by Prince Chakrapan Pensiri, in December 1951, as a new year blessing for the Thai people. ) South Southerners have three Songkran rules: Work as little as possible and avoid spending money; do not hurt other persons or animals; do not tell lies. w The calendar features the image of Songkran goddess with her vehicle and subordinates, led by Chinese zodiac animal holding a flag with Thai script for that zodiac. This corresponds to 373/800 day or 11 hours 11 minutes and 24 seconds. 15. People make merit offerings such as giving sand to the temple for construction or repair. [7][n 1] Songkran, however, was traditionally computed according to the method described in Suriyayart (Thai: สุริยยาตร์), the Thai version of Surya Siddhanta. 4. − [35] In 2018 the number of offenders arrested at 2,029 checkpoints had risen to 146,589. 638 K "[36], In 2014 "Celebrate Singapore," a large two-day Songkran-style water festival,[42] was planned for Singapore and the event was promoted as the "largest water festival party in Singapore". 800 [32]), Police statistics show that the death toll from road accidents doubles during the annual Songkran holiday. The boy remembered everything. 800 292207 Other forms of merit include releasing birds and fish. In the evening, the sri goes to the feet, so people wash their feet every evening." The songkran festival is, therefore, a celebration of the New Year in accordance with the solar calendar. In 1949, Maha Songkran was on 13 April at 12:35 and the ceremony started that day. 6. East The eastern region has activities similar to the other part of Thailand, but people in the east always make merit at the temple throughout all the days of the Songkran Festival and create sand pagodas. According to its literal meaning in Sanskrit, a songkran occurs every month. 20. n [5], Songkran is a term derived from Sanskrit संक्रान्ति saṅkrānti meaning 'to move' or 'movement'. Between 2009 and 2013 there were about 27 road deaths per day during non-holiday periods and an average of 52 road deaths per day during Songkran. He was taken into custody, fined 100 baht, then released. This is called Maha Songkran day (Thai: วันมหาสงกรานต์). [28], Songkran is celebrated by the Malaysian Siamese community, particularly in the states of Kedah, Kelantan, Penang, Perak, Perlis and Terengganu where most Siamese are located.[29][30]. "What are you going to eat tomorrow? The drunkard, who had two sons, belittled the rich man for being childless. D The female eagle asked her mate whether he knew the answer. , Lipta, OG-ANIC & NINO – Sexy Sexy [18] In 2013, Chiang Mai provincial council decided to defy the government-set holiday by rescheduling the ceremony according to the correct calculation.[19]. The cabinet later fixed this issue by shifting the holiday by one day to 13–15 April, which is still in use today. Paying reverence to ancestors is an important part of Songkran tradition. A god named Kabillaprom learned of the child and wanted to test the child's cleverness. Chinese zodiac for each year is also given since it is also used in Thai astrology. = 1732 a New Tamil Songs Download- Listen 2021 New Tamil songs free online or Download Latest Tamil Songs MP3. Sotus S, 2Gether, Dark Blue Kiss and many more.. 800 ( Every music listeners have their own music taste but right now I'm going to reveal some of the top Thai songs I've discovered, carefully reviewed, and curated in my Spotify playlist. {\displaystyle JD_{\mathrm {newyear} }} 1732 g Stream Tracks and Playlists from New Thai Song on your desktop or mobile device. The tree god asked Indra to grant the man's wish. ( [12][13][n 3] In other words, each solar year lasts 292,207 kammaja (Thai: กัมมัช, lit. , [2] The word "Songkran" comes from the Sanskrit word saṃkrānti,[3] literally "astrological passage", meaning transformation or change. A solar year lasts 292,207 kammajas or 365.25875 days every year. 373 In other words, 0 ME started at 11:11:24 of Sunday, 25 March 638 CE in proleptic Gregorian calendar. 1. According to the Buddhist scripture of Wat Pho, Songkran originated from the death of Kapila Brahma Thai: กบิลพรหม (lit. It represents purification and the washing away of one's sins and bad luck. [33], The National Council for Peace and Order (NCPO) says a total of 110,909 people were arrested and 5,772 vehicles impounded at road safety checkpoints across the country between 9–16 April 2016. In 2018 the Thai cabinet extended the festival nationwide to five days, 12–16 April, to enable citizens to travel home for the holiday. The Deputy Governor's view was supported by numerous Thai citizens on social media websites. E The announcement called Prakat Songkran (Thai: ประกาศสงกรานต์, Songkran notification), contained the information on Maha Songkran, Thaloengsok, lunisolar calendar, and religious and royal ceremonies. 3. 292207 According to Suriyayart, the sun entered Aries at 19:30 on 12 April. slowthai, A$AP Rocky (Getty Images) UK rapper slowthai has shared a new song featuring A$AP Rocky called " MAZZA." slowthai produced the track with SAMO. {\displaystyle BE} On the next day, people prepare food and useful things to offer to the monks at the temple. [20][21] Before the cut off date, astrologer uses zodiac of the last year. Water fights along the west moat, Chiang Mai, Songkran symbolic sand pagodas in temple, Wat Phu Khao Thong, Ban Maenam, Koh Samui, Group of Thai traditional dancer in Songkran festival, Bangkok, Lady Songkran parade at Songkran festival, Bangkok, Central Region People in this region clean their houses when Songkran approaches. + ( The rich man was humiliated and beseeched the Sun and the Moon gods to grant him a son. {\displaystyle JD_{\mathrm {newyear} }={\frac {\left(292207\times ME\right)+373}{800}}+1954167.5={\frac {\left(292207\times \left(CE-638\right)\right)+373}{800}}+1954167.5,}, then the number is converted back into a date using Julian day algorithm (see Julian day). r [17] Some astrologers, especially in northern Thailand, still issue their own Songkran notification containing predictions and other information. Some people, after making merit at the temple, prepare food to be given to the elderly members of their family. Here is the list of most popular Thai songs, currently trending in 2020. The integer result is the count of days at New Year's Day, while the remainder (in kammaja) suggests when the new year will start, which can be other time than midnight. 5: New. Now, welcome to Ivan Cheam official blog at IvanYolo.com. r The lady stands, sits, reclines or sleeps on the back of the animal depending on the time. [16] In 1600, 1700, 1800, 1900 and 2000, Maha Songkran was on 7 April, 9 April, 10 April, 12 April and 13 April respectively. The holiday is known for its water festival. Black Friday 2020: Best Deals on SSD & PC Gaming Parts, SQ Car Audio: Path to the Audiophile World, Yamaha Firmware Update 1.31 with Tidal & Deezel, A Little SQ Demo on Ford Fiesta Car Audio Part 1, Pepsi Rocks The Cola Industry With "PEPSI x BLACKPINK", Airbnb Celebrates the World of Korean Pop Music with Inside K-pop, Live-streaming Broadcasters Come Together at the BIGO Awards Gala 2021. 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\begin{document}
\title[Categorification of Quantum Generalized Kac-Moody Algebras and Crystal Bases] {Categorification of Quantum Generalized Kac-Moody Algebras and Crystal Bases} \author[Seok-Jin Kang]{Seok-Jin Kang $^{1,2}$}
\thanks{$^1$ This work was supported by KRF Grant \# 2007-341-C00001.} \thanks{$^2$ This work was supported by NRF Grant \# 2010-0010753.}
\address{Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, 599 Gwanak-ro, Gwanak-gu, Seoul 151-747, Korea} \email{[email protected]} \author[Se-jin Oh]{Se-jin Oh $^{3,4}$ } \thanks{$^3$ This work was supported by NRF Grant \# 2010-0019516.} \thanks{$^4$ This work was supported by BK21 Mathematical Sciences Division.} \address{Department of Mathematical Sciences, Seoul National University, 599 Gwanak-ro, Gwanak-gu, Seoul 151-747, Korea} \email{[email protected]} \author[Euiyong Park]{Euiyong Park $^{1,2}$}
\address{School of Mathematics, Korea Institute for Advanced Study, 87 Hoegiro, Dongdaemun-gu, Seoul 130-722, Korea} \email{[email protected]}
\subjclass[2000]{05E10, 16G99, 81R10} \keywords{categorification, Khovanov-Lauda-Rouquier algebras, crystal bases, quantum generalized Kac-Moody algebras}
\begin{abstract} We construct and investigate the structure of the Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^\lambda$ which give a categorification of quantum generalized Kac-Moody algebras. Let $U_\A(\mathfrak{g})$ be the integral form of the quantum generalized Kac-Moody algebra associated with a Borcherds-Cartan matrix $A=(a_{ij})_{i,j \in I}$ and let $K_0(R)$ be the Grothendieck group of finitely generated projective graded $R$-modules. We prove that there exists an injective algebra homomorphism $\Phi: U_\A^-(\mathfrak{g}) \rightarrow K_0(R)$ and that $\Phi$ is an isomorphism if $a_{ii}\ne 0$ for all $i\in I$. Let $B(\infty)$ and $B(\lambda)$ be the crystals of $U_q^-(\mathfrak{g})$ and $V(\lambda)$, respectively, where $V(\lambda)$ is the irreducible highest weight $U_q(\mathfrak{g})$-module. We denote by $\Bklr{\infty}$ and $\Bklr{\lambda}$ the isomorphism classes of irreducible graded modules over $R$ and $R^\lambda$, respectively. If $a_{ii}\ne 0$ for all $i\in I$, we define the $U_q(\mathfrak{g})$-crystal structures on $\Bklr{\infty}$ and $\Bklr{\lambda}$, and show that there exist crystal isomorphisms $\Bklr{\infty} \simeq B(\infty)$ and $\Bklr{\lambda} \simeq B(\lambda)$. One of the key ingredients of our approach is the perfect basis theory for generalized Kac-Moody algebras. \end{abstract}
\maketitle
\section*{Introduction}
In \cite{KL09, KL11} and \cite{R08}, Khovanov-Lauda and Rouquier independently introduced a new family of graded algebras $R$ which gives a {\it categorification} of quantum groups associated with symmetrizable Kac-Moody algebras. More precisely, let $U_q(\mathfrak{g})$ be the quantum group associated with a symmetrizable Kac-Moody algebra and let $U_{\A}(\mathfrak{g})$ be the integral form of $U_q(\mathfrak{g})$, where $\A = \mathbb{Z}[q, q^{-1}]$. Then it was shown that the Grothendieck group $K_{0}(R)$ of finitely generated graded projective $R$-modules is isomorphic to $U_{\A}^{-}(\mathfrak{g})$, the negative part of $U_{\A}(\mathfrak{g})$. Furthermore, for symmetric Kac-Moody algebras, Varagnolo and Vasserot proved that the isomorphism classes of principal indecomposable $R$-modules correspond to Lusztig's {\it canonical basis} (or Kashiwara's {\it lower global basis}) under this isomorphism \cite{VV09}. The algebra $R$ is called the {\it Khovanov-Lauda-Rouquier algebra} associated with $\mathfrak{g}$.
For each dominant integral weight $\lambda \in P^{+}$, the algebra $R$ has a special quotient $R^{\lambda}$ which is called the {\it cyclotomic quotient}. It was conjectured that the cyclotomic quotient $R^{\lambda}$ gives a categorification of the irreducible highest weight module $V(\lambda)$ \cite{KL09}. For type $A_\infty$ and $A_n^{(1)}$, this conjecture was proved in \cite{BK08,BK09}.
In \cite{KK11}, Kang and Kashiwara proved Khovanov-Lauda categorification conjecture for {\it all} symmetrizable Kac-Moody algebras. Webster also gave a proof of this conjecture by a completely different method \cite{Webster10}.
In \cite{LV09}, the crystal version of this conjecture was proved. That is, in \cite{LV09}, Lauda and Vazirani investigated the crystal structure on the set of isomorphism classes of irreducible graded modules over $R$ and $R^\lambda$, and showed that these crystals are isomorphic to the crystals $B(\infty)$ and $B(\lambda)$, respectively.
The purpose of this paper is to extend the study of Khovanov-Lauda-Rouquier algebras to the case of {\it generalized Kac-Moody algebras}. The generalized Kac-Moody algebras were introduced by Borcherds in his study of Monstrous Moonshine \cite{Bor88}, and they form an important class of algebraic structure behind many research areas such as algebraic geometry, number theory and string theory (see, for example, \cite{Bor92, FRS97, GN98a, GN98b,HM96, HM98, KangKwon00, Moore98, Nai95, Sch04, Sch06}). In particular, the {\it Monster Lie algebra}, a special example of generalized Kac-Moody algebras, played a crucial role in proving the Moonshine conjecture \cite{Bor92}.
Moreover, the generalized Kac-Moody algebras draw more and more attention among mathematical physicists due to their connection with string theory and other related topics.
The quantum deformations of generalized Kac-Moody algebras and their integrable highest weight modules were constructed in \cite{Kang95} and the crystal basis theory for quantum generalized Kac-Moody algebras was developed in \cite{JKK05, JKKS07}. In \cite{KO06}, the canonical bases for quantum generalized Kac-Moody algebras were realized as certain semisimple perverse sheaves, and in \cite{KKO09a,KKO09b}, a geometric construction of crystals $B(\infty)$ and $B(\lambda)$ was given using Lusztig's and Nakajima's quiver varieties, respectively.
In this paper, we construct and investigate the structure of Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^{\lambda}$ which give a categorification of quantum generalized Kac-Moody algebras. Let $U_q(\mathfrak{g})$ be the quantum generalized Kac-Moody algebra associated with a Borcherds-Cartan matrix $A=(a_{ij})_{i,j\in I}$. We first define the Khovanov-Lauda-Rouquier algebra $R$ in terms of generators and relations.
A big contrast with the case of Kac-Moody algebras is that the nil Hecke algebras corresponding to the {\it imaginary} simple roots with norm $\le 0$ may have nonconstant twisting factors for commutation and braid relations.
In this work, we choose any homogeneous polynomials ${\mathcal P}_i(u,v)$ of degree $1 - \dfrac{a_{ii}}{2}$ and their variants $\overline{\mathcal P}_{i}'$ and $\overline{\mathcal P}_{i}''$ $(i \in I)$ as these twisting factors (see Definition \ref{def:KLR}).
When $a_{ii}=2$, we are reduced to the case of Kac-Moody algebras.
The role of these twisting factors is still mysterious.
For convenience, we also give a diagrammatic presentation of the algebra $R$.
Next, we show that there exists an injective algebra homomorphism $\Phi: U_{\A}^{-}(\mathfrak{g}) \longrightarrow K_{0}(R)$, where $K_{0}(R)$ is the Grothendieck group of finitely generated graded projective $R$-modules (Theorem \ref{Thm:Phi is injective}). Thus $\text{Im} \Phi$ gives a categorification of $U_q^{-}(\mathfrak{g})$.
To do this, we need to show that the quantum Serre relations are preserved by the map $\Phi$.
In general, $\Phi$ is not surjective even for the case $A = (0)$. The whole Grothendieck group seems rather large and nontrivial. However, if $a_{ii} \neq 0$ for all $i \in I$, we can show that $\Phi$ is an isomorphism (Theorem \ref{Thm:iso of K0 and Uq}). As in the case of Kac-Moody algebras, we conjecture that, if the Borcherds-Cartan matrix $A=(a_{ij})_{i,j \in I}$ is symmetric and $a_{ii} \neq 0$ for all $i \in I$, then the isomorphism classes of graded projective indecomposable $R$-modules correspond to canonical basis elements under the isomorphism $\Phi$. We will investigate this conjecture in a forthcoming paper following the framework given in \cite{KO06, VV09}.
Now we focus on the crystal structures.
We would like to emphasize that
one of the key ingredients of our approach is the {\it perfect basis theory} for generalized Kac-Moody algebras and it can be applied to the Kac-Moody algebras setting as well.
Our work is different from \cite{LV09} in this respect.
In \cite{BerKaz07}, Berenstein and Kazhdan introduced the notion of perfect bases for integrable highest weight modules $V(\lambda)$ $(\lambda \in P^{+})$ over Kac-Moody algebras. They showed that the colored oriented graphs arising from perfect bases are all isomorphic to the crystal $B(\lambda)$. Their work was extended to the integrable highest weight modules over generalized Kac-Moody algebras in \cite{KOP09}. In this work, we define the notion of perfect bases for $U_q^{-}(\mathfrak{g})$ as a module over the {\it quantum boson algebra} $B_q(\mathfrak{g})$. The existence of perfect basis for $U_q^{-}(\mathfrak{g})$ is provided by constructing the {\it upper global basis} (or {\it dual canonical basis}) of $U_q^{-}(\mathfrak{g})$. We also show that the crystal arising from any perfect basis of $U^-_q(\mathfrak{g})$ is isomorphic to the crystal $B(\infty)$ (Theorem \ref{Thm: uniqueness of perfect graphs}).
With perfect basis theory at hand, we construct the crystal $\Bklr{\infty}$ as follows. Let $G_0(R)$ be the Grothendieck group of finite-dimensional graded $R$-modules and set $G_0(R)_{\mathbb{Q}(q)} = \mathbb{Q}(q) \otimes_\A G_0(R)$. We denote by $\Bklr{\infty}$ the set of isomorphism classes of irreducible graded $R$-modules and define the crystal operators using induction and restriction functors. Moreover, we show that $G_0(R)_{\mathbb{Q}(q)}$ has a $B_q(\mathfrak{g})$-module structure and that if $a_{ii} \neq 0$ for all $i\in I$, then $\Bklr{\infty}$ is a perfect basis of $G_0(R)_{\mathbb{Q}(q)}$. Therefore, by the main theorem of perfect basis theory, we obtain a crystal isomorphism (Theorem \ref{Thm: B(infty)}): $$\Bklr{\infty} \simeq B(\infty).$$
For a dominant integral weight $\lambda \in P^{+}$, we define the {\it cyclotomic Khovanov-Lauda-Rouquier algebra} $R^\lambda$ to be the quotient of $R$ by a certain two-sided ideal depending on $\lambda$. Let $\Bklr{\lambda}$ denote the set of isomorphism classes of irreducible graded $R^\lambda$-modules and define the crystal operators using induction/restriction functors and projection/inflation functors. It was shown in \cite{JKKS07} that there exists a strict crystal embedding $$ B(\lambda) \hookrightarrow B(\infty) \otimes T_\lambda \otimes C.$$ If $a_{ii} \neq 0$ for all $i \in I$, using the above crystal embedding, we construct a crystal isomorphism (Theorem \ref{Thm: B(lambda)}): $$ \Bklr{\lambda} \simeq B(\lambda).$$
In \cite{KKO11}, after this work was completed, Khovanov-Lauda cyclotomic conjecture was proved for all symmetrizable generalize Kac-Moody algebras.
This paper is organized as follows. Section 1 contains a brief review of quantum generalized Kac-Moody algebras and crystal bases. In Section 2, we define the Khovanov-Lauda-Rouquier algebra $R$ associated with a Borcherds-Cartan matrix $A=(a_{ij})_{i,j \in I}$, and investigate its algebraic structure and representation theory. We construct a faithful polynomial representation of $R(\alpha)$ and prove the Khovanov-Lauda-Rouquier algebra version of the quantum Serre relations. In Section 3, we show that the algebra $R$ gives a categorification of $U_\A^-(\mathfrak{g})$. We define a twisted bialgebra structure on $K_0(R)$ using induction and restriction functors, and show that there exists an injective algebra homomorphism $\Phi: U_\A^-(\mathfrak{g}) \longrightarrow K_0(R)$. In particular, we prove that $U_\A^-(\mathfrak{g}) \simeq K_0(R)$ when $a_{ii} \ne 0$ for all $i\in I$. Section 4 is devoted to the theory of perfect bases. We define the notion of perfect bases for $U_q^-(\mathfrak{g})$ as a $B_q(\mathfrak{g})$-module and show that $U_q^{-}(\mathfrak{g})$ has a perfect basis by constructing the upper global basis of $U_q^-(\mathfrak{g})$. The main theorem in Section 4 asserts that the crystals arising from perfect bases are all isomorphic to $B(\infty)$. In Section 5, we study the crystal structures on $\Bklr{\infty}$ and $\Bklr{\lambda}$. Using the theory of perfect bases, we prove that there exists a crystal isomorphism $\Bklr{\infty} \simeq B(\infty)$ when $a_{ii} \ne 0$ for $i\in I$. Furthermore, we define the cyclotomic quotient $R^\lambda$ of $R$, and investigate the basic properties of irreducible $R^\lambda$-modules. Combining the isomorphism $\Bklr{\infty} \simeq B(\infty)$ with the strict embedding $ B(\lambda) \hookrightarrow B(\infty) \otimes T_\lambda \otimes C$, we obtain a crystal isomorphism $\Bklr{\lambda} \simeq B(\lambda)$.
\vskip 3em
\section{Quantum generalized Kac-Moody algebras} \label{Sec:GKM}
Let $I$ be a countable (possibly infinite) index set. A matrix $A=(a_{ij})_{i,j \in I}$ with $a_{ij} \in \mathbb{Z}$ is called an {\it even integral Borcherds-Cartan matrix} if it satisfies (i) $a_{ii} = 2 \text{ or } a_{ii} \in 2 \mathbb{Z}_{\le 0}$, (ii) $a_{ij} \le 0 \text{ for } i \neq j$, (iii) $a_{ij}=0 \text{ if and only if } a_{ji}=0$. For
$i \in I$, $i$ is said to be \emph{real} if $a_{ii}=2$ and is said to be \emph{imaginary} otherwise. We denote by $ I^{\rm re} $ the set of all real indices and by $ I^{\rm im} $ the set of all imaginary indices. In this paper, we assume that $A$ is {\em symmetrizable}; i.e., there is a diagonal matrix $D={\rm diag}( s_i \in \mathbb{Z}_{> 0} | i \in I)$ such that $DA$ is symmetric.
A \emph{Borcherds-Cartan datum} $(A,P,\Pi,\Pi^{\vee})$ consists of \begin{enumerate} \item[(1)] a Borcherds-Cartan matrix $A$, \item[(2)] a free abelian group $P$, the \emph{weight lattice}, \item[(3)] $\Pi= \{ \alpha_i \in P \mid \ i \in I \}$, the set of \emph{simple roots},
\item[(4)] $\Pi^{\vee}= \{ h_i \ | \ i \in I \} \subset P^{\vee}:={\rm Hom}(P,\mathbb{Z})$, the set of \emph{simple coroots}, \end{enumerate} satsifying the following properties: \begin{enumerate} \item[(a)] $\langle h_i,\alpha_j \rangle = a_{ij}$ for all $i,j \in I$, \item[(b)] $\Pi$ is linearly independent, \item[(c)] for any $i \in I$, there exists $\Lambda_i \in P$ such that
$\langle h_j ,\Lambda_i \rangle =\delta_{ij}$ for all $j \in I$. \end{enumerate}
Let $\mathfrak{h} = \mathbb{Q} \otimes_\mathbb{Z} P^{\vee}$. Since $A$ is symmetrizable, there is a symmetric biliear form $( \ | \ )$ on $\mathfrak{h}^*$ satisfying
$$ (\alpha_i | \alpha_j) = s_i a_{ij} \quad (i,j \in I). $$
We denote by $P^{+} := \{ \lambda \in P | \lambda(h_i) \in \mathbb{Z}_{\ge 0}, i \in I \}$ the set of \emph{dominant integral weights}. The free abelian group $Q= \oplus_{i \in I} \mathbb{Z} \alpha_i$ is called the \emph{root lattice}. Set $Q^{+}= \sum_{i \in I} \mathbb{Z}_{\ge 0} \alpha_i$. For $\alpha = \sum k_i \alpha_i \in Q^{+}$, we denote by
$|\alpha|$ the {\it height} of $\alpha$: $|\alpha|=\sum k_i$.
Let $q$ be an indeterminate and $m,n \in \mathbb{Z}_{\ge 0}$. Set $c_i = -\frac{1}{2}a_{ii}$ and $q_i = q^{s_i}$ for $i\in I$. If $i \in I^{\rm re} $, define \begin{equation*}
\begin{aligned}
\ &[n]_i =\frac{ q^n_{i} - q^{-n}_{i} }{ q_{i} - q^{-1}_{i} },
\ &[n]_i! = \prod^{n}_{k=1} [k]_i ,
\ &\left[\begin{matrix}m \\ n\\ \end{matrix} \right]_i= \frac{ [m]_i! }{[m-n]_i! [n]_i! }.
\end{aligned} \end{equation*}
If $a_{ii} <0$, we define \begin{equation*}
\begin{aligned}
\ &\{n\}_i =\frac{ q^{c_i n}_{i} - q^{-c_i n}_{i} }{ q^{c_i}_{i} - q^{-c_i}_{i} } ,
\ &\{n\}_i! = \prod^{n}_{k=1} \{k\}_i ,
\ &\left\{ \begin{matrix}m \\ n\\ \end{matrix} \right\}_i= \frac{ \{m\}_i! }{\{m-n\}_i! \{n\}_i! }. \end{aligned} \end{equation*} If $a_{ii}=0$, we define \begin{equation*}
\begin{aligned}
\ \ &\{n\}_i =n ,
\ \ & \{n\}_i! = n!,
\ \ &\left\{ \begin{matrix}m \\ n\\ \end{matrix} \right\}_i=\left( \begin{matrix}m \\ n\\ \end{matrix} \right).
\end{aligned} \end{equation*}
\begin{Def} \label{Def: GKM} The {\em quantum generalized Kac-Moody algebra} $U_q(\mathfrak{g})$ associated with a Borcherds-Cartan datum $(A,P,\Pi,\Pi^{\vee})$ is the associative algebra over $\mathbb{Q}(q)$ with ${\bf 1}$ generated by $e_i,f_i$ $(i \in I)$ and $q^{h}$ $(h \in P^{\vee})$ satisfying following relations: \begin{enumerate}
\item $q^0=1, q^{h} q^{h'}=q^{h+h'} $ for $ h,h' \in P^{\vee},$
\item $q^{h}e_i q^{-h}= q^{\langle h, \alpha_i \rangle} e_i,
\ q^{h}f_i q^{-h} = q^{-\langle h, \alpha_i \rangle }f_i$ for $h \in P^{\vee}, i \in I$,
\item $e_if_j - f_je_i = \delta_{ij} \dfrac{K_i -K^{-1}_i}{q_i- q^{-1}_i }, \ \ \mbox{ where } K_i=q_i^{ h_i},$
\item $\displaystyle \sum^{1-a_{ij}}_{r=0} (-1)^r \left[\begin{matrix}1-a_{ij} \\ r\\ \end{matrix} \right]_i e^{1-a_{ij}-r}_i
e_j e^{r}_i =0 \quad \text{ if } i\in I^{\rm re} \text{ and } i \ne j, $
\item $\displaystyle \sum^{1-a_{ij}}_{r=0} (-1)^r \left[\begin{matrix}1-a_{ij} \\ r\\ \end{matrix} \right]_i f^{1-a_{ij}-r}_if_j
f^{r}_i=0 \quad \text{ if } i \in I^{\rm re} \text{ and } i \ne j, $
\item $ e_ie_j - e_je_i=0,\ f_if_j-f_jf_i =0 \ \ \mbox{ if }a_{ij}=0.$ \end{enumerate} \end{Def}
Let $U_q^{+}(\mathfrak{g})$ (resp.\ $U_q^{-}(\mathfrak{g})$) be the subalgebra of $U_q(\mathfrak{g})$ generated by the elements $e_i$ (resp.\ $f_i$), and let $U^0_q(\mathfrak{g})$ be the subalgebra of $U_q(\mathfrak{g})$ generated by $q^{h}$ $(h \in P^{\vee})$. Then we have the \emph{triangular decomposition} $$ U_q(\mathfrak{g}) \cong U^{-}_q(\mathfrak{g}) \otimes U^{0}_q(\mathfrak{g}) \otimes U^{+}_q(\mathfrak{g}),$$ and the {\em root space decomposition} $$U_q(\mathfrak{g}) = \bigoplus_{\alpha \in Q} U_q(\mathfrak{g})_{\alpha},$$ where $U_q(\mathfrak{g})_{\alpha}:=\{ x \in U_q(\mathfrak{g}) \mid q^{h}x q^{-h}=q^{\langle h, \alpha \rangle}x \text{ for any } h \in P^{\vee} \}$. Define a $\mathbb{Q}$-algebra automorphism \ $\bar {} : U^{-}_q(\mathfrak{g}) \to U^{-}_q(\mathfrak{g})$ by \begin{equation}\label{Eq:bar involution} e_i \mapsto e_i, \ \ f_i \mapsto f_i, \ \ q^h \mapsto q^{-h}, \ \ q \mapsto q^{-1}. \end{equation}
Let $\A =\mathbb{Z}[q,q^{-1}]$. For $n \in \mathbb{Z}_{>0}$, set $$ e_i^{(n)} = \begin{cases} \dfrac{e_i^{n}}{[n]_i!} \ \ & \text{if} \ i \in I^{\rm re} ,\\
e_i^{n} \ \ & \text{if} \ i \in I^{\rm im} , \end{cases} \quad \quad f_i^{(n)} = \begin{cases} \dfrac{f_i^{n}}{[n]_i!} \ \ & \text{if} \ i \in I^{\rm re} , \\
f_i^{n} \ \ & \text{if} \ i \in I^{\rm im} , \end{cases} $$ and denote by $U^{-}_{\A}(\mathfrak{g})$ (resp.\ $U^{+}_{\A}(\mathfrak{g})$) the $\A$-sualgebra of $U_q^{-}(\mathfrak{g})$ generated by $f_i^{(n)}$ (resp.\ $e_i^{(n)}$).
Define a twisted algebra structure on $U^{-}_q(\mathfrak{g}) \otimes U^{-}_q(\mathfrak{g})$ as follows:
$$ (x_1 \otimes x_2)(y_1 \otimes y_2)= q^{-( \beta_2 | \gamma_1)}(x_1 y_1 \otimes x_2 y_2 ),$$ where $x_i \in U^-_q(\mathfrak{g}) _{\beta_i}$ and $ y_i \in U^-_q(\mathfrak{g}) _{\gamma_i} $ ($i=1,2$).
Then there is an algebra homomorphism $\Delta_0: U^{-}_q(\mathfrak{g}) \to U^{-}_q(\mathfrak{g}) \otimes U^{-}_q(\mathfrak{g})$ satisfying \begin{align} \label{Eq:def of Delta 0}
\Delta_0(f_i) := f_i \otimes {\bf 1} + {\bf 1} \otimes f_i\ (i\in I). \end{align}
Fix $i \in I$. For any $P \in U^{-}_q(\mathfrak{g})$, there exist unique elements $Q,R \in U^{-}_q(\mathfrak{g})$ such that $$ e_i P - P e_i = \frac{K_i Q - K_i^{-1}R}{q_i -q_i^{-1}}.$$ We define the endomorphisms ${e_i'},\bse_i'': U^{-}_q(\mathfrak{g}) \to U^{-}_q(\mathfrak{g}) $ by $$ {\bse_i'}(P)=R,\ \ \mathrm{e''_i}(P)=Q .$$ Consider ${\bsf_i}$ as the endomorphism of $U^{-}_q(\mathfrak{g})$ defined by left multiplication by $f_i$. Then we have \begin{equation} \label{eq: special commute} \begin{aligned} {\bse_i'} {\bsf_j} = \delta_{ij} + q_i^{-a_{ij}}{\bsf_j}{\bse_i'}.
\end{aligned} \end{equation}
\begin{Def} The {\it quantum boson algebra} $B_{q}(\mathfrak{g})$ associated with a Borcherds-Cartan matrix $A$ is the associative algebra over $\mathbb{Q}(q)$ generated by ${\bse_i'},{\bsf_i}$ $(i \in I)$ satisfying the following relations: \begin{enumerate}
\item ${\bse_i'} {\bsf_j} = q_i^{-a_{ij}}{\bsf_j}{\bse_i'} + \delta_{ij}$,
\item $\displaystyle \sum_{r=0}^{1-a_{ij}} (-1)^{r} \left[\begin{matrix}1-a_{ij} \\ r\\ \end{matrix} \right]_i
{\bse_i'}^{1-a_{ij}-r}{\bse_j'}{\bse_i'}^{r}=0$ $\quad$ if $i \in I^{\rm re} $, $i \neq j$,
\item $\displaystyle \sum_{r=0}^{1-a_{ij}} (-1)^{r} \left[\begin{matrix}1-a_{ij} \\ r\\ \end{matrix} \right]_i
{\bsf_i}^{1-a_{ij}-r}{\bsf_j}{\bsf_i}^{r}=0$ $\quad$ if $i \in I^{\rm re} $, $i \neq j$,
\item ${\bse_i'}{\bse_j'}-{\bse_j'}{\bse_i'}=0$, ${\bsf_i}{\bsf_j}-{\bsf_j}{\bsf_i}=0$ $\quad$ if $a_{ij}=0$. \end{enumerate} \end{Def}
The algebra $U^{-}_q(\mathfrak{g})$ has a $B_q(\mathfrak{g})$-module structure from the equation $\eqref{eq: special commute}$ (\cite{JKK05,Kash91}).
\begin{Prop} \ \label{Prop:Highest vector 1} \begin{enumerate} \item If $x \in U^{-}_q(\mathfrak{g})$ and $\bse_i' x=0$ for all $i\in I$, then $x$ is a constant multiple of ${\bf 1}$. \item $U^{-}_q(\mathfrak{g})$ is a simple $B_q(\mathfrak{g})$-module. \end{enumerate} \end{Prop}
\begin{proof} The proof is almost the same as in \cite[Lemma 3.4.7, Corollary 3.4.9]{Kash91}. \end{proof}
Consider the anti-automorphism $\varphi$ on $B_q(\mathfrak{g})$ defined by $$ \varphi(\bse_i') = \bsf_i \ \text{ and } \ \varphi(\bsf_i) = \bse_i' .$$ We define the symmetric bilinear forms $( \ , \ )_K$ and $( \ , \ )_L$ on $U_q^{-}(\mathfrak{g})$ as follows (cf. \cite[Propostion 3.4.4]{Kash91}, \cite[Chapter 1]{Lus93}): \begin{equation} \label{Eq:def of ()K and ()L} \begin{aligned} & ( {\bf 1},{\bf 1} )_K=1,\ \ (b x,y)_K=(x, \varphi(b) y)_K, \\ & ( {\bf 1},{\bf 1} )_L=1, \ \ ( f_i,f_j )_L =\delta_{ij}(1-q_i^{2})^{-1},
\ \ (x,yz)_L=(\Delta_0(x), y \otimes z)_L \end{aligned} \end{equation} for $ x,y,z \in U_q^{-}(\mathfrak{g})$ and $b \in B_q(\mathfrak{g})$.
\begin{Lem} \ \label{Lem:nondegenerate pairing in GKM} \begin{enumerate} \item The bilinear form $(\ ,\ )_K$ on $U_q^{-}(\mathfrak{g})$ is nondegenerate. \item For homogeneous elements $x \in U_q^{-}(\mathfrak{g})_{-\alpha}$ and $y \in U_q^{-}(\mathfrak{g})_{-\beta}$, we have $$ (x,y)_L = \prod_{i\in I} \dfrac{1}{(1-q_{i}^{2})^{k_i}} (x,y)_K,$$ where $\alpha=\sum_{i\in I}k_i\alpha_{i} \in Q^+$. Hence $(\ ,\ )_L$ is nondegenerate. \item For any $x, y \in U_q^{-}(\mathfrak{g})$, we have $$(\bse'_i x ,y)_L = (1-q^2_i)(x, \bsf_i y)_L. $$ \end{enumerate} \end{Lem} \begin{proof} The assertion (1) is proved in \cite{JKK05}.
It was shown in \cite[(2.4)]{SV01} that the bilinear form $(\ , \ )_K$ satisfies $$ (x,yz)_K=\sum_n(x^{(1)}_{n},y )_K( x^{(2)}_{n},z)_K,$$
where $\Delta_0(x)=\sum_n x^{(1)}_{n} \otimes x^{(2)}_{n}$. Then the assertion (2) can be proved by induction on $|\alpha|$.
To prove the assertion (3), without loss of generality, we may assume that $x \in U_q^{-}(\mathfrak{g})_{-\alpha}$, where $\alpha = -\sum_{i} k_i \alpha_i \in -Q^{+}$. Then by (2) and the definition of $(\ , \ )_K$, we have \begin{align*} (\bse'_i x,y)_L & = \dfrac{1}{(1-q_i^{2})^{k_i -1}} \prod_{j \neq i } \dfrac{1}{(1-q_j^{2})^{k_j}} (\bse'_i x,y)_K \\ &= \dfrac{1-q^2_i}{(1-q_i^{2})^{k_i }} \prod_{j \neq i } \dfrac{1}{(1-q_j^{2})^{k_j}} (x, \bsf_i y)_K \\ &= (1-q^2_i) (x,\bsf_i y)_L, \end{align*} which proves the assertion (3). \end{proof}
We now briefly review the crystal basis theory of quantum generalized Kac-Moody algebras which was developed in \cite{JKK05,JKKS07}. For any homogeneous element $u \in U^{-}_q(\mathfrak{g})$, $u$ can be expressed uniquely as \begin{align} \label{eq: lowerpart i-string decomposition} u = \sum_{l \ge 0} f_i^{(l)}u_l, \end{align} where ${\bse_i'} u_l=0$ for every $l \ge 0$ and $u_l=0$ for $l \gg 0$. We call it the {\it i-string decomposition} of $u$ in $U_q^{-}(\mathfrak{g})$. We define the {\it lower Kashiwara operators} $\tilde{e}_i$, $\tilde{f}_i$ $(i \in I)$ of $U_q^{-}(\mathfrak{g})$ by $$ \tilde{e}_i u = \sum_{k \ge 1}f_i^{(k-1)}u_k, \ \ \tilde{f}_i u = \sum_{k \ge 0}f_i^{(k+1)}u_k. $$\ \ Let $\A_0=\{f/g \in \mathbb{Q}(q) \mid f,g \in \mathbb{Q}[q],g(0) \neq 0 \}$. \begin{Def} A {\it lower crystal basis} of $U^{-}_q(\mathfrak{g})$ is a pair $(L,B)$ satisfying the following conditions: \begin{enumerate} \item $L$ is a free $\A_0$-module of $U^{-}_q(\mathfrak{g})$ such that $U^{-}_q(\mathfrak{g})=\mathbb{Q}(q) \otimes_{\A_0} L$ and
$L = \bigoplus_{\alpha \in Q^+} L_{-\alpha}$, where $L_{-\alpha} := L \cap
U^{-}_q(\mathfrak{g})_{-\alpha}$, \item $B$ is a $\mathbb{Q}$-basis of $L/ q L$ such that $B = \bigsqcup_{\alpha \in Q^+} B_{-\alpha}$, where $B_{-\alpha} := B \cap (L_{-\alpha}/ q L_{-\alpha})$, \item $\tilde{e}_i B \subset B \sqcup \{0\}, \ \tilde{f}_i B \subset B $ for all $i \in I$, \item For $ b,b'\in B$ and $i \in I,$ $ b' = \tilde{f}_i b$ if and only if $b = \tilde{e}_i b'$. \end{enumerate}
\end{Def}
\begin{Prop} \cite[Theorem 7.1]{JKK05} \label{Prop: crystal bases of lowerpart } Let $L(\infty)$ be the free $\A_0$-module of $U_q^-(\mathfrak{g})$ generated by $\{\tilde{f}_{i_1} \cdots \tilde{f}_{i_r} {\bf 1} \mid r \ge 0 , i_k \in I\}$ and let $$B(\infty) = \{\tilde{f}_{i_1} \cdots \tilde{f}_{i_r}{\bf 1}+q L(\infty) \mid r \ge 0 , i_k \in I\}\setminus\{0\}.$$ Then the pair $(L(\infty),B(\infty))$ is a unique lower crystal basis of $U^{-}_q(\mathfrak{g})$. \end{Prop}
Let $\mathcal{O}_{int}$ be the abelian category of $U_q(\mathfrak{g})$-modules defined in \cite[Definition 3.1]{JKK05}. For each $\lambda \in P^+$, let $V(\lambda)$ denote the irreducible highest weight $U_q(\mathfrak{g})$-module with highest weight $\lambda$. It is generated by a unique highest weight vector $v_{\lambda}$ with defining relations: \begin{equation} \label{eq:hw module} \begin{aligned} & q^h v_{\lambda} = q^{\langle h, \lambda \rangle} v_{\lambda} \ \ \text{for all} \ h \in P^{\vee}, \\ & e_i v_{\lambda} = 0 \ \ \text{for all} \ i \in I, \\ & f_i^{\langle h_i, \lambda \rangle + 1} v_{\lambda} =0 \ \ \text{for} \ i \in I^{\rm re} , \\ & f_i v_{\lambda} = 0 \ \ \text{for} \ i\in I^{\rm im} \ \text{with} \ \langle h_i, \lambda \rangle =0. \end{aligned} \end{equation} It was proved in \cite[Theorem 3.7]{JKK05} that the category $\mathcal{O}_{int}$ is semisimple and that all the irreducible objects have the form $V(\lambda)$ for $\lambda \in P^{+}$.
Let $M$ be a $U_q(\mathfrak{g})$-module in the category $\mathcal{O}_{int}$. For any $i \in I$ and $u \in M_{\mu}$, the element $u$ can be expressed uniquely as $$ u = \sum_{k \ge 0} f_i^{(k)}u_k, $$ where $u_k \in M_{\mu + k \alpha_i}$ and $e_i u_k=0$. We call it the {\it i-string decomposition} of $u$. We define the {\it lower Kashiwara operators} $\tilde{e}_i,\tilde{f}_i \ (i \in I)$ by $$ \tilde{e}_i u = \sum_{k \ge 1}f_i^{(k-1)}u_k, \ \ \tilde{f}_i u = \sum_{k \ge 0}f_i^{(k+1)}u_k. $$
\begin{Def} A {\it lower crystal basis} of $U_q(\mathfrak{g})$-module $M$ is a pair $(L,B)$ satisfying the following conditions: \begin{enumerate} \item $L$ is a free $\A_0$-module of $M$ such that $M = \mathbb{Q}(q) \otimes_{\A_0} L$ and $L = \bigoplus_{\lambda \in P}L_{\lambda}$, where $L_{\lambda} := L \cap M_\lambda$, \item $B$ is $\mathbb{Q}$-basis of $L/qL$ such that $B = \bigsqcup_{\lambda \in P} B_{\lambda}$, where $B_{\lambda} := B \cap L_{\lambda}/qL_{\lambda}$, \item $\tilde{e}_i B \subset B \sqcup \{ 0 \} $, \ $\tilde{f}_i B \subset B \sqcup \{ 0 \} $ for all $i \in I$, \item For $b,b' \in B$ and $i\in I$, $b'=\tilde{f}_i b$ if and only if $b= \tilde{e}_i b'$. \end{enumerate} \end{Def}
\begin{Prop} \cite[Theorem 7.1]{JKK05} \label{Prop: crystal bases of integrable module } For $\lambda \in P^+$, let $L(\lambda)$ be the free $\A_0$-module of $V(\lambda)$ generated by $\{ \tilde{f_{i_1}} \cdots \tilde{f_{i_r}}v_{\lambda} \mid r \ge 0 , i_k \in I\}$ and let $$B(\lambda) = \{\tilde{f}_{i_1} \cdots \tilde{f}_{i_r}v_{\lambda}+qL(\lambda) \mid r \ge 0 , i_k \in I\}\setminus\{0\}.$$ Then the pair $(L(\lambda),B(\lambda))$ is a unique lower crystal basis of $V(\lambda)$. \end{Prop}
\vskip 3em
\section{Khovanov-Lauda-Rouquier algebra $R$} \label{Sec:KLR}
In this section, we construct the Khovanov-Lauda-Rouquier algebra $R$ associated with a Borcherds-Cartan matrix $A$, and investigate its algebraic structure and representation theory.
\subsection{The algebras $R(\alpha)$}\
Let $\F$ be a field. For $\alpha \in Q^+$ with
$|\alpha|=d$, set \begin{align*} {\rm Seq} (\alpha) &= \{ \mathbf{i}=(i_1 \ldots i_d) \in I^d \mid \alpha_{i_1} + \cdots + \alpha_{i_d} = \alpha \},\\ {\rm Seqd} (\alpha) &=\{ \mathbf{i}=(i_1^{(d_1)} \ldots i_r^{(d_r)}) \in I^d \mid d_1\alpha_{i_1} + \cdots + d_r\alpha_{i_r} = \alpha \}. \end{align*} Then the symmetric group $\sg_d = \langle r_i \mid i =1, \ldots d-1 \rangle$ acts naturally on $ {\rm Seq} (\alpha)$. For $\mathbf{i}=(i_1\ldots i_d) \in {\rm Seq} (\alpha),\ \mathbf{j}=(j_1\ldots {j_{d'}} ) \in {\rm Seq} (\beta)$, we denote by $\mathbf{i} * \mathbf{j}$ the concatenation of $\mathbf{i}$ and $\mathbf{j}$: $$ \mathbf{i} * \mathbf{j} := (i_1\ldots i_d j_1\ldots {j_{d'}}) \in {\rm Seq} (\alpha+ \beta). $$
The symmetric group $S_d$ acts on the polynomial ring $\F[x_1,\ldots,x_d]$ by \begin{align*} w \cdot f(x_1,\ldots,x_d) = f(x_{w(1)}, \ldots,x_{w(d)}) \quad\text{for $w\in S_d$ and $f(x_1,\ldots,x_d) \in \F[x_1,\ldots,x_d]$.} \end{align*} For $t=1,\ldots, d-1$, define the operator $\partial_t$ on $\F[x_1,\ldots,x_d]$ by $$ \partial_t(f) = \frac{r_t f - f}{ x_{t} - x_{t+1} } $$ for $f \in \F[x_1,\ldots,x_d]$. We take a matrix $( \mathcal{Q}_{i,j}(u,v) )_{i,j\in I}$ in $\F[u,v]$ such that $Q_{i,j}(u,v) = Q_{j,i}(v,u)$ and $Q_{i,j}(u,v)$ has the form $$ Q_{i,j}(u,v) = \left\{
\begin{array}{ll} \displaystyle
\sum_{p,q} t_{i,j;p,q}u^pv^q & \hbox{ if } i\ne j, \\
0 & \hbox{ if } i=j,
\end{array}
\right.
$$
where the summation is taken over all $p,q\in \mathbb{Z}_{\ge0}$ such that $ (\alpha_i|\alpha_j)+s_ip+s_jq=0$ and $t_{i,j;p,q} \in \F$. In particular, $t_{i,j;-a_{ij},0} \in \F^{\times}$. For each $i\in I$, choose a nonzero polynomial $\mathcal{P}_i(u,v) \in \F[u,v]$ having the form $$ \mathcal{P}_i(u,v) = \sum_{p,q } h_{i;p,q} u^pv^q , $$ where the summation is taken over all $p, q \in \mathbb{Z}_{\ge0}$ such that $2-a_{ii}-2p-2q=0$ and $h_{i;p,q} \in \F$. In particular, $h_{i;1-\frac{a_{ii}}{2},0}, h_{i;0,1-\frac{a_{ii}}{2}} \in \F^\times$.
\begin{Def} \label{def:KLR} Let $(A,P,\Pi,\Pi^\vee)$ be a Borcherds-Cartan datum. For $\alpha\in Q^+$ with height $d$, the {\em Khovanov-Lauda-Rouquier algebra $R(\alpha)$} of weight $\alpha$
associated with the data $(A,P,\Pi,\Pi^\vee)$, $(\mathcal{P}_i)_{i\in I}$ and $(\mathcal{Q}_{i,j})_{i,j\in I}$ is the associative graded $\F$-algebra generated by $1_{\mathbf{i}}\ (\mathbf{i}\in {\rm Seq} (\alpha))$, $x_k\ (1 \le k \le d)$, $\tau_t\ (1 \le t \le d-1)$ satisfying the following defining relations: \begin{equation} \label{Eq:def rel 1} \begin{aligned} & 1_{\mathbf{i}} 1_{\mathbf{j}} = \delta_{\mathbf{i},\mathbf{j}} 1_{\mathbf{i}},\ \sum_{\mathbf{i} \in {\rm Seq} (\alpha)} 1_{\mathbf{i}}=1,\ x_k 1_{\mathbf{i}} = 1_{\mathbf{i}} x_k, \ x_k x_l = x_l x_k,\\
& \tau_t 1_{\mathbf{i}} = 1_{r_t( \mathbf{i})} \tau_t,\ \tau_t \tau_s = \tau_s \tau_t \text{ if } |t - s| > 1, \\ & \tau_t^2 1_{\mathbf{i}} = \left\{
\begin{array}{ll}
\partial_t\mathcal{P}_{i_t}(x_t,x_{t+1}) \tau_t 1_{\mathbf{i}} & \hbox{ if } i_t = i_{t+1}, \\
\mathcal{Q}_{i_t, i_{t+1}}(x_t, x_{t+1}) 1_{\mathbf{i}} & \hbox{ if } i_t \ne i_{t+1},
\end{array}
\right. \\ & (\tau_t x_k - x_{r_t(k)} \tau_t ) 1_{\mathbf{i}} = \left\{
\begin{array}{ll}
- \mathcal{P}_{i_t }(x_t, x_{t+1}) 1_{\mathbf{i}} & \hbox{if } k=t \text{ and } i_t = i_{t+1}, \\
\mathcal{P}_{i_t }(x_t, x_{t+1}) 1_{\mathbf{i}} & \hbox{if } k = t+1 \text{ and } i_t = i_{t+1}, \\
0 & \hbox{otherwise,}
\end{array}
\right. \end{aligned} \end{equation} \begin{equation} \begin{aligned} \label{Eq:def rel 2} &( \tau_{t+1} \tau_{t} \tau_{t+1} - \tau_{t} \tau_{t+1} \tau_{t} ) 1_{\mathbf{i}} \\ & \qquad \qquad = \left\{
\begin{array}{ll} \mathcal{P}_{i_t }(x_t, x_{t+2}) \overline{\mathcal{Q}}_{i_t,i_{t+1}}(x_t, x_{t+1}, x_{t+2})1_{\mathbf{i}} & \hbox{if } i_t = i_{t+2} \ne i_{t+1}, \\ \overline{\mathcal{P}}_{i_t}'( x_{t}, x_{t+1}, x_{t+2}) \tau_{t}1_{\mathbf{i}} + \overline{\mathcal{P}}_{i_t}''( x_{t}, x_{t+1}, x_{t+2}) \tau_{t+1}1_{\mathbf{i}} & \hbox{if } i_t = i_{t+1} = i_{t+2},\\ 0 & \hbox{otherwise}, \end{array} \right. \end{aligned} \end{equation} where
\begin{equation} \begin{aligned} \overline{\mathcal{P}}'_i(u,v,w) &:= \frac{\mathcal{P}_{i}(v,u)\mathcal{P}_{i}(u,w)}{(u-v)(u-w)} +\frac{\mathcal{P}_{i}(u,w)\mathcal{P}_{i}(v,w)}{(u-w)(v-w)} -\frac{\mathcal{P}_{i}(u,v)\mathcal{P}_{i}(v,w)}{(u-v)(v-w)}, \\ \overline{\mathcal{P}}''_i(u,v,w) &:= - \frac{\mathcal{P}_{i}(u,v)\mathcal{P}_{i}(u,w)}{(u-v)(u-w)} - \frac{\mathcal{P}_{i}(u,w)\mathcal{P}_{i}(w,v)}{(u-w)(v-w)} + \frac{\mathcal{P}_{i}(u,v)\mathcal{P}_{i}(v,w)}{(u-v)(v-w)}, \\ \overline{\mathcal{Q}}_{i,j}(u,v,w) & := \frac{\mathcal{Q}_{i,j}(u,v) - \mathcal{Q}_{i,j}(w,v)}{u-w}. \end{aligned} \end{equation} \end{Def}
Let $R: = \bigoplus_{\alpha \in Q^{+}} R(\alpha)$.
The $\mathbb{Z}$-grading on $R(\alpha)$ is given by \begin{align} \label{Eq:degree}
\deg(1_\mathbf{i})=0, \quad \deg(x_k 1_\mathbf{i})= 2s_{i_k}, \quad \deg(\tau_t 1_\mathbf{i})= -(\alpha_{i_{t}} | \alpha_{i_{t+1}}). \end{align} Note that $\overline{\mathcal{P}}_i'$, $\overline{\mathcal{P}}_i''$ and $\overline{\mathcal{Q}}_{i,j}$ are polynomials. If $i \in I^{\rm re} $, then $\mathcal{P}_{i}(u, v)$ is a nonzero constant, which will be normalized to be 1 in this paper. If $I$ is finite and $a_{ii}=2$ for all $i\in I$, then the algebra $R$ coincides with the Khovanov-Lauda-Rouquier algebra introduced in \cite{KL09, KL11, R08}.
The algebra $R$ can be defined by using planar diagrams with dots and strands. For simplicity, we assume that $\mathcal{P}_i$ are symmetric and $t_{i,j;-a_{ij},0} = t_{i,j;0, -a_{ji}} = 1$ and $t_{i,j;p,q} = 0$ for other $p,q$. Note that $\partial_t \mathcal{P}_{i_t}(x_t,x_{t+1}) = 0$. We denote by $R$ the $\F$-vector space spanned by braid-like diagrams, considered up to planar isotropy, such that all strands are colored by $I$ and can carry dots. The multiplication $D \cdot D'$ of two diagrams $D$ and $D'$ is given by stacking of the diagram $D$ on the diagram $D'$ if the color on the top of $D'$ matches with the color at the bottom of $D$ and defined to be $0$ otherwise. It is obvious that the following elements are generators of $R(\alpha)$ $(\alpha \in Q^{+}, \mathbf{i}= (i_1 \ldots i_d) \in {\rm Seq} (\alpha))$: \begin{align*} 1_\mathbf{i} \ := \ \genOne{i_1}{i_k}{i_d} , \quad x_k 1_\mathbf{i} \ := \ \genX{i_1}{i_k}{i_d} , \quad \tau_t 1_\mathbf{i} \ := \ \genTau{i_1}{i_t}{i_{t+1}}{i_d}. \end{align*} The local relations are given as follows:
\begin{align} \label{Eq:local rel 1}
\dCross{i}{j} \quad = \quad \begin{cases} \quad \quad \quad \quad \quad \ 0 & \text{ if } i= j, \\
\quad \quad \quad \quad \ \ \twoStrands{i}{j} & \text{ if } (\alpha_i|\alpha_j)=0, \\
\quad \twoDotStrandsL{-a_{ij}}{i}{j} \ + \ \twoDotStrandsR{-a_{ji}}{i}{j} & \text{ if } (\alpha_i|\alpha_j)\neq 0,
\end{cases} \end{align}
\begin{equation} \label{Eq:local rel 2} \begin{aligned} & \CrossDR{{}}{i}{j} \ - \ \CrossUL{{}}{i}{j}\quad = \quad \begin{cases} \quad \mathcal{P}_i(x,y) \cdot \twoStrands{i}{i} & \text{ if } i = j, \\
\quad \quad \quad 0 & \text{ otherwise, } \end{cases} \\ & \CrossUR{{}}{i}{j} \ - \ \CrossDL{{}}{i}{j}\quad = \quad \begin{cases} \quad \mathcal{P}_i(x,y) \cdot \twoStrands{i}{i} & \text{ if } i = j, \\
\quad \quad \quad 0 & \text{ otherwise, } \end{cases} \\ &\ ( \text{ here, } x:=\smalltwoDotStrandsL{i}{i}\ \text{ and } \ y:=\smalltwoDotStrandsR{i}{i}\ ) \end{aligned} \end{equation}
\begin{equation} \label{Eq:local rel 3} \begin{aligned} \quad & \BraidR{i}{j}{k} \ -\ \BraidL{i}{j}{k}\ =\ \left\{
\begin{array}{ll}
\mathcal{P}_i(x,z){\displaystyle \sum_{s=0}^{-a_{ij}-1}} \threeDotStrands{s}{-a_{ij}-1-s}{i}{j}{i} & \hbox{ if } i=k \ne j, a_{ij} \ne 0, \\
\overline{\mathcal{P}}'_i(x,y,z) \left( \ \CrossL{i}{i}{i}\ -\ \CrossR{i}{i}{i} \ \right) & \hbox{ if } i=j=k, \\
0 & \hbox{otherwise.}
\end{array}
\right. \\
&\ ( \text{ here, } x:=\smallthreeDotStrandsL{i}{j}{k} , \ \ y:=\smallthreeDotStrandsM{i}{j}{k}\ \text{ and }\ z:=\smallthreeDotStrandsR{i}{j}{k}\ ) \end{aligned} \end{equation}
For $\mathbf{t}=(t_1 \ldots t_d) \in \mathbb{Z}_{\ge0}^d$ and a reduced expression $ w = r_{i_1}\cdots r_{i_t} \in \sg_d$, set $$ x^{\mathbf{t}} = x_{1}^{t_1}\cdots x_{d}^{t_d} \ \text{ and } \ \tau_{w} = \tau_{i_1}\cdots\tau_{i_t}.$$ It follows from the defining relations that $$ \{ \tau_{w} x^{\mathbf{t}} 1_\mathbf{i} \mid \mathbf{t} \in \mathbb{Z}_{\ge0}^d,\ \mathbf{i} \in {\rm Seq} (\alpha), \ w: \text{reduced in } \sg_d \} $$ is a spanning set of $R(\alpha)$.
Consider the graded anti-involution $\psi: R(\alpha) \rightarrow R(\alpha)$ which is the identity on generators. For a graded left $R(\alpha)$-module $M$, let $M^\star$ be the graded right $R(\alpha)$-module whose underlying space is $M$ with $R(\alpha)$-action given by $$ v \cdot r = \psi(r)v \quad \text{ for } v \in M^{\star}, \ r \in R(\alpha). $$
We will investigate the structure of $R(m \alpha_i)$ $(m \ge 0)$ in more detail. If $a_{ii}=2$, then the defining relations for $R(m \alpha_i)$ reduce to \begin{align*} & x_k x_l = x_l x_k, \ \ \tau_t^2 = 0 , \\
& \tau_{t}\tau_{t+1}\tau_{t} = \tau_{t+1}\tau_{t}\tau_{t+1},\ \ \tau_t \tau_s = \tau_s \tau_t\ \text{ if } |t-s|>1, \\ & \tau_{t} x_{t} = x_{t+1} \tau_{t}-1, \ \ \tau_{t} x_{t+1} = x_{t} \tau_{t} +1, \\ & \tau_t x_k = x_k \tau_t \ \ \text{if} \ k \neq t, t+1. \end{align*} Hence the algebra $R(m \alpha_i)$ is isomorphic to the {\it nil Hecke algebra} $NH_m$, which is the associative algebra generated by $\mathbf{x}_k\ (1\le k \le m)$ and $\partial_t\ (1 \le t \le m-1)$ satisfying the following relations: \begin{align*} &\mathbf{x}_k \mathbf{x}_l = \mathbf{x}_l \mathbf{x}_k, \ \ \partial_t^2 = 0 , \\ & \partial_{t}\partial_{t+1}\partial_{t} = \partial_{t+1}\partial_{t}\partial_{t+1},\ \
\partial_t \partial_s = \partial_s \partial_t\ \text{ if } |t-s|>1, \\ & \partial_t \mathbf{x}_t = \mathbf{x}_{t+1} \partial_t -1, \ \ \partial_t \mathbf{x}_{t+1}= \mathbf{x}_{t} \partial_t + 1, \\ & \partial_t \mathbf{x}_k = \mathbf{x}_k \partial_t \ \ \text{if} \ k \neq t, t+1. \end{align*} Therefore, as was shown in \cite{KL09}, the algebra $R(m \alpha_i)$ has a primitive idempotent $\tau_{w_0}x_1^{m-1} \cdots x_{m-2}^2x_{m-1}$, where $w_{0}$ is the longest element in $\sg_m$, and has a unique (up to isomorphism and degree shift) irreducible module $L(i^m)$. The irreducible module $L(i^m)$ is isomorphic to the one induced from the trivial $\F[x_1,\ldots, x_m]$-module of dimension 1 over $\bR$.
If $a_{ii}<0$, then $\mathcal{P}_i(u,v)$ is a homogeneous polynomial with degree $1-\dfrac{a_{ii}}{2} > 1$, and $\overline{\mathcal{P}}'_i(u,v,w)$ and $\overline{\mathcal{P}}''_i(u,v,w)$ have positive degree. By \eqref{Eq:degree}, $R(m \alpha_i)$ has positive grading and hence it has a unique idempotent $1_{(i\ldots i)}$. Thus there exists a unique irreducible $R(m \alpha_i)$-module $L(i^m)=\F v$ defined by \begin{align} \label{Eq:def of L in Iim}
1_{(i\ldots i)} \cdot v = v, \ \ x_k \cdot v = 0, \ \ \tau_t \cdot v = 0 . \end{align}
If $a_{ii}=0$, then in general, $R(m\alpha_i)$ has many primitive idempotents, which means that there are many irreducible $R(m\alpha_i)$-modules. For example, if $m=3$ and $\mathcal{P}_i(u,v) = u-v$, then $\tau_1\tau_2, \tau_2\tau_1$ and $1 - \tau_1\tau_2 - \tau_2\tau_1$ are orthogonal primitive idempotents. The algebra $R(m \alpha_i)$ itself, not principal indecomposable modules, will serve as one of the projective modules that give our categorification. The whole Grothendieck group of the category of finitely generated projective $R(m \alpha_i)$-modules seems rather large and nontrivial. We hope to investigate it in a later work.
We now construct a faithful polynomial representation of $R(\alpha)$. First, we define an $R(m\alpha_i)$-module structure on $\F[x_1, \ldots, x_m]$ by \begin{align*} x_k \cdot f(x_1, \ldots, x_m) &= x_k f(x_1, \ldots, x_m), \\ \tau_t \cdot f(x_1, \ldots, x_m) &= \mathcal{P}_{i}(x_t, x_{t+1}) \partial_t ( f(x_1, \ldots, x_m)) \end{align*} for $x_k, \tau_t \in R(m\alpha_i), f(x_1, \ldots, x_m) \in \F[x_1, \ldots, x_m]$.
\begin{Lem} \label{Lem:R(m_i) for i in Iim} $\F[x_1, \ldots, x_m]$ is a faithful representation of $R(m\alpha_i)$. \end{Lem} \begin{proof}
If $i \in I^{\rm re} $, our assertion was shown in \cite[Example 2.2]{KL09}. Assume that $i\in I^{\rm im} $ and let $\mathbf{x}_k$ be the endomorphism of $\F[x_1, \ldots, x_m]$ defined by \begin{align*} \mathbf{x}_k ( f(x_1, \ldots, x_m)) = x_k f(x_1, \ldots, x_m) \end{align*} for $f(x_1, \ldots, x_m) \in \F[x_1, \ldots, x_m]$. Note that $$\{ \partial_{j_1} \cdots \partial_{j_k} \mathbf{x}^\mathbf{t} \mid \mathbf{t} \in \mathbb{Z}_{\ge0}^m, \ r_{j_1} \cdots r_{j_k} \text{is a reduced expression in } \sg_m \, (k \ge 0) \}$$ is a linearly independent subset of $\End(\F[x_1, \ldots, x_m])$. Let $$\iota: R(m \alpha_i) \longrightarrow \End(\F[x_1, \ldots, x_m])$$ be the map defined by $\iota(x_k) = \mathbf{x}_k$ and $\iota(\tau_t) = \mathcal{P}_i( \mathbf{x}_t, \mathbf{x}_{t+1}) \cdot \partial_t$.
We first show that $\iota$ is well-defined. Since $\mathcal{P}_i(u,v)$ is a homogeneous polynomial, it is easy to verify that the relations $\eqref{Eq:def rel 1}$ hold. To check the relations in \eqref{Eq:def rel 2}, for simplicity, we assume that $m=3$ and let $x = x_1, y = x_2, z = x_3$, $\mathcal{P}(u,v) = \mathcal{P}_i(u,v)$.
Set $$ \mathsf{P}(u,v) = \frac{\mathcal{P}(u,v)}{u-v}. $$ By a direct computation, we have
\begin{align*} \iota(\tau_2 \tau_1 \tau_2) & = \mathsf{P}(x,y)\mathsf{P}(y,z)\mathsf{P}(x,z)(r_2r_1r_2-r_2r_1-r_1r_2+r_1)\\ & \ - \mathsf{P}(y,z)\mathsf{P}(z,y)\mathsf{P}(x,z)(1-r_2)+\mathsf{P}(x,y)\mathsf{P}(y,z)^2(r_2-1), \\ \iota(\tau_1 \tau_2 \tau_1) & = \mathsf{P}(x,y)\mathsf{P}(y,z)\mathsf{P}(x,z)(r_1r_2r_1-r_2r_1-r_1r_2+r_2)\\ & \ - \mathsf{P}(x,y)\mathsf{P}(y,x)\mathsf{P}(x,z)(1-r_1)+\mathsf{P}(x,y)^2\mathsf{P}(y,z)(r_1-1). \end{align*} As $\iota(\tau_k) = \mathsf{P}(x_k,x_{k+1}) (r_k-1) $ for $k=1,2,$ \begin{align*} \iota(\tau_2 \tau_1 \tau_2) - \iota(\tau_1 \tau_2 \tau_1) &= (-\mathsf{P}(y,x)\mathsf{P}(x,z)+\mathsf{P}(y,z)\mathsf{P}(x,z)-\mathsf{P}(x,y)\mathsf{P}(y,z))\iota(\tau_1) \\ & \quad + ( \mathsf{P}(x,y)\mathsf{P}(y,z) + \mathsf{P}(z,y)\mathsf{P}(x,z)-\mathsf{P}(x,y)\mathsf{P}(x,z))\iota(\tau_2), \end{align*} which shows that the relation $\eqref{Eq:def rel 2}$ holds. It remains to show that $\iota$ is injective. Take a nonzero element $$y = \tau_{w_1}f_1 + \cdots + \tau_{w_t}f_t \quad ( 0\ne f_k \in \F[x_1, \ldots, x_m],\ w_k \ \text{is a reduced expression in } \sg_m )$$ of $R(m\alpha_i)$ such that $w_i \ne w_j \text{ if } i \ne j$ and $ \ell( {w}_1 ) \ge \ell( {w}_k ) $ for $0 \le k \le t$. Write the reduced expression of $w_1$ as $w_1 = r_{i_1}\cdots r_{i_l}$. Then, $\iota(y)$ can be written as $$ \iota(y) = \partial_{i_1}\cdots \partial_{i_l} f' + \cdots \text{lower terms} \cdots $$ for some nonzero polynomial $f'$, which implies that $\iota(y)$ is nonzero. Therefore $\iota$ is injective. \end{proof}
Now we consider the general case $R(\alpha)$ with $\alpha \in Q^+$. Take a total order $\prec$ on $I$. Let $$ \mathfrak{Pol}(\alpha) = \bigoplus_{\mathbf{i} \in {\rm Seq} (\alpha)} \F[x_1(\mathbf{i}), \ldots, x_d(\mathbf{i})]. $$ For any polynomial $f \in \F[u_1, \ldots, u_d]$, let $f(\mathbf{i})$ be the polynomial in $\F[x_1(\mathbf{i}), \ldots, x_d(\mathbf{i})]$ obtained from $f$ by replacing $u_k$ by $x_k(\mathbf{i})$. We define an $R(\alpha)$-module structure on $\mathfrak{Pol}(\alpha)$ as follows: for $\mathbf{i} \in {\rm Seq} (\alpha)$ and $f \in \F[u_1, \ldots, u_d]$, we define \begin{equation} \label{Eq:def of faithful rep} \begin{aligned} 1_\mathbf{j} \cdot f(\mathbf{i}) &= \delta_{\mathbf{ij}} f(\mathbf{i}) \quad \ (\ \mathbf{j} \in {\rm Seq} (\alpha)\ ), \\ x_k \cdot f(\mathbf{i}) &= x_k(\mathbf{i})f(\mathbf{i}), \\ \tau_t \cdot f(\mathbf{i}) &= \left\{
\begin{array}{ll}
\mathcal{P}_{i_t}(x_t(r_t \mathbf{i}), x_{t+1}(r_t \mathbf{i})) \partial_t f(r_t \mathbf{i}) & \hbox{ if } i_t = i_{t+1}, \\
\mathcal{Q}_{i_{t+1}, i_{t}}(x_t(r_t\mathbf{i}), x_{t+1}(r_t\mathbf{i})) r_t f(r_t \mathbf{i}) & \hbox{ if } i_t \ne i_{t+1},\ i_t \succ i_{t+1}, \\
r_t f(r_t \mathbf{i}) & \hbox{ if } i_t \ne i_{t+1},\ i_t \prec i_{t+1}.
\end{array}
\right. \end{aligned} \end{equation}
\begin{Lem} \label{Lem:faithful repn} $\mathfrak{Pol}(\alpha)$ is a well-defined $R(\alpha)$-module. \end{Lem} \begin{proof} We verify the defining relations of $R(\alpha)$. The relations $\eqref{Eq:def rel 1}$ can be verified in a straightforward manner. In the proof of Lemma \ref{Lem:R(m_i) for i in Iim}, we already proved our assertion when $i_t = i_{t+1} = i_{t+2}$. Thus it suffices to consider the following three cases in $\eqref{Eq:def rel 2}$: (i) $i_t=i_{t+2} \ne i_{t+1}$, \ (ii) $i_t, i_{t+1},i_{t+2}$ are distinct, \ (iii) $i_t = i_{t+1} $ and $i_t \ne i_{t+2}$. For simplicity, let $d = 3$, $\mathbf{i} = (i,j,k)$ and $f(u,v,w)= u^a v^b w^c $. Set $x = x_1(\mathbf{i})$, $y = x_2(\mathbf{i})$ and $z = x_3(\mathbf{i})$.
Case (i): Let $\mathbf{i} = (i,j,i)$ with $i \ne j$. Without loss of generality, we may assume $i \prec j$. Then, by a direct computation, we have \begin{align*} \tau_1 \tau_2 \tau_1 (x^a y^b z^c) \ &=\ \mathcal{P}_i(x, z) \mathcal{Q}_{ij}(x,y) \frac{ x^c y^b z^a - x^a y^b z^c }{x-z}, \\ \tau_2 \tau_1 \tau_2 (x^a y^b z^c) \ &=\ \mathcal{P}_i(x, z) \frac{ \mathcal{Q}_{ij}(x,y)x^c y^b z^a - \mathcal{Q}_{ij}(z,y) x^a y^b z^c }{x-z}, \end{align*} which yield $$ (\tau_2 \tau_1 \tau_2 - \tau_1 \tau_2 \tau_1) (x^a y^b z^c) = \mathcal{P}_i(x, z) \frac{\mathcal{Q}_{ij}(x,y) - \mathcal{Q}_{ij}(z,y) }{x-z} x^ay^bz^c. $$
Case (ii): Let $\mathbf{i} = (i,j,k)$ such that $i, j,k$ are distinct. Since the other cases are similar, we will only prove our assertion when $i \succ j \succ k$. Then we have \begin{align*} \tau_1 \tau_2 \tau_1 (x^a y^b z^c) \ &=\ \mathcal{Q}_{ij}(y,z)\mathcal{Q}_{jk}(x,y) \mathcal{Q}_{ik}(x,z) x^cy^bz^a, \\ \tau_2 \tau_1 \tau_2 (x^a y^b z^c) \ &=\ \mathcal{Q}_{ij}(y,z)\mathcal{Q}_{jk}(x,y) \mathcal{Q}_{ik}(x,z) x^cy^bz^a, \end{align*} which implies that $ (\tau_2 \tau_1 \tau_2 - \tau_1 \tau_2 \tau_1) (x^a y^b z^c) = 0 $.
Case (iii): Similarly as above, we consider $\mathbf{i} = (i,i,j)$ with $i \succ j $ only. Then \begin{align*} \tau_1 \tau_2 \tau_1 (x^a y^b z^c) \ &=\ \mathcal{Q}_{ij}(x,y) \mathcal{Q}_{ij}(x,z) \mathcal{P}_i(y,z) \frac{x^cy^bz^a - x^cy^az^b }{y-z}, \\ \tau_2 \tau_1 \tau_2 (x^a y^b z^c) \ &=\ \mathcal{Q}_{ij}(x,y) \mathcal{Q}_{ij}(x,z) \mathcal{P}_i(y,z) \frac{x^cy^bz^a - x^cy^az^b }{y-z}. \end{align*} Hence we have $ (\tau_2 \tau_1 \tau_2 - \tau_1 \tau_2 \tau_1) (x^a y^b z^c) = 0 $, which completes the proof. \end{proof}
Note that $R(\alpha) =\bigoplus_{\mathbf{i},\mathbf{j} \in {\rm Seq} (\alpha)} {_\mathbf{j}}R(\alpha)_{\mathbf{i}}$, where $_{\mathbf{j}}R(\alpha)_{\mathbf{i}} := 1_\mathbf{j} R(\alpha) 1_\mathbf{i}$. Given each $w \in \sg_d$, fix a minimal representative $\underline{w}$ of $w$. For $\mathbf{i}, \mathbf{j}\in {\rm Seq} (\alpha)$, let $$ _{\mathbf{j}}\sg_{\mathbf{i}} = \{ \underline{w} \mid w \in \sg_d,\ w(\mathbf{i})=\mathbf{j} \}. $$ It follows from the defining relations that $$ _{\mathbf{j}}B(\alpha)_{\mathbf{i}}:= \{ \tau_{\underline{w}} x^{\mathbf{t}} 1_\mathbf{i} \mid \mathbf{t} \in \mathbb{Z}_{\ge0}^d,\ \underline{w} \in {_\mathbf{j} \sg_\mathbf{i}} \} $$ is a spanning set of $_\mathbf{j} R(\alpha)_\mathbf{i}$. Moreover, we have the following proposition.
\begin{Prop} \label{Prop:basis of R(alpha)} \ \begin{enumerate} \item The set $ _{\mathbf{j}}B(\alpha)_{\mathbf{i}}$ is a homogeneous basis of $_{\mathbf{j}} R(\alpha) _{\mathbf{i}}$. \item $\mathfrak{Pol}(\alpha)$ is a faithful representation of $R(\alpha)$. \end{enumerate} \end{Prop} \begin{proof}
Let $<$ be the lexicographic order of $ {\rm Seq} (\alpha)$ arising from the order $\prec$ of $I$, and let ${_{\mathbf{j}_2}}w_{\mathbf{j}_1} $ be the minimal element in $ _{{\mathbf{j}_2}}\sg_{\mathbf{j}_1}$ for $\mathbf{j}_1$, $\mathbf{j}_2 \in {\rm Seq} (\alpha)$. Let $$ \Upsilon : R(\alpha) \longrightarrow \End(\mathfrak{Pol}(\alpha))$$ be the algebra homomorphism given in $\eqref{Eq:def of faithful rep}$. We will show that $ \Upsilon ({_{\mathbf{j}}B(\alpha)_{\mathbf{i}}})$ is linearly independent, which would imply the set ${_{\mathbf{j}}B(\alpha)_{\mathbf{i}}}$ is linearly independent. The injectivity of $\Upsilon$ would also follow immediately. We prove our claim using induction on the lexicographic order $<$ on $ {\rm Seq} (\alpha)$.
Let $\mathbf{i}\in {\rm Seq} (\alpha)$, and let $$\mathbf{j} = ( \underbrace{j_1 \ldots j_1 }_{d_1} \underbrace{j_2 \ldots j_2 }_{d_2} \cdots \underbrace{j_r \ldots j_r }_{d_r} ) \in {\rm Seq} (\alpha)$$ such that $j_1 \succ j_2 \succ \cdots \succ j_r $. Note that $\mathbf{j}$ is a maximal element in $ {\rm Seq} (\alpha)$.
Let $m$ be a linear combination of ${_{\mathbf{j}}B(\alpha)_{\mathbf{i}}}$ such that $ \Upsilon(m)=0 $. Note that $m$ can be expressed as $$ m = \sum_s \tau_{w_s} \tau_{ {_\mathbf{j}}w_\mathbf{i}} x^{\mathbf{k}_s}1_{\mathbf{i}} $$ for some $ \mathbf{k}_s \in \mathbb{Z}_{\ge0}^d$ and some $w_s \in \sg_{d_1} \times \cdots \times \sg_{d_r}$. It follows from $\eqref{Eq:def of faithful rep}$ that $\Upsilon ( \tau_{ {_\mathbf{j}}w_\mathbf{i}} 1_{\mathbf{i}} ) $ can be viewed as a linear map from $\F[x_1(\mathbf{i}), \ldots, x_d(\mathbf{i})] $ to $ \F[x_1(\mathbf{j}), \ldots, x_d(\mathbf{j})] $ sending $1_{\mathbf{i}}$ to $1_{\mathbf{j}}$. Hence, $$ \Upsilon(m) = 0 \quad \text{ if and only if }\quad \Upsilon( \sum_s \tau_{w_s} x^{ {_\mathbf{j}}w_\mathbf{i} (\mathbf{k}_s) } 1_{\mathbf{j}} ) =0 .$$ Since $\Upsilon( \sum_s \tau_{w_s} x^{ {_\mathbf{j}}w_\mathbf{i} (\mathbf{k}_s) } 1_{\mathbf{j}} )$ can be regarded as a linear map in $\bigoplus_{k=1}^r \End( \F[x_1, \ldots, x_{d_k} ])$, by Lemma \ref{Lem:R(m_i) for i in Iim}, we have $$ \Upsilon( \sum_s \tau_{w_s} x^{ {_\mathbf{j}}w_\mathbf{i} (\mathbf{k}_s) } 1_{\mathbf{j}} ) =0 \quad \text{ if and only if }\quad \sum_s \tau_{w_s} x^{ {_\mathbf{j}}w_\mathbf{i} (\mathbf{k}_s) } 1_{\mathbf{j}} = 0, $$ which implies $ m = 0$. Therefore, $\Upsilon( {_{\mathbf{j}}B(\alpha)_{\mathbf{i}}} )$ is linearly independent.
We now consider the case when $\mathbf{j} $ is an arbitrary sequence in $ {\rm Seq} (\alpha)$. This step can be proved by a similar induction argument as in \cite[Theorem 2.5]{KL09}, which completes the proof. \end{proof}
For any $\alpha, \beta \in Q^+$, let \begin{align*} 1_\alpha &= \sum_{\mathbf{i} \in {\rm Seq} (\alpha)} 1_\mathbf{i}, \\ 1_{\alpha, \beta} &= \sum_{\mathbf{i} \in {\rm Seq} (\alpha),\ \mathbf{j} \in {\rm Seq} (\beta)} 1_{\mathbf{i} * \mathbf{j}} . \end{align*} Then $1_{\alpha, \beta} R(\alpha+\beta)$ has a natural graded left $R(\alpha) \otimes R(\beta)$-module structure. \begin{Cor} \label{Cor:R(alpha+beta) is free} $1_{\alpha, \beta} R(\alpha+\beta)$ is a free graded left $R(\alpha) \otimes R(\beta)$-module. \end{Cor} \begin{proof}
Let $d := | \alpha| $, $d' := |\beta|$, and $ \sg_d \times \sg_{d'} \backslash \sg_{d+d'} $ be the set of minimal right $\sg_{d} \times \sg_{d'}$-coset representatives of $\sg_{d+d'}$. For $w \in \sg_d \times \sg_{d'} \backslash \sg_{d+d'} $, set $$ \hat{\tau}_{w} = \sum_{ \mathbf{i} \in {\rm Seq} (\alpha),\ \mathbf{j} \in {\rm Seq} (\beta)} 1_{\mathbf{i}*\mathbf{j}}\ \tau_{w}\ 1_{w^{-1}( \mathbf{i}*\mathbf{j})}. $$ Then, it follows from Proposition \ref{Prop:basis of R(alpha)} that $$ \{ \hat{\tau}_{w} \mid w\in \sg_d \times \sg_{d'} \backslash \sg_{d+d'} \} $$ is a basis of $1_{\alpha, \beta} R(\alpha+\beta)$ as a left $R(\alpha) \otimes R(\beta)$-module. \end{proof}
For a graded $R(\alpha)$-module $M=\bigoplus_{i\in \mathbb{Z}}M_i$, let $M\langle k \rangle$ denote the graded $R(\alpha)$-module obtained from $M$ by shifting the grading by $k$; i.e., $M \langle k \rangle := \bigoplus_{i\in \mathbb{Z}} M_{i+k}$. Given $\alpha, \alpha', \beta, \beta' \in Q^+$ with $\alpha+\beta = \alpha' + \beta'$, let $$ _{\alpha, \beta}R_{\alpha', \beta'} := 1_{\alpha, \beta} R(\alpha+\beta) 1_{\alpha', \beta'} . $$ We write $_{\alpha}R_{\alpha', \beta'}$ (resp.\ $_{\alpha, \beta}R_{\alpha'}$) for $_{\alpha, \beta}R_{\alpha', \beta'}$ if $\beta = 0$ (resp.\ $\beta' = 0$). Note that $_{\alpha, \beta}R_{\alpha', \beta'}$ is a graded $(R(\alpha)\otimes R(\beta), R(\alpha')\otimes R(\beta'))$-bimodule. Now we obtain the Mackey's Theorem for Khovanov-Lauda-Rouquier algebras.
\begin{Prop} \label{Prop:Mackey} The graded $(R(\alpha)\otimes R(\beta), R(\alpha')\otimes R(\beta'))$-bimodule $_{\alpha,\beta}R_{\alpha', \beta'}$ has a graded filtration with graded subquotients isomorphic to $$ {_{\alpha}R_{\alpha-\gamma, \gamma}} \otimes {_{\beta}R_{\beta+\gamma-\beta', \beta'-\gamma}} \otimes_{R'} {_{\alpha-\gamma, \alpha'+\gamma-\alpha}R_{\alpha'}} \otimes
{_{\gamma, \beta-\gamma}R_{\beta}} \langle (\gamma | \beta + \gamma - \beta') \rangle , $$ where $R' = R(\alpha-\gamma)\otimes R(\gamma)\otimes R(\beta + \gamma - \beta') \otimes R(\beta' - \gamma)$ for all $\gamma \in Q^+$ such that every term above lies in $Q^{+}$. \end{Prop} \begin{proof} The proof is almost identical to that of \cite[Proposition 2.18]{KL09}. \end{proof}
For $\alpha = \sum_{i\in I} k_i \alpha_i \in Q^+$ with
$|\alpha|=d$, we define $$ \pol(\alpha) = \prod_{\mathbf{i} \in {\rm Seq} (\alpha)} \F[ x_{1,\mathbf{i}}, \ldots, x_{d,\mathbf{i}} ]. $$ Then the symmetric group $\sg_d$ acts on $\pol(\alpha)$ by $w \cdot x_{k,\mathbf{i}} := x_{w(k),w(\mathbf{i})}$ for $w\in \sg_d$. Let $$ \sym(\alpha) = \pol(\alpha)^{\sg_d}. $$ Note that $ \sym(\alpha) \simeq \bigotimes_{i\in I} \F[x_1,\ldots,x_{k_i}]^{\sg_{k_i}} $. Considering $\sym(\alpha)$ as a subalgebra of $R(\alpha)$ via the natural inclusion $\pol(\alpha) \hookrightarrow R(\alpha)$ sending $x_{k,\mathbf{i}}$ to $x_{k} 1_{\mathbf{i}}$, we have the following lemma. \begin{Lem}\ \label{Lem:center of R(alpha)} \begin{enumerate} \item $\sym(\alpha)$ is the center of $R(\alpha)$. \item $R(\alpha)$ is a free module of rank $(d!)^{2}$ over its center $\sym(\alpha)$. \end{enumerate} \end{Lem} \begin{proof} We first consider the case when $\alpha = m\alpha_i$ for $i \in I$. If $i \in I^{\rm re} $, it follows from $R(m\alpha_i) \simeq NH_m$ that $\sym(\alpha)$ is the center of $R(m\alpha_i)$. Suppose that $i\in I^{\rm im} $. By Lemma \ref{Lem:R(m_i) for i in Iim}, $R(\alpha)$ can be considered as a subalgebra of $\End(\F[x_1, \ldots, x_d])$. Let $\mathbf{x}_k$ be the endomorphism of $\F[x_1, \ldots, x_m]$ defined by multiplication by $x_k$. It is obvious that $\sym(\alpha)$ is contained in the center of $R(\alpha)$ and $ \F[ \mathbf{x}_1, \ldots, \mathbf{x}_m ] \subset R(\alpha)$.
For $ f\in \F[ \mathbf{x}_1, \ldots, \mathbf{x}_m ]$, from the defining relations, we have $$ f \tau_{i_1} \cdots \tau_{i_k} = \tau_{i_1} \cdots \tau_{i_k}( r_{i_k} \cdots r_{i_1} f) + \cdots \text{ lower terms } \cdots $$ with respect to the Bruhat order. Let $y = \sum_{i} \tau_{w_i} f_i $ be an element in the center of $R(\alpha)$. We assume $\ell(w_1) \ge \ell(w_{k})$ for all $k$. Take $j$ such that $w_1(j) \ne j$. Then $$y \mathbf{x}_j - \mathbf{x}_j y = y ( \mathbf{x}_j - \mathbf{x}_{w_1(j)} ) + \cdots \text{ lower terms }\cdots, $$ which implies $\tau_{w_i} = 1$ for all $i$. Since $y$ commutes with all $\tau_i$, $y$ should be a symmetric polynomial. Therefore, the center of $R(\alpha)$ is $\sym(\alpha)$.
We now deal with the general case when $\alpha \in Q^+$. In this case, using the fact that $\sym(m\alpha_i)$ is the center of $R(m\alpha_i)$ for $i\in I$, our assertion can be proved in the same manner as in \cite[Thoerem 2.9, Corollary 2.10]{KL09}. \end{proof}
\vskip 1em
\subsection{Quantum Serre relations} \
Let $R(\alpha)$-mod (resp.\ $R(\alpha)$-pmod, $R(\alpha)$-fmod) be the category of arbitrary
(resp.\ finitely generated projective, finite-dimensional) graded left $R(\alpha)$-modules. The morphisms in these categories are homogeneous homomorphisms. Let \begin{align*} K_0(R) = \bigoplus_{\alpha \in Q^+} K_0(R(\alpha)\text{-pmod})\ \text{ and }\ G_0(R) = \bigoplus_{\alpha \in Q^+} G_0(R(\alpha)\text{-fmod}), \end{align*} where $K_0(R(\alpha)$-pmod) (resp.\ $G_0(R(\alpha)$-fmod)) is the Grothendieck group of $R(\alpha)$-pmod (resp.\ $R(\alpha)$-fmod). Then $K_0(R)$ and $G_0(R)$ have the $\A$-module structure given by $q[M] = [M\langle -1 \rangle]$, where $[M]$ is the isomorphism classes of an $R(\alpha)$-module $M$. For $M,N \in R(\alpha)$-mod, let ${\rm Hom}(M,N)$ be the $\F$-vector space of homogeneous homomorphisms of degree $0$, and let ${\rm Hom}(M \langle k \rangle,N) = {\rm Hom}(M,N \langle -k \rangle)$ be the $\F$-vector space of homogeneous homomorphisms of degree $k$. Define $$ {\rm HOM}(M,N)= \bigoplus_{k\in \mathbb{Z}} {\rm Hom}(M,N \langle k \rangle). $$
Let $\sym^+(\alpha)$ be the maximal ideal of $\sym(\alpha)$. Since $\sym^+(\alpha)$ acts on any irreducible graded $R(\alpha)$-module trivially, the isomorphism classes of irreducible graded modules over $R(\alpha)$ are in 1-1 correspondence with the isomorphism classes of irreducible graded modules over the quotient $R(\alpha)/\sym^+(\alpha) R(\alpha)$. It follows from Lemma \ref{Lem:center of R(alpha)} that there are only finitely many irreducible $R(\alpha)$-modules, and all irreducible $R(\alpha)$-modules are finite-dimensional. Note that $R(\alpha)$ has the Krull-Schmidt unique direct sum decomposition property for finitely generated modules
since each graded part of $R(\alpha)$ is finite-dimensional. Hence irreducible $R(\alpha)$-modules form a basis of $G_0(R(\alpha)\text{-fmod})$ as an $\A$-module, which implies that the projective covers of irreducible $R(\alpha)$-modules form a basis of $K_0(R(\alpha)\text{-pmod})$ as an $\A$-module.
Let us consider the $\A$-bilinear pairing $(\ ,\ ) : K_0(R(\alpha)) \times G_0(R(\alpha)) \longrightarrow \A $ defined by \begin{equation}\label{eq:paring between K and G} ([P],[M]) = \dim_q (P^{\star} \otimes_{R(\alpha)} M), \end{equation} where $ \dim_q (N) := \sum_{i\in \mathbb{Z}} (\dim_{\bR} N_i)q^i $ for a $\mathbb{Z}$-graded module $N = \bigoplus_{i\in \mathbb{Z}} N_i$. Then, the paring $(\ ,\ )$ is perfect. Thus $K_0(R(\alpha))$ and $G_0(R(\alpha))$ are dual to each other with respect to the pairing $(\ ,\ )$.
By Lemma \ref{Lem:center of R(alpha)}, the pairing \eqref{eq:paring between K and G} can be extended to an $\A$-bilinear form $(\ ,\ ) : K_0(R(\alpha)) \times K_0(R(\alpha)) \longrightarrow \mathbb{Q}(q)$ given by \begin{equation} \label{eq:paring of K} ([P], [Q]) = \dim_q (P^{\star} \otimes_{R(\alpha)} Q). \end{equation} Since the pairing $\eqref{eq:paring between K and G}$ is perfect and $ P^{\star} \otimes_{R(\alpha)} Q \simeq Q^{\star} \otimes_{R(\alpha)} P $, we conclude that the pairing \eqref{eq:paring of K} is a nondegenerate symmetric bilinear form on $K_0(R(\alpha))$.
For a finite-dimensional $R(\alpha)$-module $M$, we define the {\em character} ${\rm ch}_q(M)$ of $M$ to be $$ {\rm ch}_{q}(M) = \sum_{\mathbf{i}\in {\rm Seq} (\alpha)} (\dim_q (1_\mathbf{i} M))\mathbf{i}.$$ For $\mathbf{i}=(i_1^{(d_1)} \ldots i_r^{(d_r)}) \in {\rm Seqd} (\alpha)$, let $$ \ie_{\mathbf{i}} := \ie_{i_1, d_1} \otimes \cdots \otimes \ie_{i_r, d_r}, $$ where $$ \ie_{i, d} := \left\{
\begin{array}{ll}
\tau_{w_0}x_1^{d-1}\cdots x_{d-1} 1_{(i\ldots i)} & \hbox{ if } i \in I^{\rm re} , \\
1_{(i\ldots i)} & \hbox{ if } i\in I^{\rm im} ,
\end{array}
\right. $$ and $w_0 = r_1r_2r_1\cdots r_{d-1}\cdots r_{1} $ is the longest element in $\sg_d$. Since each $\ie_{i_k, d_k}$ is an idempotent in $R(d_k \alpha_{i_k})$ $(k=1, \ldots, r)$, $\ie_{\mathbf{i}}$ is an idempotent.
Define an $R(\alpha)$-module $P_{\mathbf{i}}$ corresponding to $\mathbf{i}=(i_1^{(d_1)} \ldots i_r^{(d_r)}) \in {\rm Seqd} (\alpha) $ by \begin{align} \label{Eq:def of Pi}
P_{\mathbf{i}} := R(\alpha) \ie_\mathbf{i}\ \left\langle \sum_{\substack{k=1,\ldots,r, \\ i_k \in I^{\rm re} }} \frac{d_k(d_k-1)(\alpha_{i_k}|\alpha_{i_k})}{4} \right \rangle . \end{align} Note that $P_{\mathbf{i}}$ is a projective graded $R(\alpha)$-module. By construction, if $i \in I^{\rm im} $, then $$ P_{(i^{(d)})} = P_{(\underbrace{i\ldots i}_{d})}. $$ For a finitely generated graded projective $R(\alpha)$-module $P$, define \begin{align} \label{Eq:bar involution} \overline{P}= {\rm HOM}(P, R(\alpha))^\star. \end{align} Note that $\overline{P}$ is a graded projective left $R(\alpha)$-module and that $\overline{P_\mathbf{i}\langle a \rangle} \simeq P_\mathbf{i}\langle -a \rangle $ for $\mathbf{i} \in {\rm Seqd} (\alpha)$. Hence we get a $\mathbb{Z}$-linear involution $-: K_0(R) \rightarrow K_0(R)$.
We now prove the quantum Serre relations on $K_0(R)$. Suppose that $i\in I^{\rm re} , j\in I$ and $a_{ij} \ne 0$. Let $N = 1-a_{ij}$ and take nonnegative integers $a,b \ge 0 $ with $a+b = N$. Define the homogeneous elements $$ \alpha_{a,b}^+ := \alphaABpl, \quad \alpha_{a,b}^- := \alphaABmi. $$ Choose a pair of sequences $\mathbf{i}_1$ and $\mathbf{i}_2$ such that $\mathbf{i}_1 * (i^{(a)}ji^{(b)}) * \mathbf{i}_2 \in {\rm Seqd} (\alpha)$, and write $P_{ (\cdots\ i^{(a)}ji^{(b)}\cdots )}$ for $P_{ \mathbf{i}_1 * (i^{(a)}ji^{(b)}) * \mathbf{i}_2 }$. Then these elements give rise to homomorphisms of graded projective modules \begin{equation} \begin{aligned} d_{a,b}^+ &: P_{ (\cdots i^{(a)}ji^{(b)} \cdots)} \longrightarrow P_{ ( \cdots i^{(a+1)}ji^{(b-1)} \cdots) },\\
& \qquad\qquad m \quad \quad \longmapsto \quad m \cdot 1_{\mathbf{i}_1} \otimes \alpha_{a,b}^+ \otimes 1_{\mathbf{i}_2} , \\ d_{a,b}^- &: P_{( \cdots i^{(a)}ji^{(b)}\cdots) } \longrightarrow P_{(\cdots i^{(a-1)}ji^{(b+1)}\cdots) },\\
& \qquad\qquad m \quad \quad \longmapsto \quad m \cdot 1_{\mathbf{i}_1} \otimes \alpha_{a,b}^- \otimes 1_{\mathbf{i}_2}. \end{aligned} \end{equation} Set $d_{N,0}^+ = 0 $ and $d_{0,N}^- = 0 $. Then we have $$ \xymatrix{ 0\ \ \ar@<0.4ex>[r]^{} & \ar@<0.4ex>[l]^{} P_{ (\cdots i^{(0)}ji^{(N)} \cdots)} \ar@<0.4ex>[r]^{\qquad d_{0,N}^+ } & \ar@<0.4ex>[l]^{\qquad d_{1,N-1}^- }\ \ \cdots \ \ \ar@<0.4ex>[r]^{d_{a-1,b+1 }^+ \qquad } & \ar@<0.4ex>[l]^{ d_{a,b}^- \qquad} P_{ (\cdots i^{(a)}ji^{(b)} \cdots)} \ar@<0.4ex>[r]^{\qquad d_{a,b }^+ } & \ar@<0.4ex>[l]^{\qquad d_{a+1,b-1 }^-}\ \ \cdots \ \ \ar@<0.4ex>[r]^{ d_{N-1,1 }^+ \qquad} & \ar@<0.4ex>[l]^{ d_{N,0 }^- \qquad} P_{ (\cdots i^{(N)}ji^{(0)} \cdots)} \ar@<0.4ex>[r]^{} & \ar@<0.4ex>[l]^{} \ \ 0
}. $$
\begin{Lem} \ \label{Lem:Serre} \begin{enumerate} \item $ d_{a, b}^+ \circ d_{a-1, b+1}^+ = 0,\ \ d_{a, b}^- \circ d_{a+1, b-1}^- = 0 $ for $a,b > 0$. \item $ d_{N-1, 1}^+ \circ d_{N, 0}^- = t_{i,j;-a_{ij},0} \id, \ \ d_{1,N-1}^- \circ d_{0,N}^+ = (-1)^{N-1} t_{i,j;-a_{ij},0} \id $. \item For $ 1 < a, b<N$, we have $$ d_{a+1,b-1}^- \circ d_{a,b}^+ - d_{a-1,b+1}^+ \circ d_{a,b}^- = (-1)^{b-1} t_{i,j;-a_{ij},0} \id. $$ \end{enumerate} \end{Lem} \begin{proof} If $j \in I^{\rm re} $, this lemma was proved in \cite{KL11, R08}. We will prove our lemma when $j \in I^{\rm im} $.
Let $d = 2-a_{ij}$ and let $e_{a,b} = 1_{i,a} \otimes 1_{(j)} \otimes 1_{i, b}$ for $a,b \ge 0$. Since $i \in I^{\rm re} $ and $\mathcal{P}_i(u,v)=1$, it follows from \cite{KL11,R08} that \begin{align*} \alpha_{a,b}^+ &= \tau_{d-1} \cdots \tau_{a+1} e_{a+1,b-1} = e_{a,b} \tau_{d-1} \cdots \tau_{a+1} e_{a+1,b-1}, \\ \alpha_{a,b}^- &= \tau_{1} \cdots \tau_{a} e_{a-1,b+1} = e_{a,b} \tau_{1} \cdots \tau_{a} e_{a-1,b+1}. \end{align*} By a direct computation, we have \begin{align*} \alpha_{a-1,b+1}^+ \alpha_{a,b}^+ & = e_{a-1,b+1} \tau_{d-1} \cdots \tau_{a} e_{a,b} e_{a,b} \tau_{d-1} \cdots \tau_{a+1} e_{a+1,b-1}\\ &= e_{a-1,b+1} \tau_{d-1} \cdots \tau_{a} \tau_{d-1} \cdots \tau_{a+1} e_{a+1,b-1} \\ & = 0. \end{align*} In the same manner, we get $ \alpha_{a+1,b-1}^- \alpha_{a,b}^- = 0 $.
On the other hand, using the same argument as in \cite{KL11}, for $a,b>0$, we obtain \begin{align*} \alpha_{a,b}^+ \alpha_{a+1,b-1}^- &= e_{a,b} \tau_{d-1} \cdots \tau_{a+1} e_{a+1,b-1} e_{a+1,b-1} \tau_{1} \cdots \tau_{a+1} e_{a,b} \\ &= \tau_{1} \cdots \tau_{a-1} \tau_{d-1} \cdots \tau_{a+1} \tau_{a} \tau_{a+1} e_{a,b}, \\ \alpha_{a,b}^- \alpha_{a-1,b+1}^+ &= e_{a,b} \tau_{1} \cdots \tau_{a} e_{a-1,b+1} e_{a-1,b+1} \tau_{d-1} \cdots \tau_{a} e_{a,b} \\ &= \tau_{1} \cdots \tau_{a-1} \tau_{d-1} \cdots \tau_{a} \tau_{a+1} \tau_{a} e_{a,b}, \end{align*} which implies $$\alpha_{N,0}^- \alpha_{N-1,1}^+ = t_{i,j;-a_{ij},0} e_{N,0},\quad \alpha_{0,N}^+ \alpha_{1,N-1}^- = (-1)^{N-1}t_{i,j;-a_{ij},0} e_{0,N},$$ and \begin{align*} \alpha_{a,b}^+ \alpha_{a+1,b-1}^- - \alpha_{a,b}^- \alpha_{a-1,b+1}^+ &= \tau_{1} \cdots \tau_{a-1} \tau_{d-1} \cdots \tau_{a+2} (\tau_{a+1} \tau_{a} \tau_{a+1} - \tau_{a} \tau_{a+1} \tau_{a}) e_{a,b}\\ &= \tau_{1} \cdots \tau_{a-1} \tau_{d-1} \cdots \tau_{a+2} (\overline{Q}_{i,j}(x_{a},x_{a+1},x_{a+2})) e_{a,b}\\ &= (-1)^{b-1} t_{i,j;-a_{ij},0} e_{a,b}. \end{align*} Therefore, we obtain \begin{align*} & \alpha_{a-1,b+1}^+ \alpha_{a,b}^+ = 0, \quad \alpha_{a+1,b-1}^- \alpha_{a,b}^- = 0,\\ & \alpha_{N,0}^- \alpha_{N-1,1}^+ = t_{i,j;-a_{ij},0} e_{N,0},\quad \alpha_{0,N}^+ \alpha_{1,N-1}^- = (-1)^{N-1} t_{i,j;-a_{ij},0} e_{0,N}, \\ & \alpha_{a,b}^+ \alpha_{a+1,b-1}^- - \alpha_{a,b}^- \alpha_{a-1,b+1}^+ = (-1)^{b-1} t_{i,j;-a_{ij},0} e_{a,b}, \end{align*} as desired. \end{proof}
\begin{Thm} \label{Thm:Serre} \ \begin{enumerate} \item If $a_{ij}=0$, then $[P_{ (\cdots\ ij\cdots )}] = [P_{ (\cdots\ ji\cdots )}].$ \item If $i \in I^{\rm re} $ and $j \in I$ with $i \neq j$, then $$\sum_{k=0}^{1-a_{ij}} (-1)^k [ P_{ (\cdots\ i^{(k)}ji^{(1-a_{ij}-k)}\cdots )} ] = 0.$$ \end{enumerate} \end{Thm} \begin{proof} If $a_{ij}=0$, let $\tau^{-}$ (resp. $\tau^{+}$) be the element in $R$ changing $(ij)$ to $(ji)$ (resp. $(ji)$ to $(ij)$) and define $$d^-: P_{ (\cdots\ ij\cdots )} \to P_{ (\cdots\ ji\cdots )} \text{ (resp.\ } d^+: P_{ (\cdots\ ji\cdots )} \to P_{ (\cdots\ ij\cdots )} \text{)} $$ to be the map given by right multiplication by $t_{i,j;0,0} \tau^{-}$ (resp. $t_{j,i;0,0} \tau^{+}$). From the defining relation $\eqref{Eq:def rel 1}$, we see that $d^+$ and $d^-$ are inverses to each other. Hence $$[P_{ (\cdots\ ij\cdots )}] = [P_{ (\cdots\ ji\cdots )}]. $$
Suppose that $a_{ij} \ne 0$ and $i\in I^{\rm re} $. By Lemma \ref{Lem:Serre}, the complex $\left( P_{ (\cdots i^{(a)}ji^{(b)} \cdots)}, d_{a,b}^+ \right)$ becomes an exact sequence with the splitting maps $(-1)^{b-1} t_{ij;-a_{ij},0} d_{a,b}^-$. Therefore, our assertion follows from the Euler-Poincar\`{e} principle. \end{proof}
\vskip 3em
\section{Categorification of $U_q^-(\mathfrak{g})$} \label{Sec:GKM}
In this section, we show that the Khovanov-Lauda-Rouquier algebra $R$ gives a categorification of $U_\A^-(\mathfrak{g})$.
\subsection{Induction and restriction}\
For $\alpha, \beta \in Q^+$, consider the natural embedding \begin{align*} \iota_{\alpha,\beta}: R(\alpha)\otimes R(\beta) \hookrightarrow R(\alpha+\beta), \end{align*} which maps $1_\alpha \otimes 1_\beta$ to $1_{\alpha,\beta}$. For $M \in R(\alpha)\otimes R(\beta)$-mod and $N \in R(\alpha+\beta)$-mod, we define \begin{align*} {\rm Ind}_{\alpha, \beta} M &= R(\alpha+\beta) \otimes_{R(\alpha)\otimes R(\beta)}M , \\ {\rm Res}_{\alpha, \beta} N &= 1_{\alpha, \beta} N . \end{align*} Then it is straightforward to verify that the Frobenius reciprocity holds: \begin{align} \label{Eq:reciprocity} {\rm HOM}_{R(\alpha+\beta)}({\rm Ind}_{\alpha,\beta}M, N) \simeq {\rm HOM}_{R(\alpha)\otimes R(\beta)}(M, {\rm Res}_{\alpha,\beta}N). \end{align} When there is no ambiguity, we will simply write ${\rm Ind}$ and ${\rm Res} $ for ${\rm Ind}_{\alpha, \beta}$ and ${\rm Res}_{\alpha,\beta} $, respectively.
Given $\mathbf{i} \in {\rm Seq} (\alpha)$ and $\mathbf{j} \in {\rm Seq} (\beta)$, a sequence $\mathbf{k}\in {\rm Seq} (\alpha+\beta)$ is called a {\em shuffle} of $\mathbf{i}$ and $\mathbf{j}$ if $\mathbf{k}$ is a permutation of $\mathbf{i} * \mathbf{j}$ such that $\mathbf{i}$ and $\mathbf{j}$ are subsequences of $\mathbf{k}$. For a shuffle $\mathbf{k}$ of $\mathbf{i} \in {\rm Seq} (\alpha)$ and $\mathbf{j} \in {\rm Seq} (\beta)$, let $$\deg(\mathbf{i},\mathbf{j},\mathbf{k}) = \deg(\tau_{w}1_{\mathbf{i}*\mathbf{j}}),$$
where $w$ is the element in $\sg_{|\alpha| + |\beta|} /
\sg_{|\alpha| } \times \sg_{|\beta|}$ corresponding to $\mathbf{k}$. Given $X = \sum x_\mathbf{i}\ \mathbf{i}$ and $Y = \sum y_\mathbf{j}\ \mathbf{j}$, the {\em shuffle product} $X \star Y$ of $X$ and $Y$ is defined to be $$ X \star Y = \sum_{\mathbf{k}} \left(\sum_{\mathbf{i},\mathbf{j}}q^{\deg(\mathbf{i},\mathbf{j},\mathbf{k})} x_\mathbf{i} y_\mathbf{j} \right) \ \mathbf{k}, $$ where $\mathbf{k}$ runs over all the shuffles of $\mathbf{i}$ and $\mathbf{j}$. Then, by Proposition \ref{Prop:basis of R(alpha)}, we have \begin{align} \label{Eq:shuffle} {\rm ch}_{q}({\rm Ind}_{\alpha, \beta} M \boxtimes N) = {\rm ch}_{q}(M) \star {\rm ch}_{q}(N) \end{align} for $M\in R(\alpha)$-fmod and $N \in R(\beta)$-fmod.
By Corollary \ref{Cor:R(alpha+beta) is free}, ${\rm Ind}_{\alpha,\beta}$ and ${\rm Res}_{\alpha,\beta}$ take projective modules to projective modules. Since $1_{\alpha, \beta}$ is an idempotent, ${\rm Ind}_{\alpha,\beta}$ and ${\rm Res}_{\alpha,\beta}$ can be viewed as exact functors between the categories of projective modules. Hence we obtain the linear maps \begin{align*} & {\rm Ind}_{\alpha,\beta}: K_0(R(\alpha)) \otimes K_0(R(\beta)) \longrightarrow K_0(R(\alpha+\beta)),\\ & {\rm Res}_{\alpha,\beta}: K_0(R(\alpha+\beta)) \longrightarrow K_0(R(\alpha)) \otimes K_0(R(\beta)). \end{align*} It follows from Proposition \ref{Prop:Mackey} that \begin{equation} \label{Eq:ind and res of Pi} \begin{aligned} &{\rm Ind}_{\alpha, \beta}(P_\mathbf{i}\boxtimes P_\mathbf{j}) \simeq P_{\mathbf{i}*\mathbf{j}} &\text{ for }& \mathbf{i} \in {\rm Seq} (\alpha),\ \mathbf{j} \in {\rm Seq} (\beta),\\ &{\rm Res}_{\alpha, \beta} P_\mathbf{k} \simeq \bigoplus_{\mathbf{i},\mathbf{j}} P_\mathbf{i} \boxtimes P_\mathbf{j}\langle -\deg(\mathbf{i},\mathbf{j},\mathbf{k} ) \rangle
&\text{ for }& \mathbf{k} \in {\rm Seq} (\alpha+\beta), \end{aligned} \end{equation} where the sum is taken over all $\mathbf{i}\in {\rm Seq} (\alpha)$, $\mathbf{j}\in {\rm Seq} (\beta)$ such that $\mathbf{k}$ can be expressed as a shuffle of $\mathbf{i}$ and $\mathbf{j}$. We extend the linear maps ${\rm Ind}_{\alpha,\beta}$ and ${\rm Res}_{\alpha,\beta}$ to the whole space $K_0(R)$ by linearity: \begin{align*} {\rm Ind}:& K_0(R) \otimes K_0(R) \longrightarrow K_0(R)\quad \text{given by }\quad ([M],[N]) \mapsto [{\rm Ind}_{\alpha, \beta} M \boxtimes N],\\ {\rm Res}:& K_0(R) \longrightarrow K_0(R)\otimes K_0(R)\quad \text{given by }\quad [L] \mapsto \sum_{\alpha',\beta' \in Q^+ }[{\rm Res}_{\alpha', \beta'} L]. \end{align*} We denote by $[M][N]$ the product ${\rm Ind}([M],[N])$ of $[M]$ and $[N]$ in $K_0(R)$.
\begin{Prop}\ \begin{enumerate} \item The pair $(K_0(R), {\rm Ind})$ becomes an associative unital $\A$-algebra. \item The pair $(K_0(R), {\rm Res})$ becomes a coassociative counital $\A$-coalgebra. \end{enumerate} \end{Prop} \begin{proof} Our assertions on associativity and coassociativity follow from the transitivity of induction and restriction. Define \begin{align*} \iota :& \ \A \longrightarrow K_0(R) \quad \text{ by } \ \iota( \sum_k a_k q^k ) = \sum_k a_k q^k \mathbf{1},\\ \epsilon:&\ K_0(R) \longrightarrow \A \quad \text{ by } \ \epsilon(M) = \dim_q (M_0), \end{align*} where $M_0$ is the image of $M$ under the natural projection $K_0(R) \rightarrow K_0(R(0))$. Then one can verify that $\iota$ (resp.\ $\epsilon$) is the unit (resp.\ counit) of $K_0(R)$. \end{proof}
We define the algebra structure on $K_0(R)\otimes K_0(R)$ by $$ \left( [M_1] \otimes [M_2] \right) \cdot \left([N_1] \otimes [N_2] \right)
= q^{-(\beta_2|\gamma_1)}[M_1][N_1] \otimes [M_2][N_2] $$ for $M_i \in K_0(R(\beta_i))$, $N_i \in K_0(R(\gamma_i))$ $(i=1,2)$. Using Proposition \ref{Prop:Mackey}, we prove: \begin{Prop} ${\rm Res}:K_0(R) \longrightarrow K_0(R)\otimes K_0(R)$ is an algebra homomorphism. \end{Prop}
Let us recall the bilinear paring $(\ ,\ ): K_0(R) \otimes K_0(R) \longrightarrow \mathbb{Q}(q)$ given in $\eqref{eq:paring of K}$ and the projective modules $P_\mathbf{i}$ for $\mathbf{i}\in {\rm Seqd} (\alpha)$ defined in $\eqref{Eq:def of Pi}$. We denote by $\mathbf{1}$ the 1-dimensional $R(0)$-module of degree $0$.
\begin{Prop} \label{Prop:paring} The bilinear pairing $(\ ,\ ): K_0(R) \otimes K_0(R) \rightarrow \mathbb{Q}(q)$ satisfies the following properties: \begin{enumerate} \item $(\mathbf{1},\mathbf{1}) = 1$, \item $([P_{(i)}], [P_{(j)}]) = \delta_{ij} (1-q_i^{2})^{-1}$ for $i,j\in I$, \item $([L], [M][N]) = ({\rm Res} [L], [M] \otimes [N])$ for $[L],[M],[N] \in K_0(R)$, \item $([L][M],[N]) = ([L] \otimes [M], {\rm Res} [N])$ for $[L],[M],[N] \in K_0(R)$. \end{enumerate} \end{Prop} \begin{proof} The assertions (1) and (2) follow from the $\mathbb{Z}$-grading $\eqref{Eq:degree}$ on $R(\alpha)$. Suppose that $L \in R(\alpha+\beta)$-pmod, $M \in R(\alpha)$-pmod and $N \in R(\beta)$-pmod. Then we have \begin{align*} ([L], [M][N]) &= \dim_q (L^\star \otimes_{R(\alpha+\beta)} {\rm Ind}_{\alpha,\beta} M \boxtimes N) \\ &= \dim_q ( ({\rm Res}_{\alpha, \beta}L)^\star \otimes_{R(\alpha)\otimes R(\beta)} M \boxtimes N) = ({\rm Res}_{\alpha, \beta}L, M \boxtimes N), \end{align*} which yields that $([L], [M][N]) = ({\rm Res} [L], [M] \otimes [N])$.
The assertion (4) can be proved in the same manner. \end{proof}
Define a map $\Phi: U_{\A}^{-}(\mathfrak{g}) \longrightarrow K_{0}(R)$ by \begin{equation} f_{i_1}^{(d_1)} \cdots f_{i_r}^{(d_r)} \longmapsto [P_{( i_1^{(d_1)} \ldots i_r^{(d_r)}) }]. \end{equation}
\begin{Thm}\ \label{Thm:Phi is injective} The map $\Phi$ is an injective algebra homomorphism. \end{Thm}
\begin{proof} By Theorem \ref{Thm:Serre}, $\Phi$ is an algebra homomorphism. Since both of $\Delta_0$ and ${\rm Res}$ are algebra homomorphisms and $$\Delta_0 (f_i) = f_i \otimes \mathbf{1} + \mathbf{1} \otimes f_i, \ {\rm Res} (P_{(i)}) = P_{(i)} \otimes \mathbf{1} + \mathbf{1} \otimes P_{(i)}\ (i\in I),$$ by \eqref{Eq:def of ()K and ()L} and Proposition \ref{Prop:paring}, we have $$(x,y)_L = (\Phi(x), \Phi(y)) \ \ \text{for all} \ x, y \in U_{\A}^{-}(\mathfrak{g}).$$ Hence $\text{Ker} \Phi$ is contained in the radical of the bilinear form $(\ , \ )_{L}$, which is nondegenerate. The assertion follows immediately. \end{proof}
Therefore, $\text{Im} \Phi$ gives a categorification of $U_{\A}^{-}(\mathfrak{g})$. In general, the homomorphism $\Phi$ is not surjective. However, if $a_{ii} \neq 0$ for all $i \in I$, then $\Phi$ is an isomorphism as will be shown in the next subsection.
\vskip 1em
\subsection{Surjectivity of $\Phi$}\
In this subsection, we assume that $a_{ii} \ne 0$ for all $i\in I$. We have seen in Section \ref{Sec:KLR} that the algebra $R(m \alpha_i)$ has a unique irreducible graded module $L(i^m)$. If $i\in I^{\rm re} $, we have $$L(i^m) \simeq {\rm Ind}_{\F[x_1,\ldots,x_m]}^{R(m\alpha_i)} \mathbf{1},$$ where $\mathbf{1}$ is the trivial $\F[x_1,\ldots,x_m]$-module of dimension 1 over $\bR$. Note $ \dim_q (\mathbf{1})=1$. If $i\in I^{\rm im} $, then $L(i^m)$ is isomorphic to the trivial graded $R(m\alpha_i)$-module with defining relations given in $\eqref{Eq:def of L in Iim}$. We know ${\rm ch}_{q}( L(i^m) )= (i\ldots i)$.
For $M \in R(\alpha)$-mod and $i\in I$, define \begin{equation*} \begin{aligned} \Delta_{i^k} M &= 1_{k\alpha_i, \alpha- k\alpha_i} M
\in R(k\alpha_i)\otimes R(\alpha-k\alpha_i)\text{-mod}, \\ \ep_i(M) &= \max\{ k \ge 0 \mid \Delta_{i^k} M \ne 0 \}, \\ \ke_i(M) &= {\rm soc}({\rm Res}_{\alpha-\alpha_i}^{\alpha_i, \alpha-\alpha_i} \circ \Delta_{i}(M)) \in R(\alpha-\alpha_i)\text{-mod} , \\ \kf_i(M) &= {\rm hd} {\rm Ind}_{\alpha_i, \alpha} ( L(i)\boxtimes M ) \in R(\alpha+\alpha_i)\text{-mod}. \end{aligned} \end{equation*} Note that they are defined in the opposite manner to \cite{KL09,LV09}. By the Frobenius reciprocity, we have \begin{align} \label{Eq:reciprocity2} {\rm HOM}_{R(\alpha)}({\rm Ind}_{m\alpha_i, \alpha-m\alpha_i} L(i^m) \boxtimes N, M) \simeq {\rm HOM}_{R(m\alpha_i) \otimes R(\alpha-m\alpha_i)}(L(i^m) \boxtimes N,\Delta_{i^m}M) \end{align} for $N \in R(\alpha - m\alpha_i)$-mod and $M \in R(\alpha)$-mod.
\begin{Lem} \label{Lem:Kato for i in Iim}
For $i\in I^{\rm im} $, take $m_1, \ldots, m_k \in \mathbb{Z}_{> 0}$ and set $m = m_1 + \cdots +
m_k$. Then the following statements hold. \begin{enumerate} \item ${\rm Res}_{m_1 \alpha_i, \ldots, m_k \alpha_i} L(i^m) $ is isomorphic to $ L(i^{m_1}) \boxtimes \cdots \boxtimes L(i^{m_k})$. \item ${\rm Ind}_{m_1 \alpha_i, \ldots, m_k \alpha_i}( L(i^{m_1}) \boxtimes \cdots \boxtimes L(i^{m_k}))$ has an irreducible head, which is isomorphic to $L(i^m)$. \end{enumerate} \end{Lem} \begin{proof} The assertion (1) follows from the definition $\eqref{Eq:def of L in Iim}$. To prove (2), for simplicity, we assume $k=2$. Let $\mathbf{i} = (\underbrace{i \ldots i}_{m})$ and $L = {\rm Ind} L_1 \boxtimes L_2, $ where $L_j := L(i^{m_j}) \ (j=1,2)$. Set
$$L' = \{ x \in L | \deg(x) > 0 \}. $$ Then, since $ 1 \otimes (L_1 \boxtimes L_2) \nsubseteq L'$, $L'$ is a unique maximal submodule of $L$; i.e., $ L / L'\simeq L(i^m)$ as a graded module. We will show that ${\rm hd} L$ is irreducible. By a direct computation, \begin{align*}
{\rm ch}_{q}(L) &= \sum_{w\in \sg_{m_1 + m_2}/ \sg_{m_1}\times \sg_{m_2}} q^{-\ell(w)(\alpha_i|\alpha_i)}\mathbf{i} \\ &= \mathbf{i} + \text{( ...other terms with $q^{t}$...)} \ \ (t\in \mathbb{Z}_{>0}). \end{align*} Note that ${\rm ch}_{q}(L_1 \boxtimes L_2) = \mathbf{i}$. For any quotient $Q$ of $L$, by the Frobenius reciprocity $\eqref{Eq:reciprocity}$, we have an injective homomorphism of degree 0 $$ L_1 \boxtimes L_2 \hookrightarrow {\rm Res}_{m_1 \alpha_i, m_2\alpha_i} Q ,$$ which yields $${\rm ch}_{q}(Q) = \mathbf{i} + \text{( ...other terms with $q^{t}$... ) for $t \in \mathbb{Z}_{>0}$}.$$ Therefore, ${\rm hd} L$ has only one summand, and hence it is irreducible. \end{proof}
\begin{Lem} \label{Lem:epsilon and irr submodule} Let $M$ be an irreducible $R(\alpha)$-module and let $L(i^m) \boxtimes N$ be an irreducible submodule of the $R(m\alpha_i) \otimes R(\alpha-m\alpha_i)$-module $\Delta_{i^m}M$. Then $\ep_i(N) = \ep_i(M)-m$. \end{Lem} \begin{proof} If $i\in I^{\rm re} $, then the proof is the same as that of \cite[Lemma 3.6]{KL09}. If $i\in I^{\rm im} $, by the definition, we have $\ep_i(N) \le \ep_i(M)-m$. From the equation $\eqref{Eq:reciprocity2}$, we obtain $$0 \rightarrow K \rightarrow {\rm Ind} L(i^m)\boxtimes N \rightarrow M \rightarrow 0$$ for some submodule $K$ of ${\rm Ind} L(i^m)\boxtimes N$. It follows from $\eqref{Eq:shuffle}$ and the exactness of $\Delta_{i^k}$ that $\ep_i(N) \ge \ep_i(M)-m$, which yields our assertion. \end{proof}
\begin{Lem} \label{Lem:properties of L(im) bt N} Let $N$ be an irreducible $R(\alpha)$-module with $\ep_i(N)=0$ and let $M = {\rm Ind} L(i^m) \boxtimes N$. Then we have \begin{enumerate} \item $\Delta_{i^m}M \simeq L(i^m) \boxtimes N$, \item ${\rm hd} M$ is an irreducible module with $\ep_i({\rm hd} M) = m$, \item for all other composition factors $L$ of $M$, we have $\ep_i(L) < m$. \end{enumerate} \end{Lem} \begin{proof} Our assertion can be proved in the same manner as in \cite[Lemma 3.7]{KL09}. \end{proof}
\begin{Lem} \label{Lem:Delta of M} Let $M$ be an irreducible $R(\alpha)$-module and let $\ep = \ep_i(M)$. Then $\Delta_{i^\ep}M$ is isomorphic to $L(i^\ep) \boxtimes N$ for some irreducible $R(\alpha-\varepsilon\alpha_i)$-module $N$ with $\ep_i(N)=0$. \end{Lem} \begin{proof} Our assertion can be proved in the same manner as in \cite[Lemma 5.1.4]{K05} (cf.\ \cite[Lemma 3.8]{KL09}). \end{proof}
\begin{Lem} \label{Lem:hd ind for i in Iim} Suppose that $i \in I^{\rm im} $ and $N$ is an irreducible $R(\alpha)$-module with $\ep_i(N)=0$. Let $$ M = {\rm Ind} L(i^{m_1}) \boxtimes \cdots \boxtimes L(i^{m_k}) \boxtimes N $$
for some positive integers $m_{1}, \ldots m_{k} \in \mathbb{Z}_{>0}$ and set
$m= m_1 + \cdots + m_k$. Then \begin{enumerate} \item ${\rm hd} M$ is irreducible, \item $\ep_i({\rm hd} M) = m$. \end{enumerate} \end{Lem} \begin{proof} By the definition, we have $$ \Delta_{i^m} M = ( {\rm Ind} L(i^{m_1}) \boxtimes \cdots \boxtimes L(i^{m_k}) ) \boxtimes N. $$ In the Grothendieck group $G_0(R(m\alpha_i) \otimes R(\alpha - m\alpha_i))$ of the category of finite-dimensional graded $R(m\alpha_i) \otimes R(\alpha - m\alpha_i)$-modules, we have \begin{align*}
[\Delta_{i^m} M] &= \sum_w q^{-\ell(w) (\alpha_i | \alpha_i)}[L(i^m) \boxtimes N], \\
&= [L(i^m) \boxtimes N ] + ( \text{ ...other terms with $q^{t}$...
}), \end{align*} where $w$ runs over all the elements in $\sg_m / \sg_{m_1} \times \cdots \times \sg_{m_k}$. By the Frobenius reciprocity $\eqref{Eq:reciprocity2}$, for any quotient $Q$ of $M$, there is a nontrivial homomorphism of degree 0 $$ \Delta_{i^m} M = ( {\rm Ind} L(i^{m_1}) \boxtimes \cdots \boxtimes L(i^{m_k}) ) \boxtimes N \rightarrow \Delta_{i^m}Q. $$ By Lemma \ref{Lem:Kato for i in Iim} (2), we have $$ [\Delta_{i^m}Q] = [L(i^m) \boxtimes N ] + ( \text{ ...other terms with $q^{t}$... })$$ in the Grothendieck group $G_0(R(m\alpha_i) \otimes R(\alpha - m\alpha_i))$. Therefore, by the same argument as in Lemma \ref{Lem:Kato for i in Iim}, $ {\rm hd} M$ is irreducible and $\ep_i({\rm hd} M) = m$. \end{proof}
\begin{Lem} \label{Lem:irr hd} Let $N$ be an irreducible $R(\alpha)$-module and let $M = {\rm Ind} L(i^m) \boxtimes N$. \begin{enumerate} \item ${\rm hd} M$ is an irreducible module with $\ep_i( {\rm hd} M) = \ep_i(N) + m$. \item If $i \in I^{\rm re} $, then for all other composition factors $L$ of $M$, we have $\ep_i(L) < \ep_i(N)+m$. \end{enumerate} \end{Lem} \begin{proof} If $i \in I^{\rm re} $, then the proof is identical with that of \cite[Lemma 5.1.5]{K05} (cf.\ \cite[Lemma 3.9]{KL09}). Suppose that $i\in I^{\rm im} .$ Let $\ep = \ep_i(N)$. By Lemma \ref{Lem:Delta of M}, we have $$ \Delta_{i^\ep} N = L(i^\ep) \boxtimes K $$ for some irreducible $R(\alpha-m\alpha_i)$-module $K$ with $\ep_i(K)=0$. By the Frobenius reciprocity $\eqref{Eq:reciprocity2}$, there is a surjective homomorphism $$ {\rm Ind} L(i^\ep) \boxtimes K \twoheadrightarrow N ,$$ which yields $$ {\rm Ind} L(i^m) \boxtimes L(i^\ep) \boxtimes K \twoheadrightarrow {\rm Ind} L(i^m) \boxtimes N . $$ Therefore, our assertion follows from Lemma \ref{Lem:hd ind for i in Iim}. \end{proof}
\begin{Lem} \label{Lem:irr soc} Let $M$ be an irreducible $R(\alpha)$-module. Then, for $0 \le m \le \ep_i(M) $, the submodule ${\rm soc} \Delta_{i^m}M$ of $M$ is an irreducible module of the form $L(i^m) \boxtimes L$ with $\ep_i(L) = \ep_i(M)-m $ for some irreducible $R(\alpha-m\alpha_i)$-module $L$. \end{Lem} \begin{proof} If $i \in I^{\rm re} $, then the proof is the same as that of \cite[Lemma 5.1.6]{K05} (cf.\ \cite[Lemma 3.10]{KL09}). If $i \in I^{\rm im} $, let $\ep = \ep_i(M)$. Note that every summand of ${\rm soc} \Delta_{i^m}M$ has the form $L(i^m)\boxtimes L$ for some irreducible $R(\alpha - m \alpha_i)$-module $L$. It follows from Lemma \ref{Lem:epsilon and irr submodule} that $$ \ep_i(L) = \ep - m, $$ so $L(i^m) \boxtimes \Delta_{i^{\ep-m}}(L) \neq 0$. It is clear that ${\rm Res}^{ \ep \alpha_i, \alpha - \ep \alpha_i}_{ m \alpha_i, (\ep-m) \alpha_i, \alpha - \ep \alpha_i } \Delta_{i^\ep}M$ has $ L(i^m) \boxtimes \Delta_{i^{\ep-m}}(L)$ as a submodule. On the other hand, by Lemma \ref{Lem:Kato for i in Iim} and Lemma \ref{Lem:Delta of M}, there exists an irreducible $R(\alpha-\varepsilon \alpha_i)$-module $N$ such that $$ {\rm Res}^{ \ep \alpha_i, \alpha - \ep \alpha_i}_{ m \alpha_i, (\ep-m) \alpha_i, \alpha - \ep \alpha_i } \Delta_{i^\ep}M \simeq L(i^m) \boxtimes L(i^{\ep - m}) \boxtimes N,$$ which is
irreducible. Hence ${\rm soc} \Delta_{i^m}M$ is irreducible and isomorphic to $L(i^{m}) \boxtimes L$. \end{proof}
By Lemma \ref{Lem:irr hd} and Lemma \ref{Lem:irr soc}, the operators $\ke_i$ and $\kf_i$ take irreducible modules to irreducible modules or $0$, and $$ \ep_i(M) = \max \{ k \ge 0 \mid \ke_i^k M \ne 0 \}, \quad \ep_i(\kf_i M) = \ep_i(M)+1. $$
\begin{Lem} \label{Lem:soc and hd} Let $M$ be an irreducible $R(\alpha)$-module. Then we have \begin{enumerate} \item ${\rm soc} \Delta_{i^m}M \simeq L(i^m) \boxtimes (\ke_i^m M)$, \item ${\rm hd} {\rm Ind} (L(i^m) \boxtimes M) \simeq \kf_i^m M$. \end{enumerate} \end{Lem} \begin{proof} If $i \in I^{\rm re} $, then the proof is the same as in \cite[Lemma 5.2.1]{K05}. Suppose that $i \in I^{\rm im} $. We first focus on the assertion (1). Since the case $m > \ep_i(M)$ is trivial, we may assume that $m \le \ep_i(M)$. Since $L(i) \boxtimes \ke_iM \hookrightarrow \Delta_i M$, we have $$ \underbrace{L(i)\boxtimes \cdots \boxtimes L(i)}_{m} \boxtimes\ \ke_i^m M \hookrightarrow {\rm Res}_{\alpha_i, \ldots \alpha_i, \alpha-m \alpha_i }^{m\alpha_i, \alpha - m\alpha_i} \Delta_{i^m}M,$$ which implies there is a nontrivial homomorphism $$ {\rm Ind} (\underbrace{L(i)\boxtimes \cdots \boxtimes L(i)}_{m} ) \boxtimes\ \ke_i^m M \longrightarrow \Delta_{i^m} M. $$ Since any quotient of ${\rm Ind} ( L(i)\boxtimes \cdots \boxtimes L(i))$ has a 1-dimensional submodule, $\Delta_{i^m} M$ has a submodule which is isomorphic to $L(i^m)\boxtimes \ke_i^m M$. Hence the assertion (1) follows from Lemma \ref{Lem:irr soc}.
For the assertion (2), by the definition of $\kf_i$, there is a nontrivial homomorphism $$ {\rm Ind} ( {\rm Ind}( \underbrace{L(i)\boxtimes \cdots \boxtimes L(i)}_{m} ) \boxtimes M) \twoheadrightarrow \kf_i^m M. $$
Using the same argument in the proof of Lemma \ref{Lem:irr hd}, we have $$ {\rm hd} {\rm Ind} ( {\rm Ind}( L(i)\boxtimes \cdots \boxtimes L(i) ) \boxtimes M) \simeq \kf_i^m M. $$ On the other hand, the nontrivial homomorphism $$ {\rm Ind}( L(i)\boxtimes \cdots \boxtimes L(i) ) \longrightarrow L(i^m) $$ induces a nontrivial homomorphism $$ {\rm Ind} ( {\rm Ind}( L(i)\boxtimes \cdots \boxtimes L(i) ) \boxtimes M) \longrightarrow {\rm Ind} L(i^m)\boxtimes M .$$ Therefore, we conclude ${\rm hd} {\rm Ind} (L(i^m) \boxtimes M) \simeq \kf_i^m M$. \end{proof}
\begin{Lem} \label{Lem:adjoint ke and kf} Let $M $ be an irreducible $ R(\alpha)$-module and let $N$ be an irreducible $R(\alpha+\alpha_i)$-module. Then we have $$ \kf_i M \simeq N \ \text{ if and only if } \ M \simeq \ke_i N. $$ \end{Lem} \begin{proof} Using Lemma \ref{Lem:soc and hd}, our assertion can be proved in the same manner as in \cite[Lemma 5.2.3]{K05} \end{proof}
Let $\A {\rm Seq} (\alpha)$ (resp.\ $\mathbb{Q}(q) {\rm Seq} (\alpha)$) be the free $\A$-module (resp.\ $\mathbb{Q}(q)$-module) generated by $ {\rm Seq} (\alpha)$. For an irreducible $R(\alpha)$-module $M$, the character ${\rm ch}_{q}(M)$ can be viewed as an element in $\A {\rm Seq} (\alpha)$. Using the above lemmas, one can prove the following proposition in the same manner as in \cite[Theorem 5.3.1]{K05}.
\begin{Prop} \label{Prop:character map is injective} The character map $${\rm ch}_{q}: G_0(R(\alpha)) \longrightarrow \A {\rm Seq} (\alpha)$$ is injective. \end{Prop}
Let $\mathcal{F}$ be the free associative algebra over $\mathbb{Q}(q)$ generated by $f_i$ $(i\in I)$ and consider the natural projection $\pi: \mathcal{F} \to U_q^-(\mathfrak{g})$ given by $f_i \mapsto f_i$ $(i \in I)$. Then the vector space $\mathbb{Q}(q) {\rm Seq} (\alpha)$ can be regarded as the dual space of $\mathcal{F}_\alpha := \pi^{-1}( U_q^-(\mathfrak{g})_\alpha ) $ for $\alpha \in Q^+$. Set \begin{equation*} \begin{aligned} & K_{0}(R)_{\mathbb{Q}(q)} = \mathbb{Q}(q) \otimes_{\A} K_{0}(R), \quad K_{0}(R(\alpha))_{\mathbb{Q}(q)} = \mathbb{Q}(q) \otimes_{\A} K_{0}(R(\alpha)), \\ & G_0(R)_{\mathbb{Q}(q)} = \mathbb{Q}(q) \otimes_\A G_0(R), \quad G_{0}(R(\alpha))_{\mathbb{Q}(q)} = \mathbb{Q}(q) \otimes_{\A} G_{0}(R(\alpha)), \end{aligned} \end{equation*} and denote by $\Phi_{\mathbb{Q}(q)}: U_q^{-}(\mathfrak{g}) \longrightarrow K_{0}(R)_{\mathbb{Q}(q)}$ the algebra homomorphism induced by $\Phi: U_{\A}^{-}(\mathfrak{g}) \longrightarrow K_{0}(R)$. Then ${\rm ch}_{q}$ is the dual map of $ \Phi_{\mathbb{Q}(q)} \circ \pi $, which yields the following diagram: $$ \xymatrix{ \mathcal{F}_\alpha \ar[rr]^{\pi} \ar[d]^{\text{dual}} & & U_q^-(\mathfrak{g})_\alpha \ar[rr]^{\Phi_{\mathbb{Q}(q)}} & & \ar[d]^{\text{dual w.r.t. } (\ ,\ )} K_0(R(\alpha))_{\mathbb{Q}(q)} \\ \ar[u] \mathbb{Q}(q) {\rm Seq} (\alpha)& & & & \ar[llll]_{{\rm ch}_{q}} \ar[u] G_0(R(\alpha))_{\mathbb{Q}(q)} } $$ Combining Theorem \ref{Thm:Phi is injective} with Proposition \ref{Prop:character map is injective}, we conclude $$ \Phi_{\mathbb{Q}(q)}: U_q^-(\mathfrak{g}) \longrightarrow K_0(R)_{\mathbb{Q}(q)} $$ is an isomorphism.
\begin{Thm} \label{Thm:iso of K0 and Uq} The map $\Phi: U_{\A}^-(\mathfrak{g}) \longrightarrow K_0(R) $ is an isomorphism if $a_{ii} \ne 0$ for all $i\in I$. \end{Thm} \begin{proof} It suffices to show that $\Phi_{\mathbb{Q}(q)}(U_{\A}^-(\mathfrak{g})) = K_0(R)$. Choose a sequence $(i_k)_{k \ge 0}$ of $I$ such that, for each $i\in I$, $i$ appears infinitely many times in $(i_k)_{k \ge 0}$. Let $B_\alpha$ be the set of all isomorphism classes of irreducible $R(\alpha)$-modules. We fix a representative $S_b$ for each $b \in B_\alpha$. To each $b \in B_\alpha$, we assign the sequence $p_b := p_0 p_1 \cdots$ given as follows: if $M_0:= S_b$, define $$p_k = \ep_{i_{k}} (M_k)\ \text{ and } \ M_{k+1} = \ke_{i_{k}}^{p_k}(M_k)\quad (k \ge 0) $$ inductively. For $b \in B_\alpha$, let $$ P_b = P_{\mathbf{i}_b}, $$ where $\mathbf{i}_b := ( i_0 ^{(p_0)} i_1 ^{(p_1)} \ldots)$. Note that $P_{\mathbf{i}_b}$ is well-defined since $\mathbf{i}_b$ has only finitely many nonnegative integers. \ Define a total order $\prec$ on $B_\alpha$ by $$ b \prec c \text{ if and only if } p_{b} <_{\rm lex} p_{c} , $$ where $<_{\rm lex}$ is the lexicographic order. Then it follows from the definition of the pairing $\eqref{eq:paring between K and G}$ that $$ (P_{b}, S_{c}) = 0 \text{ if } b \succ c \quad \text{ and }\quad (P_{b}, S_{b}) = q^t $$ for some $t \in \mathbb{Z}$. Hence, any projective module $[P]$ in $K_0(R(\alpha))$ can be written as an $\A$-linear combination of $\{ P_b \mid b \in B_\alpha \}$, which implies $\Phi_{\mathbb{Q}(q)}(U_{\A}^-(\mathfrak{g})) = K_0(R)$. \end{proof}
\vskip 3em
\section{Crystals and Perfect bases} \label{Sec:Crystals}
In this section, we develop the theory of perfect bases for $U_q^-(\mathfrak{g})$ as a $B_q(\mathfrak{g})$-modules. We prove that the negative part $U_q^-(\mathfrak{g})$ has a perfect basis by constructing the upper global basis. We also show that the crystals arising from perfect bases of $U_q^{-}(\mathfrak{g})$ are all isomorphic to the crystal $B(\infty)$.
\subsection{Crystals}\
We review the basic theory of abstract crystals for quantum generalized Kac-Moody algebras introduced in \cite{JKKS07}.
\begin{Def}\label{Def: abstract crystal} An {\em abstract crystal} is a set $B$ together with the maps $\ {\rm wt} : B \to P,\ \varphi_{i},\varepsilon_{i} : B \to \mathbb{Z} \sqcup \{-\infty \} \mbox{ and } \ \tilde{e}_i,\tilde{f}_i : B \to B \sqcup \{0\} \ (i \in I) $ satisfying the following conditions: \begin{enumerate} \item $\varphi_{i}(b) = \varepsilon_{i}(b) + \langle h_i,{\rm wt}(b) \rangle,$ \item ${\rm wt}(\tilde{e}_i b)={\rm wt}(b)+\alpha_i, {\rm wt}(\tilde{f}_i b)={\rm wt}(b)-\alpha_i \mbox{ if } \tilde{e}_i b, \tilde{f}_i b \in B,$ \item for $ b,b^{\prime} \in B $ and $ i \in I,$ $ b'=\tilde{e}_i b$ if and only if $b = \tilde{f}_i b',$ \item for $ b \in B$, if $ \varphi_{i}(b) = -\infty$, then $ \tilde{e}_i b = \tilde{f}_i b=0,$ \item if $ b \in B $ and $\tilde{e}_i b \in B$, then
$$\varepsilon_{i}(\tilde{e}_i b)=\begin{cases} \varepsilon_{i}(b)-1 & \mbox{if} \ i \in I^{\rm re} , \\ \varepsilon_{i}(b) & \mbox{if} \ i \in I^{\rm im} , \end{cases}
\ \ \varphi_{i}(\tilde{e}_i b)=\begin{cases} \varphi_{i}(b)+1 & \mbox{if} \ i \in I^{\rm re} , \\ \varphi_{i}(b)+a_{ii} & \mbox{if} \ i \in I^{\rm im} , \end{cases}$$ \item if $ b \in B$ and $\tilde{f}_i b \in B$, then
$$\varepsilon_{i}(\tilde{f}_i b)=\begin{cases} \varepsilon_{i}(b)+1 & \mbox{if} \ i \in I^{\rm re} , \\ \varepsilon_{i}(b) & \mbox{if} \ i \in I^{\rm im} , \end{cases}
\ \ \varphi_{i}(\tilde{f}_i b)=\begin{cases} \varphi_{i}(b)-1 & \mbox{if} \ i \in I^{\rm re} , \\ \varphi_{i}(b)-a_{ii} & \mbox{if} \ i \in I^{\rm im} . \end{cases}$$ \end{enumerate} \end{Def}
\begin{Exa} \label{Exa: natural abstract crystals} \ \begin{enumerate} \item For $b \in B(\infty)$, define $ {\rm wt}, \varepsilon_i$, and $\varphi_i$ as follows: \begin{align*} \ \ & {\rm wt}(b) = -(\alpha_{i_1}+ \cdots + \alpha_{i_r}) \text{ for } b = \tilde{f}_{i_1} \cdots \tilde{f}_{i_r} {\bf 1}+ q L(\infty), \\
\ \ & \varepsilon_i(b) = \begin{cases} \max \{ k \ge 0 \ | \ \tilde{e}_i^k b\neq 0 \} & \text{ for } i \in I^{\rm re} , \\
\qquad 0 & \text{ for } i \in I^{\rm im} , \end{cases} \\ \ \ & \varphi_i(b) = \varepsilon_i(b)+ \langle h_i, {\rm wt}(b)\rangle. \end{align*} Then $(B(\infty), {\rm wt}, \tilde{e}_i, \tilde{f}_i, \varepsilon_i, \varphi_i)$ becomes an abstract crystal. \item For $b \in B(\lambda)$, define ${\rm wt}, \varepsilon_i$, and $\varphi_i$ as follows: \begin{align*} \ \ & {\rm wt}(b) = \lambda-(\alpha_{i_1}+ \cdots + \alpha_{i_r}) \text{ for } b = \tilde{f}_{i_1} \cdots \tilde{f}_{i_r}v_{\lambda}+ q L(\lambda), \\
\ \ & \varepsilon_i(b) = \begin{cases} \max \{ k \ge 0 \ | \ \tilde{e}_i^k b\neq 0 \} & \text{ for } i \in I^{\rm re} , \\
\qquad 0 & \text{ for } i \in I^{\rm im} , \end{cases} \\
\ \ & \varphi_i(b) = \begin{cases} \max \{ k \ge 0 \ | \ \tilde{f}_i^k b\neq 0 \} & \text{ for } i \in I^{\rm re} , \\
\quad \langle h_i, {\rm wt}(b)\rangle & \text{ for } i \in I^{\rm im} . \end{cases} \end{align*} Then $(B(\lambda),{\rm wt},\tilde{e}_i,\tilde{f}_i, \varepsilon_i, \varphi_i)$ becomes an abstract crystal.
\item For $\lambda \in P$, let $T_{\lambda}=\{ t_{\lambda} \}$ and define \begin{align*} \ \ & {\rm wt}(t_{\lambda}) = \lambda, \quad \tilde{e}_i t_{\lambda} =\tilde{f}_i t_{\lambda} =0 \quad
\varepsilon_i(t_{\lambda})=\varphi_i(t_{\lambda}) = - \infty \text{ for all } i \in I. \end{align*} Then $(T_{\lambda},{\rm wt},\tilde{e}_i,\tilde{f}_i, \varepsilon_i, \varphi_i)$ is an abstract crystal. \item Let $C= \{ c \}$ and define \begin{align*} \ \ &{\rm wt}(c) = 0, \quad \tilde{e}_i c =\tilde{f}_i c =0 \quad \varepsilon_i(c)=\varphi_i(c) = 0 \text{ for all } i \in I. \end{align*} Then $(C,{\rm wt},\tilde{e}_i,\tilde{f}_i, \varepsilon_i, \varphi_i)$ is an abstract crystal. \end{enumerate} \end{Exa}
\begin{Def}\label{Def: crystal morphism} \
\begin{enumerate}
\item A {\it crystal morphism} $\phi$ between abstract crystals $B_1$ and $B_2$ is a map
from $ B_1 $ to $ B_2 \sqcup \{0\}$ satisfying the following conditions: \begin{enumerate} \item if $b \in B_1$ and $\phi(b) \in B_2$, then ${\rm wt}(\phi(b))= {\rm wt}(b),\ \varepsilon_{i}(\phi(b))=\varepsilon_{i}(b)$ and $\varphi_{i}(\phi(b))=\varphi_{i}(b)$, \item if $b \in B_1$ and $ i\in I$ with $\tilde{f}_i b \in B_1$, then we have $\tilde{f}_i \phi(b) = \phi(\tilde{f}_i b).$ \end{enumerate} \item A crystal morphism $\phi: B_1 \to B_2$ is called {\it strict} if $$ \phi(\tilde{e}_i b) = \tilde{e}_i \phi(b)\quad \text{ and }\quad \phi(\tilde{f}_i b) = \tilde{f}_i \phi(b) $$ for all $i\in I$ and $b \in B_1$. \end{enumerate} \end{Def}
The tensor product of two crystals is defined as follows: for given two crystals $B_1$ and $B_2$, their tensor product
$B_1 \otimes B_2$ is the set $\{ b_1 \otimes b_2 \mid b_1 \in B_1, b_2 \in B_2 \}$ with the maps ${\rm wt}, \varepsilon_i
,\varphi_i,\tilde{e}_i$ and $\tilde{f}_i$ given by \begin{equation} \label{Eq:def of tensor product} \begin{aligned} & {\rm wt}(b_1 \otimes b_2) = {\rm wt}(b_1) \otimes {\rm wt}(b_2), \\ &\varepsilon_i(b_1 \otimes b_2)= \max \{ \varepsilon_i(b_1), \varepsilon_i(b_2)-\langle h_i, {\rm wt}(b_1) \rangle \}, \\ &\varphi_i(b_1 \otimes b_2)= \max \{ \varphi_i(b_1)+\langle h_i, {\rm wt}(b_2) \rangle, \varphi_i(b_2) \}, \\ & \tilde{f}_i(b_1 \otimes b_2)= \begin{cases} \tilde{f}_i(b_1) \otimes b_2 & \text{ if } \varphi_i(b_1)> \varepsilon_i(b_2), \\
b_1 \otimes \tilde{f}_i(b_2) & \text{ if } \varphi_i(b_1)\le \varepsilon_i(b_2), \end{cases} \\ & \text{ for } i \in I^{\rm re} ,\ \ \tilde{e}_i(b_1 \otimes b_2) =\begin{cases} \tilde{e}_i(b_1) \otimes b_2 & \text{ if } \varphi_i(b_1) \ge \varepsilon_i(b_2), \\
b_1 \otimes \tilde{e}_i(b_2) & \text{ if } \varphi_i(b_1) < \varepsilon_i(b_2), \end{cases} \\ & \text{ for } i \in I^{\rm im} ,\ \
\tilde{e}_i(b_1 \otimes b_2) =\begin{cases} \tilde{e}_i(b_1) \otimes b_2 & \text{ if } \varphi_i(b_1) > \varepsilon_i(b_2)-a_{ii}, \\
0 & \text{ if } \varepsilon_i(b_2) < \varphi_i(b_1) \le \varepsilon_i(b_2)-a_{ii},\\
b_1 \otimes \tilde{e}_i(b_2) & \text{ if } \varphi_i(b_1) \le \varepsilon_i(b_2). \end{cases} \end{aligned} \end{equation}
It was proved in \cite[Lemma 3.10]{JKKS07} that $B_1 \otimes B_2$ becomes an abstract crystal. Moreover, they proved the {\it recognition theorem} of $B(\lambda)$ $(\lambda \in P^+)$ using the abstract crystal structure of $B(\infty)$.
\begin{Prop} \cite[Theorem 5.2]{JKKS07} \label{Prop: recognition theorem of B-lambda} For $\lambda \in P^+$, the crystal $B(\lambda)$ is isomorphic to the connected component of $B(\infty) \otimes T_{\lambda} \otimes C$ containing ${\bf 1} \otimes t_{\lambda} \otimes c$. \end{Prop}
\vskip 1em
\subsection{Perfect bases}\ \label{Sec:perfect bases}
We revisit the algebra $U^{-}_q(\mathfrak{g})$. We analyze $U^{-}_q(\mathfrak{g})$ as a $B_q(\mathfrak{g})$-module and develop the perfect basis theory for $U_q^{-}(\mathfrak{g})$. The crystal structure is revealed when $e_i'$ acts on a perfect basis.
Let \begin{equation*}
\begin{aligned}
\ \ \bse_i'^{(n)} =
\begin{cases}
(\bse_i')^{n} & \text{if } i \in I^{\rm re} , \\
\dfrac{(\bse_i')^{n}}{\{n\}_{i}!} & \text{if } i \in I^{\rm im} .
\end{cases}
\end{aligned} \end{equation*} Then we obtain the following commutation relations: \begin{equation} \label{eq:commutation relation}
\begin{aligned}
\bse_i'^{(n)}\bsf_j^{(m)} =
\begin{cases}
\displaystyle \sum_{k=0}^{n} q_i^{-2nm+(n+m)k-k(k-1)/2} \left[\begin{matrix} n \\ k\\ \end{matrix} \right]_i
\bsf_i^{(m-k)} \bse_i'^{(n-k)} & \text{if } i=j \text{ and } i \in I^{\rm re} ,\\
\displaystyle \sum_{k=0}^{m} q_i^{-c_i(-2nm+(n+m)k-k(k-1)/2)} \left \{ \begin{matrix} m \\ k\\ \end{matrix} \right \}_i
\bsf_i^{(m-k)} \bse_i'^{(n-k)} & \text{if } i=j \text{ and } i \in I^{\rm im} ,\\
q_i^{-nm a_{ij}} \bsf_j^{(m)}\bse_i'^{(n)} & \text{if } i \neq j.
\end{cases}
\end{aligned} \end{equation}
For $i \in I$ and $v \in U^{-}_q(\mathfrak{g})$, let $$\ell_i(v) = \min \{n \in \mathbb{Z}_{\ge 0} \ \mid \ \bse_i'^{n+1}v=0 \}.$$ Note that $\ell_i$ is well-defined since $\bse_i'$ is locally nilpotent (see $\eqref{eq: special commute}$). Then, for $i \in I$ and $k \in \mathbb{Z}_{\ge 0}$, $$U^{-}_q(\mathfrak{g})_{i}^{< k} :=\{ v \in U^{-}_q(\mathfrak{g}) \mid \ell_i(v) < k \}$$ becomes a $\mathbb{Q}(q)$-vector space.
\begin{Def} \label{Def:perfect bases} A basis $B$ of $U^{-}_q(\mathfrak{g})$ is said to be {\it perfect} if \begin{enumerate} \item $B=\displaystyle\bigsqcup_{\mu\in Q^-} B_{\mu}$, where $B_{\mu}:= B \cap U^{-}_q(\mathfrak{g})_{\mu}$, \item for any $b \in B$ and $i \in I$ with $\bse_i'(b)\neq 0$, there exists a unique $\mathsf{e}_i(b) \in B$ such that \begin{equation} \label{eq: perfect basis} \bse_i'(b) \in c \ \mathsf{e}_i(b) + U^{-}_q(\mathfrak{g})_{i}^{< \ell_i(b)-1} \text{ for some } c \in \mathbb{Q}(q)^{\times}, \end{equation} \item if $\mathsf{e}_i(b)=\mathsf{e}_i(b')$ for $b,b' \in B$, then $b = b'\ ( i \in I)$. \end{enumerate} \end{Def}
Now, we define the {\it upper Kashiwara operators} for the $B_q(\mathfrak{g})$-module $U^{-}_q(\mathfrak{g})$. Let $u \in U^{-}_q(\mathfrak{g})$ such that $\bse_i' u=0$. Then, for $n \in \mathbb{Z}_{\ge0}$, we define the upper Kashiwara operators $ \tilde{E}_i, \tilde{F}_i $ by \begin{equation*}
\begin{aligned}
\ \ &\tilde{E}_i(f_i^{(n)}u)=
\begin{cases}
\displaystyle\frac{q_i^{-(n-1)}}{[n]_i} f_i^{(n-1)}u & \text{ if } i \in I^{\rm re} , \\
\{n\}_i q_i^{c_i(n-1)} f_i^{(n-1)}u & \text{ if } i \in I^{\rm im} ,
\end{cases} \\
\ \ &\tilde{F}_i(f_i^{(n)}u)=
\begin{cases}
\displaystyle q_i^{n}[n+1]_i f_i^{(n+1)}u & \text{ if } i \in I^{\rm re} , \\
\displaystyle\frac{1}{\{n+1\}_i q_i^{c_i n}} f_i^{(n+1)}u & \text{ if } i \in I^{\rm im} .
\end{cases}
\end{aligned}
\end{equation*} From the $i$-string decomposition $\eqref{eq: lowerpart i-string decomposition}$, the upper Kashiwara operators $ \tilde{E}_i$ and $\tilde{F}_i $ can be extended to the whole space $U^{-}_q(\mathfrak{g})$ by linearity.
\begin{Def} \label{Def:upper crystal} An {\it upper crystal basis} of $U^{-}_q(\mathfrak{g})$ is a pair $(L^{\vee},B^{\vee})$ satisfying the following conditions: \begin{enumerate} \item $L^{\vee}$ is a free $\A_0$-module of $U^{-}_q(\mathfrak{g})$ such that $U^{-}_q(\mathfrak{g})=\mathbb{Q}(q) \otimes_{\A_0}L^{\vee}$ and
$L^{\vee} = \bigoplus_{\alpha \in Q^+} L^{\vee}_{-\alpha}$, where $L^{\vee}_{-\alpha} := L^{\vee} \cap
U_q^{-}(\mathfrak{g})_{-\alpha}$, \item $B^{\vee}$ is a $\mathbb{Q}$-basis of $L^{\vee}/ q L^{\vee}$ such that $B^{\vee} = \bigsqcup_{\alpha \in Q^+} B^{\vee}_{-\alpha}$,
where $B^{\vee}_{-\alpha} := B^{\vee} \cap (L^{\vee}_{-\alpha}/ q
L^{\vee}_{-\alpha})$, \item $\tilde{E}_i B^{\vee} \subset B^{\vee} \sqcup \{0\}$, \ \
$\tilde{F}_i B^{\vee} \subset B^{\vee} $ for all $i \in I$, \item For $ b,b'\in B^{\vee}$ and $ i \in I$, $b'^{\vee} = \tilde{F}_i b^{\vee}$
if and only if $b^{\vee} = \tilde{E}_i b'^{\vee}$. \end{enumerate} \end{Def}
We have the following lemma which is the $U^{-}_q(\mathfrak{g})$-version of \cite[Lemma 4.3]{KOP09}. \begin{Lem} \label{Lem: properties of the bilnear form} For any $u, v \in U^{-}_q(\mathfrak{g})$, we have $$(\tilde{f}_i u, v)_K=(u,\tilde{E}_i v)_K, \quad ( \tilde{e}_i u, v)_K =(u, \tilde{F}_i v)_K.$$ \end{Lem}
\begin{Lem} \label{Lem: relatoin between eiip and EEi} Let $u \in U^{-}_q(\mathfrak{g})$, and $n$ be the smallest integer such that $\bse_i'^{n+1}u=0$. Then we have \begin{equation*} \bse_i'^{n}u =
\begin{cases} [n]_i! \tilde{E}_i^{n} u & \text{ if } i \in I^{\rm re} , \\
\tilde{E}_i^{n}u & \text{ if } i \in I^{\rm im} . \end{cases} \end{equation*} \end{Lem}
\begin{proof} For $u \in U_q^{-}(\mathfrak{g})$ and $i \in I$, consider the $i$-string decomposition: $u = \sum_{l=0}^{n}f_i^{(l)}u_l$, where $\bse_i' u_l=0$. If $i \in I^{\rm re} $, then by $\eqref{eq:commutation relation}$ and the definition of $\tilde{E}_i$, we have $$\bse_i'^{n}u=q_i^{- n(n-1)/2}u_n, \ \ \tilde{E}_i^{n}u=\dfrac{q_i^{- n(n-1)/2}}{[n]_i!}u_n.$$ Similarly, if $i \in I^{\rm im} $, we obtain $$\bse_i'^{(n)}u=q_i^{c_i n(n-1)/2}u_n, \ \ \tilde{E}_i^{n}u=\{n\}_i! q_i^{c_i n(n-1)/2}u_n,$$ which proves our assertion. \end{proof}
Let $(L(\infty), B(\infty))$ be the lower crystal basis of $U^{-}_q(\mathfrak{g})$. Set $$L(\infty)^{\vee} = \{ u \in U^{-}_q(\mathfrak{g}) \mid (u, L(\infty))_K \subset \A_{0} \}.$$ We also denote by $( \ , \ )_K : L(\infty)^{\vee} / q L(\infty)^{\vee} \times L(\infty) / qL(\infty) \rightarrow \mathbb{Q}$ the nondegenerate bilinear form induced by the bilinear form $( \ , \ )_K$ on $U^{-}_q(\mathfrak{g})$. Let
$$B(\infty)^{\vee} = \{b^{\vee} \mid b \in B(\infty) \}$$ be the $\mathbb{Q}$-basis of $L(\infty)^{\vee} / q L(\infty)^{\vee}$ which is dual to $B(\infty)$ with respect to $( \ , \ )_K$.
\begin{Prop} \label{prop:upper} The pair $(L(\infty)^{\vee}, B(\infty)^{\vee})$ is an upper crystal basis of $U^{-}_q(\mathfrak{g})$. \end{Prop} \begin{proof} The proof is almost the same as in \cite{Kash93b}. \end{proof}
Let $\A_{\infty}$ be the subring of $\mathbb{Q}(q)$ consisting of regular functions at $\infty$. Let $U_{\A}$ (resp.\ $L$ and $L^{-}$) be an $\A$-subalgebra (resp.\ $\A_0$-subalgebra and $\A_{\infty}$-subalgebra) of $U^{-}_q(\mathfrak{g})$.
\begin{Def} A triple $(U_{\A}, L, L^{-})$ is a {\it balanced triple} if \begin{enumerate} \item $U^{-}_q(\mathfrak{g}) \cong \mathbb{Q}(q) \otimes_{\A} U_{\A} \cong \mathbb{Q}(q) \otimes_{\A_0} L \cong \mathbb{Q}(q) \otimes_{\A_{\infty}} L^{-}$ as $\mathbb{Q}(q)$-vector spaces, \item the natural $\mathbb{Q}$-linear map $E \to L/qL$ is an isomorphism, where $ E := U_{\A} \cap L \cap L^{-}$. \end{enumerate} \end{Def} It was shown in \cite{Kash91} that the condition (2) is equivalent to saying that there are natural isomorphisms $U_\A \cong \A \otimes_{\mathbb{Q}} E$, \ $L \cong \A_0 \otimes_{\mathbb{Q}} E$, \ $L^{-} \cong \A_{\infty} \otimes_{\mathbb{Q}} E$.
Let $U^{0}_{\A}(\mathfrak{g})$ be the $\A$-subalgebra of $U_q(\mathfrak{g})$ generated by $q^{h}$, $\prod^{m}_{k=1} \dfrac{1-q^k q^h}{1-q^k}$ for all $m \in \mathbb{Z}_{\ge 0}$, $h \in P^{\vee}$ and let $U_{\A}(\mathfrak{g})$ be the $\A$-algebra generated by $U^{0}_{\A}(\mathfrak{g})$, $U^{+}_{\A}(\mathfrak{g})$ and $U^{-}_{\A}(\mathfrak{g})$.
\begin{Prop} [\cite{JKK05}] $(U_{\A}^{-}(\mathfrak{g}),L(\infty),L(\infty)^{-})$ is a balanced triple for $U^{-}_q(\mathfrak{g})$. \end{Prop}
Recall the $\mathbb{Q}(q)$-algebra automorphism $\bar {} : U^{-}_q(\mathfrak{g}) \to U^{-}_q(\mathfrak{g})$ given in $\eqref{Eq:bar involution}$. Define \begin{equation*} \begin{aligned} U^{-}_{\A}(\mathfrak{g})^{\vee} & = \{ u \in U^{-}_q(\mathfrak{g}) \mid (u,U^{-}_{\A}(\mathfrak{g}))_K \subset \A \}, \\ L(\infty)^{\vee} & = \{ u \in U^{-}_q(\mathfrak{g}) \mid (u,L(\infty))_K \subset \A_{0} \}, \\ \overline{L(\infty)}^{\vee} & = \{ u \in U^{-}_q(\mathfrak{g}) \mid (u,L(\infty)^{-})_K \subset \A_{\infty} \}. \end{aligned} \end{equation*} By the same argument as in \cite{Kash93b}, one can verify that $(U^{-}_{\A}(\mathfrak{g})^{\vee},L(\infty)^{\vee},\overline{L(\infty)}^{\vee})$ is a balanced triple for $U^{-}_q(\mathfrak{g})$. Hence there is a natural isomorphism $$E^{\vee}:=U^{-}_{\A}(\mathfrak{g})^{\vee} \cap L(\infty)^{\vee} \cap \overline{L(\infty)}^{\vee} \overset{\sim} \longrightarrow L(\infty)^{\vee} / q L(\infty)^{\vee}.$$ Let $G^{\vee}$ denote the inverse of this isomorphism and set $${\mathbb B}(\infty) =\{ G^{\vee}(b^{\vee}) \mid b^{\vee} \in B(\infty)^{\vee} \}.$$
\begin{Lem} \label{lem: Key formula} Let $b^{\vee} \in L(\infty)^{\vee} / q L(\infty)^{\vee}$ and $n \in \mathbb{Z}_{\ge 0}$. \begin{enumerate}
\item If ${\tilde{E}_i}^{n+1}b^{\vee}=0$, then ${\bse_i'}^{n}G(b^{\vee})= \begin{cases} [n]_i!G^{\vee}({\tilde{E}_i}^{n}b^{\vee}) & \text{ if } i \in I^{\rm re} ,\\
G^{\vee}({\tilde{E}_i}^{n}b^{\vee}) & \text{ if }i \in I^{\rm im} . \end{cases}$
\item ${\bse_i'}^{n+1}G^{\vee}(b^{\vee})=0$ if and only if ${\tilde{E}_i}^{n+1} b^{\vee}=0$.
\end{enumerate} \end{Lem}
\begin{proof}
We first prove the assertion (1). Let $i \in I^{\rm re} $. Since $\varphi(\dfrac{1}{[n]_i!} {\bse_i'}^{n})=\bsf_i^{(n)}$, by Lemma \ref{Lem: relatoin between eiip and EEi}, we obtain $$\dfrac{1}{[n]_i!} {\bse_i'}^{n}G^{\vee}(b^{\vee})={\tilde{E}_i}^{n}G^{\vee}(b^{\vee})\in U^{-}_{\A}(\mathfrak{g})^{\vee} \cap L(\infty)^{\vee} \cap \overline{L(\infty)}^{\vee},$$ which yields $\dfrac{1}{[n]_i!} {\bse_i'}^{n}G^{\vee}(b^{\vee})=G^{\vee}({\tilde{E}_i}^{n}b^{\vee})$.
Similarly, for $i \in I^{\rm im} $, it follows from $\varphi({\bse_i'}^{n})=\bsf_i^{(n)}$ that $${\bse_i'}^{n}G^{\vee}(b^{\vee})={\tilde{E}_i}^{n}G^{\vee}(b^{\vee})\in U^{-}_{\A}(\mathfrak{g})^{\vee} \cap L(\infty)^{\vee} \cap \overline{L(\infty)}^{\vee}.$$ Thus we have ${\bse_i'}^{n}G^{\vee}(b^{\vee})=G^{\vee}({\tilde{E}_i}^{n}b^{\vee})$.
For the assertion (2), it is obvious that ${\bse_i'}^{n+1}G^{\vee}(b^{\vee})=0$ implies ${\tilde{E}_i}^{n+1}b^{\vee}=0$. To prove the converse, suppose ${\bse_i'}^{n+1}G^{\vee}(b^{\vee})\neq 0$ and take the smallest $m > n$ such that ${\bse_i'}^{m+1}G^{\vee}(b^{\vee})=0$. By (1), we have \begin{equation*} \begin{aligned} {\bse_i'}^{m}G^{\vee}(b^{\vee})= \begin{cases} [m]_i!G^{\vee}({\tilde{E}_i}^{m}b^{\vee})=0,& \ \text{ if } i \in I^{\rm re} , \\ G^{\vee}({\tilde{E}_i}^{m}b^{\vee})=0, & \ \text{ if } i \in I^{\rm im} , \end{cases} \end{aligned} \end{equation*} which is a contradiction to the choice of $m$. Hence we conclude ${\bse_i'}^{n+1}G^{\vee}(b^{\vee})=0$. \end{proof}
For $b^{\vee}\in B(\infty)^{\vee}$, we define \begin{align*} \varepsilon_i^{\rm or}(b^{\vee}) &= \min \{ n \in \mathbb{Z}_{\ge 0} \mid {\tilde{E}_i}^{n+1}b^{\vee}=0 \}, \\
\varphi_i^{\rm or}(b^{\vee}) &= \min \{ n \in \mathbb{Z}_{\ge 0} \mid {\tilde{F}_i}^{n+1}b^{\vee}=0 \}. \end{align*}
\begin{Prop} \label{Prop:the e_i action global basis} For $b^{\vee} \in B(\infty)^{\vee}$, we have \begin{equation*} \begin{aligned} {\bse_i'} G^{\vee}(b^{\vee}) & = \begin{cases} \displaystyle [\varepsilon_i^{{\rm or}}(b^{\vee})]_i G^{\vee} ({\tilde{E}_i} b^{\vee})
+\sum_{\varepsilon_i^{{\rm or}}(b'^{\vee}) < \varepsilon_i^{{\rm or}}(b^{\vee})-1} E_{b^{\vee},b'^{\vee}}^{i} G^{\vee} (b'^{\vee}) & \
\text{if} \ i \in I^{\rm re} , \\
\displaystyle G^{\vee}({\tilde{E}_i} b^{\vee})
+\sum_{\varepsilon_i^{\rm or}(b'^{\vee}) < \varepsilon_i^{\rm or}(b^{\vee})-1} E_{b^{\vee}, b'^{\vee}}^{i} G^{\vee} (b'^{\vee}) & \ \text{if} \ i \in I^{\rm im} , \end{cases} \\ {\bsf_i} G^{\vee}(b^{\vee}) & = \begin{cases}
\displaystyle q_i^{-\varepsilon_i^{{\rm or}}(b^{\vee})} G^{\vee}({\tilde{F}_i} b^{\vee})+ \sum_{\varepsilon_i^{{\rm or}}(b'^{\vee}) \le \varepsilon_i^{{\rm or}}(b^{\vee})} F^{i}_{b^{\vee},b'^{\vee}}
G^{\vee}(b'^{\vee}) & \ \text{if} \ i \in I^{\rm re} , \\
\displaystyle \{\varepsilon_i^{\rm or}(b^{\vee})+1\}_i q_i^{c_i(\varepsilon_i^{\rm or}(b^{\vee})+1)}G^{\vee}({\tilde{F}_i} b^{\vee})
+ \sum_{\varepsilon_i^{\rm or}(b'^{\vee}) \le \varepsilon_i^{\rm or}(b^{\vee})} F^{i}_{b^{\vee},b'^{\vee}} G^{\vee} (b'^{\vee}) & \ \text{if} \ i \in I^{\rm im} . \end{cases} \end{aligned} \end{equation*} for some $E_{b^{\vee}, b'^{\vee}}^{i}, F^{i}_{b^{\vee},b'^{\vee}} \in \mathbb{Q}(q)$. \end{Prop}
\begin{proof} If $i \in I^{\rm re} $, our assertions were proved in \cite{Kash93b}. We will prove the case when $i \in I^{\rm im} $. Set $n=\varepsilon_i^{{\rm or}}(b^{\vee})$. By Lemma \ref{lem: Key formula} and Definition \ref{Def:upper crystal} (4), we have $$ {\bse_i'}^{n} G^{\vee}(b^{\vee}) = G^{\vee}({\tilde{E}_i}^{n}b^{\vee}) = G^{\vee}({\tilde{E}_i}^{n-1}{\tilde{E}_i} b^{\vee})={\bse_i'}^{n-1}G^{\vee}({\tilde{E}_i} b^{\vee}),$$ which implies $${\bse_i'} G^{\vee}(b^{\vee})-G^{\vee}({\tilde{E}_i} b^{\vee}) \in {\rm Ker}({\bse_i'}^{n-1}).$$
Using the equation $\eqref{eq:commutation relation}$, we get $${\bse_i'}^{(n+1)}\bsf_i G^{\vee}(b^{\vee})=(q_i^{2c_i(n+1)}\bsf_i{\bse_i'}^{(n+1)}+q_i^{c_i(n+1)}{\bse_i'}^{(n)})G^{\vee}(b^{\vee}).$$ Hence Lemma \ref{lem: Key formula} yields $${\bse_i'}^{(n+1)}\bsf_i G^{\vee}(b^{\vee})=\dfrac{1}{\{n\}_i!}q_i^{c_i(n+1)}{\bse_i'}^{n}G^{\vee}(b^{\vee})=\dfrac{1}{\{n\}_i!}q_i^{c_i(n+1)}G^{\vee}({\tilde{E}_i}^{n} b^{\vee}).$$ Using Lemma \ref{lem: Key formula} again, we obtain $$\dfrac{1}{\{n\}_i!}q_i^{c_i(n+1)}G^{\vee}({\tilde{E}_i}^{n+1}{\tilde{F}_i} b^{\vee})=\dfrac{1}{\{n\}_i!}q_i^{c_i(n+1)}{\bse_i'}^{n+1}G^{\vee}({\tilde{F}_i} b^{\vee}) =\{n+1\}_i q_i^{c_i(n+1)}{\bse_i'}^{(n+1)}G^{\vee}({\tilde{F}_i} b^{\vee}).$$ Thus we have $$\bsf_iG^{\vee}(b^{\vee})-\{n+1\}_i q_i^{c_i(n+1)}G^{\vee}({\tilde{F}_i} b^{\vee}) \in {\rm Ker}({\bse_i'}^{n+1})$$ as desired. \end{proof}
Combining Proposition \ref{prop:upper} and Proposition \ref{Prop:the e_i action global basis}, we obtain the existence of perfect basis for $U_q^{-}(\mathfrak{g})$.
\begin{Prop} \label{Cor perfect basis B(infty)} ${\mathbb B}(\infty)$ is a perfect basis of the $B_q(\mathfrak{g})$-module $U^{-}_q(\mathfrak{g}) $. \end{Prop}
Let $B$ be a perfect basis of $U^{-}_q(\mathfrak{g})$. For $b \in B$, define ${\rm wt}(b)= \mu$ if $b \in B_\mu$ and \begin{equation*} \begin{aligned} & \mathsf{f}_i (b) = \begin{cases} b' \ \ & \text{if} \ \mathsf{e}_i (b') = b, \\ 0 \ \ & \text{otherwise}, \end{cases} \quad \varepsilon_i(b) = \begin{cases} \ell_i(b) \ \ & \text{if} \ i\in I^{\rm re} , \\ 0 \ \ & \text{if} \ i \in I^{\rm im} , \end{cases} \\ & \varphi_i(b) = \varepsilon_i(b) + \langle h_i, {\rm wt}(b) \rangle. \end{aligned} \end{equation*} Then it is straightforward to verify that $(B, {\rm wt},\mathsf{e}_i, \mathsf{f}_i, \varepsilon_i, \varphi_i)$ is an abstract crystal. The graph obtained from the crystal $(B, {\rm wt},\mathsf{e}_i, \mathsf{f}_i, \varepsilon_i, \varphi_i)$ is called a {\it perfect graph} of $U^{-}_q(\mathfrak{g})$. The following proposition asserts that the perfect basis ${\mathbb B}(\infty)$ yields the crystal $B(\infty)$.
\begin{Prop} \label{prop:B-infinities} There exist crystal isomorphisms $${\mathbb B}(\infty) \cong B(\infty)^{\vee} \cong B(\infty).$$ \end{Prop}
\begin{proof} Let $\vee : B(\infty) \to B(\infty)^{\vee}$ defined by $b \mapsto b^{\vee}$. Then $$ \tilde{f}_i b =b' \iff (\tilde{f}_i b,b'^{\vee})_K =1 \iff (b,\tilde{E}_i b'^{\vee})_K=1 \iff b^{\vee} = \tilde{E}_ib'^{\vee} \iff \tilde{F}_i b^{\vee} =b'^{\vee}.$$ Hence we have $B(\infty)^{\vee} \cong B(\infty)$ from Lemma \ref{Lem:nondegenerate pairing in GKM} and Lemma \ref{Lem: properties of the bilnear form}.
By Proposition \ref{Prop:the e_i action global basis}, we have $$ \tilde{E}_i b^{\vee} =b'^{\vee} \iff \mathsf{e}_i G^{\vee}(b^{\vee}) = G^{\vee}(\tilde{E}_i b^{\vee}) = G^{\vee}(b'^{\vee}).$$ Hence the map $G^{\vee}$ gives a crystal isomorphism between ${\mathbb B}(\infty)$ and $B(\infty)^{\vee}$. \end{proof}
In the rest of this section, we will show that the perfect graph arising from any perfect basis of $U_q^{-}(\mathfrak{g})$ is isomorphic to the crystal $B(\infty)$. Our argument follows the outline given in \cite[Section 6]{KOP09}.
Let $B$ be a perfect basis of $U^{-}_q(\mathfrak{g})$. For each sequence ${\bf i}=(i_1,\dots,i_m) \in I^{m}$ $(m \ge 1)$, we define a binary relation $\preceq_{{\bf i}}$ on $U^{-}_q(\mathfrak{g}) \setminus \{0\} $ as follows:
\begin{equation*}
\begin{aligned}
\ \ & \mbox{ if } {\bf i}=(i), \mbox{ then } \quad v \preceq_{{\bf i}} v' \Leftrightarrow \ell_{i}(v) \le \ell_{i}(v^{\prime}),\\
\ \ & \mbox{ if } {\bf i}=(i;{\bf i}'), \mbox{ then }\quad v \preceq_{{\bf i}} v' \Leftrightarrow \begin{cases} \ell_{i}(v) < \ell_{i}(v^{\prime}) \mbox{ or} \\
\ell_{i}(v)=\ell_{i}(v^{\prime}), \bse_i'^{\ell_{i}(v)}(v) \preceq_{{\bf i}'} \bse_i'^{\ell_{i}(v^{\prime})}(v^{\prime}). \end{cases}
\end{aligned}
\end{equation*} We write $v \equiv_{{\bf i}} v'$ if $v \preceq_{{\bf i}} v'$ and $v' \preceq_{{\bf i}} v$.
For a given ${\bf i}=(i_1,\dots,i_m) \in I^{m}$, define the maps $\bse'^{ top}_{\bf i}:U^{-}_q(\mathfrak{g}) \to U^{-}_q(\mathfrak{g})$ and
${{\mathsf{e}_{{\bf i}}}^{top}}:B \to B\sqcup \{ 0 \} $ as follows:
\begin{eqnarray*}
\bse'^{top}_i(v) = \bse_i'^{\ell_{i}(v)}(v) \mbox{ for } m=1\quad & \mbox{ and } & \quad
\bse'^{top}_{{\bf i}} = \bse'^{top}_{i_m}\circ \cdots \circ \bse'^{top}_{i_1} \mbox { for } m >1, \\
\mathsf{e}_i^{top}(b) = \mathsf{e}_i^{\ell_{i}(b)}(b) \mbox{ for } m=1 \quad & \mbox{ and } &
\quad {{\mathsf{e}_{{\bf i}}}^{top}} = \mathsf{e}^{top}_{i_m}\circ \cdots \circ
\mathsf{e}^{top}_{i_1}\mbox { for } m >1.
\end{eqnarray*} By Proposition \ref{Prop:Highest vector 1}, we identify $\mathbb{Q}(q) $ with $ \{ v \in U^{-}_q(\mathfrak{g}) \mid \bse_i'(v)=0 \mbox{ for all } i \in I\}$. Note that $ \mathbb{Q}(q) \cap B =\{ {\bf 1} \}$. For each $v \in U^{-}_q(\mathfrak{g})$, there exists a sequence ${\bf i}$ such that $\bse'^{top}_{\bf i}(v) \in \mathbb{Q}(q)$. From $\eqref{eq: perfect basis}$, one can check that the following statements hold.
\begin{Lem} \label{Lem: realtions between top operators} For any sequence ${\bf i}=(i_1,\dots,i_m) \in I^{m}\ ( m\ge 1)$, we have \begin{itemize}
\item[(1)] $\bse'^{top}_{{\bf i}}(b) \in \mathbb{Q}(q)^{\times} {{\mathsf{e}_{{\bf i}}}^{top}}(b)$ for any $b\in B$,
\item[(2)] if $\bse'^{top}_{{\bf i}}(b) \in \mathbb{Q}(q)^{\times}$ for some $b\in B$, then ${{\mathsf{e}_{{\bf i}}}^{top}}(b)\in \mathbb{Q}(q)^{\times}$,
\item[(3)] if $b \equiv_{\bf i} b'$ and ${{\mathsf{e}_{{\bf i}}}^{top}}(b)={{\mathsf{e}_{{\bf i}}}^{top}}(b')$, then $b=b'$ for all $ b,b' \in B$. \end{itemize} \end{Lem}
\begin{Def} Let $B,B'$ be perfect bases of $U^{-}_q(\mathfrak{g})$. A {\em perfect morphism} $[\phi,\tilde{\phi},c]:(U^{-}_q(\mathfrak{g}),B) \to (U^{-}_q(\mathfrak{g}),B')$ is a triple $(\phi,\tilde{\phi},c)$, where \begin{itemize} \item[(1)] $\phi:U^{-}_q(\mathfrak{g}) \to U^{-}_q(\mathfrak{g})$ is a $B(\mathfrak{g})$-module endomorphism such that $0 \notin \phi(B)$, \item[(2)] $\tilde{\phi}: B \to B^{\prime}$ is a map satisfying $\tilde{\phi}({\bf 1})=\phi({\bf 1})$, \item[(3)] $c: B\setminus \{ {\bf 1} \} \to \mathbb{Q}(q)^{\times}$ is a map satisfying \begin{align*}
\phi(b)-c(b)\tilde{\phi}(b) \prec_{{\bf i}} \phi(b) \end{align*} for $b\in B \setminus \{ {\bf 1} \}$ and ${\bf i}=(i_1,\dots,i_m)$ such that $\bse'^{top}_{\bf i}(b)\in \mathbb{Q}(q)$. \end{itemize} \end{Def}
\begin{Lem} Let $\phi$ be a $B_q(\mathfrak{g})$-endomorphism of $U_q^{-}(\mathfrak{g})$. \begin{enumerate} \item If a perfect morphism $[\phi,\tilde{\phi},c]$ exists, then $\tilde{\phi}$ and $c$ are uniquely determined. \item For a given perfect morphism $[\phi,\tilde{\phi},c]: (U_q^{-}(\mathfrak{g}),B) \to (U_q^{-}(\mathfrak{g}),B')$, the map $\tilde{\phi}$ is a crystal morphism. \end{enumerate} \end{Lem}
\begin{proof} This lemma is essentially the same as \cite[Lemma 6.3, Lemma 6.4]{KOP09}. However, since our algebra $U_q^{-}(\mathfrak{g})$ is considered as a $B_q(\mathfrak{g})$-module, Proposition \ref{Prop:Highest vector 1} plays a key role in proving this lemma. Then our assertions follow by a similar argument in \cite{KOP09}. \end{proof}
Now we state and prove the main result of this section.
\begin{Thm} \label{Thm: uniqueness of perfect graphs}
Let $B$ and $B^{\prime}$ be two perfect bases of $U^{-}_q(\mathfrak{g})$.
Then the identity map ${\rm id} : U^{-}_q(\mathfrak{g}) \to U^{-}_q(\mathfrak{g}) $ induces a perfect
isomorphism from $(U^{-}_q(\mathfrak{g}) ,B)$ to $(U^{-}_q(\mathfrak{g}),B')$. That is, there exists a unique crystal isomorphism $\tilde{\phi}: B \to B^{\prime}$ and a unique map $c: B\setminus \{ {\bf 1} \} \to \mathbb{Q}(q)^{\times}$ satisfying $\tilde{\phi}( {\bf 1} )= {\bf 1} $ and $$ b-c(b)\tilde{\phi}(b)\prec_{{\bf i}} b $$ for each $b \in B\setminus \{ {\bf 1} \}$ and any sequence ${\bf i}=(i_1,\dots,i_m)$ with ${{\mathsf{e}_{{\bf i}}}^{top}}(b)= {\bf 1} $. \end{Thm}
\begin{proof} Since the proof is almost the same as \cite[Theorem 6.6]{KOP09}, we only give a sketch of proof. By a similar argument in \cite[Lemma 6.5]{KOP09}, for a given $b \in B \setminus \{ {\bf 1} \}$, one can show that there exist unique $b' \in B'$, $v \in U_q^{-}(\mathfrak{g})$ and $k \in \mathbb{Q}(q)^{\times}$ satisfying \begin{align*} \ \ & (1) \ b \equiv_{{\bf i}} b', \quad (2) \ b = v + kb', \quad (3) \ v =0 \text{ or } v \prec_{{\bf i}} b, \ v \prec_{{\bf i}} b' \end{align*} for any sequence ${\bf i}$ with $\bse'^{top}_{{\bf i}}(b) \in \mathbb{Q}(q)^{\times}$. Then the maps ${\rm id}: U_q^{-}(\mathfrak{g}) \to U_q^{-}(\mathfrak{g})$, $\tilde{\phi}:B \to B'$ and $c: B \setminus \{ {\bf 1} \}\to \mathbb{Q}(q)^{\times}$ defined by $b \mapsto b'$ and $b \mapsto k$ give rise to a perfect isomorphism. \end{proof}
\vskip 3em
\section{Construction of crystals $\Bklr{\infty}$ and $\Bklr{\lambda}$}\
In this section, we investigate the crystal structures on the sets of isomorphism classes of irreducible graded modules over $R$ and its cyclotomic quotient $R^\lambda$. We assume that $a_{ii} \ne 0$ for all $i\in I$.
\subsection{The crystal $\Bklr{\infty}$}\
Let $\Bklr{\infty}$ be the set of isomorphism classes of irreducible graded $R$-modules. In this subsection, we define a crystal structure on $\Bklr{\infty}$ and show that it is isomorphic to the crystal $B(\infty)$ using the perfect basis theory given in Section \ref{Sec:perfect bases}.
Let $\alpha \in Q^+$. For any $P \in R(\alpha)$-pmod and $M \in R(\alpha)$-fmod, we define \begin{equation}\label{Eq:def of f,e',F,E'} \begin{aligned} f_i(P) &= {\rm Ind}_{\alpha_i, \alpha} (P_{(i)} \boxtimes P), & e'_i(P)& = P^\star \otimes_{R'(\alpha_i)}L(i), \\ F_i(M) &= {\rm Ind}_{\alpha_i, \alpha} (L(i) \boxtimes M), & E'_i(M)& = {\rm Res}_{\alpha-\alpha_i}^{\alpha_i, \alpha-\alpha_i} \circ \Delta_i M, \end{aligned} \end{equation} where $R'(\alpha_i):= R(\alpha_i) \otimes 1_{\alpha-\alpha_i} \hookrightarrow R(\alpha_i) \otimes R(\alpha-\alpha_i) \subset R(\alpha)$. Here, the $(R(\alpha_i), R(\alpha - \alpha_i))$-bimodule structure of $P^\star$ is given as follows: for $ v \in P^\star, \ r \in R(\alpha - \alpha_i) $ and $ s \in R(\alpha_i)$, \begin{align*} r \cdot v := ( 1_{\alpha_i}\otimes r) \ v , \quad v \cdot s := \psi(s \otimes 1_{\alpha - \alpha_i} ) \ v . \end{align*} Since $f_i$ and $e'_i$ (resp.\ $F_i$ and $E'_i$) take projective modules to projective modules (resp.\ finite-dimensional modules to finite-dimensional modules), they induce the linear maps \begin{align*} f_i&:K_0(R) \longrightarrow K_0(R), \qquad e'_i:K_0(R) \longrightarrow K_0(R), \\ F_i&:G_0(R) \longrightarrow G_0(R), \qquad E'_i:G_0(R) \longrightarrow G_0(R). \end{align*} Then we have the following lemma, which is the Khovanov-Lauda-Rouquier algebra version of the equation $\eqref{eq: special commute}$. \begin{Lem}\ \label{Lem:relation of boson} \begin{enumerate} \item $e'_if_j = \delta_{ij} + q_i^{-a_{ij}} f_j e'_i $ on $K_0(R)$. \item $E'_iF_j = \delta_{ij} + q_i^{-a_{ij}} F_j E'_i $ on $G_0(R)$. \end{enumerate} \end{Lem} \begin{proof}
(1)
Fix $\mathbf{i} \in {\rm Seq} (\alpha)$ and let $ \mathbf{i}' = (j) * \mathbf{i} \in {\rm Seq} (\alpha+\alpha_j)$. By the equation $\eqref{Eq:ind and res of Pi}$, \begin{align*} \Delta_i P_{\mathbf{i}'} &\simeq \sum_{\mathbf{i}': \text{shuffles of $(i)$ and $\mathbf{j}$}} P_{(i)} \boxtimes P_{\mathbf{j}} \langle \deg( (i), \mathbf{j}, \mathbf{i}' ) \rangle \\ & \simeq \delta_{ij}P_{(i)}\boxtimes P_{\mathbf{i}} + \sum_{\mathbf{i}: \text{shuffles of $(i)$ and $\mathbf{k}$}} P_{(i)} \boxtimes P_{(j)*\mathbf{k}}
\langle -\deg( (i), \mathbf{k}, \mathbf{i} ) + (\alpha_i| \alpha_j) \rangle , \end{align*} which yields \begin{align*} e_i' f_j [P_\mathbf{i}] &= e_i'[P_{\mathbf{i}'}] \\ & = [ P_{\mathbf{i}'}^\star \otimes_{R'(\alpha_j)} L(i) ]\\ &= [ (\Delta_i P_{\mathbf{i}'})^\star \otimes_{R'(\alpha_i)} L(i)]\\
&= \delta_{ij} [P_{\mathbf{i}}] + q^{-(\alpha_i|\alpha_j)} f_j[({\rm Res}_{\alpha-\alpha_i}^{\alpha_i, \alpha-\alpha_i}( P_{\mathbf{i}}^\star \otimes_{R'(\alpha_i)} L(i) ))] \\
&= \delta_{ij} [P_{\mathbf{i}}] + q^{-(\alpha_i|\alpha_j)} f_j e_i'[P_\mathbf{i}]. \end{align*}
(2) For an irreducible $R(\alpha)$-module $M$, it follows from Proposition \ref{Prop:Mackey} that \begin{align*} E_i' F_j [M] &= E_i'([{\rm Ind}_{\alpha_j, \alpha}L(j) \boxtimes M])\\
&= [ E_i'L(j)] [M] + [{\rm Ind}_{\alpha_j, \alpha-\alpha_i}L(j) \boxtimes E_i'(M) \langle (\alpha_j| \alpha_i) \rangle]\\
&= \delta_{ij}[M] + q^{-(\alpha_i|\alpha_j)} F_j E_i'[M]. \end{align*} \end{proof}
We also have analogues of the equation $\eqref{Eq:def of ()K and ()L}$ and Lemma \ref{Lem:nondegenerate pairing in GKM} (3).
\begin{Lem} \ \label{Lem:duality e,f and E,F} \begin{enumerate} \item For $[P], [Q] \in K_0(R)$, we have $$ (e_i'[P], [Q]) = (1-q_i^2)([P], f_i[Q]). $$ \item For $[P] \in K_0(R)$ and $[M] \in G_0(R)$, we have $$ (f_i[P], [M]) = ([P], E'_i [M]), \quad (e'_i[P], [M]) = ([P], F_i [M]). $$ \end{enumerate} \end{Lem} \begin{proof} (1) Let $P, Q \in R(\alpha)$-mod. Then \begin{align*} ([P], f_i[Q]) & = \dim_q ( P^\star \otimes_{R(\alpha+\alpha_i)} ({\rm Ind} P_{(i)} \boxtimes Q) )\\ &= \dim_q ( (\Delta_i P)^\star \otimes_{R(\alpha_i)\otimes R(\alpha)} ( P_{(i)} \boxtimes Q) ) \\ &= (1-q_i^2)^{-1} \dim_q ( (e_i' P)^\star \otimes_{R(\alpha)} Q )\\ &= (1-q_i^2)^{-1}(e_i'[P], [Q]). \end{align*}
(2) Let $P \in R(\alpha)$-pmod and $M \in R(\alpha+\alpha_i)$-fmod. By definition, we have the first assertion: \begin{align*} (f_i[P], [M]) &= \dim_q ( ({\rm Ind} P_{(i)} \boxtimes P)^\star \otimes_{R(\alpha + \alpha_i)} M ) \\ &= \dim_q ( ( P_{(i)} \boxtimes P)^\star \otimes_{R(\alpha_i)\otimes R(\alpha)} \Delta_i M ) \\ &= \dim_q ( P^\star \otimes_{R(\alpha)} {\rm Res}_{\alpha}^{\alpha_i, \alpha} \Delta_i M ) \\ &= ( [P], E_i'[M] ). \end{align*}
In a similar manner, we have \begin{align*} (e_i'[P], [M]) &= \dim_q \left( ( P^\star \otimes_{R'(\alpha_i)}L(i) ) \otimes_{R(\alpha)} M \right) \\ &= \dim_q \left( (\Delta_i P)^\star \otimes_{R(\alpha_i) \otimes R(\alpha)} L(i) \boxtimes M \right) \\ &= \dim_q \left( P^\star \otimes_{R(\alpha+\alpha_i)} {\rm Ind} L(i) \boxtimes M \right) \\ &= ( [P], F_i [M] ). \end{align*} \end{proof}
We now define a $B_q(\mathfrak{g})$-module structure on $K_0(R)_{\mathbb{Q}(q)}$ and $G_0(R)_{\mathbb{Q}(q)}$ as follows: \begin{align*} \bse'_i \cdot [P] &:= e_i'[P], \ \ \bsf_i \cdot [P] := f_i[P] \quad \text{ for } [P] \in K_0(R)_{\mathbb{Q}(q)}, \\ \bse'_i \cdot [M] &:= E_i'[M], \ \ \bsf_i \cdot [M] := F_i[M] \quad \text{ for } [M] \in G_0(R)_{\mathbb{Q}(q)}. \end{align*} By the same argument as in the proof of \cite[Lemma 3.4.2]{Kash91}, it follows from Lemma \ref{Lem:relation of boson}, Lemma \ref{Lem:duality e,f and E,F} and Theorem \ref{Thm:Serre} that $K_0(R)_{\mathbb{Q}(q)}$ and $G_0(R)_{\mathbb{Q}(q)}$ are well-defined $B_q(\mathfrak{g})$-modules. Consider the $B_q(\mathfrak{g})$-module homomorphism $$\Phi^{\vee}_{\mathbb{Q}(q)}: U_q^-(\mathfrak{g}) \longrightarrow G_0(R)_{\mathbb{Q}(q)} $$ given by $$\Phi^{\vee}_{\mathbb{Q}(q)}(f_i) = L(i) \ \ \text{for} \ i\in I.$$ Then, by Theorem \ref{Thm:iso of K0 and Uq}, we obtain the following diagram. $$ \xymatrix{ \Phi_{\mathbb{Q}(q)}\ : \ U_q^-(\mathfrak{g}) \ar[rrr]^{\sim} \ar[d]^{\text{dual w.r.t. } (\ ,\ )_K} & & & \ar[d]^{\text{dual w.r.t. } (\ ,\ )} K_0(R)_{\mathbb{Q}(q)} \\ \Phi_{\mathbb{Q}(q)}^\vee\ : \ U_q^-(\mathfrak{g}) \ar[u] \ar[rrr]^{\sim} & & & \ar[u] G_0(R)_{\mathbb{Q}(q)} } $$ Therefore, $K_0(R)_{\mathbb{Q}(q)}$ and $G_0(R)_{\mathbb{Q}(q)}$ are well-defined $B_q(\mathfrak{g})$-modules, which are isomorphic to $U_q^-(\mathfrak{g})$.
The following lemma is the Khovanov-Lauda-Rouquier algebra version of Proposition \ref{Prop:the e_i action global basis}. \begin{Lem} \label{Lem:perfect condition of U-} Let $M$ be an irreducible $R(\alpha)$-module and $\ep = \ep_i(M)$. Then we have
\begin{align*} E_i'[M] = \left\{
\begin{array}{ll}
q^{-\ep + 1}_i [\ep]_i[\ke_i M] + \sum_{k}c_k [N_k] & \hbox{ if } i\in I^{\rm re} , \\
{[\ke_i M]} + \sum_{k}c_k' [N_k'] & \hbox{ if } i \in
I^{\rm im} ,
\end{array}
\right. \end{align*} where $c_k, c_k' \in \mathbb{Q}(q)$ and $\ep_i(N_k),\ \ep_i(N_k') < \ep - 1$. \end{Lem} \begin{proof} If $i= I^{\rm re} $, then the assertion can be proved in the same manner as \cite[Lemma 3.9]{KP10}. Suppose that $i\in I^{\rm im} $. By Lemma \ref{Lem:Delta of M}, $$ \Delta_{i^\ep}M \simeq L(i^\ep) \boxtimes N $$ for some irreducible module $N$ with $\ep_i(N)=0$. Then, from $\eqref{Eq:reciprocity2}$, we have an exact sequence \begin{align} \label{Eq:perfect for KLR eq1} 0 \rightarrow K \rightarrow {\rm Ind}_{\ep \alpha_i, \alpha-\ep \alpha_i} L(i^\ep)\boxtimes N \rightarrow M \rightarrow 0 \end{align} for some $R(\alpha)$-module $K$. Note that $\ep_i(K) < \ep$.
On the other hand, it follows from $\ep_i(N)=0$ and Lemma \ref{Lem:Kato for i in Iim} that $$ [\Delta_i {\rm Ind}_{\ep \alpha_i, \alpha-\ep \alpha_i} L(i^\ep)\boxtimes N ]
= [{\rm Ind}_{\alpha_i, (\ep-1) \alpha_i, \alpha-\ep \alpha_i}^{\alpha_i, \alpha- \alpha_i} L(i)\boxtimes L(i^{\ep-1})\boxtimes N]. $$ By Lemma \ref{Lem:properties of L(im) bt N}, Lemma \ref{Lem:soc and hd} and Lemma \ref{Lem:adjoint ke and kf}, we have $${\rm hd} ({\rm Ind}_{\alpha_i, (\ep-1) \alpha_i, \alpha-\ep \alpha_i}^{\alpha_i, \alpha- \alpha_i} L(i)\boxtimes L(i^{\ep-1})\boxtimes N) \simeq L(i)\boxtimes(\kf_i^{\ep -1}N) \simeq L(i)\boxtimes \ke_iM $$ and all the other composition factors of ${\rm Ind}_{\alpha_i, (\ep-1) \alpha_i, \alpha-\ep \alpha_i}^{\alpha_i, \alpha- \alpha_i} L(i)\boxtimes L(i^{\ep-1})\boxtimes N$ are of the form $L(i)\boxtimes L$ with $\ep_i(L) < \ep-1$. Moreover, since $\ep_i(K)<\ep$, all composition factors of $\Delta_i(K)$ are of the form $L(i)\boxtimes L'$ with $\ep_i(L')$ with $\ep_i(L') < \ep - 1$. Therefore, applying the exact functor $\Delta_i$ to $\eqref{Eq:perfect for KLR eq1}$, we have $$ E_i'[M] = {[\ke_i M]} + \sum_{k}c_k' [N_k'] $$ for some $R(\alpha)$-modules $N_k'$ with $ \ep_i(N_k') < \ep-1 $. \end{proof}
For an element $[M] \in \Bklr{\infty}$, we define \begin{align*} {\rm wt}([M]) &= -\alpha \ \ \text{ if } M \in R(\alpha)\text{-fmod}, \\ \varepsilon_i([M]) &= \left\{
\begin{array}{ll}
\max\{ k \ge 0 \mid \ke_i^k M \ne 0 \} & \hbox{ if } i\in I^{\rm re} , \\
0 & \hbox{ if } i \in I^{\rm im} ,
\end{array}
\right.\\ \varphi_i([M]) &= \varepsilon_i(b) + \langle h_i, {\rm wt}( [M]) \rangle. \end{align*} Then we have the following theorem.
\begin{Thm} \label{Thm: B(infty)} The sextuple $(\mathfrak{B}(\infty),{\rm wt}, \ke_i, \kf_i, \varepsilon_i, \varphi_i)$ becomes an abstract crystal, which is isomorphic to the crystal $B(\infty)$ of $U_q^{-}(\mathfrak{g})$. \end{Thm} \begin{proof} It follows from Lemma \ref{Lem:adjoint ke and kf} and Lemma \ref{Lem:perfect condition of U-} that the pair $(\Bklr{\infty}, \{\ke_i\}_{i\in I})$ is a perfect basis for the $B_q(\mathfrak{g})$-module $G_{0}(R)_{\mathbb{Q}(q)}$. Hence by Theorem \ref{Thm: uniqueness of perfect graphs}, $\mathfrak{B}(\infty)$ is isomorphic to $B(\infty)$. \end{proof}
\vskip 1em
\subsection{Cyclotomic quotients $R^{\lambda}$ and their crystals $\Bklr{\lambda}$ }\
In this subsection, we define the cyclotomic quotient $R^{\lambda}$ of $R$ for $\lambda \in P^+$, and investigate the crystal structure on the set of isomorphism classes of irreducible $R^{\lambda}$-modules.
For $\alpha \in Q^+$ with $|\alpha|=d$ and $\lambda \in P^+ $, let $I^{\lambda}(\alpha)$ denote the two-side ideal of $R(\alpha)$ generated by \begin{equation} \label{Eq:def of cyclotomic ideal} \begin{aligned} & \{x_d^{\langle h_{i_d}, \lambda \rangle} 1_\mathbf{i} \mid \mathbf{i}=(i_1, \ldots, i_d) \in {\rm Seq} (\alpha) \}.
\end{aligned} \end{equation} Note that it is defined in the opposite manner to \cite[Section 3.4]{KL09}. We define $$ R^\lambda(\alpha) = R(\alpha) / I^\lambda(\alpha). $$
The algebra $R^{\lambda}:= \bigoplus_{\alpha \in Q^{+}} R^{\lambda}(\alpha)$ is called the {\it cyclotomic Khovanov-Lauda-Rouquier algebra} of weight $\lambda$. For an irreducible $R(\alpha)$-module $M$, let $$ \ep^{\vee}_i (M) = \max \{ k \ge 0 \mid 1_{\alpha - k \alpha_i, k \alpha_i } M \ne 0 \}. $$ This definition is also the opposite to \cite[(5.6)]{LV09}. Combining Lemma \ref{Lem:properties of L(im) bt N} and Lemma \ref{Lem:soc and hd} with $\eqref{Eq:def of L in Iim}$ and the fact that $x_m^k L(i^m)=0$ for $k \ge m,\ i \in I^{\rm re} $, we obtain \begin{align} \label{Eq:eq condition of being in B(lamda)}
I^\lambda(\alpha) \cdot M = 0 \ \text{ if and only if }\
\left\{
\begin{array}{ll}
\ep^{\vee}_i (M) \le \langle h_i, \lambda \rangle & \hbox{ for } i \in I^{\rm re} , \\
\ep^{\vee}_i (M) = 0 & \hbox{ for } i \in I^{\rm im} \text{ with } \langle h_i, \lambda \rangle=0,
\end{array}
\right. \end{align} where $M$ is an irreducible $R(\alpha)$-module. \begin{Lem} Let $M$ be an irreducible $R(\alpha)$-module. \begin{enumerate} \item For $i \in I$, either $\ep_i^{\vee}(\kf_i M) = \ep_i^{\vee}(M)$ or $\ep_i^{\vee}(M)+1$. \item For $i,j \in I$ with $i \ne j$, we have $\ep_i^{\vee}(\kf_j M) = \ep_i^{\vee}(M)$. \end{enumerate} \end{Lem} \begin{proof} The proof is the same as that of \cite[Proposition 6.2]{LV09}. \end{proof}
For $M \in R^\lambda(\alpha)$-$ \mathrm{fmod}$ and $N \in R(\alpha)$-$ \mathrm{fmod}$, let $ {\rm infl} ^\lambda M$ be the inflation of $M$, and $ {\rm pr} ^\lambda N $ be the quotient of $ N $ by $ I^\lambda(\alpha) N$. Let $\Bklr{\lambda}$ denote the set of isomorphism classes of irreducible graded $R^\lambda$-modules. For $M \in R^\lambda(\alpha)$-$ \mathrm{fmod}$, define \begin{equation} \label{Eq:def of B(lambda)} \begin{aligned} {\rm wt}^\lambda(M) &= \lambda - \alpha, \\ \ke_i^\lambda M &= {\rm pr} ^\lambda \circ \ke_i \circ {\rm infl} ^\lambda M, \\ \kf_i^\lambda M &= {\rm pr} ^\lambda \circ \kf_i \circ {\rm infl} ^\lambda M, \\ \varepsilon_i^\lambda(M) &= \left\{
\begin{array}{ll}
\max\{ k \ge 0 \mid (\ke_i^\lambda)^k M \ne 0 \} & \hbox{ for } i\in I^{\rm re} , \\
0 & \hbox{ for } i\in I^{\rm im} ,
\end{array}
\right. \\ \varphi_i^\lambda(M) &= \left\{
\begin{array}{ll}
\max\{ k \ge 0 \mid (\kf_i^\lambda)^k M \ne 0 \} & \hbox{ for } i\in I^{\rm re} , \\
\langle h_i , {\rm wt}^\lambda(M) \rangle & \hbox{ for } i\in I^{\rm im} .
\end{array}
\right. \end{aligned} \end{equation}
We will show that $(\mathfrak{B}(\lambda),{\rm wt}^\lambda, \ke_i^\lambda, \kf_i^\lambda, \varepsilon_i^\lambda, \varphi_i^\lambda)$ is an abstract crystal. For this purpose, we need several lemmas.
\begin{Lem} \label{Lem:difference of phi} Let $i \in I^{\rm re} $ and $\lambda, \mu \in P^+$. For $[M], [N] \in \Bklr{\infty}$ with $ {\rm pr} ^\lambda M \ne \emptyset,\ {\rm pr} ^{\lambda} N \ne \emptyset,\ {\rm pr} ^\mu M \ne \emptyset,\ {\rm pr} ^{\mu} N \ne \emptyset $, we have $$ \ph_i^\lambda (M) - \ph_i^\lambda (N) = \ph_i^\mu (M) - \ph_i^\mu (N) .$$ \end{Lem} \begin{proof} The assertion can be proved in the same manner as in \cite[Proposition 6.6, Remark 6.7]{LV09}. \end{proof}
\begin{Lem} \label{Lem:structure lem for R(mi+j)}
Let $i \in I^{\rm re} $ and $ j \in I$ with $a_{ij}<0$. \begin{enumerate} \item If $ m \le -a_{ij}$, then for each $0 \le k \le m $, there exists a unique irreducible $R(m \alpha_i + \alpha_j)$-module $L(i^{k} j i^{m-k})$ with $$ \ep_i(L(i^k j i^{m-k})) = k \ \text{ and }\ \ep_i^{\vee}(L(i^k j i^{m-k})) = m-k .$$ \item If $0 \le k \le -a_{ij}$, then the module $$ {\rm Ind} L(i^s) \boxtimes L( i^k j i^{-a_{ij} - k}) \simeq {\rm Ind} L( i^k j i^{-a_{ij} - k} ) \boxtimes L(i^s) $$ is irreducible for all $s \ge 0$. \item If $0 \le k \le -a_{ij} \le c$ and $N$ is an irreducible $R(c\alpha_i + \alpha_j)$-module with $\ep_i(N)=k$, then we have $c+a_{ij} \le k \le c$ and $$ N \simeq {\rm Ind} L(i^{c+a_{ij}}) \boxtimes L( i^{k-c-a_{ij}} j i^{c-k}) . $$ \end{enumerate} \end{Lem} \begin{proof} To prove (1), we consider the induced module ${\rm Ind} L(i^k) \boxtimes L(j) \boxtimes L(i^{m-k})$ for $0 \le k \le m$. Let $$ K = {\rm Span}_\F \{ \tau_w \otimes (t \otimes u \otimes v) \mid w \in \sg_{m+1}, \ell(w)>0, t \in L(i^k), u \in L(j), v \in L(i^{m-k}) \} . $$ By the same argument as in \cite[Proposition 6.11]{LV09}, we deduce that $K$ is a proper maximal submodule of ${\rm Ind} L(i^k) \boxtimes L(j) \boxtimes L(i^{m-k})$, and that ${\rm hd} {\rm Ind} L(i^k) \boxtimes L(j) \boxtimes L(i^{m-k})$ is the quotient module ${\rm Ind} L(i^k) \boxtimes L(j) \boxtimes L(i^{m-k}) /K $ which is irreducible. We denote it by $L(i^{k} j i^{m-k})$. By the Frobenius reciprocity $\eqref{Eq:reciprocity}$, we have \begin{align} \label{Eq:epsilon of irr repn}
\ep_i(L(i^k j i^{m-k})) = k \ \text{ and }\ \ep_i^{\vee}(L(i^k j i^{m-k})) = m-k . \end{align}
On the other hand, there is a surjective homomorphism of degree 0 $$ {\rm Ind} L(i^k) \boxtimes L(j) \boxtimes L(i^{m-k}) \twoheadrightarrow \ke_i^k \ke_j \ke_i^{m-k} \mathbf{1}, $$ which implies that $ L(i^k j i^{m-k}) \simeq \ke_i^k \ke_j \ke_i^{m-k} \mathbf{1}$. By Theorem \ref{Thm: B(infty)}, $$\{ \ke_i^k \ke_j \ke_i^{m-k} \mathbf{1} \mid 0 \le k \le m \} $$ is a complete set of irreducible $R(m\alpha_i + \alpha_j)$-module. Therefore, $L(i^k j i^{m-k})$ is a unique irreducible $R(m\alpha_i + \alpha_j)$-module satisfying $\eqref{Eq:epsilon of irr repn}$.
The assertion (2), (3) can be proved by the same argument as in \cite[Theorem 6.10]{LV09}. \end{proof}
Fix $i \in I^{\rm re} $ and $ j \in I$ with $i \ne j, a_{ij} \ne 0$ and let $$\mathfrak{L}(k) = L(i^{k} j i^{-a_{ij}-k}) \ \ \text{for} \ 0 \le k \le -a_{ij}.$$
\begin{Lem} \label{Lem:surjection for R(ci+dj)} Let $c,d \in \mathbb{Z}_{\ge 0}$ with $c+d \le -a_{ij}$. \begin{enumerate} \item We have \begin{align*} {\rm hd} {\rm Ind} L(i^m) \boxtimes L(i^c j i^{d} ) & \simeq \kf_i^m L(i^cji^{d}) \simeq \kf_i^{m+c}L(ji^{d})\\ & \simeq \left\{
\begin{array}{ll}
{\rm Ind} L(i^{m+a_{ij} + c + d}) \boxtimes \mathfrak{L}(-a_{ij}-d) & \hbox{ if } m \ge -a_{ij} - c -d, \\
\mathfrak{L}(i^{m+c} j i^{d}) & \hbox{ if } m < -a_{ij} - c -d.
\end{array}
\right. \end{align*} \item Suppose that there is a nonzero homomorphism $$ {\rm Ind} L(i^m) \boxtimes \mathfrak{L}(c_1) \boxtimes \cdots \boxtimes \mathfrak{L}(c_r) \longrightarrow Q $$ where $Q$ is irreducible. Then $$ \ep_i(Q) = m+ \sum_{t=1}^r c_t\ \text{ and }\ \ep_i^{\vee}(Q) = m + \sum_{t=1}^r (-a_{ij} - c_t) .$$ \item Let $M$ and $Q$ be irreducible. Suppose that there is a nonzero homomorphism ${\rm Ind} \mathfrak{L}(k) \boxtimes M \rightarrow Q$. Then $\ep_i(Q) = \ep_i(M)+k$. \end{enumerate} \end{Lem} \begin{proof} The proof is identical to that of \cite[Lemma 6.13]{LV09}. \end{proof}
\begin{Lem}\ \label{Lem:surjection2 for R(ci+dj)} \begin{enumerate} \item If $N$ is an irreducible $R(c\alpha_i + d\alpha_j)$-module with $\ep_i(N)=0$, then there exist $r \in \mathbb{Z}_{> 0} $ and $b_t \le -a_{ij}$ for $1 \le t \le r$ such that $$ {\rm Ind} L(j i^{b_1}) \boxtimes \cdots \boxtimes L(j i^{b_r}) \twoheadrightarrow N . $$ \item Let $a := -a_{ij}$. Suppose that we have a surjective homomorphism $$ {\rm Ind} L(i^h) \boxtimes L(j i^{b_1}) \boxtimes \cdots \boxtimes L(j i^{b_r}) \twoheadrightarrow Q, $$ where $Q$ is irreducible. \begin{enumerate} \item If $h \ge \sum_{t=1}^r (a- b_t)$, then we have a surjective homomorphism $$ {\rm Ind} L(i^g) \boxtimes \mathfrak{L}(a-b_1) \boxtimes \cdots \boxtimes \mathfrak{L}(a-b_r) \twoheadrightarrow Q,$$ where $g := h- \sum_{t=1}^r (a-b_t) $. \item Otherwise, we have $$ {\rm Ind} \mathfrak{L}(a-b_1) \boxtimes \cdots \boxtimes {\rm Ind} \mathfrak{L}(a-b_{s-1}) \boxtimes L(i^{g'}ji^{b_s}) \boxtimes L(j i^{s+1}) \boxtimes \cdots \boxtimes L(ji^{b_1}) \twoheadrightarrow Q,$$ where $g' = h - \sum_{t=1}^{s-1} (a-b_t)$ and $s$ is such that $$ \sum_{t=1}^{s-1} (a-b_t) \le h < \sum_{t=1}^s (a-b_t). $$ \end{enumerate} \end{enumerate} \end{Lem} \begin{proof} The assertions can be proved in the same manner as in \cite[Lemma 6.14, Lemma 6.15]{LV09}. \end{proof}
\begin{Prop} \label{Prop:phi-ep=wt for R(ci+dj)} Let $i \in I^{\rm re} $ and $j \in I$ with $i\ne j$. Let $M$ be an irreducible $R(c\alpha_i + d\alpha_j)$-module, and $\lambda \in P^+$ such that $ {\rm pr} ^\lambda(M) \ne 0$ and $ {\rm pr} ^{\lambda}( \kf_j M) \ne 0$. Then we have $$ \ep_i^\lambda (\kf_jM) = \ep_i^\lambda(M) + a_{ij} + k, \ \ \ph_i^{\lambda}(\kf_j M) = \ph_i^\lambda(M)+k $$ for some $ 0 \le k \le -a_{ij}$. \end{Prop} \begin{proof} Using the argument in \cite[Theorem 6.19]{LV09} with Lemma \ref{Lem:difference of phi}, Lemma \ref{Lem:structure lem for R(mi+j)}, Lemma \ref{Lem:surjection for R(ci+dj)} and Lemma \ref{Lem:surjection2 for R(ci+dj)}, our assertion follows. \end{proof}
\begin{Prop} \label{Prop:ph - ep = wt} Let $i \in I^{\rm re} $, and $M$ be an irreducible $R(\alpha)$-module with $ {\rm pr} ^\lambda(M) \ne 0 $. \begin{enumerate} \item For $j\in I$ with $i \ne j$, we have $$ \ph_i^\lambda(\kf_j M) - \ep_i^\lambda(\kf_j M) = -\langle h_i, \alpha_j \rangle + \ph_i^\lambda(M) - \ep_i^\lambda(M). $$ \item Moreover, we have $$ \ph_i^\lambda(M) = \ep_i^\lambda(M) + \langle h_i, {\rm wt}^\lambda(M)\rangle. $$ \end{enumerate} \end{Prop} \begin{proof} Combining \cite[Proposition 6.20]{LV09} with Proposition \ref{Prop:phi-ep=wt for R(ci+dj)}, we obtain the assertion (1). Since $\ph_i^\lambda(\mathbf{1}) = \ep_i^\lambda(\mathbf{1}) +
\langle h_i, \lambda \rangle$, the assertion (2) follows by induction on $|\alpha|$ combined with the assertion (1). \end{proof}
Combining Proposition \ref{Prop:ph - ep = wt} with $\eqref{Eq:def of B(lambda)}$, we obtain the following proposition. \begin{Prop} \label{Prop:B(lambda) for KLR is crystal} The sextuple $(\Bklr{\lambda},{\rm wt}^\lambda, \ke_i^\lambda, \kf_i^\lambda, \varepsilon_i^\lambda, \varphi_i^\lambda)$ is an abstract crystal. \end{Prop}
We would like to show that $\Bklr{\lambda}$ is isomorphic to the crystal $B(\lambda)$.
For this purpose, we first prove the following lemma.
\begin{Lem} \label{Lem: Blambda} Let $i \in I^{\rm im} $ and $M$ be an irreducible $R^\lambda(\alpha)$-module. Then $$ \langle h_i, {\rm wt}^\lambda(M)\rangle \le 0 \ \text{ if and only if }\ \kf^\lambda_i M=0. $$ \end{Lem} \begin{proof}
Let $\alpha = \sum_{j \in I} k_j \alpha_j $ with $|\alpha|=d$. For simplicity, we identify $M$ with $ {\rm infl} ^\lambda M$.
We first assume that $\langle h_i, {\rm wt}^\lambda(M)\rangle \le 0$. Since $\langle h_i, \lambda \rangle \ge 0$ and $\langle h_i, -\alpha_j\rangle \ge 0$ for all $j \in I$, we have $$ \langle h_i, \lambda \rangle = 0 \ \ \text{ and } \ \ k_j=0 \text{ for } j\in I \text{ with } a_{ij} \ne 0. $$ Take an element $\mathbf{j} = (j_1 \ldots j_d) \in {\rm Seq} (\alpha)$ such that $ 1_\mathbf{j} M \ne 0$. Note that $a_{i j_k}=0$ for all $k=1,\ldots, d$. By the Frobenius reciprocity $\eqref{Eq:reciprocity2}$, we have an embedding $$ L(i) \boxtimes M \hookrightarrow \Delta_i \kf_i M, $$ which implies that $1_{(i)*\mathbf{j}} \ (\kf_i M) \ne 0$. Since $a_{i j_1} =0 $, it follows from the quantum Serre relations that $$1_{(j_1 i j_2 \ldots j_d) } \ (\kf_i M) \ne 0. $$ Repeating this process, we have $$ 1_{ \mathbf{j}*(i) } \ (\kf_i M) \ne 0, $$ which yields that $I^\lambda(\alpha + \alpha_i) \kf_i M \ne 0$ since $1_{ \mathbf{j}*(i) } \in I^\lambda(\alpha + \alpha_i)$. Therefore, we have the only if part of our assertion.
We now prove the converse. We will actually prove the contrapositive: $$\langle h_i, {\rm wt}^\lambda(M)\rangle > 0 \ \ \Longrightarrow \ \ \kf^\lambda_i M \ne 0 .$$ Assume that $\langle h_i, {\rm wt}^\lambda(M)\rangle > 0$.
First consider the case $\langle h_i, -\alpha \rangle = 0$. In this case, $\langle h_i, \lambda \rangle > 0$ and $k_j=0$ for $ j\in I$ with $ a_{ij} \ne 0$. Take a nonzero element $v \in L(i)$. By definition, we have $$ {\rm Ind} L(i) \boxtimes M = {\rm Span}_\F\{ \tau_t \cdots \tau_1 \otimes (v \otimes m ) \mid m \in M,\ 0 \le t \le d \}. $$ Since $I^\lambda(\alpha) M = 0 $ and $k_j=0$ for $ j\in I$ with $ a_{ij} \ne 0$, it follows from the definition $\eqref{Eq:def of cyclotomic ideal}$ that $$ I^\lambda(\alpha+\alpha_i) ({\rm Ind} L(i) \boxtimes M) = 0. $$ Hence we have $\kf^\lambda_i M \ne 0$.
Now we suppose that $\langle h_i, -\alpha \rangle > 0$. Take a nonzero element $v$ in $L(i)$, and define $N$ to be the submodule of ${\rm Ind} L(i) \boxtimes M$ generated by $$ \mathsf{N} = \{ x_{d+1}^{\langle h_i, \lambda \rangle} \tau_d \cdots \tau_1 1_{(i)*\mathbf{k}} \otimes (v \otimes m) \mid 0 \le t \le d,\ m \in M,\ \mathbf{k} \in {\rm Seq} (\alpha) \}.$$ As $\langle h_i, -\alpha \rangle > 0$, we have $$\deg( x_{d+1}^{\langle h_i, \lambda \rangle} \tau_d \cdots \tau_1 1_{(i)*\mathbf{k}} \otimes (v \otimes m)) > \deg(1 \otimes v \otimes m) .$$ Then, as $M$ is $R^\lambda(\alpha)$-module, we have $I^\lambda(\alpha+\alpha_i) ({\rm Ind} L(i) \boxtimes M)\subset N $. Hence $({\rm Ind} L(i) \boxtimes M )/ N$ is $R^\lambda(\alpha+\alpha^i)$-module. To prove $\kf^\lambda_i M \ne 0$, it suffices to show that $({\rm Ind} L(i) \boxtimes M )/ N$ is nontrivial; i.e., $N$ is proper.
Take $m_0 \in M$ such that $\deg(m_0) \le \deg(m)$ for all $m \in M$. We claim that $ 1 \otimes (v\otimes m_0) \notin N$. Suppose that $ 1 \otimes (v\otimes m_0) \in N$. Since $I^\lambda(\alpha)M=0$, it follows from the defining relations $\eqref{Eq:def rel 1}$ and $\eqref{Eq:def rel 2}$ that \begin{align*} x_r ( x_{d+1}^{\langle h_i, \lambda \rangle} \tau_d \cdots \tau_1 1_{(i)*\mathbf{k}} \otimes (v\otimes m)) &= x_{d+1}^{\langle h_i, \lambda \rangle} \tau_d \cdots \tau_1 ( x_{r+1} 1_{(i)*\mathbf{k}} \otimes (v\otimes m) ), \\ \tau_s ( x_{d+1}^{\langle h_i, \lambda \rangle} \tau_d \cdots \tau_1 1_{(i)*\mathbf{k}} \otimes (v\otimes m)) &= x_{d+1}^{\langle h_i, \lambda \rangle} \tau_d \cdots \tau_1 ( \tau_{s+1} 1_{(i)*\mathbf{k}} \otimes (v\otimes m)) \end{align*} for $m \in M$, $1 \le r \le d$ and $ 1 \le s \le d-1$. So, the element $1 \otimes (v\otimes m_0)$ can be written as $$ 1 \otimes (v\otimes m_0) = \sum_j \tau_{t_j} \tau_{t_j+1} \cdots \tau_d x_{d+1}^k n_j, $$ for some $n_j\in \mathsf{N}$, $ t_j, k \in \mathbb{Z}_{\ge0} $. Since $\langle h_i, -\alpha \rangle > 0$ and $m_0$ is minimal, we have $$ \deg (1 \otimes (v\otimes m_0)) < \deg(n_j) \le \deg(\tau_{t_j} \tau_{t_j+1} \cdots \tau_d x_{d+1}^k n_j),$$ which gives a contradiction. Therefore, $1 \otimes (v\otimes m_0)$ is not contained in $N$ and $N$ is proper. \end{proof}
We are now ready to state and prove the crystal version of categorification of $V(\lambda)$. Define a map $\Psi_\lambda: \ \Bklr{\lambda} \ \longrightarrow \ \Bklr{\infty} \otimes T_{\lambda} \otimes C$ by $$[M] \longmapsto [ {\rm infl} ^\lambda M] \otimes t_\lambda \otimes c.$$
\begin{Thm}\ \label{Thm: B(lambda)} \begin{enumerate} \item $\Psi_\lambda$ is a strict crystal embedding. \item The crystal $\mathfrak{B}(\lambda)$ is isomorphic to the crystal $B(\lambda)$. \end{enumerate} \end{Thm} \begin{proof} To prove (1), let $M $ be an irreducible $R^\lambda(\alpha)$-module and let $M_0 = {\rm infl} ^\lambda M$. Note that \begin{align*} \ep^{\lambda}_i(M) =\ep_i(M_0), \quad \varphi_i(M_0)+ \langle h_i,\lambda \rangle =\ep_i(M_0) + \langle h_i,\lambda -\alpha \rangle
= \varphi^{\lambda}_i(M) \ge 0. \end{align*} By the tensor product rule $\eqref{Eq:def of tensor product}$ and Proposition \ref{Prop:B(lambda) for KLR is crystal}, we have \begin{align*} {\rm wt}(\Psi_\lambda(M)) &= {\rm wt}( M_0 \otimes t_\lambda \otimes c ) = \lambda - \alpha = {\rm wt}^\lambda(M),\\ \ep_i(\Psi_\lambda(M)) & =\ep_i( M_0 \otimes t_{\lambda} \otimes c) =\max \{ \ep_i(M_0), -\langle h_i, \lambda-\alpha \rangle \} = \ep^{\lambda}_i(M),\\ \varphi_i(\Psi_\lambda(M)) &=\ph_i( M_0 \otimes t_{\lambda} \otimes c) =\max \{ \varphi_i(M_0)+ \langle h_i, \lambda \rangle, 0 \} = \varphi_i^\lambda(M). \end{align*}
On the other hand, it follows from Lemma \ref{Lem: Blambda} that \begin{align} \label{Eq:eq1 in B(lambda) for GKM} \langle h_i, \lambda - \alpha+\alpha_i \rangle \le 0\ \Longrightarrow \ \ke_i^\lambda M = 0. \end{align} By a direct computation, we have
\begin{equation*}
\begin{aligned}
\ \ &\tilde{f_i}(M_0 \otimes t_\lambda \otimes c) = \begin{cases} (\tilde{f_i} M_0) \otimes t_{\lambda} \otimes c & \text{ if } \varphi^{\lambda}_i(M) > 0, \\
\quad \quad \quad 0 & \text{ if } \varphi^{\lambda}_i(M) \le 0, \end{cases} \\
\ \ &\tilde{e_i}(M_0 \otimes t_\lambda \otimes c) =
\begin{cases} (\tilde{e_i} M_0) \otimes t_{\lambda} \otimes c & \text{ if } i \in I^{\rm re} , \ \varphi^{\lambda}_i(M) \ge 0, \\
(\tilde{e_i} M_0) \otimes t_{\lambda} \otimes c & \text{ if } i \in I^{\rm im} , \langle h_i, \lambda-\alpha+\alpha_i \rangle >0, \\
\quad \quad \quad 0 & \text{ if } i \in I^{\rm im} , \langle h_i, \lambda-\alpha+\alpha_i \rangle \le 0. \end{cases}
\end{aligned}
\end{equation*} By $\eqref{Eq:eq1 in B(lambda) for GKM}$ and Lemma \ref{Lem: Blambda}, we get \begin{align*} \ke_i(\Psi_\lambda(M)) = \Psi_\lambda(\ke_i^\lambda(M))\ \text{ and }\ \kf_i(\Psi_\lambda(M)) = \Psi_\lambda(\kf_i^\lambda(M)), \end{align*} which completes the proof of (1).
Since $\Psi_\lambda$ takes $\mathbf{1}$ to $\mathbf{1} \otimes t_\lambda \otimes c$, the assertion (2) follows from (1) and Proposition \ref{Prop: recognition theorem of B-lambda}. \end{proof}
\vskip 3em
\end{document} | arXiv |
\begin{document}
\title[Stability of $C^*$-algebras associated to graphs]{Stability of $\boldsymbol{C^*}$-algebras associated to graphs}
\author{Mark Tomforde
}
\address{Department of Mathematics\\ Dartmouth College\\ Hanover\\ NH 03755-3551\\ USA}
\curraddr{Department of Mathematics\\ University of Iowa\\ Iowa City\\ IA 52242\\ USA}
\email{[email protected]}
\date{\today} \subjclass{46L55}
\begin{abstract} We characterize stability of graph $C^*$-algebras by giving five conditions equivalent to their stability. We also show that if $G$ is a graph with no sources, then $C^*(G)$ is stable if and only if each vertex in $G$ can be reached by an infinite number of vertices. We use this characterization to realize the stabilization of a graph $C^*$-algebra. Specifically, if $G$ is a graph and $\tilde{G}$ is the graph formed by adding a head to each vertex of $G$, then $C^*(\tilde{G})$ is the stabilization of $C^*(G)$; that is, $C^*(\tilde{G}) \cong C^*(G) \otimes \mathcal{K}$. \end{abstract}
\maketitle
\section{Introduction}
In 1980 Cuntz and Krieger introduced a class of $C^*$-algebras generated by families of partial isometries satisfying relations determined by a finite matrix with entries in $\{0,1\}$. These Cuntz-Krieger algebras were initially studied because of their appearance in the study of topological Markov chains. Later it was found that they also have important parallels with certain kinds of dynamical systems (e.g.~shifts of finite type).
Since their inception, Cuntz-Krieger algebras have been generalized in an extraordinary number of ways. One generalization whose study has proven particularly fruitful are the $C^*$-algebras associated to directed graphs. In 1982 Watatani noted that one could view the Cuntz-Krieger algebra associated to a finite matrix $A$ as the $C^*$-algebra associated to the finite directed graph with adjacency matrix $A$ \cite{Wat}. However, these ideas were not more fully explored until the late 1990's when Kumjian, Pask, Raeburn, and Renault \cite{KPRR} introduced $C^*$-algebras associated to locally finite graphs (i.e.~possibly infinite graphs in which each vertex emits and receives a finite number of edges). Not long after this it was shown in \cite{BPRS} that many of the same results also hold for $C^*$-algebras associated to row-finite graphs (i.e.~possibly infinite graphs in which every vertex emits finitely many edges) and often the same techniques can be applied to prove these results. In the early 2000's $C^*$-algebras associated to arbitrary directed graphs were finally considered \cite{FLR}. Unlike the generalization from locally finite to row-finite graphs, it was found that extending results to $C^*$-algebras of arbitrary graphs often involved significant modifications to statements of theorems as well as the development of new techniques for their proofs.
In this paper we consider the notion of stability for $C^*$-algebras associated to arbitrary directed graphs. Recall that a $C^*$-algebra $A$ is said to be stable if $A \cong A \otimes \mathcal{K}$, where $\mathcal{K}$ denotes the compact operators on a separable infinite-dimensional Hilbert space. Furthermore, if $A$ is a $C^*$-algebra then one may form its stabilization $A \otimes \mathcal{K}$. Since $\mathcal{K} \otimes \mathcal{K} \cong \mathcal{K}$, one has that the stabilization of a $C^*$-algebra is stable.
If $G$ is a graph and $C^*(G)$ is its associated $C^*$-algebra, then we prove in Theorem~\ref{stable-thm} that the stability of $C^*(G)$ is equivalent to five other conditions. This theorem generalizes a result of Hjelmborg \cite[Theorem~2.14]{Hje2} in which a characterization of stability for $C^*$-algebras of locally finite graphs was obtained. Both of these results make use of a nontrivial characterization of stability due to R\o rdam and Hjelmborg \cite{HR}. However, our proof of Theorem~\ref{stable-thm} will involve techniques significantly different from Hjelmborg's proof of \cite[Theorem~2.14]{Hje2}. Furthermore, in addition to applying to $C^*$-algebras of arbitrary graphs, Theorem~\ref{stable-thm} is different from \cite[Theorem~2.14]{Hje2} in another respect, namely that it includes a characterization in terms of the graph traces on all of $G$, rather than on a special subgraph of $G$ as in Condition~(d) of \cite[Theorem~2.14]{Hje2}.
In Corollary~\ref{left-inf-stable} we show that there is a particularly nice characterization of stability for $C^*(G)$ when $G$ has no sources: If $G$ is a graph with no sources, then $C^*(G)$ is stable if and only if every vertex of $G$ can be reached by an infinite number of vertices. This gives an easily verifiable condition for determining the stability of the $C^*$-algebra solely in terms of the graph.
Building off this characterization, in \S\ref{stabilization-sec} we develop a method for realizing the stabilization of a graph algebra and we show that it is also a graph algebra. If $G$ is a graph, then we obtain a new graph $\tilde{G}$ by adding a ``head" $$ \xymatrix{ \cdots \ar[r] & \bullet \ar[r] & \bullet \ar[r] & \bullet \ar[r] & v\\ } $$ to each vertex $v$ in $G$. We prove in Theorem~\ref{stabilization-gr-alg-thm} that $C^*(\tilde{G})$ is the stabilization of $C^*(G)$; that is, $C^*(\tilde{G}) \cong C^*(G) \otimes \mathcal{K}$. As a corollary we have that the class of graph algebras is closed under stabilization.
\section{Preliminaries}
We provide some basic facts about graph algebras and refer the reader to \cite{KPR}, \cite{BPRS}, and \cite{BHRS} for more details. A (directed) graph $G=(G^0, G^1, r, s)$ consists of a countable set $G^0$ of vertices, a countable set $G^1$ of edges, and maps $r,s:G^1 \rightarrow G^0$ identifying the range and source of each edge. A vertex $v \in G^0$ is called a
\emph{sink} if $|s^{-1}(v)|=0$, and $v$ is called an
\emph{infinite emitter} if $|s^{-1}(v)|=\infty$. If $v$ is either a sink or an infinite emitter, then we call $v$ a \emph{singular vertex}. A graph $G$ is said to be \emph{row-finite} if it has no infinite emitters.
If $G$ is a graph we define a \emph{Cuntz-Krieger $G$-family} to be a set of mutually orthogonal projections $\{p_v : v \in G^0\}$ and a set of partial isometries $\{s_e : e \in G^1\}$ with orthogonal ranges which satisfy the \emph{Cuntz-Krieger relations}: \begin{enumerate} \item $s_e^* s_e = p_{r(e)}$ for every $e \in G^1$; \item $s_e s_e^* \leq p_{s(e)}$ for every $e \in G^1$; \item $p_v = \sum_{s(e)=v} s_e s_e^*$ for every $v \in G^0$ that is not a singular vertex. \end{enumerate} The \emph{graph algebra $C^*(G)$} is defined to be the $C^*$-algebra generated by a universal Cuntz-Krieger $G$-family.
A \emph{path} in $G$ is a sequence of edges $\alpha = \alpha_1 \alpha_2 \ldots \alpha_n$ with $r(\alpha_i) =
s(\alpha_{i+1})$ for $1 \leq i < n$, and we say that $\alpha$ has length $|\alpha| = n$. We let $G^n$ denote the set of all paths of length $n$, and we let $G^* := \bigcup_{n=0}^\infty G^n$ denote the set of finite paths in $G$. Note that vertices are considered paths of length zero. The maps $r,s$ extend to $G^*$, and for $v,w \in G^0$ we write $v \geq w$ if there exists a path $\alpha \in G^*$ with $s(\alpha)=v$ and $r(\alpha) = w$. Also for a path $\alpha := \alpha_1 \ldots \alpha_n$ we define $s_\alpha := s_{\alpha_1} \ldots s_{\alpha_n}$. It is a consequence of the Cuntz-Krieger relations that $C^*(G) = \overline{\textrm{span}} \{ s_\alpha s_\beta^* : \alpha, \beta \in G^* \text{ and } r(\alpha) = r(\beta)\}$.
We say that a path $\alpha := \alpha_1 \ldots \alpha_n$ of length $1$ or greater is a \emph{loop} if $r(\alpha)=s(\alpha)$, and we call the vertex $s(\alpha)=r(\alpha)$ the \emph{base point} of the loop. A loop is said to be \emph{simple} if $s(\alpha_i) \neq s(\alpha_1)$ for all $1 < i \leq n$. The following is an important condition for graphs to satisfy.
$\text{ }$
\noindent \textbf{Condition~(K)}: No vertex in $G$ is the base point of exactly one simple loop; that is, every vertex is either the base point of no loops or at least two simple loops.
$\text{ }$
The graph algebra $C^*(G)$ is unital if and only if $G$ has a finite number of vertices, cf.~\cite[Proposition~1.4]{KPR}, and in this case $1_{C^*(G)} = \sum_{v \in G^0} p_v$. If $G$ has an infinite number of vertices and we list them as $G^0 = \{v_1, v_2, \ldots \}$ and define $p_n := \sum_{i=1}^n p_{v_i}$, then $\{ p_n \}_{n=1}^\infty$ will be an approximate unit for $C^*(G)$.
\begin{definition} A \emph{trace} on a $C^*$-algebra $A$ is a linear functional $\tau : A \rightarrow \mathbb{C}$ with the property that $\tau(ab)=\tau(ba)$ for all $a,b \in A$. We say that $\tau$ is \emph{positive} if $\tau (a) \geq 0$ for all $a
\in A^+$. If $\tau$ is a positive trace and $\| \tau \| = 1$ we call $\tau$ a \emph{tracial state}. The set of all tracial states is denoted $T(A)$. \end{definition}
\begin{definition} If $G$ is a graph, then a \emph{graph trace} on $G$ is a function $g : G^0 \rightarrow \mathbb{R}^+$ with the following two properties: \begin{enumerate} \item \label{g-t-1} For any $v \in G^0$ with $0 <
|s^{-1}(v)| < \infty$ we have $g(v) = \sum_{s(e) = v} g(r(e))$. \item \label{g-t-2} For any infinite emitter $v \in G^0$ and any finite set of edges $e_1, \ldots, e_n \in s^{-1}(v)$ we have $g(v) \geq \sum_{i=1}^n g(r(e_i))$. \end{enumerate} \end{definition}
Because the value of $g$ at any vertex is non-negative, it follows that whenever $v$ is an infinite emitter the infinite sum $\sum_{s(e) = v} g(r(e))$ converges, and moreover $\sum_{s(e) = v} g(r(e)) \leq g(v)$.
We define the \emph{norm} of $g$ to be the (possibly infinite) value $\|g
\| := \sum_{v \in G^0} g(v)$. We shall call a graph trace
\emph{bounded} if $\| g \| < \infty$, and we shall use $T(G)$ to denote the set of all graph traces on $G$ with norm one. Also note that if $v, w \in G^0$, then $v \geq w$ implies $g(v) \geq g(w)$.
If $\tau : C^*(G) \to \mathbb{C}$ is a tracial state, then $\tau$ induces a graph trace $g_\tau$ of norm one given by $g_\tau(v) := \tau(p_v)$. If $G$ satisfies Condition~(K), then the map $\tau \mapsto g_\tau$ is a bijection (in fact, an affine homeomorphism) from $T(C^*(G))$ onto $T(G)$ \cite[\S3]{Tom6}. There are examples which show that in general this map is not injective.
\begin{definition} We say that two projections $p,q \in A$ are equivalent, written $p \sim q$, if there exists an element $v \in A$ with $p=vv^*$ and $q=v^*v$. \end{definition}
In \cite{Cun4} Joachim Cuntz introduced a notion of comparison of (positive) elements in a $C^*$-algebra for the purpose of constructing dimension functions and traces on $C^*$-algebras.
\begin{definition}[Cuntz] Let $A$ be a $C^*$-algebra, and let $a, b$ be positive elements in $A$. We write $a \lesssim b$ if there exists a sequence $\{x_k\}_{k=1}^\infty$ in $A$ with $x_k^* b x_k \to a$. \end{definition}
If $p, q$ are projections in a $C^*$-algebra $A$, then $p \lesssim q$ if and only if $p$ is equivalent to a subprojection of $q$; that is, there exists a partial isometry $v \in A$ such that $p=vv^*$ and $v^*v \leq q$. Thus the above definition agrees with usual definition of comparison of two projections.
If $e \in G^1$ then we see that $p_{r(e)} = s_es_e^*$ and $s_es_e^* \leq p_{s(e)}$. Therefore $p_{r(e)} \lesssim p_{s(e)}$. More generally we see that $v \geq w$ implies $p_w \lesssim p_v$.
\begin{definition} If $G$ is a graph, a subset $H \subseteq G^0$ is said to be \emph{hereditary} if for every $e \in G^1$ we have that $s(e) \in H$ implies $r(e) \in H$. A hereditary subset is said to be \emph{saturated} if whenever $v \in G^0$ with
$0 < | s^{-1}(v) | < \infty$ then $\{ r(e) : e \in G^1 \text{ and } s(e) = v \} \subseteq H$ implies that $v \in H$. If $H$ is a hereditary subset, then the \emph{saturation} of $H$ is the smallest saturated hereditary subset $\overline{H}$ of $G^0$ containing $H$. \end{definition}
If $H$ is a hereditary subset of $G^0$, then we can give an inductive description of the saturation $\overline{H}$. We define $H_0 := H$ and having defined $H_n$ we set $$H_{n+1} := H_n \cup \{ v \in G^0 : 0 < |
s^{-1}(v) | < \infty \text{ and } s(e) = v \text{ implies } r(e) \in H_n \}.$$ Then it is straightforward to show that $\overline{H} = \bigcup_{n=0}^\infty H_n$.
\begin{definition} Given a saturated hereditary subset $H \subseteq G^0$, we define
$$ B_H := \{v \in G^0 : |s^{-1}(v)| = \infty \text{ and } 0 <
|s^{-1}(v) \cap r^{-1}(G^0 \setminus H)| < \infty \}.$$ Since $H$ is hereditary, we see that $B_H$ is disjoint from $H$. If $\{s_e, p_v\}$ is a generating Cuntz-Krieger $G$-family in $C^*(G)$, then for $S \subseteq B_H$ we define $$I_{(H,S)} := \text{ the ideal in $C^*(G)$ generated by $\{p_v : v \in H \} \cup \{p_{v}^H : v \in S \}$},$$ where $$ p_{v}^H := p_{v} - \sum_{{s(e) = v} \atop {r(e) \notin H}} s_e s_e^*.$$ \end{definition}
\begin{definition} If $H$ is a saturated hereditary subset of $G$ and $S \subseteq B_H$, then we define a graph $G_{(H,S)}$ as follows: \begin{align*} G_{(H,S)}^0 &:= (G^0 \backslash H) \cup \{ v' : v \in B_H \backslash S \} \\ G_{(H,S)}^1 &:= \{e \in G^1 : r(e) \notin H \} \cup \{ e' : r(e) \in B_H \backslash S \} \end{align*} and we extend $r$ and $s$ to $G_{(H,S)}^1$ by $r(e') = r(e)'$ and $s(e') = s(e)$. It follows from \cite[Corollary~3.5]{BHRS} that $C^*(G) / I_{(H,S)} \cong C^*(G_{(H,S)})$. \end{definition}
\section{Stability of graph $C^*$-algebras} \label{stability-sec}
This section is devoted to proving Theorem~\ref{stable-thm}, which is a generalization of \cite[Theorem~2.14]{Hje2}.
\begin{definition} If $v$ is a vertex in a graph $G$ we define $L(v) := \{ w \in G^0 : w \geq v \}$. We say that $v$ is \emph{left infinite} if $L(v)$ contains infinitely many elements, and we say that $v$ is \emph{left finite} if $L(v)$ contains finitely many elements. \end{definition}
\begin{theorem} \label{stable-thm} If $G$ is a graph, then the following are equivalent. \begin{enumerate} \item[(a)] $C^*(G)$ is stable \item[(b)] $C^*(G)$ has no nonzero unital quotients and no tracial states \item[(c)] Every vertex in $G$ that is on a loop is left infinite and $T(G) = \emptyset$ \item[(d)] Every vertex in $G$ that is on a loop is left infinite and $G$ has no nonzero bounded graph traces \item[(e)] For every $v \in G^0$ and every finite set $F \subseteq G^0$ there exists a finite set $W \subseteq G^0$ with $W \cap F = \emptyset$ and $p_v \lesssim \sum_{w \in W} p_w$. \item[(f)] For every finite set $V \subseteq G^0$ there exists a finite set $W \subseteq G^0$ with $V \cap W = \emptyset$ and $\sum_{v \in V} p_v \lesssim \sum_{w \in W} p_w$. \end{enumerate} \end{theorem}
\begin{corollary} \label{left-inf-stable} If $G$ is a graph and every vertex of $G$ is left infinite, then $C^*(G)$ is stable. If $G$ has no sources and $C^*(G)$ is stable, then every vertex of $G$ is left infinite. \end{corollary} \begin{proof} Suppose every vertex of $G$ is left infinite. If $v \in G^0$ and $F \subseteq G^0$ is a finite set, then we may choose an element $w \in G^0$ such that $w \notin F$ and $w \geq v$. But then $p_v \lesssim p_w$ and by Theorem~\ref{stable-thm}(e) $C^*(G)$ is stable.
If $G$ has no sources then for every $v \in G^0$ there exists a sequence of edges $e_1e_2e_3 \ldots$ with $r(e_{i+1}) = s(e_i)$ and $r(e_1)=v$. If the elements of $\{s(e_i)\}_{i=1}^\infty$ are distinct, then $v$ is left infinite. If the elements of $\{s(e_i)\}_{i=1}^\infty$ are not distinct, then there exists a loop that can reach $v$. If $C^*(G)$ is stable, then by Theorem~\ref{stable-thm}(c) all vertices on loops are left infinite. Hence $v$ is also left infinite. \end{proof}
We cannot remove the condition of no sources in the converse of the above corollary. If $G$ is the graph $$ \xymatrix{ \bullet \ar[r] & \bullet \ar[r] & \bullet \ar[r] & \bullet \ar[r] & \ldots \\ } $$ then no vertex of $G$ is left infinite, but $C^*(G) \cong \mathcal{K}$ is stable.
\begin{remark} The equivalence of Conditions (a), (b), and (f) in Theorem~\ref{stable-thm} was established for locally finite graphs in \cite[Theorem~2.14]{Hje2}. We mention that Condition~(c) of Theorem~\ref{stable-thm} is often easier to verify than Condition~(b). This is because graph traces are typically easier to deal with than tracial states, and it is often easy to deduce whether $T(G)$ is empty simply by looking at $G$. Furthermore, we point out that the tracial states of $C^*(G)$ and the graph traces on $G$ of norm one are not generally in one-to-one correspondence (see \cite[\S3]{Tom6}). \end{remark}
\begin{remark} We see from Theorem~\ref{stable-thm} that a graph $C^*$-algebra is stable if and only if it has no nonzero unital quotients and no tracial states. It is always the case that any stable $C^*$-algebra will have no nonzero unital quotients and no tracial states, but in general the converse does not hold. (Interestingly, it is shown in \cite[Proposition~5.1]{HR} that the converse will hold if certain full hereditary subalgebras of the $C^*$-algebra satisfy a particular property.) \end{remark}
\begin{lemma} \label{stab-equiv} Let $A$ be a $C^*$-algebra with an increasing countable approximate unit $\{ p_n \}_{n=1}^\infty$ consisting of projections. Then the following are equivalent. \begin{enumerate} \item[(i)] $A$ is stable. \item[(ii)] For every projection $p \in A$ there exists a projection $q \in A$ such that $p \sim q$ and $p \perp q$. \item[(iii)] For all $n \in \mathbb{N}$ there exists $m > n$ such that $p_n \lesssim p_m-p_n$ \end{enumerate} \end{lemma}
\begin{proof} The equivalence of \textrm{(i)} and \textrm{(ii)} is shown in \cite[Theorem~3.3]{HR}. The equivalence of \textrm{(ii)} and \textrm{(iii)} is shown in \cite[Lemma~2.1]{Hje2}. \end{proof}
\begin{lemma} \label{zero-sat} If $G$ is a graph and $g:G^0 \to \mathbb{R}^+$ is a graph trace on $G$, then $$H := \{ v \in G^0 : g(v) = 0 \}$$ is a saturated hereditary subset. \end{lemma}
\begin{proof} If $e \in G^1$, then $g(s(e)) \geq g(r(e))$. Thus $s(e) \in H$ implies that $r(e) \in H$, and $H$ is hereditary. If $v \in G^0$ is not a singular vertex and $\{ r(e) : e \in G^1 \text{ and } s(e) = v \} \subseteq H$, then $g(v) = \sum_{s(e)=v} g(r(e)) = 0$ so $v \in H$, and $H$ is saturated. \end{proof}
\begin{lemma} \label{quotient-lift} Let $G$ be a graph, let $H$ be a saturated hereditary subset of $G^0$, and let $\pi : C^*(G) \to C^*(G) / I_{(H,\emptyset)}$ be the projection map. If $p$ is a projection in $C^*(G)$, $W \subseteq G^0 \backslash H$ is a finite set, and $\pi(p) \lesssim \sum_{w \in W} \pi (p_w)$ in $C^*(G) / I_{(H,\emptyset)}$, then there exists a finite set $X \subseteq H$ such that $p \lesssim \sum_{w \in W} p_w + \sum_{x \in X} p_x$ in $C^*(G)$. \end{lemma}
\begin{proof} Write $H = \{ v_1, v_2, \ldots \}$. If we let $p_n := \sum_{i=1}^n p_{v_i}$, then $I_{(H,\emptyset)}$ is generated by $\mathcal{P} = \{p_n\}_{n=1}^\infty$ and \cite[Lemma~2.6]{Hje2} implies that $p \lesssim \sum_{w \in W} p_w + p_n$ for some $n$. \end{proof}
\noindent \emph{Proof of Theorem~\ref{stable-thm}.} (a) $\Longrightarrow$ (b) : It is shown in \cite[Proposition~5.1]{HR} that stable $C^*$-algebras have no nonzero unital quotients and admit no nonzero traces. \\
\noindent (b) $\Longrightarrow$ (c) : We shall first show that every vertex on a loop is left infinite. Let $\alpha$ be a loop in $G$ that is based at $v$. Then $H := G^0 \backslash L(s(\alpha))$ is a saturated hereditary subset. By hypothesis $C^*(G) / I_{(H,B_{H})} \cong G_{(H,B_{H})}$ is nonunital and hence $G_{(H,B_{H})}^0 = G^0 \backslash H = L(s(\alpha))$ is infinite. Thus $s(\alpha)=v$ is left infinite.
Now we shall show that $T(G)$ is empty by supposing that there exists $g \in T(G)$ and arriving at a contradiction. Let us begin by showing that if
$v$ is a vertex on a loop, then $g(v) = 0$. From the previous paragraph every vertex on a loop is left infinite. Since $$\|g\| = \sum_{w \in G^0} g(w) \geq \sum_{w \in L(v)}g(w)$$ and since $w \geq v$ implies $g(w) \geq g(v)$ the only way that this infinite sum can be finite is if $g(v)=0$. Thus $g$ vanishes on every vertex that is on a loop.
If we now let $H := \{ v \in G^0 : g(v) = 0 \}$ then it follows from Lemma~\ref{zero-sat} that $H$ is a saturated hereditary subset. We define a graph trace $\tilde{g}$ on $G_{(H,\emptyset)}$ by $$\tilde{g} (w) := \begin{cases} g(w) & \text{ if $w \in (G^0 \backslash H) \backslash B_{H}$} \\ & \\ \displaystyle \sum_{ { s(e)=w } \atop {r(e) \notin H} } g(r(e)) & \text{ if $w \in B_{H}$} \\ g(v) - \displaystyle \sum_{ { s(e)=v} \atop {r(e) \notin H} } g(r(e)) & \text{ if $w = v'$ for some $v \in B_{H}$.}
\end{cases}$$ It is straightforward to verify that $\tilde{g}$ is a graph trace on $G_{(H,\emptyset)}$ and that $\| \tilde{g} \| = \| g \| = 1$. Now it follows from the previous paragraph that there are no loops in $G$ with vertices in $G^0 \backslash H$. Hence $G_{(H,\emptyset)}$ is a graph with no loops. Therefore \cite[\S3.3]{Tom6} implies that there exists a tracial state $\tau$ on $C^*(G_{(H,\emptyset)})$. Since $C^*(G) / I_{(H,\emptyset)} \cong C^*(G_{(H,\emptyset)})$ it follows that $\tau$ lifts to a tracial state on $C^*(G)$. But this contradicts the fact that $C^*(G)$ has no tracial states. \\
\noindent (c) $\Longrightarrow$ (d) : If $g$ was a nonzero bounded graph trace on $G$, then we could normalize to get an element $\frac{1}{\|g\|} \cdot g \in T(G)$. \\
\noindent (d) $\Longrightarrow$ (e) : Choose a vertex $v \in G^0$. Define $H := \{ w \in G^0 : \text{ $w$ is left infinite} \}$. Then $H$ is a hereditary subset, and we let $\overline{H}$ denote the saturation of $H$. Consider the following two cases:
\noindent \textsc{Case I:} $v \in \overline{H}$. Define $H_0 := H$ and for each $n \in \mathbb{N}$ set $$H_{n+1} := H_n \cup \{ w \in G^0 : 0 < | s^{-1}(w) | < \infty \text{ and } s(e) = w \text{ implies } r(e) \in H_n \}.$$ Then we see that $\overline{H} = \bigcup_{n=0}^\infty H_n$. We shall prove that the claim holds whenever $v \in \overline{H}$ by induction on $k := \min \{ n \in \mathbb{N} : v \in H_n \}$. In the base case we have $k=0$ and thus $v \in H$. Since every vertex in $H$ is left infinite for every finite set $F \subseteq G^0$ there exists $w \in G^0$ such that $w \notin F$ and $w \geq v$. But then $p_v \lesssim p_w$ and the claim holds. Now assume that the claim holds whenever $v$ is in $\overline{H}$ with $\min \{ n \in \mathbb{N} : v \in H_n \}$ strictly less than a fixed $k$. Suppose that $v \in H_k$. Then $s^{-1}(v)$ consists of a finite and nonzero number of edges $\{ e_1, \ldots, e_n \}$ with $r(e_i) \in H_{k-1}$ for all $i$. By the induction hypothesis there exists a finite set $W_1 \subseteq G^0$ such that $W_1 \cap F = \emptyset$ and $p_{r(e_1)} \lesssim \sum_{w \in W_1} p_w$. Similarly for each $1 < i \leq n$ there exists a finite set $W_i \subseteq G^0$ which is disjoint from $F \cup W_1 \cup \ldots \cup W_{i-1}$ and with $p_{r(e_i)} \lesssim \sum_{w \in W_i} p_w$. Now if we let $x = s_{e_1} + \ldots + s_{e_n}$ then we see that $x^* (\sum_{i=1}^n s_{e_i}s_{e_i}^*) x = \sum_{i=1}^n p_{r(e_i)}$, and thus $\sum_{i=1}^n s_{e_i}s_{e_i}^* \lesssim \sum_{i=1}^n p_{r(e_i)}$. Therefore if we let $W := W_1 \cup \ldots \cup W_n$ we see that $W \cap F = \emptyset$ and $p_v = \sum_{i=1}^ns_{e_i}s_{e_i}^* \lesssim \sum_{i=1}^n p_{r(e_i)} \lesssim \sum_{w \in W_1} p_w + \ldots \sum_{w \in W_n} p_w = \sum_{w \in W} p_w$.
\noindent \textsc{Case II:} $v \notin \overline{H}$. Since every vertex on a loop is left infinite, it follows that no vertices of $G^0 \backslash \overline{H}$ are on loops. Thus $G_{(\overline{H}, \emptyset)}$ contains no loops and \cite[Corollary~2.13]{DT1} implies that $C^*(G_{(\overline{H}, \emptyset)})$ is an AF-algebra. Furthermore, there are no tracial states on $C^*(G_{(\overline{H}, \emptyset)}) \cong C^*(G) / I_{(\overline{H},\emptyset)}$ since any tracial state would lift to a tracial state on $C^*(G)$ and thus induce a graph trace of norm one on $G$. Since $C^*(G_{(\overline{H}, \emptyset)})$ is an AF-algebra with no tracial states it follows from \cite[Theorem~4.10]{Bla4} that it is stable.
If we list the vertices of $G^0 \backslash \overline{H}$ as $\{ w_1, w_2, \ldots \}$ with $w_1 = v$, then the elements $p_n := \sum_{i=1}^n \pi (p_{w_i})$ form an increasing approximate unit for $C^*(G_{(\overline{H}, \emptyset)})$ consisting of projections. If $F \subseteq G^0$ is a finite set, let $n = \max \{ i \in \mathbb{N} : w_i \in F \}$. Since $C^*(G_{(H,\emptyset)})$ is stable, Lemma~\ref{stab-equiv}(c) implies that there exists $m > n$ such that $p_n \lesssim p_m-p_n$. But if we let $W_0 := \{ w_{n+1}, \ldots, w_m\}$ then $W_0 \cap F = \emptyset$ and $\pi(p_v) \lesssim p_n \lesssim p_m-p_n = \sum_{w \in W_0} \pi(p_w)$. It then follows from Lemma~\ref{quotient-lift} that there exists a finite set $X \subseteq \overline{H}$ for which $p_v \lesssim \sum_{w \in W} p_w + \sum_{x \in X} p_x$ in $C^*(G)$. Now since $X \subseteq \overline{H}$ we see from Case~I above that if $X = \{x_1 , \ldots, x_n \}$ then for each $i$ we may choose $W_i$ such that $W_i$ is disjoint from $F \cup W_0 \cup \ldots W_{i-1}$ and $p_{x_i} \lesssim \sum_{w \in W_i} p_w$. If we let $W := W_0 \cup \ldots \cup W_n$, then $W \cap F = \emptyset$ and $p_v \lesssim \sum_{w \in W_0}p_w + \sum_{x \in X}p_x \lesssim \sum_{w \in W_0}p_w + \ldots + \sum_{w \in W_n}p_w = \sum_{w \in W} p_w$. \\
\noindent (e) $\Longrightarrow$ (f) : List the elements of $V$ as $V = \{ v_1, \ldots v_n \}$. Choose $W_1$ such that $W_1 \cap V = \emptyset$ and $p_{v_1} \lesssim \sum_{w \in W_1} p_w$. Having chosen $W_k$ we may choose $W_{k+1}$ so that $W_{k+1}$ is disjoint from $V \cup W_1 \cup \ldots \cup W_k$ and $p_{v_k} \lesssim \sum_{w \in W_k} p_w$. We continue in this fashion until we produce $n$ sets $W_1, \ldots, W_n$ with these properties. If we let $W := W_1 \cup \ldots \cup W_n$, then $V \cap W = \emptyset$ and $\sum_{v \in V} p_v \lesssim \sum_{w \in W_1} p_w + \ldots + \sum_{w \in W_n} p_w = \sum_{w \in W} p_w$. \\
\noindent (f) $\Longrightarrow$ (a) : List the vertices of $G$ as $G^0 := \{ v_1, v_2, \ldots \}$. For each $n \in \mathbb{N}$ we define $p_n := \sum_{i=1}^n p_{v_i}$. Then $\{p_n \}_{n=1}^\infty$ is an increasing approximate unit consisting of projections, and by Lemma~\ref{stab-equiv} it suffices to prove that for all $n \in N$ there exists $m > n$ such that $p_n \lesssim p_m-p_n$.
Let $n \in N$, and define $V := \{ v_1, \ldots, v_n \}$. By hypothesis there exists a finite set $W \subseteq G^0$ such that $V \cap W = \emptyset$ and $\sum_{v \in V} p_v \lesssim \sum_{w \in W} p_w$. Let $m:= \max \{ k \in \mathbb{N} : v_k \in W \}$. Since $V \cap W = \emptyset$ we see that $\sum_{w \in W} p_v \leq p_m - p_n$. Thus $p_n = \sum_{v \in V} p_v \lesssim \sum_{v \in W} p_v \leq p_m-p_n$.
$\qed$
\section{The stabilization of a graph $C^*$-algebra} \label{stabilization-sec}
\begin{definition} If $G$ is a graph and $v \in G^0$ is a vertex, then by \emph{adding a head to $v$} we mean attaching a graph of the form $$ \xymatrix{ \cdots \ar[r]^{e_4} & v_3 \ar[r]^{e_3} & v_2 \ar[r]^{e_2} & v_1 \ar[r]^{e_1} & v\\ } $$ Thus we create a new graph $F$ from $G$ by defining $F^0 := G^0 \cup \{ v_1, v_2, \ldots \}$, $F^1 := G^1 \cup \{e_1, e_2, \ldots \}$, and extend $r$ and $s$ to $F^1$ by $r(e_i) = v_{i-1}$ and $s(e_i) = v_i$.
The terminology ``adding a head" is meant to complement the terminology for the analogous concept of ``adding a tail" introduced in \cite[(1.2)]{BPRS}. \end{definition}
\begin{theorem} \label{stabilization-gr-alg-thm} If $G$ is a graph, let $\tilde{G}$ be the graph obtained by adding a head to each vertex of $G$. Then $C^*(\tilde{G})$ is the stabilization of $C^*(G)$; that is, $$C^*(\tilde{G}) \cong C^*(G) \otimes \mathcal{K}.$$ \end{theorem}
\begin{proof} Following the proof of \cite[Lemma~1.2]{BPRS} one can show that $C^*(G)$ is naturally isomorphic to a full corner of $C^*(\tilde{G})$. Consequently $C^*(G)$ is Morita equivalent to $C^*(\tilde{G})$, and since Corollary~\ref{left-inf-stable} implies that $C^*(\tilde{G})$ is stable we have $C^*(G) \otimes \mathcal{K} \cong C^*(\tilde{G}) \otimes \mathcal{K} \cong C^*(\tilde{G})$. \end{proof}
\begin{corollary} The class of graph $C^*$-algebras is closed under stabilization. \end{corollary}
\begin{example} If $G$ is the graph $$ \xymatrix{ \bullet \ar[rd] & & \bullet \\ & \bullet \ar[ru] \ar[r] & \bullet \ar@(ul,ur) } $$ then $\tilde{G}$ is the graph $$ \xymatrix{ \cdots \ar[r] & \bullet \ar[r] & \bullet \ar[r] & \bullet \ar[rd] & & \bullet & \bullet \ar[l] & \bullet \ar[l] & \cdots \ar[l] \\ \cdots \ar[r] & \bullet \ar[r] & \bullet \ar[r] & \bullet \ar[r] & \bullet \ar[ru] \ar[r] & \bullet \ar@(ul,ur) & \bullet \ar[l] & \bullet \ar[l] & \cdots \ar[l] } $$ and $C^*(\tilde{G}) \cong C^*(G) \otimes \mathcal{K}$. \end{example}
\begin{example} If $G$ is the following graph with one vertex and infinitely many edges, then $C^*(G) \cong \mathcal{O}_\infty$ $$ \xymatrix{ \bullet \ar@(dr,ur)_\infty} $$ and $\tilde{G}$ is the graph $$ \xymatrix{ \cdots \ar[r] & \bullet \ar[r] & \bullet \ar[r] & \bullet \ar@(dr,ur)_\infty } $$ so that $C^*(\tilde{G}) \cong \mathcal{O}_\infty \otimes \mathcal{K}$. \end{example}
\begin{remark} To obtain the stabilization it is often unnecessary to add a head to every vertex in $G$. It suffices to add enough heads to make all vertices left infinite. For example, one could choose to add heads only at the left finite vertices of $G$. \end{remark}
\end{document} | arXiv |
Home Forums > Science > Physics & Math >
Four Dot Products and of Momenta
Discussion in 'Physics & Math' started by Anamitra Palit, Jan 11, 2021.
Anamitra Palit Registered Member
We deduce in the paper the following results
v1.v2>=c^2
[v1 and v2 are four velocities]
p1.p2>=m1m2c^2
[p1 and p2 are four velocities]
https://drive.google.com/file/d/1yTw0x5uFs1zaT9bd1n6E7jgMvpA56k6W/view?usp=sharing
By applying the reversed Cauchy Schwarz Inequality we may arrive directly at the same results
Let's consider the reversed Cauchy Schwarz inequality.
c^2 t1 t2-x1 x2 -y1 y2-z1 z2>=Sqrt[c^2t1^2-x1^2-y1^2-z1^2]Sqrt[c^2t2^2-x2^2-y2^2-z2^2]
The equality sign holds when (t1,x1,y1,z1) and (t2,x2,y2,z2) are identical vectors
Replacing x^i b y dx^i we obtain
c^2 dt1 dt2-dx1dx2 -dy1 dy2-dz1 dz2>=Sqrt[c^2dt1^2-dx1^2-dy1^2-dz1^2]Sqrt[c^2dt2^2-dx2^2-dy2^2-dz2^2]
A paper on the Reversed Cauchy Schwarz Inequality:
https://drive.google.com/file/d/1z69d0OO4WRK6CthRln0s5LBJWbDS290f/view?usp=sharing
Wikipedia Link on the Cauchy Schwarz Inequality:
https://en.wikipedia.org/wiki/Minkowski_space#Norm_and_reversed_Cauchy_inequality
Dividing both sides by dtau^2 we obtain
Four dot product v1.v2>=c^2
Multiplying both sides by m1m2[m1 and m2 being rest masses] we obtain,
m1v1.mv2>=m1 m2c^2
or,p1.p2>=m1m2 c^2
Anamitra Palit, Jan 11, 2021
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James R Just this guy, you know? Staff Member
And so? What's the point?
James R, Jan 11, 2021
The formulas v1.v2>=c^2,
p1.p2 >m1m2c^2 are important just like v,v=c^2
But alas..
c^2dtau^2=c^2dt^2-dx^2-dy^2-dz^2
c^2=c^2[dt/dtau]^2-[dx/dtau]^2-[dy/dtau]^2-[dz/dtau]^2
c^2=c^2v_t^2-v_x^2-v_y^2-v_z^2 (1)
v_i are proper speeds and as such they can exceed the speed of light without hurting or violating Special Relativity
For two proper velocities v1 and v2at the same point of the manifold.Since tensors are additive we have
c^2=c^2(v1_t+v2_t)^2-(v1_x+v2_x)^2-(v1_y+v2_y)^2-(v1_z+v2_z)^2(2)
or,c^2=c^2v1_t^2-v1_x^2-v1_y^2-v1_z^2+c^2v2_t^2-v2_x^2-v2_y^2-v2_z^2+2v1.v2
or, c^2=c^2+c^2+2v1.v2
v1.v2=-c^2 (3)
By calculations we have arrived at an untenable result.
An analogous result may be obtained in the General Relativity context
Anamitra Palit, Jan 12, 2021 at 3:36 AM
(in continuation)
Indeed v1,v2 and v=v1+v2 all satisfy(1) and hence heir existence is certified by Special relativity or even by General Relativity for that matter.
c^2dtau^2=c^2 g_tt dt^2-g_xx dx^2-g_yy dy^2-g_zz dz^2 (4)
We consider transformations g_tt dt^2=dT^2, g_xxdx^2=dX^2, g_yydy^2=dY^2,g_zzdz^2=dZ^2 (5)
Local or even transformations over infinitesimally small regions would suffice.
Equations (4) and (5) combined gives us the flat space time metric[mathematical form of it]
c^2dtau^2=c^2 dT^2- dX^2- dY^2- dZ^2 (6)
All conclusions we made earlier follow.
Incidentally, there is one point to take note of:the Lorentz transformations follow from (6) in a unique manner [Reference; Steve Wienberg,Gravitation and Cosmology,Chapter 2:Special Relativity]
In relation to the last post
We may always choose the eight unknowns unknowns:v1_i and v2_j with each i and j=1,2,3,4 , in such a manner that the next three equations hold
c^2=c^2v1_t^2-v1_x^2-v1_y^2-v1_z^2 (1)
c^2=c^2v2_t^2-v2_x^2-v2_y^2-v2_z^2+2v1.v2 (2)
and c^2=c^2(v_1+v2_t)^2_-(v1_x+v2_x)^2-(v1_y+v2_y)^2-(v1_z+v2_z)^2 (3)
Equations (1),(2) and (3) are all certified by the relation
c^2=c^2v_t^2-v_x^2-v-y^2-v_z^2 which is equivalent to the Lorentz transformations
as stated earlier.
The three equations finally lead will lead to 2v1 .v2<=-c^2
Last edited: Jan 12, 2021 at 8:39 AM
exchemist Valued Senior Member
Anamitra Palit said: ↑
What is it you want to discuss?
exchemist, Jan 12, 2021 at 9:03 AM
Michael 345 New year. PRESENT is 71 years old Valued Senior Member
exchemist said: ↑
The Inner Mind?
I sense a link
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Michael 345, Jan 12, 2021 at 10:15 AM
One has to follow the full thing
The following two formulas have bee deduced
1. v1.v2>=c^2
2. p1.p2>=m1 m2c^2
Finally we discover a contradiction
2v1.v2<=-c^2
https://drive.google.com/file/d/148q5_2x8DTPLDpa48uE4QSzZ-jJtotNV/view?usp=sharing
I would definitely try out for publication in some journal.
Latest version of the article
https://drive.google.com/file/d/1b2gBTZTYV0CC25u5H9Cd7StRW0z1sZKE/view?usp=sharing
What is the significance of this contradiction?
An important revision has been implemented[pl see "The Extra bit"]. Link to the revised file has been provided. I will keep the audience informed as I proceed with he article...
https://drive.google.com/file/d/1cFmjI3LM7-vqG0Qib9waYG2oHLppmITJ/view?usp=sharing
The very nature of the argument to bring out the contradiction has changed.
Last edited: Jan 13, 2021 at 1:16 PM
Anamitra Palit, Jan 13, 2021 at 1:03 PM
Curried Reiku, apparently.
exchemist, Jan 13, 2021 at 3:45 PM
Important revisions have been made in "the Extra Bit"
https://drive.google.com/file/d/1V_Ms1FfeiQKhMNk9lfqhxqr3GAcguBN5/view?usp=sharing
Relevant material in Latex:
\begin{equation}c^2d\tau^2=c^2dt^2-dx2-dy^2-dz^2 \end{equation} (1)
\begin{equation}c^2=c^2\left(\frac{dt}{d\tau}\right)^2-\left(\frac{dx}{d\tau}\right)^2-\left(\frac{dy}{d\tau}\right)^2-\left(\frac{dz}{d\tau}\right)^2\end{equation}
\begin{equation}c^2=c^2{v_t}^2-{v_x}^2-{v_y}^2-{v_z}^2\end{equation}(2)
We consider two proper velocities on the same manifold
\begin{equation}c^2=c^2{v_{1t}}^2-{v_{1x}}^2-{v_{1y}}^2-{v_{1z}}^2\end{equation}(3.1)
Adding (3.1) and (3.2) we obtain
\begin{equation}2c^2=c^2\left({v_{1t}}^2+{v_{2t}}^2\right)-\left({v_{1x}}^2+{v_{2x}}^2\right)-\left({v_{1y}}^2+{v_{2y}}^2\right)-\left({v_{1z}}^2+{v_{2z}}^2\right)\end{equation}
\begin{equation}2c^2=c^2\left(v_{1t}+v_{2t}\right)^2-\left(v_{1x}+v_{2x}\right)^2-\left(v_{1y}+v_{2y}\right)^2-\left(v_{1z}+v_{2z}\right)^2-2v_1\dot v_2\end{equation}
\begin{equation}2c^2+2v_1\dot v_2=c^2\left(v_{1t}+v_{2t}\right)^2-\left(v_{1x}+v_{2x}\right)^2-\left(v_{1y}+v_{2y}\right)^2-left(v_{1z}+v_{2z}\right)^2\end{equation}
Since v1.v2>=c^2 we have
\begin{equation}c^2\left({v_{1t}}+{v_{2t}}\right)^2-\left({v_{1x}}+{v_{2 x}}\right)^2-\left({v_{1y}}+{v_{1y}}\right)^2-\left({v_{1z}}+{v_{1z}}\right)^2\ge 4c^2\end{equation}(4)
\begin{equation}\left(v_1+v2)\dot (v_1+v_2)\right) \ge 4c^2\end{equation}(5)
If $v_1+v_2$ is a proper velocity then
\begin{equation}c^2=c^2\left(v_{1t}+v_{2t})\right)^2-\left(v_{1x}+v_{2x})\right)^2-\left(v_{1y}+v_{2y})\right)^2-\left(v_{1z}+v_{2z})\right)^2\end{equation} (6)
\begin{equation}c^2=c^2v_{1t}^2-v_{1x}^2-v_{1y}^2-v_{1z}^2+ c^2v_{1t}^2-v_{1x}^2-v_{1y}^2-v_{1z}^2+2v_1.v_2\end{equation}
\begin{equation}c^2=c^2+c^2+2v_1.v_2\end{equation}(7)
\begin{equation}v_1.v_2\le -½ c^2\end{equation}(8)
which is not true since
\begin{equation}v.v=c^2\end{equation}
Therefore $$v_1+v_2$$ is not a four vector if $$v_1$$ and $$v_2$$ are four vectors
Again if $$v_1-v_2$$ is a four vector then
\begin{equation}c^2=c^2\left(v_{1t}-v_{2t})\right)^2-\left(v_{1x}-v_{2x})\right)^2-\left(v_{1y}-v_{2y})\right)^2-\left(v_{1z}-v_{2z})\right)^2\end{equation} (9)
\begin{equation}c^2=c^2v_{1t}^2-v_{1x}^2-v_{1y}^2-v_{1z}^2+ c^2v_{1t}^2-v_{1x}^2-v_{1y}^2-v_{1z}^2-2v_1.v_2\end{equation}
\begin{equation}c^2=c^2+c^2-2v_1.v_2\end{equation}
\begin{equation} ½ c^2=v_1.v_2\end{equation} (10)
But the above formula is not a valid one. Given two infinitesimally close four velocities their difference is not a four velocity. Therefore the manifold has to be a perforated one. The manifold indeed is a mesh of worldlines and each world line is a train of proper velocity four vectors as tangents. A particle moves along a timelike path and therefore each point on it has a four velocity as a tangent representing the motion. The manifold is discrete and that presents difficulty an impossibility to be precise with procedure like differentiation.
Last edited: Jan 14, 2021 at 11:42 AM
Anamitra Palit, Jan 14, 2021 at 11:14 AM
[It may be necessary to refresh the page for proper viewing]
We start with the norm of proper velocity[metric signature(+,-,-,-)]
\begin{equation}c^2=c^2 v_t^2-v_x^2-v_y^2-v_z^2\end{equation}
\begin{equation}c^2=c^2\left(\frac{dt}{d\tau}\right)^2-\left(\frac{dt}{d\tau}\right)^2-\left(\frac{dt}{d\tau}\right)^2-\left(\frac{dt}{d\tau}\right)^2\end{equation}
Differentiating with respect to propertime,
\begin {equation}c^2\frac{dt}{d\tau}\frac{d^2 t}{d \tau^2}-\frac{dx}{d\tau}\frac{d^2 x}{d \tau^2}-\frac{dy}{d\tau}\frac{d^2 y}{d \tau^2}-\frac{dz}{d\tau}\frac{d^2 z}{d \tau^2}=0\end {equation} (1)
\begin{equation}\Rightarrow v.a=0\end{equation}(2)
We choose k such that [k] =T so that ka has the dimension of velocity
\begin{equation}\Rightarrow v.ka=0\end{equation}(3)
We have had earlier, \begin{equation}v.v=c^2\end{equation}(4)
From (3) and (4)
\begin{equation}\Rightarrow v. \left(v-ka\right)=c^2\end{equation} (5)
By adjusting the value [but maintaining its dimension as that of time] we always do have equation (5)
If $\left(v-ka\right)=v'$ is a proper velocity then we have $v.v'=c^2$ in opposition to $v.v'>=c^2$
If $\left(v-ka\right)=v'$ is a not a proper velocity then
\begin{equation}c^2\left(\frac{dt'}{d\tau'}\right)^2-\left(\frac{dx'}{d\tau'}\right)^2-\left(\frac{dy'}{d\tau'}\right)^2-\left(\frac{dz'}{d\tau'}\right)^2=c'^2 \ne c^2\end{equation}
We have from the reversed Cauchy Schwarz inequality,
\begin{array}{l}\left(c^2 \frac{dt}{d\tau}\frac{dt'}{d\tau'}-\frac{dx}{d\tau}\frac{dx'}{d\tau'}-\frac{dy}{d\tau}\frac{dy'}{d\tau'}-\frac{dz}{d\tau}\frac{dz'}{d\tau'}\right)^2\ge\\ \left(c^2\left(\frac{dt}{d\tau}\right)^2-\left(\frac{dx}{d\tau}\right)^2-\left(\frac{dy}{d\tau}\right)^2-\left(\frac{dz}{d\tau}\right)^2\right)\left(c^2\left(\frac{dt'}{d\tau'}\right)^2-\left(\frac{dx'}{d\tau'}\right)^2-\left(\frac{dy'}{d\tau'}\right)^2-\left(\frac{dz'}{d\tau'}\right)^2\right)\end{array}(6)
or,\begin{equation}\left(c^2 \frac{dt}{d\tau}\frac{dt'}{d\tau'}-\frac{dx}{d\tau}\frac{dx'}{d\tau'}-\frac{dy}{d\tau}\frac{dy'}{d\tau'}-\frac{dz}{d\tau}\frac{dz'}{d\tau'}\right)^2\ge c^2c'^2\end{equation} (7)
\begin{equation}c^2 \frac{dt}{d\tau}\frac{dt'}{d\tau'}-\frac{dx}{d\tau}\frac{dx'}{d\tau'}-\frac{dy}{d\tau}\frac{dy'}{d\tau'}-\frac{dz}{d\tau}\frac{dz'}{d\tau'}\ge cc'\end{equation}
\begin{equation}c^2 \frac{dt}{d\tau}\frac{dt'}{d\tau'}-\frac{dx}{d\tau}\frac{dx'}{d\tau'}-\frac{dy}{d\tau}\frac{dy'}{d\tau'}-\frac{dz}{d\tau}\frac{dz'}{d\tau'}\le -cc'\end{equation}
v.v'>=cc' or v.v'<=-cc'
But v.v'=c^2. Therefore the solution is c'=c .We have v' is a proper velocity.||But we assumed /postulated at the very outset that v' is not a proper velocity.
[One may require to refresh the page for proper viewing]
Norm of Four Acceleration
Four Acceleration
\begin{equation}\left(c\frac{d^2 t}{d\tau^2},\frac{d^2 x}{d\tau^2},\frac{d^2 y}{d\tau^2}, \frac{d^2 z}{d\tau^2}\right)\end{equation} (1)
\begin{equation}c^2N=c^2\left(\frac{d^2 t}{d\tau^2}\right)^2-\left(\frac{d^2 x}{d\tau^2}\right)^2-\left(\frac{d^2 y}{d\tau^2}\right)^2-\left( \frac{d^2 z}{d\tau^2}\right)^2\end{equation} (2)
We consider the metric
\begin{equation}c^2d\tau^2=c^2dt^2-dx^2-dy^2-dz^2 \end{equation} (4)
\begin{equation}\Rightarrow c^2=c^2\left(\frac{dt}{d\tau}\right)^2-\left(\frac{dx}{d\tau}\right)^2-
\left(\frac{dy}{d\tau}\right)^2-\left(\frac{dz}{d\tau}\right)^2
\end{equation} (5)
Differentiating (5) with respect to proper time we have,
\[ c^2\frac{dt}{d\tau}\frac{d^2 t}{d \tau^2}- \frac{dx}{d\tau}\frac{d^2 x}{d \tau^2}-\frac{dy}{d\tau}\frac{d^2 y}{d \tau^2}-\frac{dz}{d\tau}\frac{d^2 z}{d \tau^2}=0\] (6)
By applying the Cauchy Schwarz inequality we have,
\[\left(\frac{dx}{d\tau}\frac{d^2 x}{d \tau^2}+\frac{d y}{d\tau}\frac{d^2y}{d \tau^2}+\frac{dz}{d\tau}\frac{d^2 z}{d \tau^2}\right)^2 \\ \ge \left(\left(\frac{d x}{d\tau}\right)^2+\left(\frac{dy}{d\tau}\right)^2+\left(\frac{dz}{d\tau} \right)^2\right)\left(\left(\frac{d^2 x}{d \tau^2}\right)^2+\left(\frac{d^2 y}{d \tau^2}\right)^2+\left(\frac{d^2 z}{d \tau^2}\right)^2\right)\](7)
\[\left(c^2\left(\frac{d t}{d \tau}\right)^2-c^2\right)\left(c^2\left( \frac {d^2 t}{d \tau^2}\right)^2-c^2N\right) \ge\left( c^2 \frac {d^2 t}{d\tau^2}\right)^2\left(c^2\frac{d t}{d\tau}\right)^2\]
\[\left(\left(\frac{dt}{d \tau}\right)^2-1\right)\left(\left( \frac {d^2 t}{d \tau^2}\right)^2-N\right) \ge \left( \frac {d^2 t}{d\tau^2}\right)^2\left(\frac{dt}{d\tau}\right)^2\] (8)
\[ \left( \frac {d^2 t}{d\tau^2}\right)^2\left(\frac{dt}{d\tau}\right)^2-N\left(\frac{dt}{d \tau}\right)^2-\left( \frac {d^2 t}{d \tau^2}\right)^2+N\ge \left( \frac {d^2 t}{d\tau^2}\right)^2\left(\frac{dt}{d\tau}\right)^2\]
\[ \Rightarrow-N\left(\frac{dt}{d \tau}\right)^2-\left( \frac {d^2 t}{d \tau^2}\right)^2+N\ge 0\] (9)
\[N\left(1-\left(\frac{dt}{d\tau}\right)^2\right)\ge \left( \frac {d^2 t}{d \tau^2}\right)^2\]
\[N\left(1-\gamma^2\right)\ge \left( \frac {d^2 t}{d \tau^2}\right)^2\](10)
The right side of (10) is always positive or zero. Therefore the left side is also positive or zero. Therefore N<=0 since gamma[Lorentz factor] is positive[>= unity]. N cannot be positive unless the particle is moving uniformly.
If N is negative then from (1) we have
\[c^2\left(\frac{d^2 t}{d\tau^2}\right)^2\le \left(\frac{d^2 x}{d\tau^2}\right)^2-\left(\frac{d^2 y}{d\tau^2}\right)^2-\left( \frac{d^2 z}{d\tau^2}\right)^2\]
For a particle at rest (spatially) and N<0,
\[\left(\frac{d^2 t}{d\tau^2}\right)^2\le 0\](11)
Equation (11) will not hold, the left side being a [perfect square and hence positive or zero]unless
\begin{equation}\frac{d^2 t}{d\tau^2}=0\end{equation}
that is unless \begin{equation}\frac{dt}{d\tau}=constant \Rightarrow \gamma=constant\end{equation}
that is unless the particle is moving with a constant velocity. An accelerating particle will not cater to N<0.
For N=0 we have from (10)
\[\left(\frac{d^2x}{d\tau^2}\right)^2\le 0 \](12)
Equation (12) is not a valid on unless the particle moves with a constant velocity.
We conclude that the norm square of the acceleration vector c^2N cannot be positive, negative or zero unless the particle is moving uniformly that is unless it moves with a constant velocity
Anamitra Palit, Jan 16, 2021 at 12:29 PM | CommonCrawl |
Find the area in the plane contained by the graph of
\[|x + y| + |x - y| \le 4.\]
First, assume that $x \ge 0$ and $y \ge 0.$ If $y \ge x,$ then
\[|x + y| + |x - y| = x + y + y - x = 2y \le 4,\]so $y \le 2.$ If $y < x,$ then
\[|x + y| + |x - y| = x + y + x - y = 2x \le 4,\]so $x \le 2.$
Thus, the portion of the graph in the first quadrant is as follows:
[asy]
unitsize (1 cm);
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle,gray(0.7));
draw((2,0)--(2,2)--(0,2));
draw((-0.5,0)--(2.5,0));
draw((0,-0.5)--(0,2.5));
dot("$2$", (2,0), S);
dot("$2$", (0,2), W);
[/asy]
Now, suppose $(a,b)$ satisfies $|x + y| + |x - y| \le 4,$ so
\[|a + b| + |a - b| \le 4.\]If we plug in $x = a$ and $y = -b,$ then
\[|x + y| + |x - y| = |a - b| + |a + b| \le 4.\]This means if $(a,b)$ is a point in the region, so is $(a,-b).$ Therefore, the region is symmetric around the $x$-axis.
Similarly, if we plug in $x = -a$ and $y = b,$ then
\[|x + y| + |x - y| = |-a + b| + |-a - b| = |a - b| + |a + b| \le 4.\]This means $(-a,b)$ is also a point in the region. Therefore, the region is symmetric around the $y$-axis.
We conclude that the whole region is a square with side length 4.
[asy]
unitsize (1 cm);
filldraw((-2,-2)--(-2,2)--(2,2)--(2,-2)--cycle,gray(0.7));
draw((-2.5,0)--(2.5,0));
draw((0,-2.5)--(0,2.5));
dot("$2$", (2,0), SE);
dot("$2$", (0,2), NW);
dot("$-2$", (-2,0), SW);
dot("$-2$", (0,-2), SW);
[/asy]
Hence, its area is $\boxed{16}.$ | Math Dataset |
\begin{document}
\begin{abstract} We prove existence and regularity results for weak solutions of non linear elliptic systems with non variational structure satisfying $(p,q)$-growth conditions. In particular we are able to prove higher differentiability results under a dimension-free gap between $p$ and $q$. \end{abstract}
\maketitle
\section{Introduction}\label{S:1}
In this paper we focus on the existence and the regularity results for solutions $u$ to the Dirichlet problems associated with the following nonlinear system in divergence form (here $\alpha=1,\ldots,N$) \begin{equation}\label{dirichlet} \left\{\begin{array}{ll} \displaystyle \sum_{i=1}^n \frac{\partial}{\partial x_i} A_{i}^{\alpha}(Du)=0 &\text{in $ \Omega$} \\ u=u_0 &\text{on $ \partial \Omega$,} \end{array}\right. \end{equation} where the functions $A_i^{\alpha}(\xi)$ are locally Lipschitz continuous in $\mathbb{R}^{nN}$, $\Omega$ is an open bounded subset of $\mathbb{R}^n$ and $Du:\Omega \to \mathbb{R}^{nN}$ represents the gradient of a (vector-valued) function $u:\Omega \to \mathbb{R}^N$.
We equip the problem with the general $(p,q)$-growth conditions, i.e., we assume that there are $1< p\le q<\infty$ and two positive constants $m, M$ such that for all $\xi, \lambda\in \mathbb{R}^{nN}$ and for all $i,j=1,\ldots,n$, and $\alpha,\beta=1,\ldots,N$ there holds \begin{align}\label{ellitticita1}
m(1+|\xi|^2)^{\frac{p-2}{2}}|\lambda|^2&\le \sum_{i,j=1}^n\sum_{\alpha,\beta=1}^N \frac{\partial A_{i}^{\alpha}}{\partial \xi^{\beta}_j}(\xi)\lambda^{\alpha}_i\lambda_j^{\beta},\\ \label{crescita-q}
\left|\frac{\partial A_{i}^{\alpha}}{\partial \xi_j^{\beta}}(\xi)\right|&\le M (1+|\xi|^2)^{\frac{q-2}{2}}. \end{align}
Notice that \eqref{ellitticita1} is the usual ellipticity condition and \eqref{crescita-q} is the $q$-growth condition, from which the name of $(p,q)$-growth come from. Under these assumptions, one can easily observe (see Lemma \ref{stimaLp}) that $|A^{\alpha}_i(\xi)|\le C(1+|\xi|)^{q-1}$ with some generic constant $C$ and therefore we can naturally define a notion of a weak solution to \eqref{dirichlet} in the following way:
Let $u_0\in W^{1,p}(\Omega; \mathbb{R}^N)\cap W^{1,q}_{\rm loc}(\Omega;\mathbb{R}^N)$. We say that $u$ is a weak solution to \eqref{dirichlet} if \begin{equation}\label{soldebole-Dirichlet1}
u-u_0 \in W_0^{1,p} (\Omega;\mathbb{R}^N)\cap W_{\textrm{loc}}^{1,q} (\Omega;\mathbb{R}^N) \end{equation} and for all open $\Omega'$ fulfilling $\overline{\Omega'}\subset \Omega$ and for all $\varphi \in W_0^{1,q}(\Omega';\mathbb{R}^N)$ there holds \begin{equation}\label{soldebole-Dirichlet} \int_{\Omega}\sum_{i=1}^{n}\sum_{\alpha=1}^{N}A_{i}^{\alpha}(Du)\varphi^{\alpha}_{x_i}(x) \,dx=0. \end{equation} Here, and also in what follows, we use the abbreviation $\varphi^{\alpha}_{x_i}:=\frac{\partial \varphi^{\alpha}}{\partial x_i}$
Our main task in the paper is to establish the existence of such a solution and further some regularity of arbitrary weak solutions. However, contrary to the classical result, we do not in general assume any symmetry condition on the derivative of $A^{\alpha}_i$ and so we do not assume that the system is in variational form. Nevertheless, as done in \cite{mar91} in the scalar framework, we will need to compensate this lack of symmetry by the following assumption on the asymptotic behavior of the skew-symmetric part, namely,
for all $\xi, \lambda\in \mathbb{R}^{nN}$ and for all $i,j=1,\ldots,n$, and $\alpha,\beta=1,\ldots,N$ there holds \begin{equation}\label{continuita}
\left|\frac{\partial A_{i}^{\alpha}}{\partial \xi_j^{\beta}}(\xi)-\frac{\partial A_{j}^{\beta}}{\partial \xi_i^{\alpha}} (\xi)\right|\le M
(1+|\xi|^2)^{\frac{q+p-4}{4}}. \end{equation}
If $p = q$, the existence of weak solutions to (\ref{dirichlet}) can be established using the theory of coercive, monotone operators, see Leray--Lions \cite{lerlio65}, Browder \cite{browder} and Hartman--Stampacchia \cite{{hart-stamp}}. Also the regularity issue has been extensively studied, see the monographs \cite{gia2}, \cite{giusti} and the surveys \cite{Mingione} and \cite{mingione2}. Notice also, that without any further additional structural assumptions, the best\footnote{This information can be as usual slightly improved by the Gehring lemma} known regularity information about the solution is that $V(Du)\in W^{1,2}_{\rm loc}(\Omega; \mathbb{R}^{nN})$, where \begin{equation}\label{defV}
V(\xi):=(1+|\xi|^2)^{\frac{p-2}{4}}\xi. \end{equation} On the other hand, if $p < q$ the above classical existence results cannot be applied due to the lack of coercivity in $W^{ 1,q}$ . Moreover, the request $u\in W^{1,q }_{\rm loc}(\Omega; \mathbb{R}^N)$ in the definition of weak solution, needed to have a well defined integral, is an additional difficulty. Notice that such a request is a priori assumed in some regularity results under the $p, q$-growth, see for example \cite{leonetti1}, \cite{bilfuchsCalcVar} and \cite{cupmarmas4}.
The first result of the paper is that any weak solution is in fact twice weakly differentiable. \begin{theorem}\label{T:main-weak} Let $1<p\le q<\infty$ be arbitrary and $A$ satisfy \eqref{ellitticita1}, \eqref{crescita-q} and \eqref{continuita}. Then any $u\in W^{1,\max\{q,2\}}_{\rm loc}(\Omega; \mathbb{R}^N)$ fulfilling \eqref{soldebole-Dirichlet} satisfies for all $\eta\in C_{c}^{\infty}(\Omega)$ the following estimate \begin{equation}
\int_{\Omega}(1+|Du|^2)^{\frac{p-2}{2}}
|D^2u|^2\eta^{2}\, dx \le c\int_{\Omega}
(1+|Du|^2)^{\frac{q}{2}} |D\eta|^2 \,dx, \label{exdopoH3-1} \end{equation} where the constant $c$ depends only on $m$ and $M$. In particular, we also have that \begin{equation}
\int_{\Omega}|D V(Du)|^2\eta^{2}\, dx \le c\int_{\Omega}
(1+|Du|^2)^{\frac{q}{2}} |D\eta|^2 \,dx. \label{exdopoH3-V} \end{equation} \end{theorem}
The above theorem provides the existence of the second derivatives for arbitrary $1<p\le q<\infty$ but the right hand side of \eqref{exdopoH3-1} or \eqref{exdopoH3-V} still depends on the $W^{1,q}$ norm of $u$. We shall improve this estimate provided that $p$ and $q$ are sufficiently close to each other. Thus, the second main theorem of the paper is the following. \begin{theorem} \label{t:main} Let $1<p\le q<\infty$ be arbitrary and $A$ satisfy \eqref{ellitticita1}, \eqref{crescita-q} and \eqref{continuita} and $u \in W^{1,\max\{q,2\}}_{\rm loc}(\Omega;\mathbb{R}^N)$ satisfy \eqref{soldebole-Dirichlet}. Then for all open $\Omega' \subset \overline{\Omega'} \subset \Omega$ the following holds: \begin{itemize} \item[i)] If
\begin{equation}\label{ipotesip-q} q<p\frac{n+2}{n} \end{equation} then $$ \int_{\Omega'}\left(
|V(Du)|^{\frac{2q}{p}}+|D V(Du)|^2 + (1+|D u|^2)^{\frac{p-2}{2}}|D^2u|^2\right)\, dx \le C(\Omega', n, N, p, q, m, M, \|Du\|_{L^p(\Omega)}). $$ \item[ii)] If $u\in L^{\infty}(\Omega;\mathbb{R}^N)$ and
\begin{equation}\label{ipotesip-qb} q<p+2 \qquad \textrm{and} \qquad p<n \end{equation} then \begin{equation*}
\begin{split}
\int_{\Omega'}&\left(|V(Du)|^{\frac{2q}{p}}+|D V(Du)|^2 + (1+|D u|^2)^{\frac{p-2}{2}}|D^2u|^2\right)\, dx
\\ &\qquad \qquad\qquad \qquad\qquad \qquad\qquad \qquad \le C(\Omega', n, N, p, q, m, M, \|Du\|_{L^p(\Omega)},\|u\|_{L^{\infty}(\Omega)}).
\end{split} \end{equation*}
\end{itemize} In particular, in both cases we have that $V(Du)\in W^{1,2}_{\rm loc}(\Omega;\mathbb{R}^{nN})$, which, due to the embedding theorem, leads to $Du\in L^{\frac{p2^*}{2}}_{\rm loc}(\Omega;\mathbb{R}^{nN})$. \end{theorem}
Finally, we state our last main result of the paper. It is an existence result
for the Dirichlet problem \eqref{dirichlet}. For this purpose, we need
to consider a regularity assumption on the boundary datum. We shall require in what follows that \begin{equation}\label{ipotesi-dato}
u_0 \in W^{1,r} (\Omega;\mathbb{R}^N), \quad \textrm{with}\,\, r:=\max\left\{2,\frac{p(q-1)}{p-1}\right\}. \end{equation} \begin{theorem} \label{t:main2} Let $1<p\le q<\infty$ be arbitrary and $A$ satisfy \eqref{ellitticita1}, \eqref{crescita-q} and \eqref{continuita}. Moreover, let $u_0$ fulfill \eqref{ipotesi-dato}. Then there exists a weak solution to the problem \eqref{dirichlet} provided that at least one of the following conditions hold \begin{itemize} \item[i)] $p$ and $q$ satisfy \eqref{ipotesip-q}. \item[ii)] $p$ and $q$ satisfy \eqref{ipotesip-qb}, $u_0\in L^{\infty}(\partial \Omega;\mathbb{R}^N)$ {and
\begin{equation}\label{structure}
\sum_{i=1}^{n}A_{i}^{\alpha}(\xi)\xi_i^{\alpha}\ge 0, \ \ \forall\xi\in \mathbb{R}^{nN},\ \ \forall \alpha\in \{1,\ldots,N\}. \end{equation}} \end{itemize} \end{theorem}
As far as the regularity of solutions is concerned, the obstructions are essentially two: we are dealing with systems and under non-standard growth $(p < q)$. Indeed, in the vectorial case, even under the standard growth, the everywhere regularity of solutions for systems, or of minimizers of integrals, cannot be expected unless some structure conditions are assigned, and this holds also for the local boundedness, see e.g. the counterexamples by De Giorgi \cite{deg} and \v{S}ver\'ak-Yan \cite{sverak}. Since the pioneering paper by Marcellini \cite{mar89}, the theory of regularity in the framework of non- standard growth has been deeply investigated. The results and the contributions to regularity are so many, that it is a hard task to provide a comprehensive overview of the issue. For this, we refer to the survey of Mingione \cite{Mingione} for an accurate and interesting account on this subject. A common feature is that to get regularity results $p$ and $q$ must be not too far apart, as examples of irregular solutions by Giaquinta \cite{gia}, Marcellini \cite{mar87} and Hong \cite{hong} show. On the other hand, many regularity results are available if the ratio $q/p$ is bounded above by suitable constant that in general depends on the dimension $n$, and converges to $1$ when $n$ tends to infinity (\cite{bil03}, \cite{CarKriPas}, \cite {espoleomin1}, \cite {espoleomin2}, \cite {espoleomin3}). Moreover, the condition on the distance between the exponents $p$ and $q$ can usually be relaxed if the solutions/minimizers are assumed locally bounded.
Let us observe that the local higher differentiability results for bounded minimizers of integral functionals satisfying $p, q$-growth conditions is more studied than the analogous issue for systems of PDE's. In particular, recently, the Authors, in \cite{CKP}, considered integral functionals with convex integrand satisfying $p, q$-growth conditions. They proved local higher differentiability results for bounded minimizers under dimension-free conditions on the gap between the growth and the coercivity exponents; i.e., \eqref{ipotesip-qb} restricted to the case $p\geq 2$, using an improved Gagliardo-Nirenberg's inequality. We also observe that an existence result in the $(p,q)$-framework was proved in \cite{CupLeoMas} for a Dirichlet problem \eqref{dirichlet} with monotone operators possibly depending on the $x$-variable, but for $p\ge 2$ only. As a novel feature, the main results are achieved through uniform higher differentiability estimates for solutions to a class of auxiliary problems, constructed adding higher order perturbations to the integrand. Here we achieve the same result for systems with non variational structure with control on the skew-symmetric part (see (\ref{continuita})).
The plan of the paper is the following. In Section \ref{s:preliminaries} we prove some preliminar algebraic inequalities. In Sections \ref{apriori} and \ref{P-t:main} we prove the higher differentiability results Theorem \ref{T:main-weak} and Theorem \ref{t:main}, respectively. In the last section, we prove the existence result (Theorem \ref{t:main}) for the problem \eqref{dirichlet}.
\section{Auxiliary algebraic inequalities}\label{s:preliminaries}
In this part, we recall several algebraic inequalities related to the mapping $A$. Although, their proof can be in some simplified setting found in many works, see e.g. \cite[Lemma~4.4, Lemma 2.4]{mar91}, \cite[Lemma~1]{Tolksdorf}, \cite[Lemma 5.1]{cupmarmas4} or \cite[Chapter~5]{mnrr96}, we provide for the sake of clarity a detailed proof here. We start with the first auxiliary result based on the assumptions \eqref{ellitticita1}--\eqref{crescita-q}. \begin{lemma} \label{stimaLp} Let $A:\mathbb{R}^{nN}\to \mathbb{R}^{nN}$ be a continuous mapping fulfilling \eqref{ellitticita1} and \eqref{crescita-q}. Then there exists a positive constant $K$ such that for all $\xi, \eta \in \mathbb{R}^{nN}$ there hold \begin{align}\label{dis-ellitticita}
|\xi|^p &\leq K\left\lbrace (1+|\eta|^2)^{\frac{p(q-1)}{2(p-1)}} + \sum_{i=1}^{n} \sum_{\alpha=1}^NA_i^{\alpha}(\xi)(\xi^{\alpha}_i-\eta_i^{\alpha})\right\rbrace,\\
|A^{\alpha}_i(\xi)| &\leq K (1+|\xi|^2)^{\frac{q-1}{2}} \qquad \textrm{ for all } \alpha=1,\ldots, N \textrm{ and } i=1,\ldots,n. \label{crescitaai}\\
\label{casop>2}
|\xi-\eta|^p&\le K\sum_{i=1}^{n} \sum_{\alpha=1}^{N}\left( A_i^{\alpha}(\xi)-A_i^{\alpha}(\eta)\right) (\xi_i^{\alpha}-\eta_i^{\alpha}) \quad \textrm{for }p\ge 2,\\ \label{casop<2}
\left( 1+|\xi|^2+|\eta|^2\right)^{\frac{p-2}{2}}|\xi-\eta|^2 &\le K\sum_{i=1}^{n} \sum_{\alpha=1}^{N}\left(A_i^{\alpha}(\xi)-A_i^{\alpha}(\eta)\right) (\xi^{\alpha}_i-\eta^{\beta}_i) \quad \textrm{for } p \in (1,2). \end{align} \end{lemma}
\begin{proof} We start the proof with \eqref{crescitaai}. Since \begin{equation}\label{Nap1} A^{\alpha}_i(\xi)-A^{\alpha}_i(0)=\int_0^1 \sum_{j=1}^n \sum_{\beta=1}^N\frac{\partial A^{\alpha}_i(t\xi)}{\partial \xi^{\beta}_j}\xi^{\beta}_j\; dt, \end{equation} we can use the assumption \eqref{crescita-q}, to get $$
\left|A^{\alpha}_i(\xi) \right|\le \left|A^{\alpha}_i(0)\right| + M \int_0^1 \sum_{j=1}^n \sum_{\beta=1}^N(1+t^2|\xi|^2)^{\frac{q-2}{2}}
|\xi^{\beta}_j|\; dt\le \left|A^{\alpha}_i(0)\right| + M nN\int_0^1 (1+t^2|\xi|^2)^{\frac{q-2}{2}} |\xi|\; dt. $$ Thus, in case $q\ge 2$, the inequality \eqref{crescitaai} immediately follows.
If $q\in (1,2)$ we can continue with estimating the last integral in the following way $$
\int_0^1 (1+t^2|\xi|^2)^{\frac{q-2}{2}} |\xi|\; dt = \int_0^{|\xi|} (1+t^2)^{\frac{q-2}{2}} \; dt \le 2^{\frac{2-q}{2}}\int_0^{|\xi|} (1+t)^{q-2}
\; dt \le \frac{2^{\frac{2-q}{2}}}{q-1}(1+|\xi|)^{q-1} $$ and we again see that \eqref{crescitaai} follows directly.
To show \eqref{casop>2}--\eqref{casop<2}, we write $$ \begin{aligned} &\sum_{i=1}^{n}\sum_{\alpha=1}^{N}\left( A_i^{\alpha}(\xi)-A_i^{\alpha}(\eta)\right) (\xi_i^{\alpha}-\eta_i^{\alpha}) =\int_0^1 \sum_{i,j=1}^{n}\sum_{\alpha=1}^{N} \frac{\partial A^{\alpha}_i(t\xi + (1-t)\eta)}{\partial \xi^{\beta}_j} ( \xi_j^{\beta}-\eta_j^{\beta}) (\xi_i^{\alpha}-\eta_i^{\alpha})\; dt\\
&\overset{\eqref{ellitticita1}}\ge m|\xi-\eta|^2\int_0^1 (1+|t\xi+(1-t)\eta|^2)^{\frac{p-2}{2}}\; dt. \end{aligned} $$ Then, following step by step proof of Lemma~1.19 in \cite[Chapter~5]{mnrr96}, we deduce \eqref{casop>2}--\eqref{casop<2}.
To show \eqref{dis-ellitticita}, we first consider the case $p \geq 2$. Then by using \eqref{casop>2} and \eqref{crescitaai} and also the Young's inequality, we can observe that for all $\epsilon >0$ and all $\xi,\eta \in \mathbb{R}^{nN}$, we have \begin{align*}
&|\xi|^p \leq c(|\xi- \eta|^p +|\eta|^p) \leq c\left\{
\sum_{i=1}^{n}\sum_{\alpha=1}^{N} (A_i^{\alpha}(\xi)-A_i^{\alpha}(\eta)) (\xi_i^{\alpha}-\eta_i^{\alpha})+|\eta|^p\right\} \\ &\le
c \left\lbrace |\eta|^p+ \sum_{i=1}^{n}\sum_{\alpha=1}^{N} A_i^{\alpha}(\xi) (\xi_i^{\alpha}-\eta_i^{\alpha})+
\bar{C}(1+|\eta|^2)^{\frac{q-1}{2}}(|\xi|+|\eta|) \right\rbrace \\& \le
c \left\lbrace (1+|\eta|^2)^{\frac{p}{2}}+\sum_{i=1}^{n} \sum_{\alpha=1}^{N} A_i^{\alpha}(\xi) (\xi_i^{\alpha}-\eta_i^{\alpha})+
c_{\epsilon}(1+|\eta|^2)^{\frac{p(q-1)}{2(p-1)}}+\epsilon (|\xi|+|\eta|)^p \right\rbrace;
\end{align*} thus if $\epsilon$ is small enough we get \eqref{dis-ellitticita}.
In the case $1<p<2$, we proceed slightly differently. By using the Young 's inequality with complementary exponents $\frac{2}{p}$ and $ \frac{2}{2-p}$ we get for $\epsilon >0$ \begin{align*}
|\xi|^p &\leq c\left(|\xi- \eta|^p +|\eta|^p\right) \le
c\left( |\eta|^p+ (|\xi- \eta|^2)^{\frac{p}{2}}
(1+|\xi|^2 +|\eta|^2)^{ \frac{p(p-2)}{4} + \frac{p(2-p)}{4}}\right)
\\ &\le c \left\{(1+|\eta|^2)^{\frac{p}{2}}+ c_{\epsilon}
(1+ |\xi|^2+|\eta|^2)^{\frac{p-2}{2}}|\xi-\eta|^2+\epsilon (1+ |\xi|^2+|\eta|^2)^{\frac{p}{2}}\right\}. \end{align*} Therefore, by \eqref{casop<2}, with a proper choice of (small) $\epsilon>0$, we get \[
|\xi|^p \leq c \left\{ (1+|\eta|^2)^{\frac{p}{2}} +\sum_{i=1}^{n} \sum_{\alpha=1}^{N} (A_i^{\alpha}(\xi)-A_i^{\alpha}(\eta)) (\xi_i^{\alpha}-\eta_i^{\alpha})
\right\} \] and we conclude by proceeding as above. \end{proof}
The following estimate will play a crucial role for getting the information about the second derivatives of the weak solutions to \eqref{soldebole-Dirichlet}. \begin{lemma}\label{Tolsk} Let $A$ be a continuous mapping fulfilling \eqref{ellitticita1}, \eqref{crescita-q} and \eqref{continuita}. Then there exists a positive constant $K$ such that for all $\xi,\eta,\zeta\in \mathbb{R}^{nN}$ we have \begin{equation}\label{key-3} \begin{aligned}
\frac{m}{2}(1+|\zeta|^2)^{\frac{p-2}{2}}|\xi|^2 & \le \sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N \frac{\partial A^{\alpha}_i(\zeta)}{\partial \zeta^{\beta}_j}(\xi^{\alpha}_i-\eta^{\alpha}_i)\xi^{\beta}_j+K(1+|\zeta|^2)^{\frac{q-2}{2}}|\eta|^2. \end{aligned} \end{equation} \end{lemma} \begin{proof} For arbitrary $\zeta, \xi, \eta \in \mathbb{N}$, we define a bilinear form (for fixed $\zeta$) $$ (\xi,\eta)_{\zeta}:=\frac12\sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N\left(\frac{\partial A^{\alpha}_i(\zeta)}{\partial \zeta^{\beta}_j}+\frac{\partial A^{\beta}_j(\zeta)}{\partial \zeta^{\alpha}_i}\right)\eta^{\alpha}_i\xi^{\beta}_j. $$
Trivially $(\xi,\eta)_{\zeta}=(\xi,\eta)_{\zeta}$. Moreover, using the assumption \eqref{ellitticita1} we get that $$
(\xi,\xi)_{\zeta}=\sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N\frac{\partial A^{\alpha}_i(\zeta)}{\partial \zeta^{\beta}_j}\xi^{\alpha}_i\xi^{\beta}_j\ge m(1+|\zeta|^2)^{\frac{p-2}{2}}|\xi|^2, $$ and consequently, we see that for any fixed $\zeta$, the relation $(\xi,\eta)_{\zeta}$ is a scalar product on $\mathbb{R}^{nN}$ and therefore the Cauchy--Schwarz inequality holds, i.e., \begin{equation}
|(\xi,\eta)_{\zeta}|\le (\xi,\xi)^{\frac12}_{\zeta}(\eta,\eta)^{\frac12}_{\zeta}. \label{CSc} \end{equation} Thus, by assumption \eqref{ellitticita1} and taking into account that \[\sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N \left(\frac{\partial A^{\alpha}_i(\zeta)}{\partial \zeta^{\beta}_j}- \frac{\partial A^{\beta}_j(\zeta)}{\partial \zeta^{\alpha}_i}\right)\xi^{\alpha}_i\xi^{\beta}_j=0,\]
we have $$ \begin{aligned}
m&(1+|\zeta|^2)^{\frac{p-2}{2}}|\xi|^2 \le (\xi,\xi)_{\zeta} = -(\xi-\eta,\xi-\eta)_{\zeta}+ 2(\xi,\xi-\eta)_{\zeta} + (\eta,\eta)_{\zeta}\le 2(\xi,\xi-\eta)_{\zeta} + (\eta,\eta)_{\zeta}\\ &=\sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N \left(\frac{\partial A^{\alpha}_i(\zeta)}{\partial \zeta^{\beta}_j}+ \frac{\partial A^{\beta}_j(\zeta)}{\partial \zeta^{\alpha}_i}\right)(\xi^{\alpha}_i-\eta^{\alpha}_i)\xi^{\beta}_j+ \sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N \frac{\partial A^{\alpha}_i(\zeta)}{\partial \zeta^{\beta}_j}\eta^{\alpha}_i\eta^{\beta}_j\\ &=2\sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N \frac{\partial A^{\alpha}_i(\zeta)}{\partial \zeta^{\beta}_j}(\xi^{\alpha}_i- \eta^{\alpha}_i)\xi^{\beta}_j-\sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N \left(\frac{\partial A^{\alpha}_i(\zeta)}{\partial \zeta^{\beta}_j}- \frac{\partial A^{\beta}_j(\zeta)}{\partial \zeta^{\alpha}_i}\right)(\xi^{\alpha}_i-\eta^{\alpha}_i)\xi^{\beta}_j\\ &\qquad +\sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N \frac{\partial A^{\alpha}_i(\zeta)}{\partial \zeta^{\beta}_j}\eta^{\alpha}_i \eta^{\beta}_j\\ &=2\sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N \frac{\partial A^{\alpha}_i(\zeta)}{\partial \zeta^{\beta}_j}(\xi^{\alpha}_i-\eta^{\alpha}_i)\xi^{\beta}_j + \sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N \left(\frac{\partial A^{\alpha}_i(\zeta)}{\partial \zeta^{\beta}_j}-\frac{\partial A^{\beta}_j(\zeta)}{\partial \zeta^{\alpha}_i} \right)\eta^{\alpha}_i\xi^{\beta}_j\\ &\qquad +\sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N \frac{\partial A^{\alpha}_i(\zeta)}{\partial \zeta^{\beta}_j}\eta^{\alpha}_i \eta^{\beta}_j\\ &\overset{\eqref{crescita-q},\eqref{continuita}}\le 2\sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N \frac{\partial A^{\alpha}_i(\zeta)}{\partial \zeta^{\beta}_j}(\xi^{\alpha}_i-\eta^{\alpha}_i)\xi^{\beta}_j
+MnN(1+|\zeta|^2)^{\frac{q+p-4}{4}}|\eta||\xi|+MnN(1+|\zeta|^2)^{\frac{q-2}{2}}|\eta|^2.
\end{aligned} $$
Taking into account that
\[(1+|\zeta|^2)^{\frac{q+p-4}{4}}|\eta||\xi|=\left((1+|\zeta|^2)^{\frac{p-2}{4}}|\xi|\right)
\left((1+|\zeta|^2)^{\frac{q-2}{4}}|\eta|\right)\] and using the Young's inequality, we get
\[m(1+|\zeta|^2)^{\frac{p-2}{2}}|\xi|^2 \le 2\sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N \frac{\partial A^{\alpha}_i(\zeta)}{\partial \zeta^{\beta}_j}
(\xi^{\alpha}_i-\eta^{\alpha}_i)\xi^{\beta}_j+\frac{m}{2}(1+|\zeta|^2)^{\frac{p-2}{2}}|\xi|^2+C(1+|\zeta|^2)^{\frac{q-2}{2}}|\eta|^2\] for a suitable constant $C$.
Then, \eqref{key-3} easily follows. \end{proof}
\section{Proof of Theorem~\ref{T:main-weak}}\label{apriori}
We proceed via difference quotients technique. Due to the assumed regularity of the solution $u$ and thanks to \eqref{crescitaai}, it follows from \eqref{soldebole-Dirichlet} that $$ \int_{\Omega}\sum_{i=1}^n \sum_{\alpha=1}^N (A^{\alpha}_i(Du(x+he_k))-A^{\alpha}_i(Du(x)))\varphi^{\alpha}_{x_i}\; dx =0, $$ for all $\varphi\in W^{1,q}_0(\Omega_h;\mathbb{R}^N)$, all $h\in (0,1)$ and all $k=1,\ldots, n$, where $\Omega_h:=\{x\in \Omega: \; B_{2h}(x)\subset \Omega\}$ and $e_k$ is a unit vector in the $k$-th direction. Hence, setting $$ \varphi(x):=(u(x+he_k)-u(x))\tau^2(x) $$ with $\tau \in \mathcal{C}^{\infty}_c(\Omega_{2h})$ (which is an admissible choice), we obtain the starting identity \begin{equation}\label{start} \begin{split} 0&=\int_{\Omega}\sum_{i=1}^n \sum_{\alpha=1}^N (A^{\alpha}_i(Du(x+he_k))-A^{\alpha}_i(Du(x)))\tau(x)\cdot\\ &\qquad \cdot \left((u^{\alpha}_{x_i}(x+he_k)-u^{\alpha}_{x_i}(x))\tau(x)+2(u^{\alpha}(x+he_k)-u^{\alpha}(x))\tau_{x_i} \right)\; dx. \end{split} \end{equation} Since $$ \begin{aligned} &A^{\alpha}_i(Du(x+he_k))-A^{\alpha}_i(Du(x))\\ &=\int_0^t\sum_{j=1}^n \sum_{\beta=1}^N \int_0^1 \frac{\partial A^{\alpha}_i(tDu(x+he_k)+(1-t)Du(x))}{\partial \zeta^{\beta}_j}(u^{\beta}_{x_j}(x+he_k)-u^{\beta}_{x_j}(x))\; dt, \end{aligned} $$ the identity \eqref{start} can be equivalently rewritten as \begin{equation}\label{start2} \begin{split} 0&=\int_{\Omega}\sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N\int_0^1 \frac{\partial A^{\alpha}_i(tDu(x+he_k)+(1-t)Du(x))}{\partial \zeta^{\beta}_j}(u^{\beta}_{x_j}(x+he_k)-u^{\beta}_{x_j}(x))\tau(x)\cdot\\ &\qquad \cdot \left((u^{\alpha}_{x_i}(x+he_k)-u^{\alpha}_{x_i}(x))\tau(x)+2(u^{\alpha}(x+he_k)-u^{\alpha}(x))\tau_{x_i} \right)\; dt\; dx. \end{split} \end{equation} Abbreviating for the moment $$ \xi^{\alpha}_i:= \tau(x)(u^{\alpha}_{x_i}(x+he_k)-u^{\alpha}_{x_i}(x)), \quad \eta^{\alpha}_i:=-2(u^{\alpha}(x+he_k)-u^{\alpha}(x))\tau_{x_i}(x) $$ and $$ \zeta:=tDu(x+he_k)+(1-t)Du(x), $$ we can formally rewrite \eqref{start2} as \begin{equation*} \begin{split} 0&=\int_{\Omega}\int_0^1\sum_{i,j=1}^n \sum_{\alpha,\beta=1}^N \frac{\partial A^{\alpha}_i(\zeta)}{\partial \zeta^{\beta}_j}\xi^{\beta}_j\left(\xi^{\alpha}_i-\eta^{\alpha}_i\right)\; dt\; dx. \end{split} \end{equation*} Thus, using \eqref{key-3}, we obtain (here $C$ is some constant depending only on $m,M,n,N,p,q$) $$
\int_{\Omega}\int_0^1 (1+|\zeta|^2)^{\frac{p-2}{2}}|\xi|^2\; dt\; dx \le C\int_{\Omega}\int_0^1(1+|\zeta|^2)^{\frac{q-2}{2}}|\eta|^2\; dt \; dx, $$ which in terms of original variables after division by $h^2$ means that \begin{equation} \begin{split}\label{huz}
&\int_{\Omega}\int_0^1 (1+|tDu(x+he_k)+(1-t)Du(x))|^2)^{\frac{p-2}{2}}\frac{|Du(x+he_k)-Du(x)|^2}{h^2}\tau^2(x)\; dt\; dx \\
&\le 4C\int_{\Omega}\int_0^1(1+|tDu(x+he_k)+(1-t)Du(x))|^2)^{\frac{q-2}{2}}\frac{|u(x+he_k)-u(x)|^2}{h^2}|D\tau(x)|^2\; dt \; dx. \end{split} \end{equation}
Finally, we let $h\to 0_+$. First, we focus on the limit in the term on the right hand side of \eqref{huz}. In case that $q\le 2$, we use the assumption that $u\in W^{1,2}_{\rm loc}(\Omega; \mathbb{R}^N)$ and therefore, we can use the Lebesgue dominated convergence theorem to conclude that $$ \begin{aligned}
\limsup_{h\to 0}&\int_{\Omega}\int_0^1(1+|tDu(x+he_k)+(1-t)Du(x))|^2)^{\frac{q-2}{2}}\frac{|u(x+he_k)-u(x)|^2}{h^2}|D\tau(x)|^2\; dt \; dx\\
&=\int_{\Omega}(1+|Du|^2)^{\frac{q-2}{2}}|u_{x_k}|^2|D\tau|^2 \; dx\le \int_{\Omega}(1+|Du|^2)^{\frac{q}{2}}|D\tau|^2 \; dx. \end{aligned} $$ Next, if $q>2$, we use the H\"{o}lder inequality, the assumption $u\in W^{1,q}_{\rm loc}(\Omega; \mathbb{R}^N)$ and the Lebesgue dominated convergence theorem to conclude $$ \begin{aligned}
\limsup_{h\to 0}&\int_{\Omega}\int_0^1(1+|tDu(x+he_k)+(1-t)Du(x))|^2)^{\frac{q-2}{2}}\frac{|u(x+he_k)-u(x)|^2}{h^2}|D\tau(x)|^2\; dt \; dx\\ &=
\limsup_{h\to 0}\int_{\Omega}\int_0^1\left(((1+|tDu(x+he_k)+(1-t)Du(x))|^2)^{\frac{q-2}{2}}|D\tau(x)|^{2\frac{q-2}{q}}\right)\cdot \\ &\qquad
\cdot\frac{|u(x+he_k)-u(x)|^2}{h^2}|D\tau(x)|^{\frac{4}{q}}\; dt \; dx\\ &\le
\limsup_{h\to 0}\int_0^1 \left(\int_{\Omega}((1+|tDu(x+he_k)+(1-t)Du(x))|^2)^{\frac{q}{2}}|D\tau(x)|^{2}\; dx \right)^{\frac{q-2}{q}}\cdot \\ &\qquad
\left(\int_{\Omega}\frac{|u(x+he_k)-u(x)|^q}{h^q}|D\tau(x)|^{2} \; dx\right)^{\frac{2}{q}}\; dt\\
&\le \int_{\Omega}(1+|Du|^2)^{\frac{q}{2}}|D\tau|^2 \; dx. \end{aligned} $$
Consequently, substituting these limits into \eqref{huz}, we have \begin{equation} \begin{split}\label{huz2}
&\limsup_{h\to 0}\int_{\Omega}\int_0^1 (1+|tDu(x+he_k)+(1-t)Du(x))|^2)^{\frac{p-2}{2}}\frac{|Du(x+he_k)-Du(x)|^2}{h^2}\tau^2(x)\; dt\; dx \\
&\le 4C\int_{\Omega}(1+|Du|^2)^{\frac{q}{2}}|D\tau|^2 \; dx. \end{split} \end{equation} >From this estimate it immediately follows that $u\in W^{2,\min\{2,p\}}_{\rm loc}(\Omega; \mathbb{R}^N)$, in particular we know that $D^2u$ exists and that for almost all $x$
$$ \frac{Du(x+he_k)-Du(x)}{h}\to (D^2 u)_{x_k}(x) $$ where $(D^2 u)_{x_k}$ stands for $\frac{\partial Du}{\partial x_k}$. Therefore, we can use the Fatou lemma in \eqref{huz2} to conclude \begin{equation*} \begin{split}
&\int_{\Omega}(1+|Du|^2)^{\frac{p-2}{2}}| D^2 u_{x_k}(x)|^2\tau^2\; dx\le 4C\int_{\Omega}(1+|Du|^2)^{\frac{q}{2}}|D\tau|^2 \; dx. \end{split} \end{equation*} Since $k$ is arbitrary, the relation \eqref{exdopoH3-1} obviously follows. In addition, using the following algebraic inequality $$
|DV(Du)|^2\le K(1+|Du|^2)^{\frac{p-2}{2}}|D^2u|^2, $$ we see that \eqref{exdopoH3-V} holds as well. Hence the proof is complete.
\section{Proof of Theorem~\ref{t:main}}\label{P-t:main}
We shall start by recalling the definition of the Sobolev embedding exponent \begin{equation}\label{2star} 2^{*}=\left\{\begin{array}{ll}\frac{2n}{n-2}&\text{if $n\ge 3$}\\ \text{arbitrary $>2$} & \text{if $n=2$}.\end{array}\right.\end{equation} The value $2^*$ in dimension $n=2$ will be finally chosen sufficiently large. Since $u$ is assumed to be a weak solution belonging to $W^{1,\max\{q,2\}}_{\rm loc}(\Omega;\mathbb{R}^N)$, we can use Theorem~\ref{T:main-weak} and after summing \eqref{exdopoH3-V} and \eqref{exdopoH3-1}, we obtain the starting inequality valid for all $\tau \in \mathcal{C}^{\infty}_c (\Omega)$ \begin{equation}\label{split}
\int_{\Omega}\left((1+|Du|^2)^{\frac{p-2}{2}}|D^2u|^2\tau^2 + |DV(Du)|^2\tau^2\right) \; dx \le K\int_{\Omega}(1+|Du|^2)^{\frac{q}{2}}|D\tau|^2\; dx \end{equation}
Moreover, we remark that \begin{equation}\label{e:Lq}
\int_{\Omega}(1+|Du|^2)^{\frac{q}{2}}|D\tau|^2\; dx\le c\int_{\Omega}\left( (1+|Du|^2)^{\frac{p}{2}}+|V(Du)|^{\frac{2q}{p}}\right)|D\tau|^2\,dx. \end{equation}
Indeed, in $\{|Du|\le 1\}$ we have \[(1+|Du|^2)^{\frac{q}{2}}\le 2(1+|Du|^2)^{\frac{p}{2}}\]
and, in $\{|Du|> 1\}$,
\[(1+|Du|^2)^{\frac{q}{2}}= \left\{(1+|Du|^2)^{\frac{p-2}{2}}(1+|Du|^2)\right\}^{\frac{q}{p}}\le 2^{\frac{q}{p}}
|V(Du)|^{\frac{2q}{p}}.\]
Next, we split the proof for the case i) and ii).
\subsection{The case $q<p\frac{n+2}{n}$} In this case, we first use the Sobolev embedding to conclude that (with some $C$ depending on $2^*$) $$ \begin{aligned}
\|V(Du)\tau\|_{2^*}^2 &\le C\|D(V(Du)\tau)\|_2^2 \le 2C\int_{\Omega} \left(|DV(Du)|^2\tau^2 + |V(Du)|^2|D\tau|^2\right)\; dx \\
&\le 2C\int_{\Omega}\left(|DV(Du)|^2\tau^2 + (1+|Du|^2)^{\frac{p}{2}}|D\tau|^2\right)\; dx. \end{aligned} $$ Using this inequality in \eqref{split},
and taking into account \eqref{e:Lq} we get \begin{equation}\label{split2} \begin{split}
&\|V(Du)\tau\|_{2^*}^2+ \int_{\Omega}\left((1+|Du|^2)^{\frac{p-2}{2}}|D^2u|^2\tau^2 + |DV(Du)|^2\tau^2\right) \; dx \\
&\quad\le K_1\int_{\Omega}\left((1+|Du|^2)^{\frac{p}{2}}|D\tau|^2+(1+|Du|^2)^{\frac{q}{2}}|D\tau|^2\right)\; dx\\
&\quad\le K_2\int_{\Omega}\left((1+|Du|^2)^{\frac{p}{2}}|D\tau|^2+|V(Du)|^{\frac{2q}{p}}|D\tau|^2\right)\; dx. \end{split} \end{equation} In particular, we have that $V(Du)\in L_{\rm loc}^{2^*}$.
Let us now estimate the last integral on the right hand side. Since $q\in (p,p\frac{2^*}{2})$, which follows from the assumption that $q<p \frac{n+2}{n}$ (note here that the value of $2^*$ in dimension $n=2$ has to be chosen greater than $\frac{2q}{p}$), there exists a unique $\theta\in (0,1)$ such that $$ \frac{q}{2}=\frac{p}{2}(1-\theta)+\frac{p2^*}{4}\theta, \qquad \theta:=\frac{q-p}{p(\frac{2^*}{2}-1)}. $$
As we will prove below, under our assumptions on the exponents $p$ and $q$ and, if $n=2$, with a suitable choice of $2^*$, we have \begin{equation} 2>2^*\theta. \label{asss} \end{equation}
Consider $\eta\in \mathcal{C}^{\infty}_c(\Omega)$ an arbitrary nonnegative cut-off function and set $$ \tau:=\eta^{\gamma} \qquad \textrm{with } \gamma:=\frac {2}{ 2-2^*\theta}. $$ We have that \begin{equation}\label{eta-tau}
\frac{|D\tau|^2}{\tau^{2^*\theta}}= \gamma^2\eta^{\gamma(2-2^*\theta)-2}|D \eta|^2= \gamma^2|D\eta|^2. \end{equation} Then by the H\"older inequality, we have \begin{equation*} \begin{aligned} \int_{\Omega}
|V(Du)|^{\frac{2q}{p}}|D\tau|^2\,dx&=\int_{\Omega}
|V(Du)|^{2(1-\theta)}(V(Du)\tau)^{2^*\theta} \frac{|D\tau|^2}{\tau^{2^*\theta}}\,dx\\
&\le \|V(Du)\|_2^{2(1-\theta)}\|V(Du)\tau\|_{2^*}^{2^*\theta} \left\|\frac{|D\tau|^2}{\tau^{2^*\theta}}\right\|_{\infty} \end{aligned} \end{equation*} and we can apply the Young 's inequality to deduce that for arbitrary $\varepsilon>0$ we have \begin{equation}\label{ssop} \begin{aligned} \int_{\Omega}
|V(Du)|^{\frac{2q}{p}}|D\tau|^2\,dx&\le \varepsilon \|V(Du)\tau\|_{2^*}^2 + C(\varepsilon,\gamma) \|V(Du)\|_2^{2(1-\theta)\gamma}
\left\|\frac{|D\tau|^2}{\tau^{2^*\theta}}\right\|^{\gamma}_{\infty}. \end{aligned} \end{equation} Therefore, combining \eqref{split2}, \eqref{ssop}, \eqref{eta-tau} and taking into account that
$V(Du)\le (1+|Du|^2)^{\frac{p}{4}}$, with a proper choice of $\varepsilon>0$, we obtain \begin{equation*} \begin{split}
&\|V(Du)\tau\|_{2^*}^2+\int_{\Omega}\left((1+|Du|^2)^{\frac{p-2}{2}}|D^2u|^2\eta^{2\gamma} + |DV(Du)|^2\eta^{2\gamma}\right) \; dx \\
&\quad\le K(\gamma,\|D \eta\|_{\infty}) \left(\int_{\Omega}(1+|Du|^2)^{\frac{p}{2}}\; dx\right)^{\tilde{q}}, \end{split} \end{equation*} with some power $\tilde{q}$ whose value depends on $p,q$ and $\gamma$. >From this inequality the statement i) of Theorem~\ref{t:main} follows directly.
Now, we check the validity of \eqref{asss}, which, by using of definition of $\theta$,
it can be written as \[q<2p\left(1-\frac{1}{2^*}\right).\] If $n=2$, we can choose $2^*$ arbitrarily large, therefore in this case the condition \eqref{asss} reduces to $q<2p$, which is exactly the assumption \eqref{ipotesip-q} for $n=2$. If $n\ge 3$ we have $2^*=2n/(n-2)$ and the above condition is then equivalent to \[ q<p\frac{n+2}{n},\] which is nothing else than the assumption \eqref{ipotesip-q}. Hence the proof of the statement i) is finished.
\subsection{The case $q<p+2$ and $p<n$} We again start to estimate the integral on the right hand side of \eqref{split}. Using a simple inequality and the integration by parts, we find that (here $K$ is again a generic constant depending only on $q$) \begin{align}\nonumber
\int_{\Omega} &(1+|Du|^2)^{\frac{q}{2}} |D\tau|^2\; dx \le K+
\sum_{k=1}^n\sum_{\alpha=1}^N \int_{\Omega}(1+|Du|^2)^{\frac{q-2}{2}}
u^{\alpha}_{x_k}u^{\alpha}_{x_k}|D\tau|^2\,dx\\ \nonumber
&= K-\sum_{k=1}^n\sum_{\alpha=1}^N \int_{\Omega}\left((1+|Du|^2)^{\frac{q-2}{2}}
u^{\alpha}_{x_k}|D\tau|^2 \right)_{x_k}u^{\alpha}\; dx\\
&\le K+K\|u\|_{\infty} \int_{\Omega}\left((1+|Du|^2)^{\frac{q-2}{2}}
|D^2 u||D\tau|^2 +(1+|Du|^2)^{\frac{q-1}{2}}|D\tau||D^2\tau|\right)\; dx. \label{e:stima}\end{align}
Let us now set $\tau:=\eta^{\gamma}$, $\gamma\ge 2$ to be chosen later, where $\eta\in \mathcal{C}^{\infty}_c(\Omega)$ is an arbitrary nonnegative cut-off function.
By the Young 's inequality, \begin{align*}\nonumber &
K\|u\|_{\infty} \int_{\Omega}(1+|Du|^2)^{\frac{q-2}{2}}
|D^2 u||D\tau|^2\,dx \\&= \int_{ \Omega}
\left\{(1+|Du|^2)^{\frac{p-2}{4}}|D^2u|\tau
\right\}\left\{\|u\|_{\infty} (1+|Du|^2)^{\frac{2q-p-2}{4}}\frac{|D\tau|^2}{\tau} \right\} \,dx \nonumber \\&\le \varepsilon
\int_{\Omega}(1+|Du|^2)^{\frac{p-2}{2}}|D^2u|^2\tau^2\; dx
+ c_{\varepsilon,K} \|u\|^2_{\infty} \int_{\Omega}(1+|Du|^2)^{\frac{2q-p-2}{2}}\frac{|D\tau|^4}{\tau^2}\; dx.
\end{align*} Therefore,
\begin{equation}\label{inter} \begin{split}
&\int_{\Omega} (1+|Du|^2)^{\frac{q}{2}} |D\tau|^2\; dx \le \varepsilon \int_{\Omega}(1+|Du|^2)^{\frac{p-2}{2}}|D^2u|^2\tau^2\; dx \\
&\quad + K+K\|u\|^2_{\infty} \int_{\Omega}(1+|Du|^2)^{\frac{2q-p-2}{2}}\frac{|D\tau|^4}{\tau^2}\; dx +
K\|u\|_{\infty}\int_{\Omega}(1+|Du|^2)^{\frac{q-1}{2}}|D\tau||D^2\tau|\; dx, \end{split} \end{equation} with a possibly different positive constant $K$ than before.
\medbreak Let us now discuss first the case $q\in [p,p+1]$.
If $q$ belongs to this range, the above inequality immediately reduces to \begin{equation*} \begin{split}
&\int_{\Omega} (1+|Du|^2)^{\frac{q}{2}} |D\tau|^2\; dx \le \varepsilon \int_{\Omega}(1+|Du|^2)^{\frac{p-2}{2}}|D^2u|^2\tau^2\; dx \\
&\quad + K+K\|u\|^2_{\infty} \int_{\Omega}(1+|Du|^2)^{\frac{p}{2}}\frac{|D\tau|^4}{\tau^2}\; dx +K\|u\|_{\infty}\int_{\Omega}(1+|Du|^2)^{\frac{p}{2}}|D\tau||D^2\tau|\; dx. \end{split} \end{equation*}
Let us now choose $\gamma=2$, that is $\tau:=\eta^2$. Thus we get \begin{equation}\label{intera} \begin{split}
&\int_{\Omega} (1+|Du|^2)^{\frac{q}{2}} |D\tau|^2\; dx \le \varepsilon \int_{\Omega}(1+|Du|^2)^{\frac{p-2}{2}}|D^2u|^2\tau^2\; dx \\
&\quad + K+C(\|u\|_{\infty}, \|\eta\|_{2,\infty}) \int_{\Omega}(1+|Du|^2)^{\frac{p}{2}}\; dx. \end{split} \end{equation} Hence by \eqref{split} and taking a proper $\varepsilon>0$, so that we can absorb the first term on the right hand side in \eqref{intera} by the left hand side in \eqref{split}, it is not difficult to arrive to the statement ii) of Theorem~\ref{t:main} for $q\in [p,p+1]$.
\medbreak Next, we focus on the case when $q\in (p+1,p+2)$.
There exist $\theta_1,\theta_2\in (0,1)$ such that \begin{align} q-1&=p(1-\theta_1)+q\theta_1, && \theta_1:= \frac{q-1-p}{q-p},\\ 2q-p-2&=p(1-\theta_2)+ q\theta_2, && \theta_2:=\frac{2(q-1-p)}{q-p}. \end{align}
In addition, considering $\tau=\eta^\gamma$ with \begin{equation}\label{special}
\gamma= \frac{2-\theta_2}{1-\theta_2}, \end{equation}
we have that
$\frac{|D\tau|^{2-\theta_2}}{\tau}=\gamma |D\eta|^{2-\theta_2}$.
With this setting, we can now estimate the remaining integrals on the right hand side of \eqref{inter} by means of the H\"{o}lder inequality as follows $$ \begin{aligned}
&\int_{\Omega}(1+|Du|^2)^{\frac{2q-p-2}{2}}\frac{|D\tau|^4}{\tau^2}\; dx = \int_{\Omega}\left((1+|Du|^2)^{\frac{q}{2}}|D\tau|^2\right)^{\theta_2}(1+|Du|^2)^{\frac{p(1-\theta_2)}{2}}\frac{|D\tau|^{4-2\theta_2}}{\tau^2}\; dx\\
&\quad\le C\left\|\frac{|D\tau|^{4-2\theta_2}}{\tau^2}\right\|_{\infty}\|(1+|Du|)\|_p^{p(1-\theta_2)} \left(\int_{\Omega}(1+|Du|^2)^{\frac{q}{2}}|D\tau|^2\; dx\right)^{\theta_2}. \end{aligned} $$
Then the above estimate reduces to \begin{equation}\label{godot} \begin{aligned}
&\int_{\Omega}(1+|Du|^2)^{\frac{2q-p-2}{2}}\frac{|D\tau|^4}{\tau^2}\; dx \\
&\qquad \le C(\theta_2,\|\eta\|_{1,\infty})\|(1+|Du|)\|_p^{p(1-\theta_2)} \left(\int_{\Omega}(1+|Du|^2)^{\frac{q}{2}}|D\tau|^2\; dx\right)^{\theta_2}. \end{aligned} \end{equation}
We proceed similarly also with the remaining integral in \eqref{inter}, i.e., using the H\"{o}lder inequality, we have \begin{equation}\label{godot2} \begin{aligned}
&\int_{\Omega}(1+|Du|^2)^{\frac{q-1}{2}}|D\tau||D^2\tau|\; dx\\ &\quad =\int_{\Omega}(1+|Du|^2)^{\frac{p(1-\theta_1)}{2}}\left((1+|Du|^2)^{\frac{q}{2}}|D\tau|^2\right)^{\theta_1}|D\tau|^{1-2\theta_1}|D^2\tau|\; dx\\
&\quad\le K\|(1+|Du|)\|_p^{p(1-\theta_1)} \||D\tau|^{1-2\theta_1}|D^2\tau|\|_{\infty} \left(\int_{\Omega}(1+|Du|^2)^{\frac{q}{2}}|D\tau|^2\; dx \right)^{\theta_1}\\
&\quad\le C(\|\tau\|_{2,\infty},\theta_2)\|(1+|Du|)\|_p^{p(1-\theta_1)} \left(\int_{\Omega}(1+|Du|^2)^{\frac{q}{2}}|D\tau|^2\; dx \right)^{\theta_1}, \end{aligned} \end{equation} where the last inequality follows from the fact that $1-2\theta_1=1-\theta_2>0$.
Finally, using \eqref{godot} and \eqref{godot2} in \eqref{inter}, keeping in mind the special choice of $\tau$ in \eqref{special} and applying the Young's inequality (notice that $\theta_1,\theta_2<1$) we observe that \begin{equation*} \begin{split}
\int_{\Omega} (1+|Du|^2)^{\frac{q}{2}} |D\tau|^2\; dx &\le \varepsilon \int_{\Omega}(1+|Du|^2)^{\frac{p-2}{2}}|D^2u|^2\tau^2\; dx \\
&\quad + C(\varepsilon,\theta_2,\|\eta\|_{2,\infty}, \|u\|_{\infty}, \|u\|_{1,p}). \end{split} \end{equation*} Thus, going back to \eqref{split}, choosing $\varepsilon>0$ sufficiently small to absorb the term involving the second derivatives by the left hand side, we finally get the statement ii) of Theorem~\ref{t:main}.
\section{Proof of Theorem~\ref{t:main2}}
In this final section we establish the existence of a weak solution to the Dirichlet problem \eqref{dirichlet}, under the assumption \eqref{ipotesi-dato} on the boundary datum $u_0$; i.e., \[u_0 \in W^{1,r} (\Omega;\mathbb{R}^N), \qquad r:=\max\left\{2,p\frac{q-1}{p-1}\right\}.\]
We use an approximation procedure.
For arbitrary $\epsilon \in (0,1)$ we introduce the approximate problem ($\alpha=1,\ldots,N$) \begin{equation}\label{systemepsilon} \left\{\begin{array}{ll}\sum_{i=1}^n\frac{\partial }{\partial x_i}\left(A_{\epsilon,i}^{\alpha}(Du_{\epsilon})\right)=0 &\textrm{in } \Omega,\\ u_{\epsilon}=u_0 &\textrm{on }\partial \Omega, \end{array}\right. \end{equation} where $A_{\epsilon,i}^{\alpha}: \mathbb{R}^{nN}\to \mathbb{R}$ is defined as \begin{equation}
\label{Aepsilon}A_{\epsilon,i}^{\alpha}(\xi):=A_{i}^{\alpha}(\xi)+\epsilon (1+|\xi|^2)^{\frac{\max\{q,2\}-2}{2}}\xi_i^{\alpha}. \end{equation}
In addition, in case we deal with the statement ii) of the theorem, we shall require that $u_0\in L^{\infty}(\partial \Omega)$.
Due to \eqref{crescitaai}, \eqref{casop>2} and \eqref{casop<2} we have that $A_{\epsilon,i}^{\alpha}(\xi)$ satisfies the following properties:
\[\sum_{i=1}^n\sum_{\alpha=1}^N A_{\epsilon,i}^{\alpha}(\xi)\xi_i^{\alpha}\ge \epsilon |\xi|^{\max\{q,2\}}-\lambda,\] \begin{equation} \label{e:crescita-eps}
|A_{\epsilon,i}^{\alpha}(\xi)|\le M'(1+|\xi|)^{\max\{q,2\}-1}. \end{equation} for some positive $\lambda$ and $M'$ independent on $\epsilon$. We can apply the theory of monotone operators (see e.g. \cite{lerlio65,browder,hart-stamp}) to prove the existence of a unique solution to \eqref{systemepsilon}, i.e., the existence of $u_{\epsilon}\in u_0+W^{1,\max\{q,2\}}(\Omega; \mathbb{R}^N)$ fulfilling \begin{equation} \int_{\Omega}\sum_{\alpha=1}^N \sum_{i=1}^n{A_{\epsilon,i}^{\alpha}}(Du_{\epsilon})\varphi_{x_i}^\alpha\,dx=0\qquad \forall \varphi\in W_0^{1,\max\{q,2\}}(\Omega;\mathbb{R}^N).
\label{firstvariation} \end{equation}
\subsection{First a~priori estimates} We now derive estimates for $u_{\epsilon}$ independent of $\epsilon$.
Using $\varphi:=u_{\epsilon}-u_0$ as a test function in \eqref{firstvariation}, we get
\begin{equation}\label{e:begin} \begin{split} 0&=\int_{\Omega}\sum_{\alpha=1}^N \sum_{i=1}^n{A_{\epsilon,i}^{\alpha}}(Du_{\epsilon})((u_{\epsilon})_{x_i}^{\alpha}-(u_0)_{x_i}^{\alpha})\; dx\\ &=\int_{\Omega}\sum_{\alpha=1}^N \sum_{i=1}^n\left\{{A_{i}^{\alpha}}(Du_{\epsilon})((u_{\epsilon})_{x_i}^{\alpha}-(u_0)_{x_i}^{\alpha})+
\epsilon(1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}-2}{2}} (u_{\epsilon})^{\alpha}_{x_i}((u_{\epsilon})_{x_i}^{\alpha}-(u_0)_{x_i}^{\alpha})\right\}\; dx\\
&\overset{\eqref{dis-ellitticita}}\ge \int_{\Omega}\left(K^{-1}|Du_{\epsilon}|^p -(1+|Du_0|^2)^{\frac{p(q-1)}{2(p-1)}} +
\epsilon(1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}-2}{2}}|Du_{\epsilon}|(|Du_{\epsilon}|-|Du_0|)\right)\; dx. \end{split} \end{equation}
Since \begin{equation*}
\begin{split}(1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}-2}{2}}|Du_{\epsilon}|(|Du_{\epsilon}|-|Du_0|)=&
(1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}-2}{2}}|Du_{\epsilon}|^2\\ &-(1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}-2}{2}}|Du_{\epsilon}||Du_0|,\end{split} \end{equation*} then \eqref{e:begin} implies
\begin{align}\nonumber &\int_{\Omega}\left(|Du_{\epsilon}|^p +\epsilon
(1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}-2}{2}}|Du_{\epsilon}|^2\right)\; dx\\ &\le c
\int_{\Omega}\left((1+|Du_0|^2)^{\frac{p(q-1)}{2(p-1)}}+\epsilon(1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}-2}{2}}|Du_{\epsilon}||Du_0|\right) \,dx. \label{e:begin2}\end{align} We claim that \eqref{e:begin2} implies
\begin{equation} \int_{\Omega}\left(|Du_{\epsilon}|^p + \frac{\epsilon}{2}
(1+|Du_{\epsilon}|^2)^{\frac{\max\{2,q\}-2}{2}}|Du_{\epsilon}|^2\right)\; dx\le c
\int_{\Omega}(1+|Du_0|^2)^{\frac{r}{2}}\,dx \label{goal} \end{equation}
If $q\le 2$, we can conclude using Young's inequality with exponent $\frac12$ on the last term in \eqref{e:begin2}:
\begin{align*}&|Du_{\epsilon}||Du_0|\le \frac{1}{2c} |Du_{\epsilon}|^2+c' |Du_0|^2. \end{align*} Therefore, recalling that
$r=\max\{2,\frac{p(q-1)}{p-1}\}$ the inequality \eqref{goal} follows.
Otherwise, if $q>2$, the last term in \eqref{e:begin2} can be estimate as follows:
\begin{equation}
\epsilon(1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}-2}{2}}|Du_{\epsilon}||Du_0|
\le \epsilon\left\{c(1+|Du_0|^2)^{\frac{r}{2}}+c(1+|Du_{\epsilon}|^2)^{\frac{q-2}{4}} |Du_{\epsilon}|^{\frac{q}{2}}
|Du_0|\right\}.\label{e:second}
\end{equation}
Indeed, in $\{|Du_{\epsilon}|\le 1\}$ we have \[(1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}-2}{2}}|Du_{\epsilon}||Du_0|\le 2^{\frac{q-2}{2}}|Du_0|\le c(1+|Du_0|^2)^{\frac{r}{2}}\]
and, in $\{|Du_{\epsilon}|> 1\}$,
\[(1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}-2}{2}}|Du_{\epsilon}||Du_0|\le
2^{\frac{q-2}{4}}(1+|Du_{\epsilon}|^2)^{\frac{q-2}{4}}|Du_{\epsilon}|^{\frac{q}{2}}|Du_0|\] and \eqref{e:second} follows.
To estimate the last term in \eqref{e:second}, we use Young's inequality with exponents $2, q ,\frac{2q}{q-2}$. Recalling that $\epsilon<1$, we have
\begin{align}\nonumber & \epsilon c(1+|Du_{\epsilon}|^2)^{\frac{q-2}{4}} |Du_{\epsilon}|^{\frac{q}{2}}
|Du_0|=\epsilon c\left\{(1+|Du_{\epsilon}|^2)^{\frac{q-2}{4}} |Du_{\epsilon}|\right\} |Du_{\epsilon}|^{\frac{q-2}{2}}
|Du_0|\\ &\le\nonumber \frac{\epsilon}{8}
(1+|Du_{\epsilon}|^2)^{\frac{q-2}{2}}|Du_{\epsilon}|^2+
\frac{\epsilon}{8}|Du_{\epsilon}|^{q} +
\epsilon c|Du_0|^q
\\ &\le \frac{\epsilon}{4}
(1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}-2}{2}}|Du_{\epsilon}|^2 +
c(1+|Du_0|^2)^{\frac{r}{2}} \label{e:fourth}\end{align} with $c$ independent of $\epsilon$. Therefore, collecting \eqref{e:begin2}, \eqref{e:second} and \eqref{e:fourth}, the inequality \eqref{goal} follows also in the case $q>2$.
Thus, we can find a universal constant $C>0$ such that (using also the Poincar\'{e} inequality) \begin{equation}
\|u_{\epsilon}\|_{1,p} + \epsilon \|u_{\epsilon}\|^{\max\{q,2\}}_{1,\max\{q,2\}}\le C. \label{f-ap} \end{equation}
If the assumption \eqref{structure} holds, then for every $\alpha\in \{1,\ldots, N\}$ we have \begin{equation} \label{BulHLP}
\sum_{i=1}^nA_{\epsilon,i}^{\alpha}(\xi)\xi_i^{\alpha}\ge \epsilon (1+|\xi|^{2})^{\frac{\max\{2,q\}-2}{2}}|\xi^{\alpha}|^2\ge \epsilon |\xi^{\alpha}|^{\max\{q,2\}} \end{equation}
and \[|A_{\epsilon,i}^{\alpha}(\xi)|\le (K+1)(1+|\xi|^{2})^{\frac{\max\{2,q\}-1}{2}},\]
where $K$ is as in \eqref{crescitaai}. Next we denote $\tilde{M}:=\|u_0\|_{L^{\infty}(\partial \Omega)}$ and define $$ \varphi^{\alpha}:=\max \{u_{\epsilon}^{\alpha}-\tilde{M},0\} \qquad \alpha\in \{1,\ldots, N\}. $$ Evidently, $\varphi=(\varphi^1,\ldots,\varphi^N) \in W^{1,\max \{2,q\}}_0(\Omega;\mathbb{R}^N)$ and can be used as a test function in \eqref{firstvariation}. Doing so, and using the definition of $\varphi$ we obtain (here $\chi_{u_{\epsilon}^{\alpha}\ge \tilde{M}}$ denotes the characteristic function of the set, where $u^{\alpha}_{\epsilon}\ge \tilde{M}$) \begin{equation} 0=\int_{\Omega}\sum_{\alpha=1}^N \sum_{i=1}^n{A_{\epsilon,i}^{\alpha}}(Du_{\epsilon})\varphi_{x_i}^\alpha\,dx=\int_{\Omega}\sum_{\alpha=1}^N \sum_{i=1}^n{A_{\epsilon,i}^{\alpha}}(Du_{\epsilon})Du_{\varepsilon}^{\alpha}\chi_{u_{\epsilon}^{\alpha}\ge \tilde{M}}\,dx
\label{firstvariationM} \end{equation} Using finally \eqref{BulHLP}, we see that \begin{equation} \begin{split} 0&=\int_{\Omega}\sum_{\alpha=1}^N \sum_{i=1}^n{A_{\epsilon,i}^{\alpha}}(Du_{\epsilon})Du_{\varepsilon}^{\alpha}\chi_{u_{\epsilon}^{\alpha}\ge \tilde{M}}\,dx\ge \epsilon
\int_{\Omega}\sum_{\alpha=1}^N |Du_{\epsilon}^{\alpha}|^{\max\{q,2\}}\chi_{u_{\epsilon}^{\alpha}\ge \tilde{M}}\,dx\\
&=\epsilon\int_{\Omega}\sum_{\alpha=1}^N |D\varphi^{\alpha}|^{\max\{q,2\}}\,dx.
\label{firstvariationMM} \end{split} \end{equation}
Consequently, $\varphi$ is a constant function. Since it has zero trace, it must be identically zero and it directly follows from its definition that $u_{\varepsilon}^{\alpha}\le \tilde{M}=\|u_0\|_{L^{\infty}(\partial \Omega)}$ for all $\alpha\in \{1,\ldots, N\}$. The minimum principle can be obtained by repeating step by step the above procedure for a test function defined as $$ \varphi^{\alpha}:=\min \{u_{\epsilon}^{\alpha}+\tilde{M},0\} \qquad \alpha\in \{1,\ldots, N\}. $$ Therefore, we conclude that, for every $\epsilon\in (0,1)$, \begin{equation}\label{apest3}
\|u_{\epsilon}\|_{L^{\infty}(\Omega)}\le \|u_0\|_{L^{\infty}(\partial \Omega)}. \end{equation}
\subsection{Uniform higher order estimates}
Due to the proof of a~priori estimates we can use Theorem~\ref{T:main-weak} to get the existence of the second order derivatives of $u_{\epsilon}$, but with their estimates depending on $\epsilon$. Nevertheless, we can repeat step by step the estimates in Theorem~\ref{T:main-weak} to get the following inequality \begin{equation}\label{hot} \begin{aligned}
&\int_{\Omega}(1+|Du_{\epsilon}|^2)^{\frac{p-2}{2}}|D^2u_{\epsilon}|^2\tau^2\; dx \le c\int_{\Omega}(1+|Du_{\epsilon}|^2)^{\frac{q}{2}}|D\tau^2|\; dx \\ &\qquad -c\epsilon\int_{\Omega}\sum_{i,k=1}^n\sum_{\alpha=1}^N
\left((1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}-2}{2}}(u_{\epsilon})^{\alpha}_{x_i}\right)_{x_k} \left((u_{\epsilon})^{\alpha}_{x_k}\tau^2 \right)_{x_i}\; dx \end{aligned} \end{equation} for every $\tau\in \mathcal{C}_c^{\infty}(\Omega)$.
Thus, we need to bound uniformly the last integral. By a rather standard manipulation and using the Young inequality, it is not difficult to check that $$ \begin{aligned}
&\sum_{i,k=1}^n\sum_{\alpha=1}^N \left((1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}-2}{2}}(u_{\epsilon})^{\alpha}_{x_i}\right)_{x_k} \left((u_{\epsilon})^{\alpha}_{x_k}\tau^2 \right)_{x_i}\\
&\qquad \ge (1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}-2}{2}}|D^2u_{\epsilon}|^2\tau^2 - 2\max\{q,2\} (1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}-2}{2}}|D^2u_{\epsilon}|\tau |Du_{\epsilon}| |D\tau|\\
&\qquad \ge -C(1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}}{2}}|D\tau|^2 \end{aligned} $$ with $C$ independent of $\epsilon$. Substituting this into \eqref{hot}, we derive \begin{equation}\label{hot2} \begin{aligned}
&\int_{\Omega}(1+|Du_{\epsilon}|^2)^{\frac{p-2}{2}}|D^2u_{\epsilon}|^2\tau^2\; dx \le c\int_{\Omega}(1+|Du_{\epsilon}|^2)^{\frac{q}{2}}|D\tau|^2\; dx\\
&\qquad +c\epsilon \int_{\Omega}(1+|Du_{\epsilon}|^2)^{\frac{\max\{q,2\}}{2}}|D\tau|^2\; dx\\
&\qquad \overset{\eqref{f-ap}}\le c +c\int_{\Omega}(1+|Du_{\epsilon}|^2)^{\frac{q}{2}}|D\tau|^2\; dx. \end{aligned} \end{equation} Hence, we are in the same starting position as in the proof of Theorem~\ref{t:main} and due to uniform ($\epsilon$-independent) uniform bounds \eqref{f-ap} and \eqref{apest3}, we deduce that for arbitrary open $\Omega'\subset \overline{\Omega'} \subset \Omega$, \begin{equation}
\int_{\Omega'}\left(|Du_{\epsilon}|^{q}+|D V(Du_{\epsilon})|^2+ (1+|Du_{\epsilon}|^2)^{\frac{p-2}{2}}|D^2u_{\epsilon}|^2\right)\, dx \le C(\Omega', u_0). \label{exdopoH3-V2ns} \end{equation} Further, it is then not difficult to observe with the help of the H\"{o}lder inequality that \begin{equation}
\int_{\Omega'}|D^2u_{\epsilon}|^{\min\{2,p\}}\, dx \le C(\Omega', u_0). \label{exdopoH3-V2ns3} \end{equation}
\subsection{Limit $\epsilon \to 0$} Using the uniform bounds \eqref{f-ap}, \eqref{exdopoH3-V2ns} and \eqref{exdopoH3-V2ns3}, the compact Sobolev embedding and the diagonal procedure, we can find a subsequence, that we do not relabel, and it exists \[u\in (u_0+W^{1,p}(\Omega; \mathbb{R}^N))\cap W_{\rm loc}^{1,q}(\Omega; \mathbb{R}^N) \] such that for arbitrary open $\Omega'\subset \overline{\Omega'} \subset \Omega$, we have \begin{align} u^{\epsilon}&\rightharpoonup u &&\textrm{weakly in }W^{1,p}(\Omega; \mathbb{R}^N),\label{co1}\\ u^{\epsilon}&\rightharpoonup u &&\textrm{weakly in }W^{1,q}(\Omega'; \mathbb{R}^N),\label{co2}\\ Du^{\epsilon}&\to Du &&\textrm{strongly in }L^p(\Omega; \mathbb{R}^N),\label{co3}\\ Du^{\epsilon}&\to Du &&\textrm{almost everywhere in }\Omega,\label{co4}\\
\epsilon (1+|Du^{\epsilon}|^2)^{\frac{\max(2,q)-2}{2}}Du^{\epsilon}&\to 0 &&\textrm{strongly in }L^1(\Omega'; \mathbb{R}^{nN}).\label{co5} \end{align} Having \eqref{co1}--\eqref{co5}, it is easy to let $\epsilon \to 0$ in \eqref{firstvariation} with arbitrary $\varphi \in \mathcal{C}^{\infty}_c(\Omega; \mathbb{R}^N)$ to deduce \eqref{soldebole-Dirichlet} for the same class of $\varphi$'s. The density result then leads to the validity of \eqref{soldebole-Dirichlet} in the full generality. This finishes the proof.
\end{document} | arXiv |
AROUND FROBENIUS DISTRIBUTIONS AND RELATED TOPICS III
October 5-6-7, 2022
This is the third edition of a conference on the theme of Frobenius distributions. The first edition was organized by Victoria Cantoral Farfán and Seoyoung Kim in 2020; see here. The second edition was organized by Alina Carmen Cojocaru and Francesc Fité in 2021; see here.
Alina Carmen Cojocaru (University of Illinois at Chicago, USA)
Florent Jouve (Université de Bordeaux, France)
Elisa Lorenzo García (Université de Neuchâtel, Switzerland & Université de Rennes, France)
Confirmed speakers
Jonas Bergström
Philippe Michel
Dante Bonolis
Peter Sarnak
Chantal David
Kaneenika Sinha
David Kohel
Yunqing Tang
Peter Koymans
Jesse Thorner
Wanlin Li
David Zywina
In order to participate, fill in the registration form .
Note that registration is free, but required in order to be admitted in the conference.
Schedule of talks
Chicago time Wednesday, October 5 Thursday, October 6 Friday, October 7 Paris time
8:30am-9:20am Sinha David Kohel 3:30pm-4:20pm
9:25am-10:15am Zywina Li Michel 4:25pm-5:15pm
10:20am-11:10am Bergström Thorner Bonolis 5:20pm-6:10pm
11:10am-1:00pm Break 6:10pm-8:00pm
1:00pm-1:50pm Tang 8:00pm-8:50pm
1:55pm-2:45pm Sarnak 8:55pm-9:45pm
2:50pm-3:40pm Koymans 9:50pm-10:40pm
Jonas Bergström (Stockholms universitet, Sweden)
Lower bounds on the maximal number of points on curves over finite fields (slides)
Abstract: In this talk I will present three approaches to finding lower bounds on the maximal number of points on curves over finite fields. We will focus on the one involving the cohomology of moduli spaces of curves. Using a variant of this approach we will also get information on Serre's obstruction problem (which concerns the asymmetry in the distribution of traces of Frobenius for curves of genus at least three).
This is joint work with E. Howe, E. Lorenzo García and C. Ritzenthaler.
Dante Bonolis (University of Basel, Switzerland)
On the density of rational points on some quadric bundle threefolds
Abstract: In this talk, we present a proof of the Manin-Peyre conjecture for the number of rational points of bounded height outside of a thin subset on a family of Fano threefolds of bidegree $(1,2)$. This is a joint work with Tim Browning and Zhizhong Huang.
Chantal David (Concordia University, Canada)
On the vanishing of twisted $L$-functions of elliptic curves over function fields
Joint work with A. Comeau-Lapointe (Concordia University), M. Lalin (Université de Montréal) and W. Li (Washington University).
Abstract: Let $E$ be an elliptic curve over $\mathbb Q$, and let $\chi$ be a Dirichlet character of order $\ell$ for some prime $\ell\geq 3$. Heuristics based on the distribution of modular symbols and random matrix theory have led to conjectures predicting that the vanishing of the twisted $L$-functions $L(E,\chi,s)$ at $s=1$ is a very rare event (David-Fearnley-Kisilevsky and Mazur-Rubin). In particular, it is conjectured that there are only finitely many characters of order $\ell>5$ such that $L(E,\chi,1)=0$ for a fixed curve $E$. We investigate in this talk the case of elliptic curves over function fields. For Dirichlet $L$-functions over function fields, Li and Donepudi-Li have shown how to use the geometry to produce infinitely many characters of order $\ell\geq 2$ such that the Dirichlet $L$-function $L(\chi,s)$ vanishes at $s=1/2$, contradicting (the function field analogue of) Chowla's conjecture. We show that their work can be generalized to isotrivial curves $E/\mathbb F_q(t)$, and we show that if there is one Dirichlet character $\chi$ of order $\ell$ such that $L(E,\chi,1)=0$, then there are infinitely many, leading to some specific examples contradicting (the function field analogue of) the number field conjectures on the vanishing of twisted $L$-functions. Such a dichotomy does not seem to exist for general (non-isotrivial) curves over $\mathbb F_q(t)$, and we produce empirical evidence which suggests that the conjectures over number fields also hold over function fields for non-isotrivial $E/\mathbb F_q(t)$.
David Kohel (Aix-Marseille Université, France)
On Sato-Tate groups ${\rm SO}(2n+1)$ and the exceptional group ${\rm UG}_2$ (slides)
Abstract: The character method, developed by Yih-Dar Shieh in his thesis, recognizes a Sato-Tate from an associated Frobenius distribution. Previous algorithms used moments of coefficients of a characteristic polynomial of Frobenius. The higher moments are degrees of the tensor product characters, which are direct sums with high multiplicities, hence the moment sequences converge (slowly, with sufficient precision) to large integers. The character method replaces the moments with a precomputed list of irreducible characters. From the orthogonality relations of characters, a Sato-Tate group $G$ is recognized by inner products yielding 0 or 1 (for which only one bit of precision is required to determine its value). We describe the character theory of the orthogonal groups ${\rm SO}(2n+1)$, with a view to characterizing orthogonal Sato-Tate groups. In particular, we specialize the character theory method to ${\rm SO}(7)$ and its subgroup ${\rm UG}_2$, the unitary subgroup ${\rm UG}_2$, of the exceptional Lie group $G_2$. In particular, we demonstrate its effectiveness with certain character sums associated to abelian factors of families of Jacobians known to give rise to the Sato-Tate group ${\rm UG}_2$.
Peter Koymans (University of Michigan, USA)
The negative Pell equation and applications (slides)
Abstract: In this talk we will study the negative Pell equation, which is the conic $C_D : x^2 - D y^2 = -1$ to be solved in integers $x, y \in \mathbb{Z}$. We shall be concerned with the following question: as we vary over squarefree integers $D$, how often is $C_D$ soluble? Stevenhagen conjectured an asymptotic formula for such $D$. Fouvry and Klüners gave upper and lower bounds of the correct order of magnitude. We will discuss a proof of Stevenhagen's conjecture, and potential applications of the new proof techniques. This is joint work with Carlo Pagano.
Wanlin Li (Washington University in St Louis, USA)
A generalization of Elkies's theorem
Abstract: Elkies proved that for a fixed elliptic curve $E$ defined over $\mathbb Q$, there exist infinitely many primes at which the reductions of $E$ are supersingular. In this talk, we give the first generalization of Elkies's theorem to curves of genus $>2$. We consider families of cyclic covers of the projective line ramified at $4$ points parametrized by a Shimura curve. This is joint work in progress with Elena Mantovan, Rachel Pries, and Yunqing Tang.
Philippe Michel (Ecole Polytechnique Fédérale de Lausanne, Switzerland)
Equidistribution of CM points on products (slides)
Abstract: In this talk we will discuss several results concerning the equidistribution of CM elliptic curves mapped on various product of arithmetic quotients. The main ingredient towards the proofs is a special case of a general classification theorem for joinings on product of locally homogeneous spaces due to Einsiedler and Lindenstrauss. We will explain which additional information are needed to verify the assumptions of the EL classification theorem.
Peter Sarnak (Institute for Advanced Study and Princeton University, USA)
An underdetermined moment problem for eigenvalues of matrices in classical groups and its application to computing root numbers and zeros of $L$ functions (slides , and murmurations mentioned during the talk)
Abstract: We describe thresholds for the recovery of the determinant and the exact count of eigenvalues in certain intervals, of random matrices in classical groups of dimension $n$ which share the same traces of their powers up to $k$ (less than $n$). Key to this is the study of the real algebraic geometry and shapes of semialgebraic sets that are associated with compact moment curves. This study is applied to give subexponential in the conductor, algorithms to compute the root numbers and exact counts of zeros of $L$-functions coming from arithmetical algebraic geometry. Joint work with Michael Rubinstein.
Kaneenika Sinha (Indian Institute of Science, Education, and Research - Pune, India)
Questions about error terms in Sato-Tate distributions (slides)
Abstract A sequence that is equidistributed with respect to a probability measure such as the Sato-Tate measure often motivates us to ask finer questions. Can we find explicit bounds for the discrepancy in these sequences? By varying the sequences over a suitable family, is the discrepancy estimate better upon averaging? How do the discrepancies fluctuate? What do we know about the small scale statistics and spacing statistics of these sequences? We explore these questions in the context of the Sato-Tate distribution law for the Hecke eigenvalues with respect to modular cusp forms.
Yunqing Tang (University of California at Berkeley, USA)
Reductions of abelian varieties and $K3$ surfaces
Abstract: For a $K3$ surface $X$ over a number field, we prove that there are infinitely many primes modulo which the reduction of $X$ has larger geometric Picard rank than that of the generic fiber $X$. There is also analogous result for $K3$ surface over global function field (under certain assumptions). In this talk, I will sketch the ideas in the proofs via the (arithmetic) intersection theory on good integral models of ${\rm GSpin}$ Shimura varieties and its consequences on certain abelian varieties. This talk is based on joint work with Davesh Maulik, Ananth Shankar, Arul Shankar, and Salim Tayou, and also the work of Tayou removing the good reduction assumptions.
Jesse Thorner (University of Illinois at Urbana-Champaign, USA)
Extremal class groups
Abstract: Fix an integer $n \geq 2$, and let $K_n$ be the set of number fields $F$ with $[F:\mathbb{Q}]=n$ whose Galois closure (over $\mathbb{Q}$) has as its Galois group the full symmetric group $S_n$. Conditional on the generalized Riemann hypothesis and Artin's holomorphy conjecture, Duke proved that there are infinitely many number fields $F\in K_n$ whose ideal class group has maximal order (as a function of the absolute discriminant). The result is now known unconditionally for $n=2,3,4$, and it is known conditionally on the strong Artin conjecture for $n\geq 5$. I will report on joint work with Robert Lemke Oliver and Asif Zaman wherein we prove Duke's theorem for all $n\geq 2$ without any unproven hypotheses.
David Zywina (Cornell University, USA)
Computing images of Galois representations for elliptic curves over $\mathbb{Q}$.
Abstract: Consider a non-CM elliptic curve $E/\mathbb{Q}$. The natural Galois action on the torsion points of $E(\overline{\mathbb{Q}})$ can be encoded by a Galois representation $\rho_E : Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\widehat{\mathbb{Z}})$. A famous theorem of Serre says that the image of $\rho_E$ is an open, and hence finite index, subgroup of $GL_2(\widehat{\mathbb{Z}})$. The image of $\rho_E$ is an important invariant for studying the distribution of the traces of Frobenius $a_p(E)$ for a fixed $E/\mathbb{Q}$ and varying prime $p$.
We shall describe recent results that allow us to actually compute the image of $\rho_E$. As an application, we explain how to compute the constants occurring in the conjecture of Lang and Trotter on the distribution of primes $p$ for which $a_p(E)$ is equal to a fixed integer.
Support for the conference comes from Université de Bordeaux, Université de Neuchâtel, Université de Rennes, the University of Illinois at Chicago, and the Simons Foundation. | CommonCrawl |
\begin{document}
\title{Generalized Laplacian decomposition of vector fields on fractal surfaces}
\author{Daniel Gonz\'alez-Campos$^{(1)}$, Marco Antonio P\'erez-de la Rosa$^{(2)}$\\and\\ Juan Bory-Reyes$^{(3)}$}
\date{ \small $^{(1)}$ Escuela Superior de F\'isica y Matem\'aticas. Instituto Polit\'ecnico Nacional. CDMX. 07738. M\'exico. \\ E-mail: daniel\[email protected] \\
$^{(2)}$ Department of Actuarial Sciences, Physics and Mathematics, Universidad de las Am\'ericas Puebla.
San Andr\'es Cholula, Puebla. 72810. M\'exico. \\ Email: [email protected] \\
$^{(3)}$ ESIME-Zacatenco. Instituto Polit\'ecnico Nacional. CDMX. 07738. M\'exico. \\ E-mail: [email protected] }
\maketitle
\begin{abstract} We consider the behavior of generalized Laplacian vector fields on a Jordan domain of $\mathbb{R}^{3}$ with fractal boundary. Our approach is based on properties of the Teodorescu transform and suitable extension of the vector fields. Specifically, the present article addresses the decomposition problem of a H\"older continuous vector field on the boundary (also called reconstruction problem) into the sum of two generalized Laplacian vector fields in the domain and in the complement of its closure, respectively. In addition, conditions on a H\"older continuous vector field on the boundary to be the trace of a generalized Laplacian vector field in the domain are also established. \end{abstract} \small{ \noindent \textbf{Keywords.} Quaternionic analysis; vector field theory; fractals.\\ \noindent \textbf{Mathematics Subject Classification (2020).} 30G35, 32A30, 28A80.}
\section{Introduction} Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy-Riemann equations to the quaternion skew field $\mathbb{H}$, which would somehow resemble the classical complex one-dimensional function theory.
An ordered set of quaternions $\psi:=(\psi_1, \psi_2, \psi_3)\in \mathbb{H}^{3}$, which form an orthonormal (in the usual Euclidean sense) basis in $\mathbb{R}^{3}$ is called a structural $\mathbb{H}$-vector.
The foundation of the so-called $\psi$-hyperholomorphic quaternion valued function theory, see \cite{NM, VSMV, MS} and elsewhere, is that the structural $\mathbb{H}$-vector $\psi$ must be chosen in a way that the factorization of the quaternionic Laplacian holds for $\psi$-Cauchy-Riemann operators. This question goes back at least as far as N\^{o}no's work \cite{Nono1, Nono2}.
The use of a general orthonormal basis introducing a generalized Moisil-Teodorescu system is the cornerstone of a generalized quaternionic analysis, where the generalized Cauchy-Riemann operator with respect to the standard basis in $\mathbb{R}^3$ are submitted to an orthogonal transformation. Despite the fact that some of the results in the present work can be obtained after the action of an orthogonal transformation on the standard basis; we keep their proofs in the work for the sake of completeness.
The $\psi$-hyperholomorphic functions theory by itself is not much of a novelty since it can be reduced by an orthogonal transformation to the standard case. In the face of this, the picture changes entirely by studying some important operators involving a pair of different orthonormal basis.
Moreover, the possibility to study simultaneously several conventional known theories, which can be embedded into a corresponding version of $\psi$-hyperholomorphic functions theory, again cannot be reduced to the standard context and reveal indeed the relevance of the $\psi$-hyperholomorphic functions theory.
The advantageous idea behind the unified study of particular cases of a generalized Moisil-Teodorescu system in $\psi$-hyperholomorphic functions theory simultaneously is considered in the present work.
The special case of structural $\mathbb{H}$-vector $\psi^\theta:=\{\textbf{i},\, \textbf{i}e^{\textbf{i}\theta}\textbf{j},\, e^{\textbf{i}\theta}\textbf{j}\}$ for $\theta\in[0,2\pi)$ fixed and its associated $\psi^\theta$-Cauchy-Riemann operator
\begin{equation*}
{^{\psi^{\theta}}}D:=\displaystyle\frac{\partial}{\partial x_{1}}\textbf{i}+\frac{\partial}{\partial x_{2}}\textbf{i}e^{\textbf{i}\theta}\textbf{j}+\frac{\partial}{\partial x_{3}} e^{\textbf{i}\theta}\textbf{j},
\end{equation*} are used in \cite{BAPS} to give a quaternionic treatment of inhomogeneous case of the system
\begin{equation}\label{sedi}
\left\{
\begin{array}{rcl}
-\displaystyle \frac{\partial f_{1}}{\partial x_{1}}+\left(\frac{\partial f_{2}}{\partial x_{2}}-\frac{\partial f_{3}}{\partial x_{3}}\right)\sin\theta-\left(\frac{\partial f_{3}}{\partial x_{2}}+\frac{\partial f_{2}}{\partial x_{3}}\right)\cos\theta & = & 0,
\\ {}\\ \displaystyle {\left(\frac{\partial f_{3}}{\partial x_{3}}-\frac{\partial f_{2}}{\partial x_{2}}\right)}\cos\theta-\left(\frac{\partial f_{3}}{\partial x_{2}}+\frac{\partial f_{2}}{\partial x_{3}}\right)\sin\theta & = & 0,
\\ {}\\ \displaystyle {-\frac{\partial f_{3}}{\partial x_{1}}+\frac{\partial f_{1}}{\partial x_{3}}\sin\theta+\frac{\partial f_{1}}{\partial x_{2}}\cos\theta} & = & 0, \\ {}\\
\displaystyle {\frac{\partial f_{2}}{\partial x_{1}}-\frac{\partial f_{1}}{\partial x_{3}}\cos\theta+\frac{\partial f_{1}}{\partial x_{2}}\sin\theta} & = & 0,
\end{array}
\right.
\end{equation} wherein the unknown well-behaved functions $f_m: \Omega \rightarrow \mathbb{C}, m=1,2,3$ are prescribed in an smooth domain $\Omega\subset\mathbb{R}^{3}$.
From now on, an smooth vector field $\vec{f}=(f_{1}, f_{2}, f_{3})$ that satisfies \eqref{sedi}, will said to be a generalized Laplacian vector field.
We will consider complex quaternionic valued functions (a detailed exposition of notations and definitions will be given in Section 2) to be expressed by \begin{equation}
\notag
f=f_{0}+f_{1}\textbf{i}+f_{2}\textbf{j}+f_{3}\textbf{k}, \end{equation} where $\textbf{i}$, $\textbf{j}$ and $\textbf{k}$ denote the quaternionic imaginary units.
On the other hand, the one-to-one correspondence \begin{equation}\label{corre} \mathbf{f}=f_1\mathbf{i}+f_2\mathbf{j}+f_3\mathbf{k}\, \longleftrightarrow \vec{f}=(f_{1}, f_{2}, f_{3}) \end{equation} makes it obvious that $\eqref{sedi}$ can be obtained from the classical Moisil-Theodorescu system after the action of some element in $O(3)$ as: $${^{\psi^{\theta}}}D[\mathbf{f}]= 0.$$
System \eqref{sedi} contains as a particular case the well-known solenoidal and irrotational, or harmonic system of vector fields (see \cite{ABS, ABMP} and the references given there). Indeed, under the correspondence $\mathbf{f}=f_1\mathbf{i}+f_3\mathbf{j}+f_2\mathbf{k}\, \longleftrightarrow \vec{f}=(f_{1}, f_{2}, f_{3})\,$ we have for $\theta=0$: \begin{equation}\label{equi} {}{^{\psi^{0}}}D[\mathbf{f}]=0\,\Longleftrightarrow \, \begin{cases} \text{div} \vec{f}=0,\cr \text{rot} \vec{f}=0. \end{cases} \end{equation}
Besides, the system \eqref{sedi} includes other partial differential equations systems (see \cite{BAPS} for more details): A particular case of the inhomogeneous Cimmino system (\cite{C}) when one looks for a solution $(f_1,f_2,f_3)$, where each $f_m,\,m=1,2,3$ does not depend on $x_0$. This system is obtained from \eqref{sedi} for $\theta=\frac{\pi}{2}$. Also, an equivalent system to the so-called the Riesz system \cite{Riesz} studied in \cite{Gur, Gur2}, which can be obtained from \eqref{sedi} for $\theta=\pi$ and the convenient embedding in $\mathbb{R}^3$.
In order to get more generalized results than those of \cite{ABMP}, it is assumed in this paper that $\Omega\subset \mathbb{R}^{3}$ is a Jordan domain (\cite{HN}) with fractal boundary $\Gamma$ in the Mandelbrot sense, see \cite{FKJ, FJ}.
Let us introduce the temporary notations $\Omega_{+}:=\Omega$ and $\Omega_{-}:=\mathbb{R}^{3}\setminus \{\Omega_{+}\cup\Gamma\}$. We are interested in the following problems: Given a continuous three-dimensional vector field $\vec{f}: \Gamma \rightarrow \mathbb{C}^{3}$:
\begin{itemize}
\item [$(I)$]
(Problem of reconstruction) Under which conditions can $\vec{f}$ be decomposed on $\Gamma$ into the sum:
\begin{equation} \label{des}
\vec{f}(t)=\vec{f}^{+}(t)+\vec{f}^{-}(t), \quad \forall \, t\in\Gamma,
\end{equation} where $\vec{f}^{\pm}$ are extendable to generalized Laplacian vector fields $\vec{F}^{\pm}$ in $\Omega_{\pm}$, with $\vec{F}^{-}(\infty)=0$?
\item [$(II)$] When $\vec{f}$ is the trace on $\Gamma$ of a generalized Laplacian vector field $\vec{F}^{\pm}$ in $\Omega_{\pm}\cup\Gamma$?
\end{itemize}
In what follows, we deal with problems $(I)$ and $(II)$ using the quaternionic analysis tools and working with $\mathbf{f}$ instead of $\vec{f}$ under the one-to-one correspondence (\ref{corre}). It will cause no confusion if we call $\mathbf{f}$ also vector field.
In the case of a rectifiable surface $\Gamma$ (the Lipschitz image of some bounded subset of $\mathbb{R}^{2}$) these problems have been investigated in \cite{GPB}. \section{Preliminaries.} \subsection{Basics of $\psi^{\theta}$-hyperholomorphic function theory.} Let $\mathbb{H}:=\mathbb{H(\mathbb{R})}$ and $\mathbb{H(\mathbb{C})}$ denote the sets of real and complex quaternions respectively. If $a\in\mathbb{H}$ or $a\in\mathbb{H(\mathbb{C})}$, then $a=a_{0}+a_{1}\textbf{i}+a_{2}\textbf{j}+a_{3}\textbf{k}$, where the coefficients $a_{k}\in\mathbb{R}$ if $a\in\mathbb{H}$ and $a_{k}\in\mathbb{C}$ if $a\in\mathbb{H(\mathbb{C})}$. The symbols
$\textbf{i}$, $\textbf{j}$ and $\textbf{k}$ denote different imaginary units, i. e. $\textbf{i}^{2}=\textbf{j}^{2}=\textbf{k}^{2}=-1$ and they satisfy the following multiplication rules $\textbf{i}\textbf{j}=-\textbf{j}\textbf{i}=\textbf{k}$; $\textbf{j}\textbf{k}=-\textbf{k}\textbf{j}=\textbf{i}$; $\textbf{k}\textbf{i}=-\textbf{i}\textbf{k}=\textbf{j}$. The unit imaginary $i\in\mathbb{C}$ commutes with every quaternionic unit imaginary.
It is known that $\mathbb{H}$ is a skew-field and $\mathbb{H(\mathbb{C})}$ is an associative, non-commutative complex algebra with zero divisors.
If $a\in\mathbb{H}$ or $a\in\mathbb{H(\mathbb{C})}$, $a$ can be represented as $a=a_{0}+\vec{a}$, with $\vec{a}=a_{1}\textbf{i}+a_{2}\textbf{j}+a_{3}\textbf{k}$,
$\text{Sc}(a):=a_{0}$ is called the scalar part and $\text{Vec}(a):=\vec{a}$ is called the vector part of the quaternion $a$.
Also, if $a\in\mathbb{H(\mathbb{C})}$, $a$ can be represented as $a=\alpha_{1}+i\alpha_{2}$ with $\alpha_{1},\,\alpha_{2}\in\mathbb{H}$.
Let $a,\,b\in\mathbb{H(\mathbb{C})}$, the product between these quaternions can be calculated by the formula:
\begin{equation} \label{pc2}
ab=a_{0}b_{0}-\langle\vec{a},\vec{b}\rangle+a_{0}\vec{b}+b_{0}\vec{a}+[\vec{a},\vec{b}],
\end{equation} where
\begin{equation} \label{proint}
\langle\vec{a},\vec{b}\rangle:=\sum_{k=1}^{3} a_{k}b_{k}, \quad
[\vec{a},\vec{b}]:= \left|\begin{matrix}
\textbf{i} & \textbf{j} & \textbf{k}\\
a_{1} & a_{2} & a_{3}\\
b_{1} & b_{2} & b_{3}
\end{matrix}\right|.
\end{equation} We define the conjugate of $a=a_{0}+\vec{a}\in\mathbb{H(\mathbb{C})}$ by $\overline{a}:=a_{0}-\vec{a}$.
The Euclidean norm of a quaternion $a\in\mathbb{H}$ is the number $\abs{a}$ given by:
\begin{equation}\label{normar}
\abs{a}=\sqrt{a\overline{a}}=\sqrt{\overline{a}a}.
\end{equation}
We define the quaternionic norm of $a\in\mathbb{H(\mathbb{C})} $ by:
\begin{equation}
\abs{a}_{c}:=\sqrt{{{\abs {a_{0}}}_{\mathbb{C}}}^{2}+{{\abs {a_{1}}}_{\mathbb{C}}}^{2}+{{\abs {a_{2}}}_{\mathbb{C}}}^{2}+{{\abs {a_{3}}}_{\mathbb{C}}}^{2}},
\end{equation} where ${\abs {a_{k}}}_{\mathbb{C}}$ denotes the complex norm of each component of the quaternion $a$.The norm of a complex quaternion $a=a_{1}+ia_{2}$ with $a_{1}, a_{2} \in \mathbb{H}$ can be rewritten in the form
\begin{equation} \label{nc2}
{\abs{a}_{c}}=\sqrt{\abs{\alpha_{1}}^2+\abs{\alpha_{2}}^2}.
\end{equation}
If $a \in \mathbb{H}$, $b \in \mathbb{H(\mathbb{C})}$, then
\begin{equation}
{\abs{ab}}_{c}=\abs{a}{\abs{b}}_{c}.
\end{equation}
If $a\in\mathbb{H(\mathbb{C})}$ is not a zero divisor then $\displaystyle a^{-1}:=\frac{\overline{a}}{a\overline{a}}$ is the inverse of the complex quaternion $a$. \begin{subsection}{Notations}
\begin{itemize}
\item We say that $f:\Omega \rightarrow \mathbb{H(\mathbb{C}})$ has properties in $\Omega$ such as continuity and real differentiability of order $p$ whenever all $f_{j}$ have these properties. These spaces are usually denoted by $C^{p}(\Omega,\, \mathbb{H(\mathbb{C})})$ with $p\in \mathbb{N}\cup\{0\}$.
\item Throughout this work, $\text{Lip}_{\mu}(\Omega,\, \mathbb{H(\mathbb{C})})$, $0<\mu\leq 1$, denotes the set of H\"older continuous functions $f:\Omega \rightarrow \mathbb{H(\mathbb{C}})$ with H\"older exponent $\mu$. By abuse of notation, when $f_{0}=0$ we write $\mathbf{Lip}_{\mu}(\Omega,\, \mathbb{C}^{3})$ instead of $\text{Lip}_{\mu}(\Omega,\, \mathbb{H(\mathbb{C})})$.
\end{itemize} \end{subsection}
In this paper, we consider the structural set $\psi^\theta:=\{\textbf{i},\, \textbf{i}e^{\textbf{i}\theta}\textbf{j},\, e^{\textbf{i}\theta}\textbf{j}\}$ for $\theta\in[0,2\pi)$ fixed, and the associated operators ${^{\psi^\theta}}D$ and $D{^{\psi^\theta}}$ on $C^{1}(\Omega,\, \mathbb{H(\mathbb{C})})$ defined by \begin{equation} {^{\psi^{\theta}}}D[f]:=\textbf{i}\frac{\partial f}{\partial x_{1}}+\textbf{i}e^{\textbf{i}\theta}\textbf{j}\frac{\partial f}{\partial x_{2}}+e^{\textbf{i}\theta}\textbf{j}\frac{\partial f}{\partial x_{3}}, \end{equation} \begin{equation} D{^{\psi^\theta}}[f]:=\frac{\partial f}{\partial x_{1}}\textbf{i}+\frac{\partial f}{\partial x_{2}}\textbf{i}e^{\textbf{i}\theta}\textbf{j}+\frac{\partial f}{\partial x_{3}} e^{\textbf{i}\theta}\textbf{j}, \end{equation} which linearize the Laplace operator $\Delta_{\mathbb{R}^{3}}$ in the sense that \begin{equation} {^{\psi^{\theta}}}D^{2}= \left[D{^{\psi^\theta}}\right]^{2}=-\Delta_{\mathbb{R}^{3}}. \end{equation} All functions belong to $\ker \left({^{\psi^{\theta}}}D\right) := \left\{f : {^{\psi^{\theta}}}D[f]=0\right\}$ are called left-$\psi^{\theta}$-hyperholomorphic in $\Omega$. Similarly, those functions which belong to $\ker \left(D{^{\psi^{\theta}}}\right):= \left\{f : D{^{\psi^{\theta}}}[f]=0\right\}$ will be called right-$\psi^{\theta}$-hyperholomorphic in $\Omega$. For a deeper discussion of the hyperholomorphic function theory we refer the reader to \cite{KVS}.
The function \begin{equation} \label{kernel} \mathscr{K}_{\psi^{\theta}}(x):=-\frac{1}{4\pi}\frac{(x)_{\psi^{\theta}}}{\abs{x}^3}, \quad x\in\mathbb{R}^{3}\setminus\{0\}, \end{equation} where \begin{equation} (x)_{\psi^{\theta}}:=x_{1}\textbf{i}+x_{2} \textbf{i}e^{\textbf{i}\theta}\textbf{j}+x_{3}e^{\textbf{i}\theta}\textbf{j}, \end{equation} is a both-side-$\psi^{\theta}$-hyperholomorphic fundamental solution of $^{\psi^{\theta}}D$. Observe that $\abs{(x)_{\psi^{\theta}}}=\abs{x}$ for all $ x \in \mathbb{R}^{3}$.
For $f=f_{0}+\mathbf{f}\in C^1(\Omega,\mathbb{H(\mathbb{C})})$ let us define \begin{equation} {^{\psi^{\theta}}}\text{div}[\mathbf{f}]:=\frac{\partial f_{1}}{\partial x_{1}}+\left({\frac{\partial f_{2}}{\partial x_{2}}-\frac{\partial f_{3}}{\partial x_{3}}}\right)\textbf{i}e^{\textbf{i}\theta}, \end{equation} \begin{equation} {^{\psi^{\theta}}}\text{grad}[f_{0}]:=\frac{\partial f_{0}}{\partial x_{1}}\textbf{i}+\frac{\partial f_{0}}{\partial x_{2}}\textbf{i}e^{\textbf{i}\theta}\textbf{j}+\frac{\partial f_{0}}{\partial x_{3}}e^{\textbf{i}\theta}\textbf{j}, \end{equation} \begin{equation} \begin{split} {^{\psi^{\theta}}}\text{rot}[\mathbf{f}]:=\left({-\frac{\partial f_{3}}{\partial x_{2}}-\frac{\partial f_{2}}{\partial x_{3}}}\right)e^{\textbf{i}\theta}+\left({-\frac{\partial f_{1}}{\partial x_{3}}\textbf{i}e^{\textbf{i}\theta}-\frac{\partial f_{3}}{\partial x_{1}}}\right)\textbf{j} +\left({\frac{\partial f_{2}}{\partial x_{1}}-\frac{\partial f_{1}}{\partial x_{2}}\textbf{i}e^{\textbf{i}\theta}}\right)\textbf{k}. \end{split} \end{equation} The action of ${^{\psi^{\theta}}}D$ on $f\in C^1(\Omega, \, \mathbb{H(\mathbb{C})})$ yields \begin{equation} {^{\psi^{\theta}}}D[f]=-{^{\psi^{\theta}}}\text{div}[\mathbf{f}]+{^{\psi^{\theta}}}\text{grad}[f_{0}]+ {^{\psi^{\theta}}}\text{rot}[\mathbf{f}], \end{equation} which implies that $f \in \ker ({^{\psi^{\theta}}}D) $ is equivalent to \begin{equation} \label{eq1} -{^{\psi^{\theta}}}\text{div}[\mathbf{f}]+{^{\psi^{\theta}}}\text{grad}[f_{0}]+ {^{\psi^{\theta}}}\text{rot}[\mathbf{f}]=0. \end{equation} If $f_{0}=0$, \eqref{eq1} reduces to \begin{equation} \label{eq2} -{^{\psi^{\theta}}}\text{div}[\mathbf{f}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}]=0. \end{equation} We check at once that \eqref{sedi} is equivalent to \eqref{eq2}.
Similar considerations apply to $D^{\psi^{\theta}}$, for this case one obtains \begin{equation} \label{eq3} D^{\psi^{\theta}}[f]=-{^{\overline{\psi^{\theta}}}}\text{div}[\mathbf{f}]+{^{\psi^{\theta}}}\text{grad}[f_{0}]+{^{\overline{\psi^{\theta}}}}\text{rot}[\mathbf{f}], \end{equation} where \begin{equation} {^{\overline{\psi^{\theta}}}}\text{div}[\mathbf{f}]:=\frac{\partial f_{1}}{\partial x_{1}}+\left({\frac{\partial f_{2}}{\partial x_{2}}-\frac{\partial f_{3}}{\partial x_{3}}}\right)\overline{\textbf{i}e^{\textbf{i}\theta}}, \end{equation} \begin{equation} \begin{split} {^{\overline{\psi^{\theta}}}}\text{rot}[\mathbf{f}]:=\left({-\frac{\partial f_{3} }{\partial x_{2}}-\frac{\partial f_{2}}{\partial x_{3}}}\right) \overline{e^{\textbf{i}\theta}}-{\frac{\partial f_{1}}{\partial x_{3}}\overline{\textbf{i}e^{\textbf{i}\theta}\textbf{j}}+\frac{\partial f_{3}}{\partial x_{1}}}\textbf{j} -\frac{\partial f_{2}}{\partial x_{1}}\textbf{k}-\frac{\partial f_{1}}{\partial x_{2}}\overline{\textbf{i}e^{\textbf{i}\theta}\textbf{k}}. \end{split} \end{equation} If $f_{0}=0$, \eqref{eq3} reduces to \begin{equation} \label{eq4} D^{\psi^{\theta}}[f]=-{^{\overline{\psi^{\theta}}}}\text{div}[\mathbf{f}]+{^{\overline{\psi^{\theta}}}}\text{rot}[\mathbf{f}]. \end{equation} It follows easily that \begin{equation} \label{eq5} -{^{\overline{\psi^{\theta}}}}\text{div}[\mathbf{f}]+{^{\overline{\psi^{\theta}}}}\text{rot}[\mathbf{f}]=0, \end{equation} is also equivalent to \eqref{sedi}.
\begin{lemma} \label{two-sided} Let $f=f_{0}+\mathbf{f}\in C^{1}(\Omega, \, \mathbb{H(\mathbb{C})})$. Then $f$ is both-side-$\psi^\theta$-hyperholomorphic in $\Omega$ if and only if ${^{\psi^{\theta}}}\text{grad}[f_{0}](x)\equiv 0$ in $\Omega$ and $\mathbf{f}$ is a generalized Laplacian vector field in $\Omega$.
\begin{proof} The proof is based on the fact that \eqref{eq2} and \eqref{eq5} are equivalent to \eqref{sedi}.
\end{proof} \end{lemma}
\subsection{Fractal dimension and the Whitney operator} Let $E$ a subset in $\mathbb{R}^{3}$, we denote by $\mathcal{H}_{\lambda}(E)$ the $\lambda$-Hausdorff measure of $E$ (\cite{GJ}).
Assume that $E$ is a bounded set, the Hausdorff dimension of $E$ (denoted by $\lambda(E)$) is the infimum $\lambda$ such that $\mathcal{H}_{\lambda}(E)<\infty$.
Frequently, the Minkowski dimension of $E$ (also called box dimension and denoted by $\alpha(E)$) is more appropriate than the Hausdorff dimension to measure the roughness of E (\cite{ABMP,ABS}).
It is known that Minkowski and Hausdorff dimensions can be equal, for example, for rectifiable surfaces (the Lipschitz image of some bounded subset of $\mathbb{R}^{2}$). But in general, if $E$ is a two-dimensional set in $\mathbb{R}^{3}$ \begin{equation} 2\leq \lambda(E)\leq \alpha(E)\leq3. \end{equation} If $2<\lambda(E)$, $E$ is called a fractal set in the Mandelbrot sense. For more information about the Hausdorff and Minkowski dimension, see \cite{FKJ,FJ}.
Let $f\in \text{Lip}_{\mu}(\Gamma, \mathbb{H(\mathbb{C})})$, then $f=f_{1}+if_{2}$ with $f_{k}\in \text{Lip}_{\mu}(\Gamma, \mathbb{H(\mathbb{R})})$ and $\mathcal{E}_{0}(f):=\mathcal{E}_{0}(f_{1})+i\mathcal{E}_{0}(f_{2})$. Write
\begin{equation}
f^{w}:=\mathcal{X}\mathcal{E}_{0}(f),
\end{equation} where $\mathcal{E}_{0}$ is the Whitney operator and $\mathcal{X}$ denotes the characteristic function in $\Omega_{+}\cup\Gamma$.
For completeness, we recall the main lines in the construction of the Whitney decomposition $\mathcal W$ of the Jordan domain $\Omega$ with boundary $\Gamma$ by squares $Q$ of diameter $||Q||_{\mathbb{R}^{3}}$ and the notion of Whitney operator. This can be found in \cite[Ch VI]{SEM}.
Consider the lattice $\mathbb Z^{3}$ in $\mathbb R^{3}$ and the collection of closed unit cubes defined by it; let $\mathcal{M}_1$ be the mesh consisting of those unit cubes having a non-empty intersection with $\Omega$. Then, we recursively define the meshes $\mathcal{M}_k$, $k=2,3,\ldots$, each time bisecting the sides of the cubes of the previous one. The cubes in $\mathcal{M}_k$ thus have side length $2^{-k+1}$ and diameter $||Q||_{\mathbb{R}^{3}} = (\sqrt{3})\, 2^{-k+1}$. Define, for $k=2,3,\ldots$, \begin{eqnarray*}
\mathcal{W}^1 & := & \left \{ Q\in \mathcal{M}_1 \, | \, \mbox{$Q$ and every cube of $\mathcal{M}_1$ touching $Q$ are contained in $\Omega$} \right \}, \\
\mathcal{W}^k & := & \left \{ Q\in \mathcal{M}_k \, | \, \mbox{$Q$ and every cube of $\mathcal{M}_k$ touching $Q$ are contained in $\Omega$} \right .\\
& & \hspace*{50mm} \left . \mbox{and}\,\not \exists \, Q^\ast \in \mathcal{W}^{k-1}: Q \subset Q^\ast \right \}, \end{eqnarray*} for which it can be proven that $$ \Omega = \bigcup_{k=1}^{+\infty} \mathcal{W}^k = \bigcup_{k=1}^{+\infty} \bigcup_{Q \in \mathcal{W}^k} Q \equiv \bigcup_{Q \in \mathcal{W}} Q, $$ all cubes $Q$ in the Whitney decomposition $\mathcal{W}$ of $\Omega$ having disjoint interiors.
We denote by $Q_{0}$ the unit cube with center at the origin and fix a $C^{\infty}$ function with the properties: $0\leq \varphi \leq 1$; $\varphi(x)=1$ if $x\in Q_{0}$; and $\varphi(x)=0 $ if $x\notin Q^*_{0}$.
Let $\varphi_{k}$ the function $\varphi(x)$ adjusted to the cube $Q_{k}\in\mathcal{W}$, that is \begin{equation} \varphi_{k}(x):=\varphi\bigg(\frac{x-x^{k}}{l_{k}}\bigg), \end{equation} where $x^{k}$ is the center of $Q_{k}$ and $l_{k}$ the common length of its sides.
Function $\varphi_{k}$ satisfies that $0\leq \varphi_{k} \leq 1$, $\varphi_{k}(x)=1$ if $x\in Q_{k}$ and $\varphi_{k}(x)=0 $ if $x\notin Q^*_{k}$. Let ${\varphi_{k}^*}(x)$ be defined for $x\in \Omega$ by \begin{equation} \label{pdu} {\varphi_{k}^*}(x):=\frac{\varphi_{k}(x)}{\Phi (x)}, \end{equation} with \begin{equation} \Phi(x):=\sum_{k}^{}\varphi_{k}(x) \end{equation} and $\sum_{k}^{}\varphi_{k}^{*}(x)=1$ for $x\in \Omega$.
For each cube $Q_{k}$ let $p_{k}$ be a point fixed in $\Gamma$ such that $dist(Q_{k}, \Gamma)=dist(Q_{k}, p_{k})$. Then the Whitney operator is defined as follows \begin{equation} \mathcal{E}_{0}(f)(x):=f(x), \quad \text{if} \quad x\in \Gamma, \end{equation} \begin{equation} \label{suma} \mathcal{E}_{0}(f)(x):=\sum_{k}f(p_{k})\varphi_{k}^{*}(x), \quad \text{if} \quad x\in \Omega. \end{equation} Similar construction may be made for the domain $\mathbb{R}^{3}\setminus \{\Omega\cup\Gamma\}$.
The operator $\mathcal{E}_{0}$ extends functions $f$ defined in $\Gamma$ to functions defined in $\mathbb{R}^{3}$. Its main properties are given as follows: \begin{itemize} \item Assume $f\in\text{Lip}_{\mu}(\Omega\cup\Gamma, \mathbb{H(\mathbb{C})})$. Then $\mathcal{E}_{0}(f) \in \text{Lip}_{\mu}(\mathbb{R}^{3}, \mathbb{H(\mathbb{C})})$ and in fact is $C^{\infty}$ in $\mathbb{R}^{3}\setminus\Gamma$, see \cite[Proposition, pag. 172]{SEM}. \item The following quantitative estimate holds (see \cite[(14), pag. 174]{SEM}) \begin{equation} \absol{\frac{\partial{\mathcal{E}_{0}(f)}}{ { \partial x_{i} } } (x)}\leq c (dist(x, \Gamma))^{\mu-1}, \, \text{for}\, x \in \mathbb{R}^{3}\setminus\Gamma. \end{equation} \end{itemize} It is necessary to go further and to express the essential fact that under some specific relation between $\mu$ and $\alpha(\Gamma)$ we have that \begin{equation}\label{integrability} {^{\psi^{\theta}}D}[f^{w}]\in L_{p}(\mathbb{R}^{3}, \mathbb{H(\mathbb{R})})\ \mbox{for}\; \displaystyle p<\frac{3-\alpha(\Gamma)}{1-\mu}. \end{equation} This follows in much by the same methods as \cite[Proposition 4.1]{AB}. \section{Auxiliary results on $\psi^{\theta}$-hyperholomorphic function theory.} It is a well-known fact that in proving the existence of the boundary value of the Cauchy transform via the Plemelj-Sokhotski formulas, the solvability of the jump problem is an easy task whenever the data is a H\"older continuous function and the boundary of the considered domain is assumed sufficiently smooth. But by far much more subtle is the case where it can be thought of as a fractal surface. Then the standard method is no longer applicable, and it is necessary to introduce an alternative way of defining Cauchy transform, where a central role is played by the Teodorescu operator involving fractal dimensions. This is the idea behind the proofs of the following auxiliary results.
\begin{theorem} \label{thm 6} Let $f\in \text{Lip}_{\mu}(\Gamma,\, \mathbb{H(\mathbb{C})})$, $\displaystyle \frac{\alpha(\Gamma)}{3}<\mu \leq 1$. Then the function $f$ can be represented as $f=\left.F^{+}\right|_{\Gamma}-\left.F^{-}\right|_{\Gamma}$, where $F^{\pm}\in \text{Lip}_{\nu}(\Omega_{\pm}\cup\Gamma)\cap \ker\left( {^{\psi^{\theta}}}D\right)$ for some $\nu<\mu$, $F^{\pm}$ are given by
\begin{equation}\label{tc}
F^{\pm}(x):=-{^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[f^{w}]\right](x)+ f^{w}(x), \quad x\in\big({\Omega}_{\pm}\cup\Gamma\big),
\end{equation}
where
\begin{equation}
{^{\psi^{\theta}}}T[v](x):=\int_{\Omega_{+}}{\mathscr{K}_{\psi^{\theta}}(x-\xi) \,v(\xi) }\,dm(\xi), \quad x\in \mathbb{R}^{3}.
\end{equation}
is the well-defined Teodorescu transform for the $\mathbb{H(\mathbb{C})}$-valued function $v$, see \cite{KVS}.
\end{theorem} \begin{proof}
Since ${f}^{w}={f}_{1}^{w}+i {f}_{2}^{w}$ with ${f}_{k}^{w}:\Omega\cup\Gamma\to\mathbb{H}$, $\displaystyle \mu>\frac{\alpha(\Gamma)}{3}$, and by (\ref{integrability}) ${^{\psi^{\theta}}}D[{f}_{k}^{w}]\in L_{p}(\Omega,\, \mathbb{H})$ for some $p\in\left( 3, \,\displaystyle\frac{3-\alpha(\Gamma)}{1-\mu}\right)$. Then the integral on the right side of \eqref{tc} exists and represents a continuous function in the whole $\mathbb{R}^{3}$ (see \cite[Theorem 2.8]{GPB}). Hence, the functions $F^{\pm}$ possess continuous extensions to the closures of the domains $\Omega_{\pm}$ and they satisfy that $\left.F^{+}\right|_{\Gamma}-\left.F^{-}\right|_{\Gamma}=f$. By the property of the Teodorescu operator to still being a right inverse to the Cauchy-Riemann operator (see \cite{KVS}, p. 73), ${^{\psi^{\theta}}}D[F^{+}]=0$ and ${^{\psi^{\theta}}}D[F^{-}]=0$ in the domains $\Omega_{\pm}$, respectively. \end{proof} \begin{remark} Uniqueness in the statement of Theorem 3.1 could be ensured introducing an additional requirement analogous to that in \cite[Theorem 6.6]{ABJ} \end{remark}
In the remainder of this section we assume that $\displaystyle \frac{\alpha(\Gamma)}{3}<\mu \leq 1$.
The following results are related to the problem of extending $\psi^{\theta}$-hyperholomorphically a $\mathbb{H(\mathbb{C})}$-valued H\"older continuous function.
\begin{theorem} \label{t1}
Let $f\in \text{Lip}_{\mu}(\Gamma,\mathbb{H(\mathbb{C})})$ the trace of $F\in \text{Lip}_{\mu}(\Omega_{+}\cup\Gamma,\mathbb{H(\mathbb{C})})\cap \ker\left(\left.^{\psi^{\theta}}D\right|_{\Omega_{+}}\right).$ Then
\begin{equation}\label{c1}
{^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f^{w}]\right]\right|_{\Gamma}=0.
\end{equation}
Conversely, if \eqref{c1} is satisfied, then $f$ is the trace of $F\in \text{Lip}_{\nu}(\Omega_{+}\cup\Gamma,\mathbb{H(\mathbb{C})})\cap \ker\left(\left.^{\psi^{\theta}}D\right|_{\Omega_{+}}\right)$ for some $\nu<\mu$.
\begin{proof}
Sufficiency. As we can write $f=f_{1}+if_{2}$ and $F=F_{1}+iF_{2}$ with $f_{r}\in \text{Lip}_{\mu}(\Gamma,\mathbb{H(\mathbb{R})}), r=1,2$ and $F_{r}\in \text{Lip}_{\nu}(\Omega_{+}\cup\Gamma,\mathbb{H(\mathbb{R})})\cap \ker\left(\left.^{\psi^{\theta}}D\right|_{\Omega_{+}}\right)$. Then $f^{w}=f^{w}_{1}+if^{w}_{2}$ and
\begin{equation}
{^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[f^{w}]\right]={^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[f_{1}^{w}]\right]+i\;{^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[f_{2}^{w}]\right].
\end{equation} Following \cite[Theorem 3.1]{ABMT}, let $F_{r}^*=f_{r}^{w}-F_{r}$, $\tilde{Q}_{k}$ the union of cubes of the mesh $\mathcal{M}_{k}$ intersecting $\Gamma$, $\Omega_{k}=\Omega_{+}\setminus \tilde{Q}_{k}$, $\Delta_{k}=\Omega_{+}\setminus\Omega_{k}$ and denote by $\Gamma_{k}$ the boundary of $\Omega_{k}$. Applying the definition of $\alpha(\Gamma)$, given $\varepsilon>0$ there is a constant $C(\varepsilon)$ such that $\mathcal{H}^{2}(\Gamma_{k})$ (the Hausdorff measure of $\Gamma_{k}$) is less or equal than $6C(\varepsilon)2^{k(\alpha(\Gamma)-2+\varepsilon)}$.
Since $F_{r}^*\in\text{Lip}_{\mu}(\Gamma,\mathbb{H(\mathbb{C})})$, $F_{r}^*|_{\Gamma}=0$ and any point of $\Gamma_{k}$ is distant by no more than $C_{1}2^{-k}$, then
\begin{equation*}
\text{max}_{\xi\in\Gamma_{k}}\abs{F_{r}^*(\xi)}\leq C_{2}2^{-\mu k}
\end{equation*}
where $C_{1}$, $C_{2}$ denoted absolute constants.
Therefore, for $x\in\Omega_{-}$, let $s=dist(x,\Gamma)$
\begin{equation*}
\abso{\int_{\Gamma_{k}}\mathscr{K}_{\psi^{\theta}}(\xi-x){^{\psi^{\theta}}}D[F_{r}^{*}](\xi)dS(\xi)}\leq C_{2}C(\varepsilon)\frac{6}{s^{2}}2^{(\alpha(\Gamma)-2-\mu+\varepsilon)}.
\end{equation*} As $\displaystyle \frac{\alpha(\Gamma)}{3}<\mu \leq 1$ the right-hand side of the previous inequality tends to zero as $k\to \infty$. By the Stokes formula, we have that \begin{equation*}
\begin{split}
&\int_{\Omega_{+}}\mathscr{K}_{\psi^{\theta}}(\xi-x){^{\psi^{\theta}}}D[F_{r}^{*}](\xi)dm(\xi)=\lim_{k\to\infty}\bigg( \int_{\Delta_{k}}+\int_{\Omega_{k}}\bigg)\mathscr{K}_{\psi^{\theta}}(\xi-x){^{\psi^{\theta}}}D[F_{r}^{*}](\xi)dm(\xi)\\ &=\lim_{k\to\infty}\bigg( \int_{\Delta_{k}}\mathscr{K}_{\psi^{\theta}}(\xi-x){^{\psi^{\theta}}}D[F_{r}^{*}](\xi)dm(\xi)-\int_{\Gamma_{k}}\mathscr{K}_{\psi^{\theta}}(\xi-x){^{\psi^{\theta}}}D[F_{r}^{*}](\xi)dS(\xi)\bigg)=0.
\end{split}
\end{equation*} Then \begin{equation}
{^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f_{r}^{w}]\right]\right|_{\Gamma}={^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[F_{r}]\right]\right|_{\Gamma}=0. \end{equation}
Necessity. If \eqref{c1} is satisfied we have
\begin{equation}
{^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f^{w}]\right]\right|_{\Gamma}={^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f_{1}^{w}]\right]\right|_{\Gamma}+i\;{^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f_{2}^{w}]\right]\right|_{\Gamma}=0,
\end{equation}
and we take
\begin{equation}
\begin{split}
F(x):=-{^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[f^{w}]\right](x)+ f^{w}(x), \quad x\in \Omega_{+}\cup\Gamma.
\end{split}
\end{equation}
\end{proof}
\end{theorem} In the same manner next theorem can be verified
\begin{theorem}
Let $f\in \text{Lip}_{\mu}(\Gamma,\mathbb{H(\mathbb{C})})$. If $f$ is the trace of a function $F\in \text{Lip}_{\mu}(\Omega_{-}\cup\Gamma,\mathbb{H(\mathbb{C})})\cap \ker\left(\left.^{\psi^{\theta}}D\right|_{\Omega_{-}}\right)$
\begin{equation}\label{c2}
{^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f^{w}]\right]\right|_{\Gamma}=-f.
\end{equation}
Conversely, if \eqref{c2} is satisfied, then $f$ is the trace of a function $F\in \text{Lip}_{\nu}(\Omega_{-}\cup\Gamma,\mathbb{H(\mathbb{C})})\cap \ker\left(\left.^{\psi^{\theta}}D\right|_{\Omega_{-}}\right)$ for some $\nu<\mu$.
\end{theorem}
These two results generalize those of\cite[Theorem 3.1, Theorem 3.2]{ABMT}.
\begin{remark}
Similar results can be drawn for the case of right $\psi^{\theta}$-hyperholomorphic extensions. The only necessity being to replace in both theorems $\ker\left(\left.^{\psi^{\theta}}D\right|_{\Omega_{\pm}}\right)$ by $\ker\left(\left.D^{\psi^{\theta}}\right|_{\Omega_{\pm}}\right)$ and ${^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f^{w}]\right]\right|_{\Gamma}$ by $\left[D^{\psi^{\theta}}[f^{w}]\right]\, \left.{{^{\psi^{\theta}}}T}\right|_{\Gamma}$, where for every $\mathbb{H(\mathbb{C})}$-valued function $v$ we have set
\begin{equation}
[v]\, {{^{\psi^{\theta}}}T}=\int_{\Omega_{+}}{ v(\xi)\, \mathscr{K}_{\psi^{\theta}}(x-\xi) }\,dm(\xi), \quad x\in \mathbb{R}^{3}.
\end{equation}
The following theorem presents a result connecting two-sided $\psi^{\theta}$-hyperholomorphicity in the domain $\Omega_{+}$ and it is obtained by application of the previous results
\begin{theorem}
If $F\in \text{Lip}_{\mu}(\Gamma,\mathbb{H(\mathbb{C})})\cap \ker\left(\left.^{\psi^{\theta}}D\right|_{\Omega_{+}}\right)$ has trace $\left.F\right|_{\Gamma}=f$, then the following assertions are equivalent:
\begin{itemize}
\item [1.] F is left and right $\psi^{\theta}$-hyperholomorphic in $\Omega_{+}$,
\item [2.] ${^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f^{w}]\right]\right|_{\Gamma}= \left[D^{\psi^{\theta}}[f^{w}]\right]\, \left.{{^{\psi^{\theta}}}T}\right|_{\Gamma}$.
\end{itemize}
\begin{proof}
The proof is obtained reasoning as in \cite[Theorem 3.3]{ABMP}.
\end{proof}
\end{theorem}
\end{remark}
\section{Main results} In this section our main results are stated and proved. They give sufficient conditions for solving the Problems $(I)$ and $(II)$.
Let $\mathscr{M}_{\psi^{\theta}}^{*}$ be the subclass of vector fields $\mathbf{f}\in C^{1}(\Omega, \mathbb{C}^{3})\cap\mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ defined by
\begin{equation} \label{set}
\mathscr{M}_{\psi^{\theta}}^{*}:=\left\{\mathbf{f}: \int_{\Omega_{+}}{\left\langle\mathscr{K}_{\psi^{\theta}}(x-\xi)\,,\,\mathbf{f}(\xi)\right\rangle}\,dm(\xi)=0, \; x\in\Gamma \right\},
\end{equation}
where $m$ denotes the Lebesgue measure in $\mathbb{R}^{3}$. The set $\mathscr{M}_{\psi^{\theta}}^{*}$ can be seen as a fractal version of the corresponding class in \cite{ZMS}, which can be described in purely physical terms.
\begin{theorem} \label{TH1}
Let $\mathbf{f}\in \mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ such that $\displaystyle \mu>\frac{\alpha(\Gamma)}{3}$. Then the problem (I) is solvable if
\begin{equation}
\begin{split}
\text{Vec}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)&:=\left({\left(\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{3}}-\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{2}}\right)}\cos\theta-\left(\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{2}}+\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{3}}\right)\sin\theta\right)\textbf{i}\\ & +\left(\displaystyle {-\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{1}}+\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{3}}\sin\theta+\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{2}}\cos\theta}\right)\textbf{j}\\ & +\left(\displaystyle {\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{1}}-\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{3}}\cos\theta+\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{2}}\sin\theta}\right)\textbf{k}\in\mathscr{M}_{\psi^{\theta}}^{*}.
\end{split}
\end{equation}
\begin{proof}
It is enough to prove that
\begin{equation}
\mathbf{F^{\pm}}(x):=-{^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right](x)+ \mathbf{f}^{w}(x), \quad x\in\big({\Omega}_{\pm}\cup\Gamma\big),
\end{equation}
are vector fields.
Observe that
\begin{equation}
\notag
\text{Sc}\left({^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right]\right)(x)=-\int_{\Omega_{+}}{\left\langle \mathscr{K}_{\psi^{\theta}}(x-\xi),\text{Vec}\left( -{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right) \right\rangle }\,dm(\xi), \quad x\in \Omega_{\pm},
\end{equation}
\begin{equation}
\notag
\Delta\left(\text{Sc}\left({^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right]\right)\right)(x)=0, \quad x\in \Omega_{\pm}
\end{equation}
and
\begin{equation}
\notag
\text{Sc}\left.\left({^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right]\right)\right|_{\Gamma}=0,
\end{equation}
because $\text{Vec}\left( -{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)\in\mathscr{M}_{\psi^{\theta}}^{*}$. Therefore $\text{Sc}\left({^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right]\right)\equiv 0$ in $\Omega_{\pm}$. Then $\mathbf{F^{\pm}}(x)$
are vector fields.
\end{proof}
\end{theorem}
\begin{theorem} \label{TH2}
Let $\mathbf{f}$ $\in \mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ such that $\displaystyle \mu>\frac{\alpha(\Gamma)}{3}$ and suppose that\\ $\text{Vec}\left( -{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)\in\mathscr{M}_{\psi^{\theta}}^{*}$. If $\bf{f}$ is the trace of a generalized Laplacian vector field in $\mathbf{Lip}_{\mu}(\Omega_{+}\cup\Gamma,\, \mathbb{C}^{3})$, then
\begin{equation}\label{c3}
\begin{split}
&\int_{\Omega_{+}}{\mathscr{K}_{\psi^{\theta}}(t-\xi)\; \text{Sc}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right) }dm(\xi)\\ &=\int_{\Omega_{+}}{\left[\mathscr{K}_{\psi^{\theta}}(t-\xi)\, ,\,\text{Vec}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)\right] }dm(\xi), \quad t\in \Gamma,
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}
\text{Sc}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)&=-\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{1}}+\left(\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{2}}-\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{3}}\right)\sin\theta-\left(\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{2}}+\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{3}}\right)\cos\theta.
\end{split}
\end{equation}
Conversely, if \eqref{c3} is satisfied, then $\bf{f}$ is the trace of a generalized Laplacian vector field in $\mathbf{Lip}_{\nu}(\Omega_{+}\cup\Gamma,\, \mathbb{C}^{3})$ for some $\nu<\mu$.
\begin{proof}
Suppose that $\mathbf{f}$ $\in \mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ is the trace of a generalized Laplacian vector field in $\mathbf{Lip}_{\mu}(\Omega_{+}\cup\Gamma,\, \mathbb{C}^{3})$. Therefore
\begin{equation*}
{^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right]\right|_{\Gamma}=0,
\end{equation*}
by Theorem \ref{t1}.
Of course
\begin{equation*}
\begin{split}
&\int_{\Omega_{+}}{\mathscr{K}_{\psi^{\theta}}(t-\xi)\; \text{Sc}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right) }\,dm(\xi)\\ &=\int_{\Omega_{+}}{\left[\mathscr{K}_{\psi^{\theta}}(t-\xi)\, ,\,\text{Vec}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)\right] }\,dm(\xi), \quad t\in \Gamma,
\end{split}
\end{equation*}
as is easy to check.
Now, if \eqref{c3} is satisfied. Set
\begin{equation}
\mathbf{F^{+}}(x):=-{^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right](x)+ \mathbf{f}^{w}(x), \quad x\in\big(\Omega_{+}\cup\Gamma\big).
\end{equation}
As $\text{Vec}\left( -{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)\in\mathscr{M}_{\psi^{\theta}}^{*}$, $\mathbf{F^{+}}$ is a generalized Laplacian vector field in $\Omega_{+}$. By Theorem \ref{thm 6}, $\left.\mathbf{F^{+}}\right|_{\Gamma}=\mathbf{f}$, which completes the proof.
\end{proof}
\end{theorem}
The method of proof carries to domain $\Omega_{-}$. Indeed, we have
\begin{theorem} \label{TH3}
Let $\mathbf{f}\in \mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ such that $\displaystyle \mu>\frac{\alpha(\Gamma)}{3}$ and suppose that\\ $\text{Vec}\left( -{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)\in\mathscr{M}_{\psi^{\theta}}^{*}$. If $\bf{f}$ is the trace of a generalized Laplacian vector field in $\mathbf{Lip}_{\mu}(\Omega_{-}\cup\Gamma,\, \mathbb{C}^{3})$ which vanishes at infinity, then
\begin{equation}\label{c4}
\begin{split}
&\int_{\Omega_{+}}{\mathscr{K}_{\psi^{\theta}}(t-\xi)\; \text{Sc}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right) }\,dm(\xi)\\ &-\int_{\Omega_{+}}{\left[\mathscr{K}_{\psi^{\theta}}(t-\xi)\, ,\,\text{Vec}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)\right] }\,dm(\xi)=-\mathbf{f}(t), \quad t\in \Gamma.
\end{split}
\end{equation}
Conversely, if \eqref{c4} is satisfied, then $\bf{f}$ is the trace of a generalized Laplacian vector field in $\mathbf{Lip}_{\nu}(\Omega_{-}\cup\Gamma,\, \mathbb{C}^{3})$ for some $\nu<\mu$, which vanishes at infinity.
\end{theorem} \begin{remark} The mains results of this paper are generalizations of those in \cite{ABMP}, where is considered the operator Moisil-Teodorescu
\begin{equation}
D_{MT}:=\textbf{i}\frac{\partial }{\partial x_{1}}+\textbf{j}\frac{\partial }{\partial x_{2}}+\textbf{k}\frac{\partial }{\partial x_{3}}.
\end{equation}
Applying the operator $D_{MT}$ to $\mathbf{h}^{w}:=\mathbf{f}^{w}_{1}\textbf{i}+\mathbf{f}^{w}_{2}\textbf{j}+\mathbf{f}^{w}_{3}\textbf{k}\in C^{1}(\Omega, \mathbb{C}^{3})\cap\mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ we get
\begin{equation}
\begin{split}
D_{MT}[\mathbf{h}^{w}]&=-\text{div}[\mathbf{h}^{w}]+\text{rot}[\mathbf{h}^{w}]\\ &=-\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{1}}-\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{2}}-\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{3}}+\left(\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{2}}-\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{3}}\right)\textbf{i}\\ & +\left(\displaystyle {\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{3}}-\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{1}}}\right)\textbf{j}+\left(\displaystyle {\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{1}}-\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{2}}}\right)\textbf{k}.
\end{split}
\end{equation}
For abbreviation, we let $D_{MT}[\mathbf{h}^{w}]$ stand for
\begin{equation} \label{2}
\begin{split}
D_{MT}[\mathbf{h}^{w}]=\left[D_{MT}[\mathbf{h}^{w}]\right]_{0}+\left[D_{MT}[\mathbf{h}^{w}]\right]_{1}\textbf{i} +\left[D_{MT}[\mathbf{h}^{w}]\right]_{2}\textbf{j}+\left[D_{MT}[\mathbf{h}^{w}]\right]_{3}\textbf{k}.
\end{split}
\end{equation} On the other hand, setting $\mathbf{f}^{w}:=\mathbf{f}^{w}_{1}\textbf{i}+\mathbf{f}^{w}_{3}\textbf{j}+\mathbf{f}^{w}_{2}\textbf{k}\in C^{1}(\Omega, \mathbb{C}^{3})\cap\mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ it follows that
\begin{equation}
\begin{split}
{^{\psi^{0}}D}[\mathbf{f}^{w}]&=-\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{1}}-\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{2}}-\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{3}}+\left({\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{3}}-\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{2}}}\right)\textbf{i}\\ & +\left(\displaystyle {\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{2}}-\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{1}}}\right)\textbf{j}+\left(\displaystyle {\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{1}}-\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{3}}}\right)\textbf{k}.
\end{split}
\end{equation}
The above expression may be written as
\begin{equation} \label{1}
\begin{split}
{^{\psi^{0}}D}[\mathbf{f}^{w}]=\left[{^{\psi^{0}}D}[\mathbf{f}^{w}]\right]_{0}+\left[{^{\psi^{0}}D}[\mathbf{f}^{w}]\right]_{1}\textbf{i}+\left[{^{\psi^{0}}D}[\mathbf{f}^{w}]\right]_{2}\textbf{j}+\left[{^{\psi^{0}}D}[\mathbf{f}^{w}]\right]_{3}\textbf{k}.
\end{split}
\end{equation} It is worth noting that under the correspondence $\left(\mathbf{f}^{w}_{1},\,\mathbf{f}^{w}_{2},\,\mathbf{f}^{w}_{3}\right)\, \leftrightarrow \, \left(\mathbf{f}^{w}_{1},\,\mathbf{f}^{w}_{3},\,\mathbf{f}^{w}_{2}\right)$ we can assert that \begin{equation}\label{equiv} D_{MT}[\mathbf{h}^{w}]=0\,\Longleftrightarrow \, {}{^{\psi^{0}}D}[\mathbf{f}^{w}]=0, \end{equation} which follow from
\begin{align*}
\left[D_{MT}[\mathbf{h}^{w}]\right]_{0} &=\left[{^{\psi^{0}}D}[\mathbf{f}^{w}]\right]_{0} ,\\ \left[D_{MT}[\mathbf{h}^{w}]\right]_{1} & =- \left[{^{\psi^{0}}D}[\mathbf{f}^{w}]\right]_{1},\\ \left[D_{MT}[\mathbf{h}^{w}]\right]_{2} & =-\left[{^{\psi^{0}}D}[\mathbf{f}^{w}\right]_{3}, \\
\left[D_{MT}[\mathbf{h}^{w}]\right]_{3} & =-\left[{^{\psi^{0}}D}[\mathbf{f}^{w}]\right]_{2}.
\end{align*} \end{remark} \begin{remark}
In \cite{ABMP} is defined
\begin{equation}
\mathscr{M}^{*}:=\left\{\mathbf{f}: \frac{1}{4\pi}\int_{\Omega_{+}}{\left\langle \text{grad}\;\frac{1}{\abs{t-\xi}}\, ,\,\mathbf{f}(\xi)\right\rangle}\,dm(\xi)=0, \, t\in\Gamma \right\}.
\end{equation}
For $\mathbf{h}:=\mathbf{f_{1}}\textbf{i}+\mathbf{f_{2}}\textbf{j}+\mathbf{f_{3}}\textbf{k} \in \mathscr{M}^{*}$ it is clear that
\begin{equation}
\begin{split}
\frac{1}{4\pi}\int_{\Omega_{+}}{\left\langle \text{grad}\;\frac{1}{\abs{t-\xi}}\, ,\,\mathbf{h}(\xi)\right\rangle}\,dm(\xi)=\int_{\Omega_{+}}{\left\langle\mathscr{K}_{\psi^{0}}(t-\xi)\, ,\,\mathbf{f}(\xi)\right\rangle}\,dm(\xi)=0,
\end{split}
\end{equation} where $\mathbf{f}:=\mathbf{f}_{1}\textbf{i}+\mathbf{f}_{3}\textbf{j}+\mathbf{f}_{2}\textbf{k} \in \mathscr{M}^{*}_{\psi^{0}}$. Hence $$\mathbf{h}:=\mathbf{f}_{1}\textbf{i}+\mathbf{f}_{2}\textbf{j}+\mathbf{f}_{3}\textbf{k}\in \mathscr{M}^{*} \iff \mathbf{f}:=\mathbf{f}_{1}\textbf{i}+\mathbf{f}_{3}\textbf{j}+\mathbf{f}_{2}\textbf{k} \in \mathscr{M}^{*}_{\psi^{0}}.$$
\end{remark} From Theorems \ref{TH1}, \ref{TH2}, \ref{TH3} and the previous remarks the followings corollaries are obtained. \begin{corollary} \cite[Theorem 2.2]{ABMP}.
Let $\mathbf{f}\in \mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ such that $\displaystyle \mu>\frac{\alpha(\Gamma)}{3}$. Then the reconstruction problem for the div-rot system is solvable if $\text{rot}[\mathbf{f}^{w}]\in\mathscr{M}^{*}$. \end{corollary}
\begin{corollary} \cite[Theorem 2.3]{ABMP}.
Let $\mathbf{f}\in \mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ such that $\displaystyle \mu>\frac{\alpha(\Gamma)}{3}$ and suppose that $\text{rot}[\mathbf{f}^{w}]\in\mathscr{M}^{*}$. If $\bf{f}$ is the trace of a Laplacian vector field in $\mathbf{Lip}_{\mu}(\Omega_{+}\cup\Gamma,\, \mathbb{C}^{3})$, then
\begin{equation}\label{c31}
\begin{split}
&\frac{1}{4\pi}\int_{\Omega_{+}}{ \text{grad}\;\frac{1}{\abs{t-\xi}}\; \text{div}[\mathbf{f}^{w}]}\,dm(\xi)\\ &=\frac{1}{4\pi}\int_{\Omega_{+}}{\left[ \text{grad}\;\frac{1}{\abs{t-\xi}}\,,\, \text{rot}[\mathbf{f}^{w}]\right] }\,dm(\xi), \quad t\in \Gamma.
\end{split}
\end{equation}
Conversely, if \eqref{c31} is satisfied, then $\bf{f}$ is the trace of a Laplacian vector field in $\mathbf{Lip}_{\nu}(\Omega_{+}\cup\Gamma,\, \mathbb{C}^{3})$ for some $\nu<\mu$. \end{corollary}
\begin{corollary} \cite[Theorem 2.4]{ABMP}.
Let $\mathbf{f}\in\mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ such that $\displaystyle \mu>\frac{\alpha(\Gamma)}{3}$ and suppose that $\text{rot}[\mathbf{f}^{w}]\in\mathscr{M}^{*}$. If $\bf{f}$ is the trace of a Laplacian vector field in $\mathbf{Lip}_{\mu}(\Omega_{-}\cup\Gamma, \, \mathbb{C}^{3})$ which vanishes at infinity, then
\begin{equation}\label{c42}
\begin{split}
&\frac{1}{4\pi}\int_{\Omega_{+}}{ \text{grad}\;\frac{1}{\abs{t-\xi}}\;\text{div}[\mathbf{f}^{w}]}\,dm(\xi)\\ &-\frac{1}{4\pi}\int_{\Omega_{+}}{\left[ \text{grad}\;\frac{1}{\abs{t-\xi}}\,,\,\text{rot}[\mathbf{f}^{w}]\right] }\,dm(\xi)=-\mathbf{f}(t), \quad t\in \Gamma.
\end{split}
\end{equation}
Conversely, if \eqref{c42} is satisfied, then $\bf{f}$ is the trace of a Laplacian vector field in $\mathbf{Lip}_{\nu}(\Omega_{-}\cup\Gamma,\, \mathbb{C}^{3})$ for some $\nu<\mu$, which vanishes at infinity. \end{corollary}
\section*{Appendix. Criteria for the generalized Laplacianness of a vector field}
We continue to assume that $\Omega\subset \mathbb{R}^{3}$ is a Jordan domain with a fractal boundary $\Gamma$. Our interest here is to find necessary and sufficient conditions for the generalized Laplacianness of an vector field $\mathbf{F}\in\mathbf{Lip}_{\nu}(\Omega\cup\Gamma,\, \mathbb{C}^{3})$ in terms of its boundary value $\mathbf {f}:=\left.\mathbf {F}\right|_\Gamma$.
The inspiration for the following definition is that in \cite[Definition 2.1]{ARBR}.
\begin{defi} \label{dtc1}
Let $\Omega$ a Jordan domain with fractal boundary $\Gamma$. Then we define the Cauchy transform of $\mathbf{f}\in \mathbf{Lip}_{\mu}(\Gamma,\,\mathbb{C}^{3})$ by
\begin{equation}\label{tc2}
K_{\Gamma}^{*}[\mathbf{f}](x):=-{^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right](x)+ \mathbf{f}^{w}(x), \quad x\in \mathbb{R}^{3}\setminus\Gamma.
\end{equation}
\end{defi} Under condition $\displaystyle \frac{\alpha(\Gamma)}{3}<\mu \leq 1$ the Cauchy transform $K_{\Gamma}^{*}[\mathbf{f}]$ has continuous extension to $\Omega\cup\Gamma$ for every vector field $\mathbf {f}\in \mathbf{Lip}_{\mu}(\Gamma,\,\mathbb{C}^{3})$ (take a fresh look at Theorem \ref{thm 6}). On the other hand, using the properties of the Theodorescu operator (see \cite{KVS}, p. 73) we obtain that $K_{\Gamma}^{*}[\bf{f}]$ is left-$\psi^{\theta}$-hyperholomorphic in $\mathbb{R}^{3}\setminus\Gamma$. Note that $K_{\Gamma}^{*}[\mathbf{f}](x)$ vanishes at infinity.
Let us introduce the following fractal version of the Cauchy singular integral operator
\begin{equation*}
\mathcal{S}_{\Gamma}^{*}[\mathbf{f}](x):=2K^{*}_{\Gamma}[\mathbf{f}]^{+}(x)-f(x), \quad x\in\Gamma.
\end{equation*}
Here and subsequently, $K^{*}_{\Gamma}[\mathbf{f}]^{+}$ denotes the trace on $\Gamma$ of the continuous extension of $K^{*}_{\Gamma}[\mathbf{f}]$ to $\Omega\cup\Gamma$.
Let us now establish and prove the main result of this appendix, which gives necessary and sufficient conditions for the generalized Laplacianness of a vector field in terms of its boundary value.
\begin{theorem}
Let $\mathbf{F}\in\mathbf{Lip}_{\mu}(\Omega\cup\Gamma,\mathbb{C}^{3})$ with trace $\mathbf{f}=\left.\mathbf{F}\right|_{\Gamma}$. Then the following sentences are equivalent:
\begin{itemize}
\item [(i)] $\mathbf{F}$ is a generalized Laplacian vector field.
\item [(ii)] $\mathbf{F}$ is harmonic in $\Omega$ and $\mathcal{S}_{\Gamma}^{*}[\mathbf{f}]=\mathbf{f}$.
\end{itemize}
\begin{proof} Let $\mathbf{F}^{w}$ be the Whitney extension of $\mathbf{F}$ in $\mathbf{Lip}_{\mu}(\Omega\cup\Gamma,\mathbb{C}^{3})$. Suppose that $\mathbf{F}$ is a generalized Laplacian vector field in $\Omega$. Since ${^{\psi^{\theta}}}D[\mathbf{F}]=0$ in $\Omega$, it follows that $\mathbf{F}$ is harmonic. Also $\mathbf{F}^{w}$ is a Whitney extension of $\mathbf{f},$ i.e. $\mathbf{f}=\left.\mathbf{F}^{w}\right|_{\Gamma}$.
According to Definition \ref{dtc1}, with $\mathbf{f}^{w}$ replaced by $\mathbf{F}^{w}$, we get
\begin{equation*}
K_{\Gamma}^{*}[\mathbf{f}](x)=-\int_{\Omega}{\mathscr{K}_{\psi^{\theta}}(x-\xi)\, {{^{\psi^{\theta}}}D}[\mathbf{F}^{w}](\xi) }\,dm(\xi)+ \mathbf{F}^{w}(x)=\mathbf{F}(x), \quad x\in\Omega,
\end{equation*} which imply that $K_{\Gamma}^{*}[\mathbf{f}]^{+}=\mathbf{f}$ and $\mathcal{S}_{\Gamma}^{*}[\mathbf{f}]=\mathbf{f}$.
Conversely, assume that $(ii)$ holds and define
\begin{equation}
\Psi(x):= \left\{
\begin{array}{ll}
K_{\Gamma}^{*}[\mathbf{f}](x), & x \in\Omega, \\
\mathbf{f}(x), & x \in \Gamma.
\end{array}
\right.
\end{equation}
Note that $\Psi(x)$ is left-$\psi^{\theta}$-hyperholomorphic function, hence harmonic in $\Omega$. Since $\mathcal{S}_{\Gamma}^{*}[\mathbf{f}]=\mathbf{f}$ in $\Gamma$, it follows that $K_{\Gamma}^{*}[\mathbf{f}]^{+}=\mathbf{f}$. Therefore $K_{\Gamma}^{*}[\mathbf{f}]$ is also continuous on $\Omega\cup\Gamma$.
As $\mathbf{F}-\Psi$ is harmonic in $\Omega$ and $\left.(\mathbf{F}-\Psi)\right|_{\Gamma}=0$ we have that $\mathbf{F}(x)=K_{\Gamma}^{*}[\mathbf{f}](x)$ for all $x \in\Omega,$ which follows from the harmonic maximum principle. Lemma \ref{two-sided} now forces $\mathbf{F}$ to be a generalized Laplacian vector field in $\Omega,$ and the proof is complete.
\end{proof}
\end{theorem}
\end{document} | arXiv |
\begin{document}
\title[Hessian equations and systems with weights]{A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights} \author[D.-P. Covei]{Dragos-Patru Covei} \address{Department of Applied Mathematics, The Bucharest University of Economic Studies, Piata Romana, 1st district, postal code: 010374, postal office: 22, Romania} \email{\texttt{[email protected]}} \keywords{{\small Existence; Keller-Osserman condition; k-Hessian equation and system.}\\ \phantom{aa} 2010 AMS Subject Classification: Primary: 35J60; 35J65; 35J66; Secondary: \ 35J96; 35J99.}
\begin{abstract} {\footnotesize In this article we consider the existence} {\footnotesize of positive radial solutions for\textbf{\ }Hessian equations and systems with weights and we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation. Our results complete and improve a recently work published by Zhang and Zhou (\textit{Existence of entire positive k-convex radial solutions to Hessian equations and systems with weights}, Applied Mathematics Letters, Volume 50, December 2015, Pages 48--55).} \end{abstract}
\maketitle \tableofcontents
\section{Introduction}
Let $D^{2}u$ be the Hessian matrix of a $C^{2}$ (i.e., a twice continuously differentiable) function $u$ defined over $\mathbb{R}^{N}$ ($N\geqslant 3$) and $ \lambda \left( D^{2}u\right) =\left( \lambda _{1},...,\lambda _{N}\right) $ the vector of eigenvalues of $D^{2}u$. For $k=1,2,...,N$ is defined the $k$ -Hessian operator as follows \begin{equation*} S_{k}\left( \lambda \left( D^{2}u\right) \right) =\underset{1\leqslant i_{1}<...<i_{k}\leqslant N}{\sum }\lambda _{i_{1}}\cdot ...\cdot \lambda _{i_{k}} \end{equation*} i.e., it is the $k^{th}$ elementary symmetric polynomial of the Hessian matrix of $u$. In other words, $S_{k}\left( \lambda \left( D^{2}u\right) \right) $ it is the sum of all $k\times k$ principal minors of the Hessian matrix $D^{2}u$ and so is a second order differential operator, which may also be called the $k$-trace of $D^{2}u$. Especially, it is easily to see that the $N$-Hessian is the Monge-Amp\'{e}re operator and that the $1$ -Hessian is the well known classical Laplace operator. Hence, the $k$ -Hessian operators form a discrete collection of partial differential operators which includes both the Laplace and the Monge-Amp\'{e}re operator.
In this paper we study the existence of radial solutions for the following Hessian equation \begin{equation} S_{k}^{1/k}\left( \lambda \left( D^{2}u\right) \right) =p\left( \left\vert x\right\vert \right) h\left( u\right) \text{ in }\mathbb{R}^{N}\text{ ,} \label{11} \end{equation} and system \begin{equation} \left\{ \begin{array}{c} S_{k}^{1/k}\left( \lambda \left( D^{2}u\right) \right) =p\left( \left\vert x\right\vert \right) f\left( u,v\right) \text{ in }\mathbb{R}^{N}\text{ ,} \\ S_{k}^{1/k}\left( \lambda \left( D^{2}v\right) \right) =q\left( \left\vert x\right\vert \right) g\left( u,v\right) \text{ in }\mathbb{R}^{N}\text{ ,} \end{array} \right. \label{11s} \end{equation} where $k\in \left\{ 1,2,...,N\right\} $, the continuous functions $p$, $q$ \textit{\ }$:\left[ 0,\infty \right) \rightarrow \left( 0,\infty \right) $, $ h:\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) $ and $f$, $g: \left[ 0,\infty \right) \times \left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) $ satisfy some of the conditions:
(P1)\quad $p$, $q$ is a spherically symmetric function (i.e.\textit{\ }$ p\left( x\right) =p\left( \left\vert x\right\vert \right) $, $q\left( x\right) =q\left( \left\vert x\right\vert \right) );$
(P2)\quad $r^{N+\frac{N}{k}-2}p^{k}\left( r\right) $\textit{\ }is nondecreasing for large $r$;
(P3)\quad $r^{N+\frac{N}{k}-2}\left[ p^{k}\left( r\right) +q^{k}\left( r\right) \right] $\textit{\ }is nondecreasing for large $r$\textit{;}
(C1)\quad $h$ is monotone non-decreasing, $h(0)=0$ and $h\left( s\right) >0$ for all $s>0$;
(C2)\quad $f$, $g$ are monotone non-decreasing in each variable, $ f(0,0)=g\left( 0,0\right) =0$ and $f(s,t)>0,$ $g\left( s,t\right) >0$ for all $s,t>0$;
(C3)\quad $\int_{1}^{\infty }\frac{1}{\sqrt[k+1]{\left( k+1\right) H(t)}} dt=\infty $ for\ $H(t)=\int_{0}^{t}h^{k}(z)dz;$
(C4)\quad $\int_{1}^{\infty }\frac{1}{\sqrt[k+1]{\left( k+1\right) F(t)}} dt=\infty $ for\ $F(t)=\int_{0}^{t}\left( f^{k}(z,z)+g^{k}(z,z)\right) dz.$
The properties of the $k$-Hessian operator was well discussed in a numerous papers written as a first author by Ivochkina (see \cite{III}-\cite{IF} and others). Moreover, this operator appear as an object of investigation by many remarkable geometers. For example, Viaclovsky (see \cite{V}, \cite{VI}) observed that the $k-$Hessian operator is an important class of fully nonlinear operators which is closely related to a geometric problem of the type (\ref{11}), where we cite the work of Bao-Ji-Li \cite{BAOII} for a more detailed discussion. Moreover, equation (\ref{11}) arises via the study of the quasilinear parabolic problem ( see for example the introduction of Moll-Petitta \cite{MP}). In the present work we will limit ourselves to the development of mathematical theory for (\ref{11}) and (\ref{11s}). The main difficulty in investigating problems, such as (\ref{11}) or (\ref{11s}), in which appear the $k$-Hessian operator is related to the fact that their properties change depending on the subset of $C^{2}$ from where the solution is taken. Our main objective here is to find functions in $C^{2}$ that are strictly $k$-convex and verifies the problems (\ref{11}), (\ref{11s}), where by strictly $k$-convex function $u$ we mean that all eigenvalues $\lambda _{1},...,\lambda _{N}$ of the symmetric matrix $D^{2}u$ are in the so called G\aa rdding open cone $\Gamma _{k}$ which is defined by \begin{equation*} \Gamma _{k}\left( N\right) =\left\{ \lambda \in \mathbb{R}^{N}\left\vert S_{1}\left( \lambda \right) >0,....,S_{k}\left( \lambda \right) >0\right. \right\} \end{equation*} In the next we adopt the notation from Bao-Li \cite{BAO} for the space of all admissible functions \begin{equation*} \Phi ^{k}\left( \mathbb{R}^{N}\right) :=\left\{ u\in C^{2}\left( \mathbb{R} ^{N}\right) \left\vert \lambda \in \Gamma _{k}\left( N\right) \text{ for all }x\in \mathbb{R}^{N}\right. \right\} . \end{equation*} In our direction, there are some recently papers resolving existence for blow-up solutions of (\ref{11}) and (\ref{11s}). Here we wish to mention the works of Bao-Ji-Li \cite{BAOII}, Jacobsen \cite{J}, Bao-Li \cite{BAO}, Lazer and McKenna \cite[(the case $k=N$)]{LM}, Salani \cite{S} and Zhang-Zhou \cite {ZZ} which will be useful in our proofs. It is interesting to note that in our results the dimension of the space $\mathbb{R}^{N}$ affect the properties of the solution of the equation and system which in the case of the classical Laplace operator and the Monge-Amp\'{e}re operator this condition doesn't appear in any works.
Motivated by the recent work of Zhang-Zhou \cite{ZZ} we are interested in proving the following theorems:
\begin{theorem} \label{th1}Let $k\in \left\{ 1,2,...,\left[ N/2\right] \right\} $ if $N$ is odd or $k\in \left\{ 1,2,...,\left[ N/2\right] -1\right\} $ if $N$ is even. Suppose that \textrm{(P1), (P2)}, \textrm{(C1)}, \textrm{(C3)} are satisfied. If there exists a positive number $\varepsilon $ such that \begin{equation} \quad \text{ }\int_{0}^{\infty }t^{1+\varepsilon +\frac{2\left( k-1\right) }{ k+1}}\left( p\left( t\right) \right) ^{\frac{2k}{k+1}}dt<\infty , \label{5} \end{equation} then system \textrm{(\ref{11})} has a nonnegative nontrivial radial bounded solution $u\in \Phi ^{k}\left( \mathbb{R}^{N}\right) $. \end{theorem}
\begin{theorem} \label{th2}\textit{If }$p$ satisfy \textit{\textrm{(P1)}} and\ $f$ \textit{ satisfy \textrm{(C1)}, \textrm{(C3)}, then the problem \textrm{(\ref{11})} has a nonnegative nontrivial entire radial solution }$u\in \Phi ^{k}\left( \mathbb{R}^{N}\right) $\textit{. Suppose furthermore that \textrm{(P2)}} holds.\textit{\ }If $p$ satisfies \begin{equation} \int_{0}^{\infty }\left( \frac{k}{t^{N-k}}\int_{0}^{t}\frac{s^{N-1}}{ C_{N-1}^{k-1}}p^{k}\left( s\right) ds\right) ^{1/k}dt=\infty ,\text{ } \label{12} \end{equation} then any nonnegative nontrivial radial solution $u\in \Phi ^{k}\left( \mathbb{R}^{N}\right) $ of \textrm{(\ref{11})} is large. Conversely, \textit{ if }\textrm{(\ref{11})} has a nonnegative entire large radial solution $u\in \Phi ^{k}\left( \mathbb{R}^{N}\right) $, then one or both of the following \begin{equation} \begin{array}{ll} 1. & \int_{0}^{\infty }t^{1+\varepsilon +\frac{2\left( k-1\right) }{k+1} }\left( p\left( t\right) \right) ^{\frac{2k}{k+1}}dr=\infty \text{ for every }\varepsilon >0\,; \\ 2. & k\in \left\{ \left[ N/2\right] +1,...,N\right\} \text{ if }N\text{ is odd or }k\in \left\{ \left[ N/2\right] ,...,N\right\} \text{ if }N\text{ is even,} \end{array} \label{13} \end{equation} hold. \end{theorem}
Regarding existence of solution to (\ref{11s}), we have the following results.
\begin{theorem} \label{th3}Let $k\in \left\{ 1,2,...,\left[ N/2\right] \right\} $ if $N$ is odd or $k\in \left\{ 1,2,...,\left[ N/2\right] -1\right\} $ if $N$ is even. Suppose that \textrm{(P1), (P3)}, \textrm{(C2)}, \textrm{(C4)} are satisfied. If there exists a positive number $\varepsilon $ such that \begin{equation} \text{ }\int_{0}^{\infty }t^{1+\varepsilon +\frac{2\left( k-1\right) }{k+1} }\left( p^{k}\left( t\right) +q^{k}\left( t\right) \right) ^{\frac{2}{k+1} }dt<\infty \text{ }, \label{5s} \end{equation} then system \textrm{(\ref{11s})} has a nonnegative nontrivial radial bounded solution $\left( u,v\right) \in \Phi ^{k}\left( \mathbb{R}^{N}\right) \times \Phi ^{k}\left( \mathbb{R}^{N}\right) $. \end{theorem}
\begin{theorem} \label{th4}\textit{If }$p,$ $q$ satisfy \textit{\textrm{(P1)}} and\textit{\ } \ $f,$ $g$ \textit{satisfy \textrm{(C2)}, \textrm{(C4)}, then the problem \textrm{(\ref{11})} has a nonnegative nontrivial entire radial solution. Suppose furthermore that \textrm{(P3)} holds. }If $p$ satisfies \begin{equation} \int_{0}^{\infty }\left( \frac{k}{t^{N-k}}\int_{0}^{t}\frac{s^{N-1}}{ C_{N-1}^{k-1}}p^{k}\left( s\right) ds\right) ^{1/k}dt=\infty \text{ and } \int_{0}^{\infty }\left( \frac{k}{t^{N-k}}\int_{0}^{t}\frac{s^{N-1}}{ C_{N-1}^{k-1}}q^{k}\left( s\right) ds\right) ^{1/k}dt=\infty , \label{12s} \end{equation} then any nonnegative nontrivial solution $\left( u,v\right) \in \Phi ^{k}\left( \mathbb{R}^{N}\right) \times \Phi ^{k}\left( \mathbb{R} ^{N}\right) $ of \textrm{(\ref{11s})} is large. Conversely, if \textrm{(\ref {11s})} has a nonnegative entire large radial solution $\left( u,v\right) \in \Phi ^{k}\left( \mathbb{R}^{N}\right) \times \Phi ^{k}\left( \mathbb{R} ^{N}\right) $, then one or both of the following \begin{equation} \begin{array}{ll} 1. & \int_{0}^{\infty }t^{1+\varepsilon +\frac{2\left( k-1\right) }{k+1} }\left( p^{k}\left( t\right) +q^{k}\left( t\right) \right) ^{\frac{2}{k+1} }dr=\infty \text{ for every }\varepsilon >0\,; \\ 2. & k\in \left\{ \left[ N/2\right] +1,...,N\right\} \text{ if }N\text{ is odd or }k\in \left\{ \left[ N/2\right] ,...,N\right\} \text{ if }N\text{ is even,} \end{array} \label{13s} \end{equation} hold. \end{theorem}
For the readers' convenience, we recall the radial form of the $k$-Hessian operator.
\begin{remark} (see, for example, \cite{BAO}, \cite{S}) If $u:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is radially symmetric then a calculation show \begin{equation*} S_{k}\left( \lambda \left( D^{2}u\left( r\right) \right) \right) =r^{1-N}C_{N-1}^{k-1}\left[ \frac{r^{N-k}}{k}\left( u^{^{\prime }}\left( r\right) \right) ^{k}\right] ^{\prime }\text{ ,} \end{equation*} where the prime denotes differentiation with respect to $r=\left\vert x\right\vert $ and $C_{N-1}^{k-1}=(N-1)!/\left[ (k-1)!(N-k)!\right] $. \end{remark}
\section{Proofs of the main results}
In this section we give the proofs of Theorems \ref{th1} - \ref{th4}. The main references for proving Theorems \ref{th1} - \ref{th2} is the work of Lair \cite{LA} and Delano\"{e} \cite{DEL} see also Afrouzi-Shokooh \cite{AF}.
\subparagraph{Proof of the Theorem \protect\ref{th1}}
Assume that (\ref{5}) holds. We prove the existence of $w\in \Phi ^{k}\left( \mathbb{R}^{N}\right) $ to the problem
\begin{equation} \begin{array}{l} S_{k}^{1/k}\left( \lambda \left( D^{2}w\left( \left\vert x\right\vert \right) \right) \right) =p\left( \left\vert x\right\vert \right) h\left( w\left( \left\vert x\right\vert \right) \right) \text{ in }\mathbb{R}^{N} \text{. } \end{array} \label{6} \end{equation} Observe that we can rewrite (\ref{6}) as follows: \begin{equation*} \left[ \frac{r^{N-k}}{k}\left( w^{^{\prime }}(r)\right) ^{k}\right] ^{\prime }=\frac{r^{N-1}}{C_{N-1}^{k-1}}p^{k}\left( r\right) h^{k}\left( w\left( r\right) \right) ,\text{ }r=\left\vert x\right\vert . \end{equation*} Then radial solutions of (\ref{6}) are any solution $w$ of the integral equation \begin{equation*} w\left( r\right) =1+\int_{0}^{r}\left( \frac{k}{t^{N-k}}\int_{0}^{t}\frac{ s^{N-1}}{C_{N-1}^{k-1}}p^{k}\left( s\right) h^{k}\left( w\left( s\right) \right) ds\right) ^{1/k}dt.\text{ } \end{equation*} To establish a solution to this problem, we use successive approximation. Define sequence $\left\{ w^{m}\right\} ^{m\geqslant 1}$ on $\left[ 0,\infty \right) $ by \begin{equation*} \left\{ \begin{array}{l} w^{0}=1,\text{ }r\geqslant 0, \\ w^{m}\left( r\right) =1+\int_{0}^{r}\left( \frac{k}{t^{N-k}}\int_{0}^{t} \frac{s^{N-1}}{C_{N-1}^{k-1}}p^{k}\left( s\right) h^{k}\left( w^{m-1}\left( s\right) \right) ds\right) ^{1/k}dt. \end{array} \right. \end{equation*} We remark that, for all $r\geqslant 0$ and $m\in N$ \begin{equation*} w^{m}\left( r\right) \geqslant 1\text{.} \end{equation*} Moreover, proceeding by induction we conclude $\left\{ w^{m}\right\} ^{m\geqslant 1}$ are non-decreasing sequence on $\left[ 0,\infty \right) $. We note that $ \left\{ w^{m}\right\} ^{m\geqslant 1}$ satisfies \begin{equation*} \left\{ \frac{r^{N-k}}{k}\left[ \left( w^{^{m}}(r)\right) ^{\prime }\right] ^{k}\right\} ^{\prime }=\frac{r^{N-1}}{C_{N-1}^{k-1}}p^{k}\left( r\right) h^{k}\left( w^{m-1}\left( r\right) \right) . \end{equation*} By the monotonicity of $\left\{ w^{m}\right\} ^{m\geqslant 1}$ we have the inequalities \begin{equation} \left\{ \frac{r^{N-k}}{k}\left[ \left( w^{^{m}}(r)\right) ^{\prime }\right] ^{k}\right\} ^{\prime }=\frac{r^{N-1}}{C_{N-1}^{k-1}}p^{k}\left( r\right) h^{k}\left( w^{m-1}\left( r\right) \right) \leqslant \frac{r^{N-1}}{C_{N-1}^{k-1}} p^{k}\left( r\right) h^{k}\left( w^{m}\left( r\right) \right) . \label{8} \end{equation} Choose $R>0$ so that $r^{N+\frac{N}{k}-2}p^{k}\left( r\right) $ are non-decreasing for $r\geqslant R$. We are now ready to show that $w^{m}\left( R\right) $ and $\left( w^{m}\left( R\right) \right) ^{\prime }$, both of which are nonnegative, are bounded above independent of $m$. To do this, let \begin{equation*} \phi ^{R}=\max \{p^{k}\left( r\right) :0\leqslant r\leqslant R\}\text{.} \end{equation*} Using this and the fact that $\left( w^{m}\right) ^{\prime }\geqslant 0$, we note that (\ref{8}) yields \begin{eqnarray*} r^{N-k}\left[ \left( w^{^{m}}(r)\right) ^{\prime }\right] ^{k-1}\left( w^{^{m}}(r)\right) ^{\prime \prime } &\leqslant &\frac{N-k}{k}r^{N-k-1}\left[ \left( w^{^{m}}(r)\right) ^{\prime }\right] ^{k}+r^{N-k}\left[ \left( w^{^{m}}(r)\right) ^{\prime }\right] ^{k-1}\left( w^{^{m}}(r)\right) ^{\prime \prime } \\ &\leqslant &\phi _{1}^{R}\frac{r^{N-1}}{C_{N-1}^{k-1}}h^{k}\left( w^{m}\left( r\right) \right) , \end{eqnarray*} and moreover \begin{equation*} r^{N-k}\left[ \left( w^{^{m}}(r)\right) ^{\prime }\right] ^{k-1}\left( w^{^{m}}(r)\right) ^{\prime \prime }\leqslant \phi ^{R}\frac{r^{N-k+k-1}}{ C_{N-1}^{k-1}}h^{k}\left( w^{m}\left( r\right) \right) \leqslant R^{k-1}\phi ^{R} \frac{r^{N-k}}{C_{N-1}^{k-1}}h^{k}\left( w^{m}\left( r\right) \right) , \end{equation*} from which we have \begin{equation*} \left[ \left( w^{^{m}}(r)\right) ^{\prime }\right] ^{k-1}\left( w^{^{m}}(r)\right) ^{\prime \prime }\leqslant R^{k-1}\phi ^{R}\frac{1}{ C_{N-1}^{k-1}}h^{k}\left( w^{m}\left( r\right) \right) . \end{equation*} Multiply this by $\left( w^{^{m}}(r)\right) ^{\prime }$ we obtain \begin{equation} \left\{ \left[ \left( w^{^{m}}(r)\right) ^{\prime }\right] ^{k+1}\right\} ^{\prime }\leqslant \frac{\left( k+1\right) R^{k-1}\phi ^{R}}{C_{N-1}^{k-1}} h^{k}\left( w^{m}\left( r\right) \right) \left( w^{^{m}}(r)\right) ^{\prime }. \label{ec} \end{equation} Integrate (\ref{ec}) from $0$ to $r$ to get \begin{equation} \left[ \left( w^{^{m}}(r)\right) ^{\prime }\right] ^{k+1}\leqslant \frac{\left( k+1\right) R^{k-1}\phi ^{R}}{C_{N-1}^{k-1}}\int_{0}^{r}h^{k}\left( w^{m}\left( s\right) \right) \left( w^{^{m}}(s)\right) ^{\prime }ds=\frac{ \left( k+1\right) R^{k-1}\phi ^{R}}{C_{N-1}^{k-1}} \int_{1}^{w^{^{m}}(r)}h^{k}\left( s\right) ds \label{9} \end{equation} for $0\leqslant r\leqslant R$, which yields \begin{equation*} \int_{1}^{w^{^{m}}(R)}\left[ \int_{1}^{t}h^{k}\left( s\right) ds\right] ^{-1/\left( k+1\right) }dt\leqslant \sqrt[k+1]{\frac{\left( k+1\right) \phi ^{R}}{ C_{N-1}^{k-1}}}\cdot R^{\frac{2k}{k+1}}. \end{equation*} It follows from the above relation and by the assumption (C2) that $ w_{1}^{m}\left( R\right) $ is bounded above independent of $m$. Using this fact in (\ref{9}) shows that the same is true of $\left( w^{m}\left( R\right) \right) ^{\prime }$. Thus, the sequences $w^{m}\left( R\right) $ and $\left( w^{m}\left( R\right) \right) ^{\prime }$ are bounded above independent of $m$.
Finally, we show that the non-decreasing sequences $w^{m}$ is bounded for all $r\geqslant 0$ and all $m$. Multiplying the equation (\ref{8}) by $r^{N+\frac{ N}{k}-2}\left( w^{^{m}}(r)\right) ^{\prime }$, we get \begin{equation} \left\{ \left[ r^{\frac{N}{k}-1}\left( w^{m}\left( r\right) \right) ^{\prime }\right] ^{k+1}\right\} ^{\prime }=\frac{k+1}{C_{N-1}^{k-1}}p^{k}\left( r\right) h^{k}(w^{m}\left( r\right) )r^{N+\frac{N}{k}-2}\left( w^{m}\left( r\right) \right) ^{\prime }. \label{ma14} \end{equation} Integrating from $R$ to $r$ gives \begin{equation*} \left[ r^{\frac{N}{k}-1}\left( w^{m}\left( r\right) \right) ^{\prime }\right] ^{k+1}=\left[ R^{\frac{N}{k}-1}\left( w^{m}\left( R\right) \right) ^{\prime } \right] ^{k+1}+\frac{k+1}{C_{N-1}^{k-1}}\int_{R}^{r}p^{k}\left( s\right) h^{k}(w^{m}\left( s\right) )s^{N+\frac{N}{k}-2}\left( w^{m}\left( s\right) \right) ^{\prime }ds, \end{equation*} for $r\geqslant R$. Noting that, by the monotonicity of $s^{N+\frac{N}{k} -2}p^{k}\left( s\right) $ for $r\geqslant s\geqslant R$, we get \begin{equation*} \left[ r^{\frac{N}{k}-1}\left( w^{m}\left( r\right) \right) ^{\prime }\right] ^{k+1}\leqslant C+\frac{k+1}{C_{N-1}^{k-1}}r^{N+\frac{N}{k}-2}p^{k}\left( r\right) H\left( w^{m}\left( r\right) \right) \end{equation*} where $C=\left[ R^{\frac{N}{k}-1}\left( w^{m}\left( R\right) \right) ^{\prime }\right] ^{k+1}$, which yields \begin{equation*} r^{\frac{N}{k}-1}\left( w^{m}\left( r\right) \right) ^{\prime }\leqslant C^{\frac{ 1}{k+1}}+\left( \frac{k+1}{C_{N-1}^{k-1}}\right) ^{\frac{1}{k+1}}r^{\frac{N}{ k}-\frac{2}{k+1}}p^{\frac{k}{k+1}}\left( r\right) H^{\frac{1}{k+1}}\left( w^{m}\left( r\right) \right) \end{equation*} or, equivalently \begin{equation*} \left( w^{m}\left( r\right) \right) ^{\prime }\leqslant C^{\frac{1}{k+1}}r^{1- \frac{N}{k}}+\left( \frac{k+1}{C_{N-1}^{k-1}}\right) ^{\frac{1}{k+1}}r^{1- \frac{2}{k+1}}p^{\frac{k}{k+1}}\left( r\right) H^{\frac{1}{k+1}}\left( w^{m}\left( r\right) \right) \end{equation*} and hence \begin{equation} \frac{d}{dr}\int_{w^{m}\left( R\right) }^{w^{m}\left( r\right) }\left[ H\left( t\right) \right] ^{-1/\left( k+1\right) }dt\leqslant C^{\frac{1}{k+1} }r^{1-\frac{N}{k}}H^{-\frac{1}{k+1}}\left( w^{m}\left( r\right) \right) +\left( \frac{k+1}{C_{N-1}^{k-1}}\right) ^{\frac{1}{k+1}}\left( r^{k-1}p^{k}\left( r\right) \right) ^{\frac{1}{k+1}}. \label{in} \end{equation} Inequality (\ref{in}) combined with \begin{eqnarray*} \frac{1}{\sqrt{2}}\sqrt{2\cdot \left( s^{k-1}p^{k}\left( s\right) \right) ^{ \frac{2}{k+1}}} &=&\frac{1}{\sqrt{2}}\sqrt{2\cdot s^{\frac{1+\varepsilon }{2} }\left( s^{k-1}p^{k}\left( s\right) \right) ^{\frac{2}{k+1}}s^{\frac{ 1-\varepsilon }{2}}} \\ &\leqslant &\frac{1}{\sqrt{2}}\left[ s^{1+\varepsilon }\left( s^{k-1}p^{k}\left( s\right) \right) ^{\frac{2}{k+1}}+s^{-1-\varepsilon }\right] \end{eqnarray*} gives \begin{equation*} \begin{array}{ll} \int_{w^{m}\left( R\right) }^{w^{m}\left( r\right) }\left[ H\left( t\right) \right] ^{-1/\left( k+1\right) }dt & \leqslant C^{\frac{1}{k+1}}\int_{R}^{r}t^{1- \frac{N}{k}}H^{-\frac{1}{k+1}}\left( w^{m}\left( t\right) \right) dt \\ & +\frac{1}{\sqrt{2}}\left( \frac{k+1}{C_{N-1}^{k-1}}\right) ^{\frac{1}{k+1}} \left[ \int_{R}^{r}t^{1+\varepsilon +\frac{2\left( k-1\right) }{k+1}}\left( p\left( t\right) \right) ^{\frac{2k}{k+1}}dt+\int_{R}^{r}t^{-1-\varepsilon }dt\right] \\ & \leqslant C^{\frac{1}{k+1}}H^{-\frac{1}{k+1}}\left( w^{m}\left( R\right) \right) \int_{R}^{r}t^{1-\frac{N}{k}}dt \\ & +\frac{1}{\sqrt{2}}\left( \frac{k+1}{C_{N-1}^{k-1}}\right) ^{\frac{1}{k+1}} \left[ \int_{R}^{r}t^{1+\varepsilon +\frac{2\left( k-1\right) }{k+1}}\left( p\left( t\right) \right) ^{\frac{2k}{k+1}}dt+\frac{1}{\varepsilon R^{\varepsilon }}\right] \text{.} \end{array} \end{equation*} The above relation is needed in proving the bounded of the function $\left\{ w^{m}\right\} ^{m\geqslant 1}$ in the following. Indeed, since for each $ \varepsilon >0$ the right side of this inequality is bounded independent of $ m$ (note that $w^{m}\left( t\right) \geqslant 1$), so is the left side and hence, in light of (C2), the sequence $\left\{ w^{m}\right\} ^{m\geqslant 1}$ is a bounded sequence and so $\left\{ w^{m}\right\} ^{m\geqslant 1}$ are bounded sequence. Thus $\left\{ w^{m}\right\} ^{m\geqslant 1}\rightarrow w$ as $ m\rightarrow \infty $ and the limit functions $w$ are positive entire bounded solutions of equation (\ref{6}).
\subparagraph{Proof of the Theorem \protect\ref{th2}.}
We know that for any $a_{1}>0$ a solution of \begin{equation*} v\left( r\right) =a_{1}+\int_{0}^{r}\left( \frac{k}{t^{N-k}}\int_{0}^{t} \frac{s^{N-1}}{C_{N-1}^{k-1}}p^{k}\left( s\right) h^{k}\left( v\left( s\right) \right) ds\right) ^{1/k}dt, \end{equation*} exists, at least, small $r$. Since $v^{\prime }\geqslant 0$, the only way that the solution can become singular at $R$ is for $v\left( r\right) \rightarrow \infty $ as $r\longrightarrow \infty $. Thus, we can show that, for each $R>0 $, there exists $C_{R}>0$ so that $v\left( R\right) \leqslant C_{R}$, we have existence. To this end, let $M_{R}=\max \left\{ p\left( r\right) \left\vert 0\leqslant r\leqslant R\right. \right\} $ and consider the equation \begin{equation*} w\left( r\right) =a_{2}+M_{R}\int_{0}^{r}\left( \frac{k}{t^{N-k}}\int_{0}^{t} \frac{s^{N-1}}{C_{N-1}^{k-1}}h^{k}\left( v\left( s\right) \right) ds\right) ^{1/k}dt \end{equation*} where $a_{2}>a_{1}$. We next observe that the solution to this equation exists for all $r\geqslant 0$ and of course, it is a solution to $ S_{k}^{1/k}\left( \lambda \left( D^{2}w\left( r\right) \right) \right) =M_{R}h\left( w\right) $ on $\mathbb{R}^{N}$ which is treated in \cite[ (Theorem 1.1, p. 177)]{BAO}. We now show that $v\left( r\right) \leqslant w\left( r\right) $ for all $0\leqslant r\leqslant R$ and hence we conclude the proof of existence. Clearly $v\left( 0\right) <w\left( 0\right) $ so that $v\left( r\right) <w\left( r\right) $ for at least all $r$ near zero. Let \begin{equation*} r_{0}=\sup \left\{ r\left\vert v\left( s\right) <w\left( s\right) \text{ for all }s\in \left[ 0,r\right] \right. \right\} \text{.} \end{equation*} If $r_{0}=R$, then we are done. Thus assume that $r_{0}<R$. It follows from assumption $a_{2}>a_{1}$ that \begin{eqnarray*} v\left( r_{0}\right) &=&a_{1}+\int_{0}^{r_{0}}\left( \frac{k}{t^{N-k}} \int_{0}^{t}\frac{s^{N-1}}{C_{N-1}^{k-1}}p^{k}\left( s\right) h^{k}\left( v\left( s\right) \right) ds\right) ^{1/k}dt \\ &<&a_{2}+M_{R}\int_{0}^{r_{0}}\left( \frac{k}{t^{N-k}}\int_{0}^{t}\frac{ s^{N-1}}{C_{N-1}^{k-1}}h^{k}\left( v\left( s\right) \right) ds\right) ^{1/k}dt=w\left( r_{0}\right) . \end{eqnarray*} Thus there exists $\varepsilon >0$ so that $v\left( r\right) <w\left( r\right) $ for all $\left[ 0,r+\varepsilon \right) $, contradicting the definition of $r_{0}$. Thus we conclude that $v<w$ on $\left[ 0,R\right] $ for all $R>0$ and hence $v$ is a nontrivial entire solution of (\ref{11}). Now let $u$ be any nonnegative nontrivial entire solution of (\ref{11}) and suppose $p$ satisfies \begin{equation*} \int_{0}^{\infty }\left( \frac{k}{t^{N-k}}\int_{0}^{t}\frac{s^{N-1}}{ C_{N-1}^{k-1}}p^{k}\left( s\right) ds\right) ^{1/k}dt=\infty . \end{equation*} Since $u$ is nontrivial and non-negative, there exists $R>0$ so that $ u\left( R\right) >0$. On the other hand since $u^{\prime }\geqslant 0$, we get $ u\left( r\right) \geqslant u\left( R\right) $ for $r\geqslant R$ and thus from \begin{equation*} u\left( r\right) =u\left( 0\right) +\int_{0}^{r}\left( \frac{k}{t^{N-k}} \int_{0}^{t}\frac{s^{N-1}}{C_{N-1}^{k-1}}p^{k}\left( s\right) h^{k}\left( u\left( s\right) \right) ds\right) ^{1/k}dt,\text{ } \end{equation*} since $u$ will satisfy that equation for all $r\geqslant 0,$ we get \begin{equation*} \begin{array}{l} u\left( r\right) =u\left( 0\right) +\int_{0}^{r}\left( \frac{k}{t^{N-k}} \int_{0}^{t}\frac{s^{N-1}}{C_{N-1}^{k-1}}p^{k}\left( s\right) h^{k}\left( u\left( s\right) \right) ds\right) ^{1/k}dt \\ \text{ \ \ \ \ \ \ }\geqslant u\left( R\right) +h\left( u\left( R\right) \right) \int_{R}^{r}\left( \frac{k}{t^{N-k}}\int_{R}^{t}\frac{s^{N-1}}{C_{N-1}^{k-1}} p^{k}\left( s\right) ds\right) ^{1/k}dt\rightarrow \infty \text{ as } r\rightarrow \infty . \end{array} \end{equation*} Conversely, assume that\ $h$ satisfy (C1), (C3) and that $w$ is a nonnegative entire large solution of (\ref{11}). Note also, that $w$ satisfies \begin{equation*} \left[ \frac{r^{N-k}}{k}\left( w^{^{\prime }}(r)\right) ^{k}\right] ^{\prime }=\frac{r^{N-1}}{C_{N-1}^{k-1}}p^{k}\left( r\right) h^{k}\left( w\left( r\right) \right) . \end{equation*} Using the monotonicity of \textit{\ }$r^{N+\frac{N}{k}-2}p\left( r\right) $ \textit{\ } we can apply similar arguments used in obtaining Theorem \ref {th1} to get \begin{equation*} \left( w\left( r\right) \right) ^{\prime }\leqslant C^{\frac{1}{k+1}}r^{1-\frac{N }{k}}+\left( \frac{k+1}{C_{N-1}^{k-1}}\right) ^{\frac{1}{k+1}}r^{1-\frac{2}{ k+1}}p^{\frac{k}{k+1}}\left( r\right) H^{\frac{1}{k+1}}\left( w\left( r\right) \right) , \end{equation*} which we may rewrite as \begin{equation} \begin{array}{ll} \int_{w\left( R\right) }^{w\left( r\right) }\left[ H\left( t\right) \right] ^{-1/\left( k+1\right) }dt & \leqslant C^{\frac{1}{k+1}}\int_{R}^{r}t^{1-\frac{N}{ k}}H^{-\frac{1}{k+1}}\left( w\left( t\right) \right) dt \\ & +\frac{1}{\sqrt{2}}\left[ \int_{R}^{r}t^{1+\varepsilon +\frac{2\left( k-1\right) }{k+1}}\left( p\left( t\right) \right) ^{\frac{2k}{k+1} }dt+\int_{R}^{r}t^{-1-\varepsilon }dt\right] \\ & \leqslant C^{\frac{1}{k+1}}H^{-\frac{1}{k+1}}\left( w\left( R\right) \right) \int_{R}^{r}t^{1-\frac{N}{k}}dt \\ & +\frac{1}{\sqrt{2}}\left( \frac{k+1}{C_{N-1}^{k-1}}\right) ^{\frac{1}{k+1}} \left[ \int_{R}^{r}t^{1+\varepsilon +\frac{2\left( k-1\right) }{k+1}}\left( p\left( t\right) \right) ^{\frac{2k}{k+1}}dt+\frac{1}{\varepsilon R^{\varepsilon }}\right] \\ & \leqslant C_{R}\int_{R}^{r}t^{1-\frac{N}{k}}dt+\frac{1}{\sqrt{2}}\left( \frac{ k+1}{C_{N-1}^{k-1}}\right) ^{\frac{1}{k+1}}\int_{R}^{r}t^{1+\varepsilon + \frac{2\left( k-1\right) }{k+1}}\left( p\left( t\right) \right) ^{\frac{2k}{ k+1}}dt \end{array} \label{final} \end{equation} where \begin{equation*} C_{R}=C^{\frac{1}{k+1}}H^{-\frac{1}{k+1}}\left( w\left( R\right) \right) + \frac{1}{\sqrt{2}}\frac{1}{\varepsilon R^{\varepsilon }}\left( \frac{k+1}{ C_{N-1}^{k-1}}\right) ^{\frac{1}{k+1}}. \end{equation*} By taking $r\rightarrow \infty $ in (\ref{final}) we obtain (\ref{13}) since $w$ is large and $h$ satisfies (C3). These observations completes the proof of the theorem.
\subparagraph{\textbf{Proof of the Theorem \protect\ref{th3} and \protect\ref {th4}.}}
In order, to obtain the conclusion, combine the proof of \textbf{Theorem \ref {th1} }and\textbf{\ \ref{th2}} with some technical results from \cite{COV} and \cite{COVS}.
\qed
\end{document} | arXiv |
\begin{document}
\title{Generation of multiphoton entangled quantum states with a single silicon nanowire}
\author{Ming Zhang\footnote[1]} \affiliation{State Key Laboratory for Modern Optical Instrumentation, Centre for Optical and Electromagnetic Research, Zhejiang Provincial Key Laboratory for Sensing Technologies, Zhejiang University, Zijingang Campus, Hangzhou 310058, China.} \author{Lan-Tian Feng\footnote[1]{These authors contributed equally to this work.}} \author{Zhi-Yuan Zhou} \author{Yang Chen} \affiliation {Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, People's Republic of China.} \affiliation{Synergetic Innovation Center of Quantum Information $\&$ Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China.} \author{Hao Wu} \author{Ming Li} \affiliation{State Key Laboratory for Modern Optical Instrumentation, Centre for Optical and Electromagnetic Research, Zhejiang Provincial Key Laboratory for Sensing Technologies, Zhejiang University, Zijingang Campus, Hangzhou 310058, China.} \author{Guo-Ping Guo} \author{Guang-Can Guo} \affiliation {Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, People's Republic of China.} \affiliation{Synergetic Innovation Center of Quantum Information $\&$ Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China.} \author{Dao-Xin Dai\footnote[3]{[email protected]}} \affiliation{State Key Laboratory for Modern Optical Instrumentation, Centre for Optical and Electromagnetic Research, Zhejiang Provincial Key Laboratory for Sensing Technologies, Zhejiang University, Zijingang Campus, Hangzhou 310058, China.} \author{Xi-Feng Ren\footnote[2]{[email protected]}} \affiliation {Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, People's Republic of China.} \affiliation{Synergetic Innovation Center of Quantum Information $\&$ Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China.}
\maketitle \textbf{Multiphoton entanglement plays a critical role in quantum information processing, and greatly improves our fundamental understanding of the quantum world. Despite tremendous efforts in either bulk media or fiber-based devices, nonlinear interactions in integrated circuits show great promise as an excellent platform for photon pair generation with its high brightness, stability and scalability \cite{Caspani2017}. Here, we demonstrate the generation of bi- and multiphoton polarization entangled qubits in a single silicon nanowire waveguide, and these qubits directly compatible with the dense wavelength division multiplexing in telecommunication system. Multiphoton interference and quantum state tomography were used to characterize the quality of the entangled states. Four-photon entanglement states among two frequency channels were ascertained with a fidelity of $0.78\pm0.02$. Our work realizes the integrated multiphoton source in a relatively simple pattern and paves a way for the revolution of multiphoton quantum science.}
Optical quantum states with entanglement shared among several modes are critical resources for studies in quantum communication \cite{Kimble2008}, computation \cite{Walther2005,Humphreys2013}, simulation \cite{Aspuru-Guzik2012}, and metrology \cite{Afek2010}. Therefore, the controllable and scalable realization of multiphoton quantum states would enable a practical and powerful implementation of quantum technologies. For generating and manipulating photon pairs, nonlinear interactions in integrated circuits show great promise as an excellent platform with its high brightness, stability and scalability. Recently, optical integrated kerr frequency combs in high-refractive-index glass platform were used to generate time-bin bi- and multiphoton entangled qubits \cite{Reimer2016}, which started a new time to exploit multiphoton entangled states with the help of quantum photonic integrated circuits.
The silicon-on-insulator (SOI) photonic circuit is a very promising platform for realizing complex quantum states for its strong third-order optical nonlinearity, low nonlinear noise, small footprint, mature fabrication technics and compatibility with complementary metal oxide semiconductor (CMOS) electronics \cite{Bogaerts2005} as well as telecom techniques. Multiple biphoton entanglement sources have been manipulated and further used for large-scale integration and quantum information processings \cite{Li2017,Silverstone2014,Harris2014,Wang2016,Wang2017,Paesani2017}. Nevertheless, whether the silicon photonic circuit can be used to manipulate multiphoton entangled source is still unexplored.
In this study, we use the silicon-on-insulator (SOI) photonics platform to explore the generation of multiphoton entangled states. A single silicon nanowire waveguide, which has a length of $1\ cm$ and transverse dimension of $\sim450\ nm\times220\ nm$, is employed to generate photon pairs. The external pump laser is coupled to the waveguide with grating couplers. The waveguide has a propagation loss of $1\ dB$, and the total insertion loss of this structure is $11\ dB$, including $5\ dB$ coupling loss each at the input and output grating coupler.
Polarization encoding was chosed to demonstrate the multiphoton entanglement generation, since it is popular for various quantum information applications \cite{Kok2007}. The experimental setup for generation and measurement of the multiphoton polarization entangled states is shown in Fig. 1. A pulsed erbium laser with a repetition frequency $100\ MHz$ was involved for the degenerate spontaneous four wave mixing process in the silicon nanowire. After forward and backward filtering with $100\ GHz$ bandwidth, the pump light was passed through a polarization controller (PC) and a fiber circulator, and then went into a Sagnac loop \cite{Takesue2008}. The Sagnac loop consists of half-wave plates (HWPs), quarter-wave plates (QWPs), a polarization beam splitter (PBS) and the chip of silicon nanowire. The combination of the HWP and QWP is used to modulate the optical polarization for maximum grating coupling. In the Sagnac loop, pump beam was split into clockwise and counterclockwise circulation directions by the PBS, and the HWP before the PBS is used for optical power modulation. In each direction, time correlated signal and idler photons are generated and their frequencies are equally separated from the central pump frequency. In the clockwise circulation direction, the photon pair generated is $\left|V_sV_i\right\rangle$, while in the counterclockwise circulation direction, the photon pair generated is $\left|H_sH_i\right\rangle$ after the PBS. The biphoton quantum state at the output port of the loop can be expressed as \begin{equation}\label{1}
\left|\Phi\right\rangle=\frac{1}{\sqrt{1+\eta^2}}(\left|H_sH_i\right\rangle+\eta e^{i\delta}\left|V_sV_i\right\rangle), \end{equation} where $\eta$ is determined by the ratio of the pump power in the two circulation directions, and $\delta$ depends on the initial phase of the pump beam and the birefringence experienced by the pump beam in the $H$ and $V$ polarizations. By modulating the pump power ratio and relative phase, a maximally polarization entangled Bell state \begin{equation}\label{2}
\left|\Phi\right\rangle=\frac{1}{\sqrt{2}}(\left|H_sH_i\right\rangle+\left|V_sV_i\right\rangle) \end{equation} can be produced.
At the output port of the fiber circulator, the pump laser was blocked by post-filters with $200\ GHz$ bandwidth. One 40-channel dense wavelength division multiplexing (DWDM) system was used to separate the signal and idler photons, whose frequencies are equally separated from the central pump frequency.. Each channel has a $100\ GHz$ bandwidth, then we can select any combination of photon pairs with a frequency detuning about $2\ THz$ from the central pump frequency. The generated photon pairs were finally detected by superconducting nanowire single-photon detectors (SCONTEL, dark count rate $100\ Hz$, detector efficiency $85\%$ at C band).
The entangled photon pairs are frequency multiplexed and generated over all DWDM frequency channels \cite{Li2017}. We selected five signal-idler channels to ascertain the effectiveness and stability of our system (Supplementary Table 1). Through inputting $120\ \mu W$ pump light into the single cycle of the Sagnac loop, we recorded two-photon coincidences between different signal-idler channel combinations (Supplementary Fig. 1). It shows that the crosstalk is negligible between different frequency channels, even for the adjacent ones.
Next, we modulated the polarization of the pump light before the PBS of the Sagnac loop to obtain the maximally entangled states (Eq. 2). Combinations of QWP, HWP and PBS after the DWDM were used to constitute a normal polarization state tomography architecture \cite{James2001} for characterizing the degree of entanglement. We took the signal-idler channels 1 for testing the quality of the entanglement. We first measured the two-photon polarization interference fringes. The interference is expected to be proportional to $1+Vsin(2\pi(x-x_c)/T)$, where $V$ is the fringe visibility, $x_c$ is the initial phase, and $T$ is the oscillation period. The fringe visibility $V$ is defined as $V=(d_{max}-d_{min})/(d_{max}+d_{min})$, where $d_{max}$, $d_{min}$ are the maximum and minimum fitting data, respectively. Through setting the angle of the HWP in the signal channel as $0^\circ$ or $45^\circ$, and modulating the angle of the HWP in the idler channel, we obtained two interference fringes (Fig. 2a). The raw visibilities in the $0^\circ$ (solid red line) and $45^\circ$ (solid black line) bases were $96.1\pm3.2\%$ and $93.0\pm3.2\%$, respectively, which, being greater than $\frac{1}{\sqrt{2}}\approx70.7\%$, confirming entanglement through the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. In this case, we estimated the photon pair generation rate of $270\ kHz$ per channel (0.0027 pairs per double pulse), accounting for system and detection losses of $16\ dB$ (see methods).
Now we consider the multiphoton cases. In principle, the generated multiphoton entanglement state for our experimental setup can be expressed as \begin{equation}\label{3}
\left|\Phi^{2n}\right\rangle=\frac{1}{\sqrt{2^n}}(\left|H_{s1}H_{i1}\right\rangle+\left|V_{s1}V_{i1}\right\rangle)\otimes\cdots\otimes(\left|H_{sn}H_{in}\right\rangle+\left|V_{sn}V_{in}\right\rangle). \end{equation}
That is, combination of different correlated signal-idler channels could be used to produce multiphoton entangled states. For example, by selecting the signal-idler channels 1 and 5, we can get two two-photon entangled states, given by $\left|\Phi_1\right\rangle=\frac{1}{\sqrt{2}}(\left|H_{s1}H_{i1}\right\rangle+\left|V_{s1}V_{i1}\right\rangle)$ and $\left|\Phi_5\right\rangle=\frac{1}{\sqrt{2}}(\left|H_{s5}H_{i5}\right\rangle+\left|V_{s5}V_{i5}\right\rangle)$, respectively. By post-selecting four-photon events with one photon on each frequency channel, these two states are multiplied, resulting in four-photon polarization entangled states, which is given by \begin{equation}\label{4}
\left|\Phi^{4}\right\rangle=\frac{1}{2}(\left|H_{s1}H_{i1}\right\rangle+\left|V_{s1}V_{i1}\right\rangle)\otimes(\left|H_{s5}H_{i5}\right\rangle+\left|V_{s5}V_{i5}\right\rangle). \end{equation} To prove this state, we measured the four-photon quantum interference fringes, which generally cannot be present for two completely independent two-photon qubit states. By setting the pump power to $600\ \mu W$, we obtained a four-fold coincidence rate of $0.34\ Hz$, which corresponds to a calculated generation rate of $340\ kHz$, taking into account the system and detection losses of $15\ dB$ (see methods). The interference is expected to be proportional to $1+(1-\sqrt{1-V^2})/V)sin^2(2\pi(x-x_c)/T$. By setting the angle of the HWPs in the signal channels as $0^\circ$ or $45^\circ$, and modulating the angle of the HWPs in the idler channels simultaneously, we obtained two four-photon interference fringes (Fig. 2b). The raw visibilities in the $0^\circ$ (solid red line) and $45^\circ$ (solid black line) bases were $96.5\pm1.5\%$ and $99.1\pm0.4\%$, respectively. Four-photon entangled states unfold totally different interference patterns from the two-photon entangled states and the high interference visibilities prove the existence of the multiphoton entanglement. Note that, for the measurement of quantum interference fringes, the QWPs were moved out from the setup.
Quantum state tomography \cite{James2001} was also used to fully characterize the entangled states. Using this method, we could obtain the experimental state density matrix and thus study how close of the measured states to the ideal entangled states. Firstly, we performed the two-photon quantum state tomography for the frequency channels 1. 16 data at different measurement bases were acquired to construct the state density matrix. The ideal density matrix of the maximally entangled states (Eq. 2) and the measured density matrix of the output states were displayed in Figs. 3a, and 3b, respectively. We used the maximum-likelihood-estimation method to construct the density matrix with the experimental results. The fidelity is defined as $F=Tr(\widehat{\rho}_{exp}\widehat{\rho}_{th})$, where $Tr$ is the trace, $\widehat{\rho}_{exp}$ is the measured density matrix, and $\widehat{\rho}_{th}$ is the ideal density matrix. We estimated the fidelity of $0.94\pm0.01$, confirming that the generated quantum states were of high quality and very close to the ideal maximally entangled states. The error of the fidelity was obtained by 100 times Monto Carlo calculation with the experimental data subject to Gaussian statistics. The deviation of the fidelity from unity was mainly due to the unideal rotating the angle of the wave plates. It is worth pointing out that the quantum states are frequency multiplexed and entanglement is in any signal-idler channels. As a test, we also performed the two-photon quantum state tomography for signal-idler channels 3 and 5, and found the fidelities of $0.97\pm0.01$ and $0.95\pm0.01$, respectively (Supplementary Fig. 1).
For the four-photon entangled states, 256 data of different measurement bases were obtained to construct the state density matrix. The ideal density matrix of the maximally entangled states (Eq. 4) and the measured density matrix of the output states from signal-idler channels 1 and channel 5 were displayed in Figs. 4c, and 4d, respectively. We obtained a fidelity of $0.78\pm0.02$ without background and accidence subtraction. The high visibility clearly proved the entanglement and is satisfied for further practical applications.
Since it was demonstrated recently that the frequency encoding can be used for single-photon frequency shifting \cite{Fan2016} and construction of high-fidelity quantum gates \cite{Lukens2017,Lu2018} for high-dimensional quantum information processing, our presented multiphoton entangled source is fully compatible with these post-processing systems and it will constitutes an indispensable block for the frequency-based quantum information processing. Furthermore, the multiphoton entangled state (Eq. 3) can be converted to the Greenberger-Horne-Zeilinger (GHZ) state through post operation and selection, which is usually used as the resource for linear-optical quantum computation \cite{WangX2016}.
In addition, the entangled photon numbers can be increased \cite{Reimer2016}. Further integration of multiphoton manipulation block in one integrated circuits \cite{Matthews2009} will also lead to more compact and stable systems with higher performance, resulting better detection rates and higher fidelity. A scheme to decrease coupling loss by using high coupling efficiency etched facet tapers \cite{Cardenas2014} and low-loss filter system would increase the brightness of four-fold coincidence greatly, even to the useful kilohertz range.
In conclusion, we have experimentally shown that a single silicon nanowire can be used for multiphoton source manipulation. Due to its strong third-order nonlinearity, we achieved a high brightness photon pair source with a very low pump power. The fidelity of the four-photon entangled states is high enough for practical applications. The multiphoton entangled source is directly compatible with the dense wavelength division multiplexing communication system and frequency-based post-processing system, thus provided a scalable and practical platform for quantum information processing.
\section*{Methods}
\noindent {\bf System efficiency.} We ascertained the efficiencies of all departments with laser light measurements. The grating coupler has coupling loss of $5\ dB$. The system for state manipulation and state measurement has loss of $4.3\ dB$. The post-filters and WDMs have inherent loss of $6\ dB$ and both detectors have efficiency of $85\%\ (-0.7\ dB)$. The total loss is $16\ dB$ for two-photon state measurements and $15\ dB$ for four-photon state measurements where we decreased the loss of $1\ dB$ in post-processing system through finer adjustment.
\noindent {\bf Optical apparatus.} We used pulsed erbium laser (repetition frequency $100\ MHz$) to generate the pump light (Supplementary Fig. 2). After filtered, the left laser was amplified by an erbium-doped fiber amplifier and filtered again before input into the source manipulation setup. Fiber alignment was maintained using a piezo-controlled four-dimensional displacement table for position and coupling angle adjustment. The coupling angle was set as $10^\circ$ for fiber-chip coupling. Two cascaded off-chip post-filters ($100\ dB$ extinction ratio) were used to remove the pump photons, and one DWDM system ($30\ dB$ extinction ratio for adjacent channel and $50\ dB$ for non-adjacent channel) was used to separate the signal and idler photons. The correlated photons were recorded by superconducting nanowire single-photon detectors. The electrical signals were collected and analyzed through time-correlated single photon counting (TCSPC) system, and the coincidence window was set as $0.8\ ns$ for two-photon state measurements. For four-photon state measurements, the electrical signals were analyzed by the UQD-Logic and the coincidence window was set as $1\ ns$.
\section*{Acknowledgments} This work was supported by the National Natural Science Foundation of China (NSFC) (Nos. 61590932, 11774333, 61725503, 61431166001), Anhui Initiative in Quantum Information Technologies (No. AHY130300), the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDB24030600), the National Key R \& D Program (No. 2016YFA0301700), Zhejiang Provincial Natural Science Foundation of China (Z18F050002), and the Fundamental Research Funds for the Central Universities. This work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication.
\section*{Author contributions}
All authors contributed extensively to the work presented in this paper. M.Z., Z.Y.Z., H.W., M.L. and D.X.D. prepared the samples, L.T.F., M.Z., D.X.D. and X.F.R. performed the measurements, data analyses and discussions. Y.C., G.P.G. and G.C.G. conducted theoretical analysis. X.F.R. and D.X.D. wrote the manuscript and supervised the project.
\section*{Additional information}
Supplementary information is available in the online version of the paper. Correspondence and requests for materials should be addressed to D.X.D or X.F.R.
\section*{Competing financial interests} The authors declare no competing financial interests.
\end{document} | arXiv |
\begin{definition}[Definition:Basis (Hilbert Space)]
Let $H$ be a Hilbert space.
A '''basis for $H$''' is a maximal orthonormal subset of $H$.
Thus, $B$ is a '''basis''' for $H$ {{iff}} for all orthonormal subsets $B'$ of $H$:
:$B \subseteq B' \implies B = B'$
\end{definition} | ProofWiki |
\begin{document}
\nocite{*} \newtheorem{thm}{Theorem} \newtheorem{maintheorem}{Main Theorem} \renewcommand{\themaintheorem}{} \newtheorem{Thm}{Theorem} \newtheorem{prop}[thm]{Proposition} \newtheorem{Prop}{Proposition} \newtheorem{lemma}[thm]{Lemma} \newtheorem{fact}[thm]{Fact} \newtheorem{corollary}[thm]{Corollary} \newtheorem{cor}[thm]{Corollary} \newtheorem{step}{Step} \newtheorem{stp}{Step} \newtheorem{defn}[equation]{Definition}
\newcommand\st{\mbox{$\ :\ $}} \newcommand\Ind{{\rm Ind}} \newcommand\lan{{\langle}} \newcommand\ran{{\rangle}} \newcommand{\order}[1]{\vert {#1} \vert} \newcommand\ud{{\underline{d}}} \newcommand\ue{{\underline{e}}} \newcommand\ux{{\underline{x}}} \newcommand\uC{{\underline{C}}} \newcommand\uy{{\underline{y}}} \newcommand\eps{{\epsilon}} \newcommand\heps{{\hat\epsilon}} \newcommand\mustar{{\mu_*}} \newcommand\F{{\bf F}} \newcommand\bZ{{\bf Z}} \newcommand\bR{{\bf R}} \newcommand\N{{\bf N}} \newcommand\bn{{\bf n}} \newcommand\bs{{\bf s}} \newcommand\cX{{\cal X}} \newcommand\C{{\Bbb C}} \newcommand\Q{{\Bbb Q}} \newcommand\PD{{\bf PD}} \newcommand\ccc{v} \newtheorem{definition}{Definition}
\renewcommand{\thedefinition}{} \newcommand\cE{{\cal E}} \newcommand\cC{{\cal C}} \newcommand\cEg{{\cal E}_g} \newcommand\cM{{\cal M}} \newcommand\cN{{\cal N}} \newcommand\tG{{\widetilde{G}}} \newcommand\hG{{\widehat{G}}} \newcommand\hx{{\widehat{x}}} \newcommand\tH{{\widetilde{H}}} \newcommand\tV{{\widetilde{V}}} \newcommand\sig{\mathop{\mathrm{sig}}} \newcommand\codim{\mathop{\mathrm{codim}}} \newcommand\fpr{\mathop{\mathrm{fpr}}} \newcommand\Fix{\mathop{\mathrm{Fix}}} \newcommand{{\Bbb Z}}{{\Bbb Z}} \newcommand{\Bbb{P}^1}{\Bbb{P}^1} \newcommand{{\mathcal{M}}}{{\mathcal{M}}} \newcommand{{\mathcal{H}}}{{\mathcal{H}}}
\newcommand\rad{\mathop{\mathrm{rad }}} \newcommand\Mon{\mathop{\mathrm{Mon }}} \newcommand\Aut{\mathop{\mathrm{Aut }}}
\newcommand{\floor}[1]{\lfloor {#1} \rfloor} \newcommand{\ceil}[1]{\lceil {#1} \rceil} \newcommand\dv{{\ \mid\ }} \newcommand\lcm{{\rm lcm}} \newcommand\pf{\noindent {\rm Proof.\ \ \/}} \newcommand\eop{{
$\Box$ \vskip 2ex}} \newcommand\ppp\rho \newcommand\zetao{\zeta^\ast} \newcommand\zzeta{\pi}
\newcommand\U{W} \newcommand\W{W} \newcommand\WW{W}
\title{Primitive Monodromy Groups of Genus at most Two} \author{ Daniel Frohardt \\ Robert Guralnick\footnote{The second author was supported by NSF grant DMS 0653873 } \\ Kay Magaard}
\maketitle \begin{abstract} We show that if the action of a classical group $G$ on a set $\Omega$ of $1$-spaces of its natural module
is of genus at most two, then $|\Omega| \leq 10,000$. \end{abstract}
\section*{Introduction}
Let $X$ be a compact, connected Riemann surface of genus $g$, and let $\phi: X \rightarrow \Bbb{P}^1\C$ be meromorphic of degree $n$. Let
$B := \{ x \in \Bbb{P}^1\C \ : \ |\phi^{-1}(x)| < n \}$ be the set of branch points of $\phi$. It is well known that $B$ is a finite set and that if $b_0 \in \Bbb{P}^1\C \setminus B$, then the fundamental group $\pi_1(\Bbb{P}^1\C \setminus B,b_0)$ acts transitively on $F:= \phi^{-1}(b_0)$ via path lifting. The image of the action of $\pi_1(\Bbb{P}^1\C \setminus B,b_0)$ on $\phi^{-1}(b_0)$ is called the {\it monodromy group} of $(X,\phi)$ and is denoted by $\Mon(X,\phi)$. \\
We are interested in the structure of the monodromy group when the genus of $X$ is less than or equal to two and $\phi$ is indecomposable in the sense that there do not exist holomorphic functions $\phi_1 : X \rightarrow Y$ and $\phi_2 : Y \rightarrow \Bbb{P}^1\C $ of degree less than the degree of $\phi$ such that $\phi = \phi_2 \circ \phi_1$. The condition that $X$ is connected implies that $\Mon(X,\phi)$ acts transitively on $F$ whereas the condition that $\phi$ is indecomposable implies that the action of $\Mon(X,\phi)$ on $F$ is primitive.
Our question is closely related to a conjecture made by Guralnick and Thompson \cite{GT} in 1990. By $cf(G)$ we denote the set of isomorphism types of the composition factors of $G$. In their paper Guralnick and Thompson \cite{GT} defined the set $${\mathcal E}^*(g) = (\bigcup_{(X,\phi)} cf\Mon(X,\phi)) \ \setminus \ \{A_n, {\Bbb Z}/p{\Bbb Z} \ : \ n > 4 \ , \ p \ \mbox{a prime} \}$$
where $X $ is a compact connected Riemann surface of genus $g$, and $\phi : X \longrightarrow \Bbb{P}^1(\C)$ is meromorphic, and conjectured that ${\mathcal E}^*(g)$ is finite for all $g \in \N$. Building on work of Guralnick-Thompson \cite{GT}, Neubauer \cite{N92}, Liebeck, Saxl \cite{LSa}, and Liebeck, Shalev \cite{LSh}, the conjecture was established in 2001 by Frohardt and Magaard \cite{FM2}.\\
The set ${\mathcal E}^*(0)$ is distinguished in that it is contained in ${\mathcal E}^*(g)$ for all $g$. Moreover the proof of the Guralnick-Thompson conjecture shows that is possible to compute ${\mathcal E}^*(0)$ explicitly and indeed to describe the
minimal covers $\phi:\Bbb{P}^1(\C) \rightarrow |^1(\C)$ (at least those whose monodromy group is not an alternating or symmetric of the same degree as the cover).
The idea of the proof of the Guralnick-Thompson conjecture is to employ Riemann's Existence Theorem to translate the geometric problem to a problem in group theory as follows. If $\phi: X \rightarrow \Bbb{P}^1\C$ is as above with branch points $B = \{b_1,\dots, b_r\}$, then the set of elements $\alpha_i \in \pi_1(\Bbb{P}^1\C \setminus B,b_0)$ each represented by a simple loop around $b_i$ forms a standardized set of generators of $\pi_1(\Bbb{P}^1\C \setminus B,b_0)$. We denote by $\sigma_i$ the image of $\alpha_i$ in $\Mon(X,\phi)\subset S_F \cong S_n$. Thus we have that $$\Mon(X,\phi) = \langle \sigma_1,\dots,\sigma_r\rangle \subset S_n $$ and that $$ \Pi_{i = 1}^r \sigma_i = 1.$$ Moreover the conjugacy class of $\sigma_i$ in $\Mon(X,\phi)$ is uniquely determined by $\phi$. Recall that the index of a permutation $\sigma \in S_n$ is equal to the minimal number of transpositions needed to express $\sigma$ as a product of such. The Riemann-Hurwitz formula asserts that $$ 2(n+g-1) = \sum_{i=1}^r \Ind(\sigma_i), $$ where $g$ is the genus of $X$.
\begin{defn} If $\tau_1,\dots,\tau_r \in S_n$ generate a transitive subgroup $G$ of $S_n$ such that $\Pi_{i=1}^r \tau_i = 1$ and $ 2(n+g-1) = \sum_{i=1}^r \Ind(\tau_i) $ for some $g \in \N_0$, then we call $(\tau_1,\dots,\tau_r)$ a {\it genus $g$-system} and $G$ a genus $g$-group. We call a genus $g$-system $(\tau_1,\dots,\tau_r)$ primitive if the subgroup of $S_n$ it generates is primitive. \end{defn}
If $X,\phi$ are as above, then we say that $(\sigma_1,\dots.\sigma_r)$ is the genus $g$-system induced by $\phi$.
\begin{thm}[Riemann's existence theorem] For every genus $g$-system $(\tau_1,\dots,\tau_r)$ of $S_n$ there exists a Riemann surface $Y$ and a cover $\phi': Y \longrightarrow \Bbb{P}^1\C$ with branch point set $B$ such that the genus $g$-system induced by $\phi'$ is $(\tau_1,\dots,\tau_r)$. \end{thm}
\begin{defn} Two covers $Y_i,\phi_i$, $i = 1,2$ are equivalent if there exist holomorphic maps $\xi_1:Y_1\longrightarrow Y_2$ and $\xi_2:Y_2\longrightarrow Y_1$ which are inverses of one another such that $\phi_1 = \xi_1\circ\phi_2$ and $\phi_2 = \xi_2\circ\phi_1$. \end{defn}
The Artin braid group acts via automorphisms on $\Pi_1(\Bbb{P}^1\C \setminus B,b_0)$. We have that all sets of canonical generators of $\Pi_1(\Bbb{P}^1\C \setminus B,b_0)$ lie in the same braid orbit. Also the group $G$ acts via diagonal conjugation on genus $g$-generating sets. The diagonal and braiding actions on $g$-generating sets commute and preserve equivalence of covers; that is if two genus $g$-generating sets lie in the same orbit under either the braid or the diagonal conjugation action, then the corresponding covers given by Riemann's existence theorem are equivalent. We call two genus $g$-generating systems {\it braid equivalent} if they are in the same orbit under the group generated by the braid action and diagonal conjugation. We have, see for example \cite{V} Proposition 10.14,
\begin{thm} Two covers are equivalent if and only if the corresponding genus $g$-systems are braid equivalent. \end{thm}
Suppose now that $(\tau_1,\dots,\tau_r)$ is a primitive genus $g$-system of $S_n$. Express each $\tau_i$ as a product of a minimal number of transpositions; i.e. $\tau_i := \prod_j \sigma_{i,j}$. The system $(\sigma_{1,1},\dots,\sigma_{r,s})$ is a primitive genus $g$-system generating $S_n$ consisting of precisely $2(n+g-1)$ transpositions. By a famous result of Clebsch, see Lemma 10.15 in \cite{V}, any two primitive genus $g$-systems of $S_n$ are braid equivalent. Thus we see that every genus $g$- system can be obtained from one of $S_n$ which consists entirely of transpositions.
So generically we expect primitive genus $g$-systems of $S_n$ to generate either $A_n$ or $S_n$.
We define $P{\mathcal E}^*(g)_{n,r}$ to be the braid equivalence classes of primitive genus $g$-systems $(\tau_1,\dots,\tau_r)$ of $S_n$ such that $G:=\langle \tau_1,\dots,\tau_r\rangle$ is a primitive subgroup of $S_n$ with $A_n \not \leq G$. We also define $G{\mathcal E}^*(g)_{n,r}$ to be the conjugacy classes of primitive subgroups of $S_n$ which are generated by members of $P{\mathcal E}^*(g)_{n,r}$.
We also define $$P{\mathcal E}^*(g) := \cup_{(n,r) \in \N^2} P{\mathcal E}^*(g)_{n,r},$$ and similarly $$G{\mathcal E}^*(g) := \cup_{(n,r) \in \N^2} G{\mathcal E}^*(g)_{n,r}.$$
We note that the composition factors of elements of $G{\mathcal E}^*(g)$ are elements of ${\mathcal E}^*(g)$.
Our assumption that $G = \Mon(X,\phi)$ acts primitively on $F$ is a strong one and allows us to organize our analysis along the lines of the Aschbacher-O'Nan Scott theorem exactly as was done in the original paper of Guralnick and Thompson \cite{GT}. We recall the statement of the Aschbacher-O'Nan-Scott theorem from \cite{GT}
\begin{thm} Suppose $G$ is a finite group and $H$ is a maximal subgroup of $G$ such that $$\bigcap_{g \in G} H^g = 1.$$ Let $Q$ be a minimal normal subgroup of $G$, let $L$ be a minimal normal subgroup of $Q$, and let $\Delta = \{L=L_1, L_2, \dots, L_t\} $ be the set of $G$-conjugates of $L$. Then $G= HQ$ and precisely one of the following holds:
\begin{itemize} \item[(A)] $L$ is of prime order $p$. \item[(B)] $F^*(G) = Q \times R$ where $Q \cong R$ and $H \cap Q = 1$. \item[(C1)] $F^*(G) = Q$ is nonabelian, $H \cap Q = 1$. \item[(C2)] $F^*(G) = Q$ is nonabelian, $H \cap Q \neq 1 = L \cap H$. \item[(C3)] $F^*(G) = Q$ is nonabelian, $H \cap Q = H_1 \times \dots \times H_t$, \\ where $H_i = H \cap L_i \neq 1$, $1 \leq i \leq t.$ \end{itemize} \end{thm}
We summarize briefly what is known about $G{\mathcal E}^*(0)$ and $P{\mathcal E}^*(0)$. The members of $G{\mathcal E}^*(0)$ that arise in case (C2) were determined by Aschbacher \cite{A}. In all such examples $Q = A_5 \times A_5$. Shih \cite{Shi} showed that no elements of $G{\mathcal E}^*(0)$ arise in case (B) and Guralnick and Thompson \cite{GT} showed the same in case (C1). In his thesis Neubauer \cite{N} showed that in case (A) either $G''=1$ and $G/G'$ is an abelian subgroup of $GL_2(p)$, or that $n \leq 256$. Recently Magaard, Shpectorov and Wang \cite{MSW}, determined all elements of $P{\mathcal E}^*(0)_{n,r}$ with $n \leq 256$. The elements $G$ of $G{\mathcal E}^*(0)$ arising in case (C3) have generalized Fitting subgroups with fewer than $5$ components; i.e. $t \leq 5$. This was shown by Guralnick and Neubauer \cite{GN} and later strengthened by Guralnick \cite{G} to $t \leq 4$. Moreover Guralnick showed that the action of $L_i$ on the cosets of $H_i$ is a member of $G{\mathcal E}^*(0)$. In case (C3) where $L_i$ is of Lie type of rank one all elements of $G{\mathcal E}^*(0)$ and $G{\mathcal E}^*(1)$ were determined by Frohardt, Guralnick, and Magaard \cite{FGM1}, moreover they show that $t \leq 2$. In \cite{Kong} Kong shows that if $G$ is an almost simple group of type $L_3(q)$, then $G \in G{\mathcal E}_{(q^2+q+1,r)}^*(g)$ with $g \leq 2$ only if $q \leq 13$, and $G \in G{\mathcal E}_{(q^2+q+1,r)}^*(0)$ if and only if $q \leq 7$. Combining the results of Frohardt, Magaard \cite{FM3} with those of Liebeck, Seitz \cite{LSe} we have that if $F^*(G)$ is exceptional of Lie type and $G \in G{\mathcal E}^*(0)_{n,r}$, then $n \leq 65$. In \cite{GS} Guralnick and Shareshian show that $G{\mathcal E}^*(0)_{n,r} = \empty$ if $r \geq 9$. Moreover they show that if $G \in G{\mathcal E}^*(0)_{n,r}$ with $F^*(G)$ alternating of degree $d < n$, then either $r \leq 4$ or $r = 5$ and $n = d(d-1)/2$. In \cite{M} Magaard showed that if $F^*(G)$ is sporadic and $G \in G{\mathcal E}^*(0)_{n,r}$ then $n \leq 280$. We would like to take this opportunity to point out that $\Aut(HS) \in G{\mathcal E}^*(0)_{100,3}$ which was missed in cite \cite{M}. Furthermore we thank the referee for pointing out that $\Aut(HS)$ possesses four genus zero systems in its action on $100$ points with signatures $(2, 4, 10), (2, 5, 6), (2, 4, 5)$, and $(2, 4, 6).$ The referee has further pointed out that first two of these genus zero systems are rational and rigid. This is because in both of these cases the involution has an odd number of transpositions, and therefore the corresponding genus 0 field is rational. Thus there exists $\phi: \Bbb{P}^1\Q\rightarrow \Bbb{P}^1\Q$ of a degree $100$ with monodromy group $\Aut(HS)$.
This leaves open the cases $F^*(G) = A_d^t$, $t,r \leq 4$ and the cases $F^*(G) = L^t$, $t \leq 4$ where $L$ is a classical group of Lie type. In light of the results of \cite{AGM} we suspect that if $G$ is in the second case and $G \in G{\mathcal E}^*(0)$, then $L_i/H_i$ is a point action, i.e. equivalent to an action of $L_i$ acting on an orbit of one-spaces of its natural module. Hence they are the focus of this paper.
Another problem closely related to the Guralnick-Thompson conjecture is the description of the monodromy groups from the generic Riemann surface of genus $g$ to $\Bbb{P}^1(\C)$ of degree $n$. This is related to Zariski's thesis where he answered a conjecture of Enrique by showing that the generic Riemann surface of genus $g > 6$ does not admit a solvable map of fixed degree $n$ to $\Bbb{P}^1(\C)$ (i.e. where the monodromy group is solvable). The condition on $n$ being fixed was removed in \cite{GN}. Note that any Riemann surface of genus at $6$ admits a degree $4$ map to $\Bbb{P}^1(\C)$ (and so is solvable). Interestingly, Zariski's methods were mostly group theoretic.
Recall that the images of the canonical generators of $\pi_1(\Bbb{P}^1\C \setminus B,b_0)$ are determined uniquely up to conjugacy in $G$. We say that a $G$-cover of $\Bbb{P}^1\C$ has \emph{ramification type} $C_1,\dots, C_r$ if the $i$'th canonical generator lies in conjugacy class $C_i$ of $G$. The moduli space of $G$-covers of $\Bbb{P}^1\C$ with ramification type $C_1,\dots, C_r$ is a \emph{Hurwitz space} and is denoted by ${\mathcal{H}}(G,0,C_1,\dots,C_r)$. Via the Riemann-Hurwitz formula we see that every $G$-cover $X \in {\mathcal{H}}(G,0,C_1,\dots,C_r)$ has the same genus $g$. So the forgetful functor ${\mathcal F} : {\mathcal{H}}(G,0,C_1,\dots,C_r) \rightarrow {\mathcal{M}}_g$ is well defined and so the problem of determining maps of degree $n$ from the generic Riemann surface of genus $g$ can be rephrased as follows:
For which groups $G$ and which ramification types $C_1, \dots, C_r$ of $G$ is the forgetful functor ${\mathcal F} : {\mathcal{H}}(G,0,C_1,\dots,C_r) \rightarrow {\mathcal{M}}_g$ dominant; i.e. is the image of ${\mathcal{H}}(G,0,C_1,...,C_r)$ dense in ${\mathcal{M}}_g$?
Now Theorem 2 of Guralnick Magaard \cite{GM} shows that if the image of ${\mathcal{H}}(G,0,C_1,...,C_r)$ under the forgetful functor is dense in ${\mathcal{M}}_g$, then one of the following holds \begin{enumerate} \item $g \leq 2$, \item $g = 3$ and $G$ is affine of degree $8$ or $16$, \item $g = 3$ and $G \cong L_3(2)$, \item $g \geq 3$ and $G \cong S_n, n \ge (g+2)/2$ or $A_n, n > 2g$. \end{enumerate}
It is well known that $S_n$ does cover $\Bbb{P}^1\C$ generically. However it was only in 2006 when Magaard and V\"olklein \cite{MV} proved that $A_n$ and $L_3(2)$ also cover $\Bbb{P}^1\C$ generically. It was later shown by Magaard, V\"olklein and Wiesend \cite{MVW} that $AGL_3(2)$ and $AGL_4(2)$ cover $\Bbb{P}^1\C$ generically. This leaves only the first possibility, and is a reason why our ultimate goal is to determine $P{\mathcal E}^*(g)$ where $g \leq 2$.
Our two primary results here are Theorem \ref{main.result}, which shows that if $n > 10^4$ then the elements of $P{\mathcal E}_{n,r}^*(g)$ with $g \leq 2$ are not point actions of classical groups, and Theorem \ref{basic.result} which is more technical but can be applied to a wider class of actions. Combining Theorem \ref{main.result} with the main theorem of \cite{FGM2} shows that if $n > 10^4$, then the elements of $P{\mathcal E}_{n,r}^*(g)$ with $g\leq 2$ are generally not subspace actions of classical groups. The potential exceptions to the statement are also explicitly given in \cite{FGM2}. These potentially exceptional actions are precisely those actions whose permutation modules do not contain the permutation module of the action on singular points as a submodule. The main result of \cite{AGM} determines all classes of maximal subgroups of classical groups whose permutation module does not contain the permutation module of the action on singular points. For these classes of maximal subgroups we hope to establish the hypotheses of Theorem \ref{main.result} which would then show that if $n > 10^4$, then the elements of $G{\mathcal E}_{n,r}^*(g)$ with $g \leq 2$ are either cyclic of prime order $n$ or contain the alternating group $A_n$.
To establish Theorem \ref{basic.result} we show that for any pair $(G,\Omega)$, where $G$ is a classical group acting primitively on a set $\Omega$ such that the hypotheses of Theorem \ref{basic.result} are satisfied, and any generating $r$-tuple $(\tau_1, \dots, \tau_r)$ of $G$ which satisfies the product $1$ condition, then the expression $\sum_{i=1}^r \Ind(\tau_i)$ is greater than $(2+\epsilon)n$ for some positive constant $\epsilon.$ We achieve this by proving effective lower bounds on $\Ind(\tau_i)$ using Scott's Theorem \ref{generation.fact} and the technique of translating tuples, see Lemma \ref{magnus}.
\section{Statement of Results}
\begin{definition} $\ux = (x_1, x_2, \ldots, x_r)$ is a {\em normalized generating $r$-tuple\/} for $G$ provided \begin{enumerate} \item $G = \lan x_1, x_2, \ldots, x_r \ran$ \item $x_1 x_2 \ldots x_r = 1$ \item $x_i \neq 1$, $i = 1, 2, \ldots, r$ \end{enumerate}
If, in addition, $G$ is a transitive permutation group of degree $N$ and $$\sum \Ind(x_i) = 2(N+g -1)$$ then $\ux$ has genus $g$. \end{definition}
The formula above is the Riemann-Hurwitz Formula (RH). The Riemann Existence Theorem \cite{GT} guarantees that given a normalized generating tuple $\ux$ for a permutation group $G$ there is a surface $X$ and a covering $\rho : X \mapsto \Bbb{P}^1(\C)$ such that $G \cong \Mon(X,\rho)$ and the genus of $X$ is the genus of the tuple $\ux$, written $g(\ux)$.
The primary result of this paper is the following.
\begin{thm} \label{main.result} If $(G,\Omega)$ is a primitive classical point action of degree at least $10^4$, then the action has genus larger than 2. \end{thm}
The case of point actions will lead almost all the examples (indeed using \cite{FGM} and some ongoing work of {AGM}, one can eliminate most other situations).
The proof of Theorem~\ref{main.result} uses inequalities based on RH and estimates for the fixed point ratios of elements of $G$.
\begin{definition} For $x$ a permutation of the finite set $\Omega$, let $F_\Omega(x)$ (or $F(x)$) denote the fixed points of $x$ on $\Omega$ and let $f_\Omega(x)$ (or $f(x)$) denote the {\em fixed point ratio\/} of this permutation. That is, $f(x) = \order{F(x)}/\order{\Omega}$. \end{definition}
\begin{definition}
Let $V$ be a vector space and let $x \in \Gamma L(V)$. If $x$ acts as a permutation on the set $\Omega$ then the triple $(x,V,\Omega)$ satisfies Grassmann Condition $\eps$ provided $$f_\Omega(x) < \frac{\order{\W}}{\order{V}} + \eps$$ for some eigenspace $\W$ for the action of $x$ on $V$.
A classical group $G$ with natural module $V$ acting as a permutation group on the set $\Omega$ satisfies Grassmann Condition $\eps$ provided $(x,V,\Omega)$ sastifies Grassmann Condition $\eps$ for every $x \in G$. \end{definition}
Note: For the purposes of the previous definition, an eigenspace for the action of $x$ on $V$ is a set $\W \subset V$ which is a subspace of $V$ over some (possibly proper) subfield of the field of definition
on which $x$ acts as a scalar. Note that $|\W|$ does not depend on its field of scalars.
The role of Grassmann Conditions in the proof of Theorem~\ref{main.result} is apparent in the statement of the following technical results which together yield Theorem~\ref{main.result}. The key feature of the point actions is that with known exceptions they satisfy Grassmann Condition $1/100$. Theorem~\ref{basic.result} also applies to other actions that satisfy this condition.
\begin{thm} \label{basic.result}
Let $G$ be a linear group with module $V$ where $V$ contains at least $10^4$ projective points and no constituent for the action of $G$ on $V$ has dimension $1$. If $\ux$ is a normalized generating $r$-tuple for $G$ in some primitive permutation action, then one of the following is true. \begin{enumerate} \item $g(\ux) > 2$. \item $G$ does not satisfy Grassmann Condition $1/100$. More specifically, for some $i \in \{1, \ldots, r\}$, the group $\langle x_i \rangle$ contains an element $y$ that violates Grassmann Condition $1/100$. \item The characteristic of $V$, the dimension of $V$ over its prime field, and the signature of $\ux$ are given in Table~\ref{pnd.table}. \end{enumerate} \end{thm}
Note: The {\em signature\/} $\sig(\ux)$ of the $r$-tuple $\ux = (x_1, x_2, \ldots, x_r)$ is the $r$-tuple $(d_1,d_2, \ldots, d_r)$, where $d_i = o(x_i)$.
\begin{thm} \label{grassmann.exceptions}
Let $G$ be a classical group with natural module $V$. Let $\Omega$ be a primitive point action for $G$ with $\order{\Omega} \geq 10^4$ and assume that $G$ does not satisfy Grassmann Condition $1/100$. Then $g(\ux) > 2$ for every normalized generating tuple $\ux$ such that $\langle x_i \rangle$ contains an element $y$ violating Grassmann Condition $1/100$.
\end{thm}
\begin{thm} \label{touch.up}
Let $G$ be a classical group with natural module $V$ Assume $\ux$ is a normalized generating tuple for $G$ and that $\Omega$ is a primitive point action for $G$ with $\order{\Omega} \geq 10^4$. If the characteristic of $V$, the dimension of $V$ over its prime field, and the signature of $\ux$ are given in Table~\ref{pnd.table} then $g(\ux) > 2$. \end{thm}
\begin{table} \caption{Characteristic, Dimension and Signature of Exceptional Cases in Theorem~\ref{basic.result}} \label{pnd.table} \begin{center} \begin{tabular}{ccc} $p$ & $\dim_{\F_p}(V)$ & $\sig(\ux)$ \\ \hline $11$ & $5,6$ & $(2,3,7)$\\ $7$ & $6$ & $(2,3,7)$\\ $5$ & $7,8,9$ & $(2,3,7)$\\ $3$ & $12$ & $(2,3,7)$\\ $2$ & $14,15, \ldots, 21$ & $(2,3,7)$\\ $3$ & $10$ & $(2,3,8)$\\ $2$ & $16$ & $(2,4,5)$ \end{tabular} \end{center} \end{table}
\begin{definition} The almost simple classical group $G$ has a {\em point action\/} on $\Omega$ provided $G$ has a natural module $V$ of dimension $n$ over $\F_q$ where $(G,\Omega,n,V)$ satisfy one of the following conditions.
\begin{enumerate} \item[$L$]: $F^\ast(G) \cong L_n(q)$, and $\Omega$ is the set of all points in $V$. $n \geq 2$.
\item[$O^\eps,\bs$]: $F^\ast(G) \cong O^\eps_{n}(q)$, $V$ is a non-degenerate orthogonal space of type $\eps$, and $\Omega$ is the set of singular points in $V$. $n$ is even, $n \geq 6$, $\eps = +$ or $-$.
\item[$O^\eps,\bn$]: $F^\ast(G) \cong O^\eps_{n}(q)$, $V$ is a non-degenerate orthogonal space of type $\eps$, and $\Omega$ is the set of $+$-type points in $V$. $n$ is even, $n \geq 6$, $\eps = +$ or $-$.
\item[$O,\bs$]: $F^\ast(G) \cong O_n(q)$, $V$ is a non-degenerate orthogonal space, and $\Omega$ is the set of singular points in $V$. $n$ is odd, $n \geq 5$, and $q$ is odd.
\item[$O,\delta$]: $F^\ast(G) \cong O_n(q)$, $V$ is a non-degenerate orthogonal space, and $\Omega$ is the set of $\delta$-type points in $V$. $n$ is odd, $n \geq 5$, $\delta = +$ or $-$, and $q$ is odd.
\item[$Sp$]: $F^\ast(G) \cong Sp_{n}(q)$, $V$ is a non-degenerate symplectic space and $\Omega$ is the set of points in $V$. $n$ is even, $n \geq 4$.
\item[$Sp,\delta$]: $F^\ast(G) \cong Sp_{n}(q)$, $V$ is a symplectic space, and $\tV$ is an orthogonal space of dimension $n+1$ such that $\rad \tV$ is anisotropic of dimension $1$ and $V \cong \tV/\rad \tV$, and $\Omega$ is the set of all complements to $\rad \tV$ in $\tV$ of type $\delta$. $n$ is even, $n \geq 4$, $\delta = +$ or $-$, and $q$ is even.
\item[$U,\bs$]: $F^\ast(G) \cong U_n(q^{1/2})$, $V$ is a non-degenerate hermitian space, and $\Omega$ is the set of singular points in $V$. $n \geq 3$, $q$ is a square,
\item[$U,\bn$]: $F^\ast(G) \cong U_n(q^{1/2})$, $V$ is a non-degenerate hermitian space, and $\Omega$ is the set of nonsingular points in $V$. $n \geq 3$, $q$ is a square, \end{enumerate} \end{definition}
We prove Theorems~\ref{basic.result}, \ref{grassmann.exceptions}, and \ref{touch.up} in the subsequent sections. Since the action of a classical group $G$ on its natural module $V$ satisfies the hypotheses of Theorem~\ref{basic.result}, it is evident that Theorem~\ref{main.result} follows from these theorems.
\section{Proof of Theorem~\ref{basic.result}}
\subsection{Notation and preliminary results} \label{notation.section}
Let $G$ be an almost simple classical group with natural module $V$ of dimension $n_q$ over $\F_q$ and let $p$ be the characteristic of $\F_q$. Then $V_{\F_p}$ is an $\F_p$-vector space and all elements of $G$ correspond to $\F_p$-linear maps. We have $G = \hG/Z$ where $\hG \subseteq GL(V_{\F_p})$ and $Z$ acts as scalars on $V_{\F_p}$. Set $n_p = \dim_{\F_p} V_{\F_p}$, so that $n_p = n_q \log_p(q) $.
\begin{definition} $v_q(y)$ [resp., $v_p(y)$] is the codimension of the largest eigenspace of the action of an associate of $y$ on $V$ [resp., $V_{\F_p}$]. \end{definition} Regarding $V$ as an $\F_p$-space, $v_p(x) = \max(\codim C_V(\hx) \ : \ \hx \mapsto x$ under $\hG \rightarrow G)$.
Let $\Omega$ be a primitive $G$-set of order $N$. Let $\ux = (x_1,\ldots, x_r)$ be a normalized generating tuple for $G$.
Let $g = g(\ux)$, and let $$\ud = (d_1, \ldots, d_r)$$ be the signature of $\ux$, so that $d_i = o(x_i), i = 1, \ldots, r$.
When the context is clear, we will write $n$ instead of $n_q$ or $n_p$ and $v$ instead of $v_q$ or $v_p$.
The Cauchy-Frobenius Formula says that if $x \in G$ has order $d$, then $$\Ind(x) = N - \frac1d \sum_{y \in \lan x \ran} F(y).$$
Combining this with (RH), we have \begin{equation} \label{CF.consequence} \sum_{i=1}^r \frac1{d_i}
\left( 1 + \sum_{y \in \lan x_i \ran^\sharp} f(y) \right)
= r - 2 - 2\left(\frac{g-1}N\right). \end{equation}
\begin{definition} $$\eps_0 = 2\left(\frac{g-1}N\right),$$ $$A(\ud) = \sum\frac{d_i-1}{d_i}.$$ \end{definition}
\begin{definition} For $x \in G$, with $o(x) = d$, set $$\kappa(x) = \frac1d\left(1+ \sum_{y \in \lan x \ran^\sharp} p^{-v(y)}\right)$$ \end{definition}
\begin{fact} \label{kappa.inequality} If $G$ satisfies Grassmann condition $\eps$ then $$\sum \kappa(x_i) > r-2 - A(\ud)\eps - \eps_0$$ \end{fact}
\pf
Since $p^{-v(y)} = \frac{|W|}{|V|}$ where $W$ is the largest eigenspace for $V$, we have $f(y) < p^{-v(y)} + \eps$ for all $y \in G$. Therefore, $$r - 2 - \eps_0 < \displaystyle \sum_{i=1}^r \frac1{d_i} \left( 1 + (d_i-1) \eps - \sum_{y \in \lan x \ran^\sharp} p^{-v(y)}\right) < A(\ud) \eps + \sum \kappa(x_i).$$ \eop
The relevance of this result can be seen from the main result of \cite{FM1}. \begin{thm}[Grassmann Theorem] There is a function $\heps : \N \rightarrow \bR^+$ such that
\begin{enumerate}
\item $(G,\Omega)$ satisfies Grassmann condition $\heps(m)$
whenever $(G,\Omega)$ is a classical subspace action of degree $m$,
and
\item
$\displaystyle \lim_{m\rightarrow \infty} \heps(m) = 0$.
\end{enumerate} \end{thm}
In the balance of this subsection we obtain upper bounds for $\kappa(x)$ that will be used in the proof of Theorem~\ref{basic.result}.
Set
$$\zeta(d) = \zeta(d,p) = \frac1d\left( 1 + \sum_{m|d,m>1} \phi(m) p^{-1}\right),$$
where $\phi$ is the Euler $\phi$-function on integers. When $a$ is not an integer we take $\phi(a) = 0$. Since \begin{eqnarray*}
\kappa(x) & = &\frac1d\left(1 + \sum_{m|d,m > 1} \phi(m) p^{-v(x^{d/m})} \right) \\
& = &\frac1d\left(1 + \sum_{m|d, m<d} \phi(\frac{d}{m})p^{-v(x^m)}\right), \end{eqnarray*} it follows that if $x$ has order $d$, then \begin{equation} \kappa(x) \leq \zeta(d) = \frac1d + \frac1p - \frac1{dp}. \end{equation} Note that $\zeta$ is a decreasing function of both $d$ and $p$.
For each positive integer $s \geq 1$, set $$\zeta_s(d) = \frac1d\left( 1 + \phi(d)\cdot p^{-s}
+ \sum_{m|d, 1 < m < d} \phi(m) p^{-1}\right).$$ More generally, for a finite sequence $s_1,s_2,\ldots, s_l$ of positive integers, let $$\zeta_{s_1,s_2,\ldots,s_l}(d) = \frac1d\left( 1 + \sum_{i=1}^l \phi(d/i) p^{-s_i}
+ \sum_{m|d,1 < m < d/l} \phi(m) p^{-1}\right).$$
The following statement is evident. \begin{fact} \label{zeta.sub.s} If $x$ has order $d$ and $v(x^{i}) \geq s_i, i = 1, \ldots, l$, then $\kappa(x) \leq \zeta_{s_1,\ldots,s_l}(d)$. \end{fact}
The estimates for $\kappa(x)$ can be further refined by taking into consideration the possible actions of elements of a given order on a vector space over $\F_p$.
\begin{definition} For each prime $p$ and integer $d \geq 2$ let $\mustar(d,p)$ be the smallest positive integer $\mu$ such that $\mu = \dim{([V,x])}$ for some linear operator $x$ of order $d$ acting on a vector space $V$ over $\F_p$. \end{definition}
Note that each $x \in G$ is the image of some element $\hx$ in $\hG$ with $\dim C_{V_p}(\hx) = v(x)$.
If $y \in G$ has order $m$ then $v(y) \geq \mustar(m,p)$. This inequality holds in particular when
$m | d$, $o(x) = d$, and $y = x^{d/m}$. Set $$ \zetao(d) = \zetao(d,p) =
\frac1d \left( 1 + \sum_{m | d, m > 1} \phi(m) p^{-\mustar(m,p)}\right). $$ Then \begin{equation} \kappa(x) \leq \zetao(d). \end{equation}
Similarly, if $$ \zetao_{s_1,\ldots,s_l}(d) =
\frac1d \left( 1 + \sum_{m | d, m > 1} \phi(m) p^{-\alpha(d/m)}\right), \ \ \alpha(i) = \max(s_i,\mustar(d/i,p)), $$ then \begin{equation} \kappa(x) \leq \zetao_{s_1,\ldots, s_l}(d) \end{equation} whenever $v(x^i) \geq s_i$, $i =1, \ldots, l$.
\begin{lemma} \label{Lemma5} \begin{enumerate} \item If $p > 2$ then $\zetao(d) < \frac3d + .04$. \item If $p = 2$ then $\zetao(d) < \frac4d + .032$. \end{enumerate} \end{lemma}
\pf Suppose $p > 3$. Then $\mustar(d) = 1$ if and only if $d=p$ or $d | p-1$, and $\mustar(d) > 1$ for all other $d$. Since at most $p-1$ nontrivial powers of an element have order $p$ and at most $p-2$ nontrivial powers of an element have order dividing $p-1$ this implies that $\zetao(d,p) \leq \frac1d(1 + (2p-3)p^{-1} + (d-1-(2p-3))p^{-2}) < \frac1d(1 + 2 + d/p^2) = 3/d + 1/p^2 \leq 3/d + 1/5^2$. If $p = 3$, then $\mustar(m,p) = 1$ if and only if $m = 2$ or $3$, and $\mustar(m,p) = 2$ if and only if $m = 4,6$, or $8$. This implies that
$\displaystyle \sum_{m|d,\mustar(m,p) = 2} \phi(m) \leq \phi(4) + \phi(6) + \phi(8) = 8$. Therefore $\zetao(d,3) \leq \frac1d(1 + 3 \cdot 3^{-1} + 8 \cdot3^{-2} + (d-12) \cdot 3^{-3}) < 3/d + 1/27$.
For $p=2$, we note that $\mustar(m,2) = 1$ if and only if $m=2$; $\mustar(m,2) = 2$ if and only if $m=3$ or $4$; $\mustar(m,2) = 3$ if and only if $m=6$ or $7$; and $\mustar(m,2) = 4$ if and only if $m=5,8,12,14$, or $15$. It follows from this that $\zetao(d,2) \leq 4/d + 1/32$. \eop
\begin{corollary} \label{zeta.lemma} Let $x \in G$ have order $d$, and let $k$ be a real number. \begin{enumerate} \item \label{3d.bound} If $p > 2$ and $\zeta(d) \geq k > .04$ then $d \leq \displaystyle \frac3{k-.04}$. \item If $p = 2$ and $\zeta(d) \geq k > .032$ then $d \leq \displaystyle \frac4{k-.032}$. \end{enumerate} \end{corollary}
The precise value of $\mustar(d,p)$, the smallest possible commutator dimension for an element of order $d$ over $\F_p$, can be computed using the following statement.
\begin{fact} \label{mu.fact} \begin{enumerate} \item \label{decomp.statement} If $d_p$ is the largest power of $p$ dividing $d$ and $d_{p'} = d/d_p$, then $\mustar(d,p) = \mustar(d_p,p) + \mustar(d_{p'},p)$. \item \label{jordan.statement} For $a \geq 1$, $\mustar(p^a,p) = p^{a-1}$. \item If $(d,p) = 1$ then either $\mustar(d,p)$ is the exponent of $p \pmod d$ or $\mustar(d,p) = \mustar(a,p) + \mustar(b,p)$ for some integers $a,b$ with $ab = d$, $a, b > 1$, and $(a,b) = 1$. \end{enumerate} \end{fact}
\pf We may assume that $d > 1$. Suppose $x$ is an operator of order $d$ on $V$ that achieves the minimum commutator dimension. Without loss, assume that $\dim V$ is minimal.
$V$ is a direct sum of indecomposable $\F_p\lan x\ran$-submodules $V_i$. Setting $x_i = x\!\!\mid_{V_i}$ we have $o(x) = \gcd(\{o(x_i)\})$ and $\dim ([V,x]) = \sum \dim([V_i,x_i])$. Since $\dim ([V_i,x_i^{m}]) \leq \dim ([V_i,x_i])$ for all $m \in \N$, by minimality of $\dim ([V,x])$ we may assume that $o(x_i)$ is relatively prime to $o(x_j)$ when $i \neq j$.
To prove statement~\ref{jordan.statement}, suppose $d = p^a$. Then $V$ consists of a single Jordan block with eigenvalue $1$. The order of a Jordan block of size $b$ with eigenvalue $1$ is $p^a$ where $p^{a-1} < b \leq p^a$. [Proof: $(y-1)^{b-1} \neq 0$ and $(y-1)^b = 0$ imply $y^{p^k} = 1$ exactly when $p^k \geq b$.] Therefore $p^a \geq \dim V > p^{a-1}$, whence $\dim V = p^{a-1} + 1$ by minimality, and $\mustar(a,p) = \dim ([V,x]) = \dim V - 1 = p^{a-1}$.
To prove \ref{decomp.statement}, note that since $ab \geq a-1+b$ for positive integers $a$ and $b$, for unipotent $u$ and semisimple $s$ the commutator dimension of $u \otimes s$ is always at least as large as the commutator dimension of $u \oplus s$.
The last statement follows easily from the fact that if $x$ acts irreducibly and semisimply on $V$ then $\dim V$ is the exponent of $p \pmod d$. This completes the proof of ({\bf\ref{mu.fact}}). \eop
\subsection{System bounds} \label{scottbound.section}
The results of the previous subsection apply to individual elements. We shall require stronger bounds, which depend on the system, not merely the individual generating elements. As in \cite{FM2}, we use a result of L.~Scott on linear groups together with a fact about group generation to control the contributions of elements with large fixed point ratios to the index sum.
\begin{thm} [Scott] \label{generation.fact} Suppose $\hG$ is a group of linear operators on $V$ with $[V,\hG] = V$ and $C_V(\hG) = 1$. If $\hG = \lan g_1, \dots, g_r \ran$ where $\prod g_i = 1$, then $\sum \dim ([V,g_i]) \geq 2 \dim V$. \end{thm}
\pf See \cite{Scott}. \eop
\begin{lemma} \label{magnus} Assume that $\ue$ is an ordered $r$-tuple that is a permutation of one of the following. \begin{enumerate} \item $(m,m,1, \ldots, 1)$, $m \geq 1$. \item $(2,2,m,1, \ldots, 1)$, $m \geq 2$. \item $(2,3,m,1, \ldots, 1)$, $m = 3, 4$, or $5$. \end{enumerate} Set $\displaystyle C_i = C_i(\ue) = \frac2{e_i ( 2 - A(\ue) ) }$.
Let $H$ be a group with generators $\{y_1, \ldots, y_r \}$ where $y_1y_2\cdots y_r =1$. Then there is an ordered $M$-tuple $(z_1, \ldots, z_M)$, of elements of $H$, where $M = \sum_j C_j$ such that the following conditions hold. \begin{enumerate} \item $z_1 z_2 \cdots z_M = 1$ \item $\displaystyle \{1, \ldots, M \} = \cup_{i=1}^n {\cal C}_i$ (disjoint union) where
$|{\cal C}_i| = C_i$ and $z_j$ is conjugate to $y_i^{e_i}$ for all $j \in {\cal C}_i$.
\item The group $K$ generated by $\{ z_j \}$ is normal of index $2/(2-A(\ue))$ in $H$, and $H/K$ is cyclic, dihedral, or isomorphic to $Alt_4$, $Sym_4$, or $Alt_5$. \end{enumerate} \end{lemma}
\pf By well-known properties of generators and relations (see
\cite{Magnus}, for example), if $\ue$ is one of the specified tuples, then the group $\lan y_i, i = 1, \ldots, r \ | \ y^{e_i} = \prod_i y_i = 1 \ran$ is, in the respective cases, cyclic of order $m$, dihedral of order $2m$, or isomorphic to $Alt_4$, $Sym_4$, or $Alt_5$. In each case, this group has order $2/(2-A(\ue))$. The statements follow from the proof of Lemma~3.2 in \cite{FM1} or from \cite{GN}. \eop
Note that if $\uC(\ue) = (C_1(\ue),\dots,C_r(\ue))$ then $$\uC(m,m,1, \ldots, 1) = (1,1,m, \ldots, m)$$ $$\uC(2,2,m,1, \ldots, 1) = (m,m,2,2m \ldots, 2m)$$ $$\uC(2,3,3,1, \ldots, 1) = (6,4,4,12, \ldots, 12)$$ $$\uC(2,3,4,1, \ldots, 1) = (12,8,6,24, \ldots, 24)$$ $$\uC(2,3,5,1, \ldots, 1) = (30,20,12,60, \ldots, 60)$$
Assume now that $\ux$ is a normalized generating $r$-tuple for $G$, a classical group with natural module $V$ with $\dim (V/C_\hG(V)) = n$.
\begin{lemma} \label{translation.lemma} If $\ue$ and $\uC$ are as above, then, for each $i^\ast$ in $\{ 1, \dots, r\}$, $$(C_{i^\ast}-1)v(x_{i^\ast}^{e_{i^\ast}})
+ \sum_{i\neq {i^\ast}} C_i v(x_i^{e_i}) \geq n.$$ If $p = 2$, then $$\sum C_i v(x_i^{e_i}) \geq 2n.$$ \end{lemma}
\pf
We apply Theorem~\ref{generation.fact} to the preimages ${\hat{z}_j}$ of the elements $z_j$ under the homomorphism $\hG \rightarrow G$. In general, we can choose $M-1$ preimages $\hat{z}_j$ so that
$\dim ([V,\hat{z}_j]) = v(x_i^{e_i})$, when $j \in \cC_i$. If $j^\ast$ is the remaining subscript and $j^\ast \in \cC_{i^\ast}$, then $\dim([V,\hat{z}_{i^\ast}]) \leq n$, and we have the first statement.
If $p= 2$, then
$\dim([V,\hat{z}_j]) = v(x_i^{e_i})$
whenever $j \in \cC_i$ because $|{\bf F}^\times| = 1$. \eop
\begin{fact} \label{zetas} Suppose $r = 3$ and $d_1 \leq d_2 \leq d_3$. \begin{enumerate}
\item If $n > d_1$, then $v(x_i) \geq 2$ for $i \geq 2$.
\item If $n > d_2$, then $v(x_1) \geq 2$ for all $i$.
\item If $n \geq 4$ and $d_1 \leq 3$, then $\kappa(x_i) < \zeta_2(d_i)$ for $i > 1$.
\item If $n \geq 4$ and $d_2 \leq 3$, then $\kappa(x_i) < \zeta_2(d_i)$ for all $i$. \end{enumerate} \end{fact}
\pf Setting $\ue = (d_1,1,d_1)$ and $i^\ast = 3$, Lemma~\ref{translation.lemma} implies that $d_1 v(x_2) = v(x_1^{d_1}) + d_1 v(x_2) \geq n$. Therefore $v(x_2) \geq n/d_1 > 1$, so the first statement holds for $i = 2$. Using $\ue = (d_1,d_1,1)$ and $i^\ast = 2$ establishes the statement for $i = 3$. To establish the second statement, use $\ue = (1,d_2,d_2), i^\ast = 3$. The remaining two statements follow from {\bf\ref{zeta.sub.s}}. \eop
Set $\displaystyle \zeta^t(d) =
\frac1d\left(1 + \sum_{m|d,m<d} \phi(d/m) p^{-\max(1,n-mt)}\right)$.
Note that $$\zeta^t(d) = \zeta_{n-t,n-2t, \ldots}(d).$$
\begin{lemma} \label{LemmaX} If $j \neq k$ and $\sum_{i \neq j,k} v(x_i) \leq t$, then $\kappa(x_j) \leq \zeta^t(d_j)$ and $d_j \geq n/t$. \end{lemma}
\pf Without loss, $j = 1$ and $k = 2$. From Lemma~\ref{translation.lemma} with $\ue = (m,m,1,\ldots,1)$ and $i^\ast = 2$ we have $v(x_1^m) \geq n - mt$. The total contribution of the $\phi(d_1/m)$ generators of $\lan x_1^m \ran$ to $\kappa(x_1)$ is therefore at most $\phi({d_1}/m) \cdot \frac1{d_1} \cdot p^{-\max(1,n-mt)}$. This implies the inequality for $\kappa(x_j)$. Since $v(x_1^{d_1}) = 0$, it also follows that $d_1 \geq n/t$. \eop
\begin{lemma} \label{LemmaX4} If $j, k, l$ are distinct, $d_k = d_l = 2$, and $\sum_{i \neq j,k,l} v(x_i) \leq t$, then $\kappa(x_j) \leq \zeta^{2t}(d_j)$ and $d_j \geq n/2t$. \end{lemma}
\pf Argue as in the proof of Lemma~\ref{LemmaX}. Assume $j = 1$, $k = 2$, $l = 3$, and use Lemma~\ref{translation.lemma} with $\ue = (m,2,2,1,\dots,1)$ and $i^\ast = 1$ to get $v(x_1^m) \geq n - 2mt$. \eop
\begin{lemma} \label{LemmaY} Suppose $\ud = (2,d_2,d_3)$ and $v(x_2^2) = v$. \begin{enumerate} \item $\kappa(x_3) \leq \zeta^v(d_3)$ and $d_3 \geq n/v + 1$. \item If $p = 2$ then $\kappa(x_3) \leq \zeta^{v/2}(d_3)$ and $d_3 \geq 2n/v$. \end{enumerate} \end{lemma}
\pf Using $\ue = (2,2,k)$, $i^\ast = 3$, in Lemma~\ref{translation.lemma}, we have $v(x_3^k) \geq n - kv$ in general, and $v(x_3^k) \geq n - kv/2$ when $p = 2$. Using $\ue = (2,2,d_3)$, $i^\ast = 2$, we have $(d_3-1)v \geq n$ in general and $d_3 v \geq 2n$ when $p=2$. \eop
\begin{lemma} \label{Scott3} If $r = 3$ and $i \neq j$, then $d_i v(x_j) \geq n$. In particular, $\kappa(x_j) \leq \zeta_{\ceil{n/d_i}}(d_j)$, where $\ceil{x}$ is the smallest integer not less than $x$. \end{lemma}
\pf Without loss, $i = 1$ and $j = 2$. The first statement follows from Lemma~\ref{translation.lemma} with $\ue - (d_1,1,d_1)$ and $i^\ast = 3$. The second statement follows from the first. \eop
\begin{lemma} \label{23d} Assume that $\ud = (2,3,d)$. If $p$ is odd, set $s_2 = \ceil{n/2}$, $s_3 = \ceil{n/3}$, $s_4 = \ceil{n/5}$, and $s_5 = \ceil{n/11}$. If $p = 2$, set $s_2 = \ceil{2n/3}$, $s_3 = \ceil{n/2}$, $s_4 = \ceil{n/3}$, and $s_5 = \ceil{n/6}$. Then $$v(x_3^k) \geq s_k, \ \ d = 2,3,4,5.$$ In particular $\kappa(x_3) \leq \zeta^\ast_{s_2,s_2,s_3,s_4,s_5}(d)$. \end{lemma}
\pf Lemma~\ref{translation.lemma} with $\ue = (2,3,e)$ and $e = 2,3,4,5$, with $i^\ast = 3$ shows that $(C_3(\ue) -1)v(x_3^e) \geq n$ in general and $C_3(\ue) v(x_3^e) \geq 2n$ when $p = 2$. We have $C_3(\ue) = 3,4,6,12$ in the respective situations, and the result follows immediately. \eop
\begin{lemma} \label{24d} Assume that $\ud = (2,4,d)$. If $p$ is odd, set $s_2 = \ceil{n/3}$ and $s_3 = \ceil{n/7}$. If $p = 2$, set $s_2 = \ceil{n/2}$ and $s_3 = \ceil{n/4}$. Then $$v(x_3^k) \geq s_k, \ \ d = 2,3.$$ In particular $\kappa(x_3) \leq \zeta^\ast_{s_2,s_2,s_3}(d)$. \end{lemma}
\pf Use Lemma~\ref{translation.lemma} with $\ue = (2,4,e)$ and $e = 2,3$, with $i^\ast = 3$ for the general case. We have $C_3(\ue) = 4,8$ in the respective situations. \eop
\begin{lemma} \label{zeta5*} Suppose $p=2$, $n \geq 14$, $r=3$, and $\{i,j,k\} = \{1,2,3\}$. Then \begin{enumerate} \item \label{part1} $v(x_i^2) + v(x_j^2) \geq 28/d_k$. \item \label{part2} If $d_i = d_j = 3$, then $v(x_k^2) \geq 5$. \item \label{part3} If $d_i = 3$ and $d_j = 4$, then $v(x_k^2) \geq 3$. \end{enumerate} \end{lemma}
\pf Without loss, $i = 1$, $j = 2$, and $k = 3$. Use Lemma~\ref{translation.lemma} with $\ue = (2,2,d_3)$, $(3,3,2)$, and $(3,4,2)$, respectively. \eop
\subsection{Initial reductions}
The proof of Theorem~\ref{basic.result} uses routine, but extensive, calculations based on the results of the previous subsections. We have verified these calculations using GAP4 \cite{GAP4}.
Assume that \begin{enumerate} \item $G$ is a classical group with natural module $V$ and $\F_p$ dimension $n$. \item $V$ contains at least $10^4$ points. \item $\ux$ is a normalized generating $r$-tuple
for $G$ in a primitive action. \item Every power of every element of $\ux$ satisfies
Grassmann Condition $1/100$. \item $g(\ux) \leq 2$. \end{enumerate} To prove Theorem~\ref{basic.result} it suffices to show that the characteristic of $V$, the dimension of $V$ over its prime field, and the signature of $\ux$ are given in Table~\ref{pnd.table}.
Unless stated otherwise, we assume that $d_1 \leq d_2 \leq \dots \leq d_r$ and that $v(x_i) \leq v(x_{i+1})$ whenever $d_i = d_{i+1}$. Also recall that $\epsilon_0$ and $A(d)$ were defined just before Fact {\bf \ref{kappa.inequality}}.
We have $\eps_0 < 2 \cdot 10^{-4}$ and $\eps < 10^{-2}$. Combining Fact {\bf \ref{kappa.inequality}} with the inequality $\kappa(x_i) \leq \frac1{d_i} + \frac1p - \frac1{d_ip}$, we have the following inequalities.
\begin{fact} \label{Ap.fact} $A(\ud) > (.99A(\ud) - 2.0002)p$. Consequently \begin{enumerate} \item \label{p.inequality} $\displaystyle p < \frac{A(\ud)}{.99A(\ud) - 2.0002}$ \item \label{A.inequality} $\displaystyle A(\ud) < \frac{2.0002p}{.99p - 1}$ \end{enumerate} \end{fact}
\begin{lemma} \label{n.bounds} $n \geq 3$. \begin{enumerate} \item If $p \leq 97$ then $n \geq 4$. \item If $p \leq 19$ then $n \geq 5$. \item If $p = 7$ then $n \geq 6$. \item If $p = 5$ then $n \geq 7$. \item If $p = 3$ then $n \geq 10$. \item If $p = 2$ then $n \geq 14$. \end{enumerate} \end{lemma}
\pf The enumerated statements are immediate consequences of the inequality $(p^n-1)/(p-1) \geq 10000$.
If $n = 2$, then $F^\ast(G) \cong L_2(p)$, and $F(x) \leq 2$ for all $x \in G^\sharp$. It follows that $f(x) \leq 1/5000$ for all $x \in G^\sharp$, so equation (\ref{CF.consequence}) cannot hold for $g \leq 2$. \eop
Set $S = S(\ud) = r-2 - .01A(\ud) - .0002$, the right hand side of the inequality in statement {\bf \ref{kappa.inequality}}. For $i = 1, \ldots, r$, set $\kappa_i = \kappa(x_i)$. Set $\Sigma = \sum \kappa_i$. Then $\Sigma > S$ by {\bf\ref{kappa.inequality}} and assumptions on $\ux$.
\begin{lemma} \label{p.prop} \begin{enumerate} \item If $p \geq 17$, then $r = 3$. \item If $p \geq 7$, then $r \leq 4$. \item If $p = 7$, then $r \leq 4$ and $S \geq (r-3) + .9761$. \item If $p = 5$, then $r \leq 5$ and $S \geq (r-3) + .9744$. \item If $p = 3$, then $r \leq 6$ and $S \geq (r-3) + .9693$. \item If $p = 2$, then $r \leq 8$ and $S \geq (r-3) + .9589$. \end{enumerate} \end{lemma}
\pf Since $\zeta$ is a decreasing function, we have $\zeta(d) \leq \zeta(2) = (p+1)/2p$, so $\kappa(x_i) \leq (p+1)/2p$ for all $i$. Therefore $r \cdot \frac{p+1}{2p} > r - 2 - .01A(\ud) - .0002 > .99r -2.0002$, whence $$r < \frac{4.0004p}{.98p - 1}.$$ All assertions about $r$, except the first, follow from this.
If $r = 4$, then $A(\ud) \geq 13/6$, so $p < 17$ by {\bf\ref{Ap.fact}.\ref{p.inequality}}.
The statements concerning $S$ follow from {\bf \ref{Ap.fact}.\ref{A.inequality} }. \eop
\begin{lemma} \label{S.bounds} \begin{enumerate} \item If $r = 3$, then $S \geq .9698$. \item If $r=3$ and $d_1 = 2$ then $S \geq .9748$. \item If $r=3$, $d_1 = 2$, and $d_2 = 3$ then $S \geq .9781$. \item If $\ud = (2,3,7)$, then $S \geq .9795$. \end{enumerate} \end{lemma}
\pf These statements follow from straightforward computations. \eop
\subsection{Completion of the Proof}
\begin{lemma} $n \geq 4$. \end{lemma}
\pf Suppose $n = 3$. Then $\Omega$ is the set of points in the natural module for $F^\ast(G) \cong L_3(p)$. We have $N = p^2 + p + 1$. By Lemma~\ref{n.bounds}, $p > 100$, so $A(\ud) < 2.02$ by {\bf\ref{Ap.fact}.\ref{A.inequality}}. It follows that $\ud = (2,3,7)$.
Since $x_1$ is an involution in $G$, we have $\Fix(x_1) \leq p+2$, and $\Ind(x_1) \geq \frac12(p^2-1)$. By Lemma~\ref{Scott3}, $v(x_i) \geq 2$, for $i = 2,3$. This implies that $\Fix(x_i) \leq 3$, $i = 2, 3$, whence $\Ind(x_i) \geq (d_i-1)/d_i \cdot (p^2 + p -2)$. It follows from the Riemann-Hurwitz equation that $g > 2$, a contradiction. \eop
\begin{lemma} \label{prop.lt.23} $p \leq 19$ \end{lemma}
\pf Suppose $p \geq 23$. Then $A(\ud) \leq 2.0002p/(.99p-1) < 2.114$ by {\bf \ref{Ap.fact}.\ref{A.inequality} }. This implies that $\ud$ is one of the following: $(2,3,d)$, $(2,4,\leq7)$, $(2,5,5)$, or $(3,3,4)$. Also, $S > .9787$ by {\bf\ref{kappa.inequality}}.
If $\ud = (2,3,d)$, $d \geq 8$, then {\bf \ref{zetas}} implies that $\sum \kappa(x_i) \leq \zeta_2(2) + \zeta_2(3) + \zeta_2(d)$. Since $\phi(d) \geq 4$, it follows that $\zeta_2(d) \leq \frac1d(1 + (d-5)/p + 4/p^2) = \frac1p + (1 + \frac4{p^2} + \frac5p)\cdot\frac1d \leq \zeta_2(8) < .1423$, whence $\sum\zeta_2(d_i) \leq .9778$, a contradiction.
In the remaining six cases, we have $\kappa_i \leq \zeta_2(d_i), i = 2,3$ and $\kappa_1 \leq \zeta(d_1)$ in all cases, and $\kappa_1 \leq \zeta_2(2)$ in the $(2,3,7)$ case. By inspection, either $\Sigma < S$ or $p=23$ and $\ud = (2,4,5)$ or $(2,3,7)$.
Suppose $\ud = (2,4,5)$. Then $v(x_3) \geq \mustar(5,23) = 4$. If $v(x_1) \geq 2$, then $\sum \kappa_i < .9628$, so we must have $v(x_1) =1$. Therefore $v(x_2) \geq n - v(x_1) \geq 3$. Furthermore, $n = 4$, by Lemma~\ref{Scott3}. This implies that $v(x_2^2) \geq 2$ since every involution $t$ in $PGL(4,23)$ with $v(t) = 1$ is not a square in that group. It follows that $\sum \kappa_i < .974$, a contradiction.
We must have $\ud = (2,3,7)$, whence $v(x_3) \geq \mustar(7,23) =3$, and $\kappa_3 \leq \zeta_3(7)$. This implies that $\sum \kappa_i < .9786$, which is not so. \eop
\begin{prop} \label{large.p} If $p > 7$ then $p = 11$, $\ud = (2,3,7)$, and $n = 5$ or $6$. \end{prop}
\pf By Lemmas~\ref{n.bounds} and~\ref{prop.lt.23}, $n \geq 5$. Suppose $p > 7$. Then $p \geq 11$, and for purposes of estimation with $\zeta(d)$ and $\zeta_k(d)$ we may assume that $p = 11$.
Since $A(\ud) \leq 2.2246$ by {\bf\ref{Ap.fact}}, we have $S \geq (r - 3) + .9775$.
If $r > 3$, then $\ud = (2,2,2,3)$ by the condition on $A(\ud)$. Since $\sum_{i \neq j} v(x_i) \geq n \geq 5$ for $j = 3, 4$, we have $v(x_3) > 1$ and either $v(x_2) > 1$ or $v(x_4) > 1$. Therefore $\sum \kappa_i \leq \max(2 \zeta(2) + \zeta_2(2) + \zeta_2(3), \zeta(2) + 2\zeta_2(2) + \zeta(3))< 1.95$, a contradiction.
Thus $r = 3$. Since $\zeta(d_1) \geq S/3 > \zeta(4)$, it follows that $d_1 = 2$ or $3$.
Suppose $d_1 = 3$. Then $\zeta(d_2) > (S-\zeta(3))/2 > \zeta(5)$, so $d_2 \leq 4$, and $\kappa_1 \leq \zeta_2(3)$ by {\bf \ref{zetas}}. Since $\zeta_2(4) < \zeta_2(3)$, this implies that $\kappa_3 > S - 2 \zeta_2(3) > \zeta_2(4) > \zeta(5)$, whence $d_3 = 3$, which is impossible because $\ud \neq (3,3,3)$. This shows that $d_1 = 2$.
Since $\kappa_3 \leq \zeta(d_3) \leq \zeta(d_2)$ and $\kappa_2 \leq \zeta(d_2)$, we must have $\zeta(d_2) > (S- \zeta(2))/2 > \zeta(8)$ so $d_2 \leq 7$. If $d_2 = 5, 6$, or $7$, then $\kappa_2 \leq \max_{5 \leq d \leq 7}(\zeta_2(d)) \leq \zeta_2(6)$. [Recall that $p=11$ for the purpose of calculation.] Since $\zeta(8) < \zeta_2(6)$ and $\zeta(d) < \zeta(8)$ for $d > 8$, we have $\kappa_3 \leq \zeta_2(6)$ and $\sum \kappa_i \leq \zeta(2) + 2 \zeta_2(6) < S$. Therefore $d_2 \leq 4$.
Suppose $d_2 = 4$. Then $\kappa_3 \geq S - \zeta_2(2) - \zeta_3(4) > .2002 > \zeta_3(d)$ for $d > 6$, so $d_3 \leq 6$. If $d_3 = 5$, then $A(\ud) = 2.05$, so $S \geq .9793$ and $\Sigma \leq \zeta_2(2) + \zeta_3(4) + \zeta_3(5) < .9781$. It follows that $d_3 = 6$. From Lemma~\ref{translation.lemma} with $\ue = (2,2,2)$, either $v(x_2^2) > 1$ or $v(x_3^2) > 1$. If $v(x_2^2) > 1$, then $\kappa_2 \leq \zeta_{3,2}(4)$. If $v(x_3^2) > 1$, then $\kappa_3 \leq \zeta_{3,2}(6)$. In either case, $\Sigma < .97 < S$.
Suppose $d_2 = 3$. Then $S \geq .9781$ and $\kappa_1 + \kappa_2 \leq \zeta_2(2) + \zeta_3(3) < .8426$, so $\kappa_3 > .1355$. If $d \geq 21$, then $\zeta(d) < \zeta(21) < .135$. Therefore $d_3 \leq 20$. By inspection, if $d_3 = 9$ or $d_3 \geq 11$, then $\zeta_3(d_3) < .137$, and the inequality cannot hold. Therefore $d_3$ is one of $7, 8$, or $10$. If $d_3 = 8$, or $10$, then $\kappa_3 \leq \zeta_{3,3}(d_3)$ by Lemma~\ref{23d} and $\sum \kappa_i < S$.
Therefore $d_3 = 7$, so $S \geq .9795$ and the condition $\zeta_2(2) + \zeta_3(3) + \zeta_3(7) \geq S$ implies that $p = 11$ or $13$. If $p = 13$, then $v(x_3)$ is necessarily even, so $\kappa_3 \leq \zeta_4(7)$ and $\sum \kappa_i < S$. Therefore $p = 11$. It follows that $\kappa_2$ is even, so $\kappa_2 \leq \zeta_4(3)$. If $v(x_1) > 2$, then $\sum \kappa_i \leq \zeta_3(2) + \zeta_4(3) + \zeta_3(7) < S$. Therefore $v(x_1) = 2$ and $n = 5$ or $6$. \eop
\begin{prop} \label{p=7.prop} If $p = 7$, then $n = 6$ and $\ud = (2,3,7)$. \end{prop}
\pf By Lemma~\ref{p.prop} $n \geq 6$, $r \leq 4$, and $S \geq (r-3) + .9761$.
Suppose $r = 4$. If $v(x_1) + v(x_2) = 2$, then $d_j \geq 3$, $j > 2$, and $\sum \kappa_i \leq 2 \zeta(2) + 2 \zeta^2(3) < 1.9$ by Lemma~\ref{LemmaX}. Therefore $v(x_1) + v(x_2) \geq 3$, and in fact $v(x_i) \geq 2$ for at least $3$ choices of $i$. It follows from inspection of values of $\zeta(d)$ and $\zeta_2(d)$ that $\sum \kappa_i < S$, a contradiction.
Therefore $r = 3$. If $v(x_1) = 1$, then $\kappa_2, \kappa_3 \leq \zeta^1(d) < .168$ by Lemma~\ref{LemmaX}. Since $\kappa_1 \leq \zeta(2) < .572$, we have $\sum \kappa_i < S$, a contradiction. Therefore $v(x_i) \geq 2$ and $\kappa_i \leq \zeta_2(d_i)$ for all $i$. Since $\zeta_2(d) < .3$ for $d > 3$, we have $d_1 \leq 3$.
Suppose $d_1 = 3$. Then, by inspection of $\zeta_2(d)$, $d \geq 3$, we have $\ud = (3,3,4)$. Either $v(x_1) = 2$, in which case $\sum \kappa_i \leq \zeta_2(3) + \zeta_4(3) + \zeta_4(4)$, or $v(x_1) \geq 3$, in which case $\sum \kappa_i \leq 2 \zeta_3(3) + \zeta_2(4)$. In either case, $\sum \kappa_i < .97$, a contradiction. We conclude that $d_1 = 2$.
We have $\kappa_2 + \kappa_3 \geq S - \zeta_2(2) \geq .465$. Also $\kappa_i \leq \zeta_3(d_i), i > 1$ by Lemma~\ref{Scott3}. By inspection, $\zeta_3(d) < .2$ for $d > 6$, so $d_2 \leq 6$.
Suppose $v(x_1) = 2$. Then $\kappa_j \leq \zeta^2(d_j), j \geq 2$, whence $d_2 \leq 4$ because $\zeta^2(d) < .21$ for $d > 4$. If $d_2 = 4$, then $d_3 \geq 5$ because $A(\ud) > 2$, so $\sum \kappa_i \leq \zeta_2(2) + \zeta^2(4) + \zeta^2(5) < .97$, a contradiction. Therefore $d_2 = 3$ and $d_3 \geq 7$, so $S \geq .9781$, and $\kappa_2 + \kappa_3 \geq .4678$, whence $\kappa_3 \geq .4678 - \zeta^2(3) > .1341$. By inspection, $d_3 \in \{7,8,9,12\}$. By Lemma~\ref{23d}, $\kappa_3 < \zeta_{4,3,2,2}(d_3)$, and we conclude that $\ud = (2,3,7)$. Note that $n = 6$ by Lemma~\ref{Scott3}.
We may assume henceforth that $v(x_1) \geq 3$, so $\kappa_1 \leq .5015$ and $\kappa_2 + \kappa_3 > .4747$.
If $d_2 = 6$, then $d_3 = 6$ by inspection of the values of $\zeta_3(d)$, $d \geq 6$. From Lemma~\ref{translation.lemma} with $\ue = (2,2,2)$ we have $v(x_j^2) > 1$ for some $j > 1$, so $\kappa_2 + \kappa_3 \leq \zeta_3(6) + \zeta_{3,2}(6) < S -\kappa_1$. This implies that $d_2 < 6$.
By inspection, $d_2 \neq 5$. If $d_2 = 4$, then $\kappa_2 \leq \zeta_3(4) < .2872$, so $\kappa_3 > .1875$. This implies that $d_3 \leq 6$. From Lemma~\ref{translation.lemma} with $\ue = (2,2,d_3)$ we have $v(x_2^2) > 1$, so $\kappa_2 \leq \zeta_{3,2}(4) < .257$. When $d_3 = 6$, the same argument shows that $\kappa_3 \leq \zeta_{3,2}(6)< .2$. In each case, $\sum \kappa_i < S$.
So $d_2 \neq 4$, and we have $d_2 = 3$. Also, $\kappa_1 + \kappa_2 \leq \zeta_3(2) + \zeta_3(3) < .8368$. so $\kappa_3 \geq S - \kappa_1 - \kappa_2 > .1413$. By inspection of $\zeta_3(d)$, we have $d_3 \leq 18$. By Lemma~\ref{23d}, $\kappa_3 \leq \zeta_{3,3,2,2}(d_3)$, so by inspection $d_3 = 7$. If $n > 6$, then $v(x_1) \geq 3$ and $v(x_j) \geq 4$, $j > 1$, so $\sum \kappa_i \leq \zeta_3(2) + \zeta_4(3) + \zeta_4(7) < .9781 < S$. Therefore $n = 6$. \eop
\begin{prop} \label{p=5.prop} If $p = 5$, then $\ud = (2,3,7)$, $n = 7$, $8$, or $9$, $v(x_1) = 3$, and $v(x_3) = 6$. \end{prop}
\pf By Lemma~\ref{p.prop}, $n \geq 7$, $r \leq 5$, and $S \geq (r-3) + .9744$.
If $r = 5$, then $\sum \kappa_i \leq 3 \zeta(2) + 2 \zeta_2(2) < S$ because $v(x_i) > 1$ for at least two choices of $i$. Therefore $r \leq 4$.
Suppose $r = 4$. If $v(x_1) + v(x_2) \leq 3$, then Lemma~\ref{LemmaX} implies that $d_i \geq 7/3 > 2$ for $i = 3, 4$, and $\kappa_i \leq \zeta^3(d_i) \leq \zeta^3(3) = .3344$. Since $\kappa_1 + \kappa_2 \leq 2 \zeta(2) = 1.2$, it follows that $\Sigma < S$. Therefore $v(x_1) + v(x_2) \geq 4$. Moreover, $v(x_i) + v(x_j) \geq 4$ whenever $i \neq j$. If $v(x_1) = 1$, then $\sum \kappa_i \leq \zeta(2) + 2 \zeta_3(2) + \zeta_3(3) < 1.95$. If $v(x_1) = 2$, then $\sum \kappa_i \leq 3\zeta_2(2) + \zeta_2(3) = 1.92$. Therefore $v(x_1) \geq 3$. If $d_3 > 2$, then we have$\sum \kappa_i < 2 \zeta_3(2) + 2 \zeta(3) < 1.95$, noting that $\kappa_1, \kappa_2 \leq \zeta_3(2)$ since $\zeta(3) < \zeta_3(2)$. So $d_3 = 2$ and $\kappa_1 + \kappa_2 + \kappa_3 \leq 3 \zeta_3(2) = 1.512$. From Lemma~\ref{translation.lemma} with $\ue = (2,2,2,1)$ we have $v(x_4) \geq 3$, so $\kappa_4 \leq \zeta_3(d)$ and $\sum \kappa_i \leq 3 \zeta_3(2) + \zeta_3(d) < 1.9$. We conclude that $r \neq 4$. Thus, $r = 3$.
If $v(x_1) = 1$, then Lemma~\ref{LemmaX} shows that $d_2 \geq 7$ and $\kappa_i \leq \zeta^1(d_i)$, $i = 2, 3$. So $\sum \kappa_i \leq \zeta(2) + 2 \zeta^1(d) < .9$. Therefore $v(x_1) \geq 2$, and in fact $\kappa_i \leq \zeta_2(d_i)$ for all $i$. Since $\zetao(d) \leq .29$ for $d \geq 12$ by Lemma~\ref{Lemma5} and $\zeta_2(d) \leq .32$ for $4 \leq d \leq 11$ by inspection it follows that $d_1 \leq 3$, whence $\kappa_i \leq \zeta_3(d_i)$, $i = 2,3$ by Lemma~\ref{Scott3}.
Suppose $d_1 = 3$. Then $\kappa_1 \leq \zeta_2(3) = .36$. If $d_2 \geq 4$, then $\kappa_i \leq \zeta_3(d) \leq.304$ for $i > 1$, and $\sum \kappa_i \leq .968$. Therefore $d_2 = 3$, so $v(x_1) \geq 3$ and $\kappa_1 + \kappa_2 \leq 2 \zeta_3(3) < .6774$. If $d_3 > 4$, then $\kappa_3 \leq \zeta_3(d_3) \leq .27$, so $d_3 = 4$. From Lemma~\ref{translation.lemma} with $\ue = (3,3,2)$ we have $v(x_3^2) \geq 2$ so $\kappa_3 \leq \zeta_{3,2}(4) = .264 < S - \kappa_1 - \kappa_2$. We conclude that $d_1 = 2$.
We have shown that $v(x_1) > 1$. If $d_3 > 23$, then $\kappa_3 < \zetao(d_3) < .165$. Suppose $v(x_1) = 2$. Then $d_2 \geq 4$ and $\kappa_i \leq \zeta^2(d_i)$, $i= 2,3$. Since $\kappa_1 \leq .52$ and $\zeta^2(d) \leq .203$ for $d > 4$, we must have $d_2 = 4$, whence $d_3 > 4$. If $d_3 \neq 6$, then $\sum \kappa_i \leq \zeta_2(2) + \zeta^2(4) + \zeta^2(5) < .973 < S$. Therefore $d_3 = 6$ and $A(\ud) < 2.09$, so $S > .9789$. We have $\sum \kappa_i \leq \zeta_2(2) + \zeta^2(4) + \zeta^2(6) \leq .975$, a contradiction. This shows that $v(x_1) > 2$.
We have $\kappa_1 \leq \zeta_3(2) = .504$ so $\kappa_2 + \kappa_3 > .47$. Also, $\kappa_i \leq \zeta_4(d_i), i = 1,2$, Since $\zetao(d) < .2$ for $d \geq 20$ and $\zeta_4(d) < .21$ for $6 < d < 20$, we have $d_2 \leq 6$. From Lemma~\ref{translation.lemma} with $\ue = (2,d_2,2)$ and $i^\ast = 3$ it follows that $v(x_3^2) \geq 7/(d_2-1) > 1$. This implies that $\kappa_3 \leq \zeta_{4,2}(d_3)$. If $d_2 = 6$, then $\kappa_2 \leq .268$. We have $d_3 = 6$ as otherwise $\kappa_3 \leq \min(\zetao(d_3),\zeta_{4,2}(d_3)) < .174$, so $\kappa_2, \kappa_3 \leq \zeta_{4,2}(6) < .214$, and $\Sigma < S$. Therefore $d_2 < 6$. If $d_2 = 5$, then $\kappa_2 + \kappa_3 \leq \zeta_4(5) + \zeta_{4,2}(6) \leq .47$. If $d_2 = 4$, then $\kappa_2 \leq \zeta_4(4) = .3008$. By Lemma~\ref{24d}, $\kappa_3 \leq \zeta_{4,3}(d_3) $. If $d_3 > 23$, then $\kappa_3 < \zetao(d_3) < .165$. It follows from inspection that $\zeta_{4,3}(d)\leq .214$ for $6 < d < 24$. Therefore $d_3 \leq 6$, whence $v(x_2^2) \geq 2$ and $\kappa_2 \leq \zeta_{4,2}(4) = .2608$. If $d_3 = 5$, then $\sum\kappa_i < S$. If $d_3 = 6$, then $\kappa_3 \leq .2032$, and $\sum \kappa_i < S$. It follows from this paragraph that $d_2 \neq 4$. Therefore $d_2 = 3$.
We have $\kappa_1 + \kappa_2 \leq \zeta_3(2) + \zeta_4(3) = .8384$. By Lemma~\ref{p.prop}, $S \geq .9781$, so $\kappa_3 \geq .1397$. Since $\kappa_3 < 3/d_3 +.04$ we may assume that $d_3 \leq 30$. By Lemma~\ref{23d}, $\kappa_3 \leq \zeta_{4,4,3,2}(d_3)$. By inspection, $d_3 = 7$.
If $v(x_1) \geq 4$, then $\sum \kappa_i \leq \zeta_4(2) + \zeta_4(3) + \zeta_4(7) < S$. So $v(x_1) = 3$ and $n \leq d_2 v(x_1) = 9$. \eop
\begin{prop} \label{p=3.prop} If $p = 3$, then either \begin{enumerate} \item $\ud = (2,3,7)$, $n = 12$, $v(x_1) = 4$, $v(x_2) = 8$, and $v(x_3) = 12$ or \item $\ud = (2,3,8)$, $n = 10$, $v(x_1) = 4$, $v(x_2) = 6$, and $v(x_3^4) = 2$. \end{enumerate} \end{prop}
\pf By Lemma~\ref{p.prop}, $n \geq 10$, $r \leq 6$, and $S \geq (r-3) + .9693$.
We note that $\zetao(d) < .11$ for $d > 42$ by Lemma~\ref{Lemma5} and $\zetao(d) < .11$ by direct computation for $24 < d \leq 42$. Also, $\zetao(d) < .2$ for $d > 12$. Thus, statements bounding $\kappa_i$ with weaker bounds need only be verified for a finite number of possible values of $d_i$. We shall use this implicitly in the following argument.
Since $n > r$, we have $\kappa_i \leq \zeta_2(d_i)$ for at least two choices of $i$. If $r= 6$, then $\sum \kappa_i \leq 4 \zeta(2) + 2 \zeta_2(2) < 3.8$, a contradiction, so $r \leq 5$.
Suppose $r= 5$. If $v(x_1)+ v(x_2) + v(x_3) = 3$, then $d_i \geq 4$ and $\kappa_i \leq \zeta^3(d_i) < .3$ for $i = 4, 5$ by Lemma~\ref{LemmaX}. If $v(x_1)+ v(x_2) + v(x_3) = 4$, then $d_i \geq 3$ and $\kappa_i \leq \zeta^4(d_i) < .35$ for $i = 4,5$ Since $\kappa_1 + \kappa_2 + \kappa_3 \leq 3 \zeta(2) = 2$ we have $\Sigma < S$ in this case. Therefore $v(x_i) + v(x_j) + v(x_k) \geq 5$ for any choice of distinct $i,j,k$. If $v(x_i) = 1$ for two values of $i$, then $v(x_i) \geq 3$ for three values and $\sum \kappa_i \leq 2\zeta(2) + 3 \zeta_3(2) < 2.9$. Therefore $v(x_i) = 1$ for at most one value of $i$, and $\sum \kappa_i \leq \zeta(2) + 4\zeta_2(2) < 2.9$. We conclude that $r \leq 4$.
Suppose $r= 4$. If $v(x_1) + v(x_2) = 2,3,4$, respectively, then $\kappa_1 + \kappa_2$ is respectively at most $1.3334$, $1.2223$, $1.1852$, while Lemma~\ref{LemmaX} implies that for $i = 3$ or $4$, $\kappa_i \geq \zeta^2(d_i)$ and $d_i \geq 5$, $\kappa_i \geq \zeta^3(d_i)$ and $d_i \geq 4$, $\kappa_i \geq \zeta^4(d_i)$ and $d_i \geq 3$, in the respective cases. By inspection, $\kappa_3 + \kappa_4$ is respectively at most $.401$, $.511$, $.67$, whence $\sum \kappa_i < S$. It follows that $v(x_1) + v(x_2) \geq 5$. Since the same is true of $v(x_i) + v(x_j)$, $i \neq j$, it follows that $v(x_i) \geq 3$ for at least $3$ choices of $i$. Since $\zeta(2) < .67$, $\zeta_3(2) < .52$, and $\zeta(d) < .56$, $\zeta_3(d) < .36$ when $d > 2$, we have $d_3 = 2$, else $\sum \kappa_i < 1.96 < S$. Set $v = v(x_1)$. If $v= 1$, then $\kappa_1 + \kappa_2 + \kappa_3 \leq \zeta(2) + 2 \zeta_4(2) < 1.68$ and, by Lemma~\ref{LemmaX4}, $\kappa_4 \leq \zeta^2(d_4)$ where $d_4 \geq 5$, so $\kappa_4 < .21$. If $v= 2$, then $\kappa_1 + \kappa_2 + \kappa_3 \leq \zeta_2(2) + 2 \zeta_3(2) < 1.6$ and, by Lemma~\ref{LemmaY}, and inspection of $\zeta^2$ values, $\kappa_4 \leq \zeta^4(d_4)$ where $d_4 \geq 4$, so $\kappa_4 < .28$. If $v > 2$, then $\kappa_1 + \kappa_2 + \kappa_3 \leq 3 \zeta_3(2) < 1.56$. From Lemma~\ref{translation.lemma} with $\ue = (2,2,2,1)$ and $i^\ast = 4$ we have $v(x_4) \geq 4$ and $\kappa_4 \leq \zeta_4(3) < .35$. In all cases, $\sum \kappa_i < S$. Therefore $r \neq 4$.
We have $r=3$. By inspection, $d > 6$ implies $\zetao(d) \leq .25$. Therefore $d_1 \leq 6$ and $A(\ud) \leq 3 \cdot 5/6 < 2.84$, so $S > .9714$. By inspection, $\kappa_1 \leq \zeta(2) < .67$.
If $v(x_1) = 1$, then, by Lemma~\ref{LemmaX}, $d_i \geq 10$ and $\kappa_i \leq \zeta^1(d_i) < .11$, $i = 2,3$. If $v(x_1) = 2$, then $\kappa_1 \leq \zeta_2(2) < .556$. Also, by Lemma~\ref{LemmaX}, $d_i \geq 5$ and $\kappa_i \leq \zeta^2(d_i) < .201$, $i = 2,3$. It follows that $v(x_1) > 2$, so $\kappa_i \leq \zeta_3(d_i)$ for all $i$.
Suppose $d_1 \geq 4$. Then $\kappa_i \leq \zeta_3(d_i) \leq \zeta_3(4) < .3519$ for all $i$, so $\sum_{j \neq i} \kappa_j \geq S - \zeta_3(4) > .619$, $i = 1,2,3$. If $v(x_i) = 3$ for some $i$, then Lemma~\ref{LemmaX} shows that $\kappa_j \leq \zeta^3(d_j) \leq .254$ for $j \neq i$. If $v(x_i) = 4$ for some $i$, then $\kappa_i \leq \zeta_4(d_i) \leq \zeta_4(4) < .34$ and $\kappa_j \leq \zeta^4(d_j) < .28$, $j \neq i$. It follows that $v(x_i) \geq 5$ for all $i$, so $\kappa_i \leq \zeta_5(4) < .336$. Since $\zetao(d_3) \leq .25$ for all $d > 4$ with $d \neq 6$ we conclude that $d_i = 4$ or $6$ for all $i$. From Lemma~\ref{translation.lemma} with $\ue = (2,2,2)$ we have $v(x_i^2) \geq 2$ for some $i$. Therefore $\kappa_i \leq \zeta_{5,2}(d_i) < .28$ for some $i$. Since $\kappa_j \leq \zeta_5(d_j) < .34$ for all $j$ it follows that $\Sigma < .96 < S$.
Suppose $d_1 = 3$. Then $\kappa_1 \leq \zeta_3(3) < .36$. If $v(x_1) = 3$, then $d_i \geq 4$ and $\kappa_i \leq \zeta^3(d_i) < .26$, $i= 2,3$, by Lemma~\ref{LemmaX}, whence $\sum \kappa_i < S$. Therefore $v(x_1) \geq 4$ and $\kappa_1 \leq \zeta_4(3) < .342$, so $\kappa_2 + \kappa_3 \geq S - \kappa_1 > .6295$. If $d > 3$ and $d$ is odd, then $\zetao(d) < .21$. For all $d\geq 3$ we have $\zeta_4(d) \leq \zeta_4(3) < .342$. It follows that $d_i$ is even whenever $d_i > 3$. If $d_2 > 3$, then Lemma~\ref{translation.lemma} with $\ue = (3,2,2)$ implies that $v(x_i^2) > 1$ for some $i > 1$. Therefore $\kappa_2 + \kappa_3 \leq \zeta_{4,2}(d_i) + \zeta_4(d_{5-i}) \leq \zeta_{4,2}(4) + \zeta_4(4) < S- \kappa_1$. This implies that $d_2 = 3$, so $d_3 > 3$. From Lemma~\ref{translation.lemma} with $\ue = (3,3,2)$ and $i^\ast = 3$ we have $v(x_3^2) \geq 2$. Therefore $\sum \kappa_i \leq 2 \zeta_4(3) + \zeta_{4,2}(d_3) \leq 2 \zeta_4(3) + \zeta_{4,2}(4) < S$, a contradiction.
We have $d_1 = 2$ and $\kappa_1 \leq \zeta_3(2) < .5186$. Since $\zetao(d) < .22$ for $d > 8$ it follows that $d_2 \leq 8$. By Lemma~\ref{Scott3}, $v(x_i) \geq 5$ for $i = 2,3$.
We claim that if $i = 2$ or $3$ and $d_i > 4$, then $\kappa_i \leq .236$ and furthermore, either $\kappa_i < .204$ or $d_i = 6$ and $v(x_i^2) \geq 3$. Since $\zetao(d) < .2$ for $d \geq 13$ and $\zetao_5(d) < .204$ for $d$ odd with $4 \leq d < 12$, it suffices to assume that $d_i$ is even and $d_i \leq 12$. We have $\kappa_i \leq \zeta_5(d_i) < .236$. Suppose $v(x_i^2) = 1$. By Lemma~\ref{LemmaY}, $\kappa_{5-i} \leq \zeta^1(d_{5-i})$ and $d_{5-i} \geq 11$, so $\kappa_{5-i} < .11$. It follows that $\sum \kappa_i < .97 < S$. Therefore $v(x_i^2) > 1$. Suppose $v(x_i^2) = 1$. Then $\kappa_i \leq \zeta_{5,2}(d_i) < .281$. By Lemma~\ref{LemmaY}, $\kappa_{5-i} \leq \zeta^2(d_{5-i})$ and $d_{5-i} \geq 6$, so $\kappa_{5-i} < .17$. This also implies that $\sum \kappa_i < .97 < S$. Therefore $v(x_i^2) \geq 3$ and $\kappa_i \leq \zetao_{5,3}(d_i)$. The claim follows.
It follows from the claim that if $d_2 > 4$ then $\ud = (2,6,6)$ and $v(x_i^2) \geq 3$ for $i= 2, 3$. By Lemma~\ref{translation.lemma} with $\ue = (2,3,3)$ we have $v(x_i^3) >1$ for some $i > 1$, so $\kappa_2 + \kappa_3 \leq \zeta_{5,3}(6) + \zeta_{5,3,2}(6) < .435 < S - \kappa_1$. This shows that $d_2 \leq 4$.
Suppose $d_2 = 4$. Set $v = v(x_2^2)$. If $v= 1$, then $\kappa_2 \leq \zeta_5(4) < .336$ and, as above, $\kappa_3 < .11$. If $v = 2$, then $\kappa_2 \leq \zeta_{5,2}(4) \leq .28$ and $\kappa_3 \leq \zeta^2(d_3) \leq \zeta^2(6) < .17$. In either case, $\kappa_2 +\kappa_3 < .45 < S - \kappa_1$. Therefore $v \geq 3$ and we have $\kappa_2 \leq \zeta_{5,3}(4) < .2614$. If $d_3 \neq 5,6,8,9,12$, then $\kappa_3 < \zetao(d_3) < .15$, so we may assume that $d_3 \in \{5,6,8,9,12\}$. By Lemma~\ref{24d} and the condition that $v(x_3) \geq 5$, $\kappa_3 \leq \zeta_{5,4,2}(d_3)$. By inspection, this is at most $.191$ for $d_3 > 5$, so $\sum \kappa_i < .971 < S$ in this case. We must have $\ud = (2,4,5)$. Thus, $A(\ud) = 2.05$ and $S = .9793$. If $v(x_1) = 3$, then $\Sigma \leq \zeta_3(2) + \zeta^3(4) + \zeta^3(5) < .975 < S$. Therefore $v(x_1) \geq 4$ and $\kappa_1 \leq \zeta_4(2) < .507$. We have $\kappa_2 \leq \zeta_{5,3}(4) < .262$ and $\kappa_3 \leq \zeta_5(5) \leq .204$, so $\sum \kappa_i < .973 < S$, a contradiction. This shows that $d_2 \neq 4$, so $d_2 = 3$.
We have $S > .9781$ by Lemma~\ref{S.bounds}. By Lemma~\ref{Scott3}, $v(x_1) \geq 4$ and $v(x_2) \geq 5$. Since $v(x_1) + v(x_2) \geq 10$, we have $\kappa_1 + \kappa_2 \leq \max(\zeta_4(2) + \zeta_6(3), \zeta_5(2) + \zeta_5(3)) < .8405$. By Lemma~\ref{23d}, $\kappa_3 \leq \zeta_{5,5,4,2}(d_3)$. If $d_3 > 8$, then $\kappa_3 < .137 < S-\kappa_1 -\kappa_2$. Therefore $d_3 = 7$ or $8$.
If $d_3 = 7$, then $S > .9795$. If $n > 12$, then $v(x_1) \geq 5$, $v(x_2) \geq 7$, and $v(x_3) \geq 7$, so $\sum \kappa_i \leq \zeta_5(2) + \zeta_7(3) + \zeta_7(7) < S$. Therefore $n \leq 12$. Since $\zeta_6(2) + 1/3 + 1/7 > S([2,3,7])$ we must have $v(x_1) \leq 5$. We have $n \geq 10$. Therefore $v(x_1) \leq n-v(x_1)$. From the strong form of Scott's Theorem we have $\max(v(x_1), n-v(x_1)) + v(x_2) + v(x_3) \geq 2n$. Therefore $v(x_2) + v(x_3) \geq n + v(x_1) \geq 4n/3$. Since $p=3$, we have $v(x_2) \leq 2n/3$, so $v(x_3) \geq 2n/3$. Since $3$ has multiplicative order $6$ modulo $d_3 = 7$, $v(x_3)$ is necessarily a multiple of $6$. Since $10 \leq n \leq 12$ we must have $v(x_3) = n = 12$. If $v(x_1) \geq 5$, then $v(x_2) \geq 7$ and $\sum \kappa_i \geq \zeta_5(2) + \zeta_7(3) + \zeta_{12}(7) > S([2,3,7])$, a contradiction. Therefore $v(x_1) = 4$ and $v(x_2) = 8$.
Suppose $d_3 = 8$. If $n > 10$, then $v(x_1) \geq 4$, $v(x_2) \geq 6$, $v(x_2^2) \geq 6$, and $v(x_2^4) \geq 3$, so $\Sigma \leq \zeta_4(2) + \zeta_5(3) + \zeta_{6,6,1,3}(8) < S$. Therefore $n=10$. If $d_1 > 4$, then $d_1 = 5$, $5 \leq d_2 \leq 6$, and $d_3 \geq 8$ by the strong form of Scott's Theorem, so $\Sigma \leq \zeta_5(2) + \zeta_5(3) + \zeta_{8,5,1,2}(8) < S$. Therefore $d_1 = 4$, whence $d_2 = 6$. Since $\Sigma \leq \zeta_4(2) + \zeta_6(3) + \zeta_{6,5,1,3}(8) < S$, we also have $v(x_3^4) = 2$. \eop
\begin{prop} \label{2prop} If $p= 2$, then $14 \leq n \leq 21$ and one of the following is true. \begin{enumerate} \item $\ud = (2,3,7)$ \item $n=16$, $\ud = (2,4,5)$, $v(x_1) = 4$, $v(x_2) = 12$, and $v(x_3) = 16$. \end{enumerate} \end{prop}
\pf Assume that $p=2$. By Lemma~\ref{p.prop}, $n \geq 14$, $r \leq 8$, and $S > (r-3) + .9589$.
\begin{step} \label{step1} \begin{enumerate} \item $\zetao(2) = .75$. \item If $d > 2$, then $\zetao(d) \leq .5$. \item If $d > 4$, then $\zetao(d) \leq .375$. \item If $d > 6$, then $\zetao(d) < .282$. \item If $d > 8$, then $\zetao(d) \leq .25$. \item If $d > 12$, then $\zetao(d) < .19$. \item If $d > 14$, then $\zetao(d) \leq .15$. \item If $d > 30$, then $\zetao(d) < .094$. \item If $d > 42$, then $\zetao(d) < .08$. \end{enumerate} \end{step}
In view of Lemma~\ref{Lemma5}, the assertions follows immediately from inspection of the values of $\zetao(d)$ for $d < 100$.
\begin{step} $r < 5$. \end{step}
If $r = 8$, then $v(x_i) \geq 2$ for at least $2$ choices of $x_i$, so $\sum \kappa_i \leq 6 \zeta(2) + 2 \zeta_2(2) = 5.75$. If $r = 7$, then $v(x_i) \geq 3$ for at least $2$ choices of $x_i$ since $v(x_1) + \ldots + v(x_6) \geq 14 > 6 \cdot 2$. Therefore $\sum \kappa_i \leq 5 \zeta(2) + 2 \zeta_3(2) \leq 4.875 < S$. This shows that $r \leq 6$.
Suppose $r = 6$. Set $w = v(x_1) + v(x_2) + v(x_3) + v(x_4)$. If $w \leq 6$, then Lemma~\ref{LemmaX} implies that $d_5, d_6 > 2$ and $\kappa_i \leq \zeta^6(d_i) < .34$, $i = 5,6$, so $\sum \kappa_i \leq 4 \zeta(2) + 2 \cdot .34 < 3.7$. Therefore $v(x_1) + v(x_2) + v(x_3) + v(x_4) \geq 7$, and the same is true for any other choice of $4$ distinct subscripts. If $v(x_i) = 1$ for $3$ values of $i$, then $v(x_j) \geq 4$ for all other values and $\sum \kappa_i \leq 3 \zeta(2) + 3 \zeta_4(2) < 3.9$. If $v(x_i) = 1$ for exactly $2$ values of $i$, then $v(x_j) \geq 3$ for at least $3$ values of $j$ and $\sum \kappa_i \leq 2 \zeta(2) + \zeta_2(2) + 3 \zeta_3(2) < 3.9$. It follows that $v(x_i) = 1$ for at most $1$ choice of $i$, and $\sum \kappa_i \leq \zeta(2) + 5\zeta_2(2) < 3.9$. Therefore $r < 6$.
Suppose $r = 5$. We claim that if $i, j$, and $k$ are distinct, then $v(x_i) + v(x_j) + v(x_k) \geq 7$. Assume that $v(x_i) + v(x_j) + v(x_k) \leq 6$. Then, by Lemma~\ref{LemmaX}, $d_l > 2$ for $l \neq i,j,k$ and $\kappa_l \leq \zeta^6(d_l)$. If $d_l > 6$, then $\kappa_l < .3$ by Step~\ref{step1}. If $3 \leq d_l \leq 6$, then $\zeta^6(d_l) < .34$ by inspection. This implies that $\sum \kappa_i < 3 \zeta(2) + 2 \cdot .34 = 2.93 < S$, and the claim follows.
We claim further that if $v(x_i) + v(x_j) \leq 4$ for distinct $i,j$, then $d_k = 2$ for all $k \neq i,j$. For the purpose of establishing this claim we remove the running assumption on the ordering of $x_i$ for the balance of this paragraph and show that if $v(x_1) + v(x_2) \leq 4$ then $d_k = 2$ for $k > 2$. If $v(x_1) + v(x_2) = 2$, then $v(x_k) \geq 5$ for $k > 2$ by the previous paragraph, and $\sum \kappa_i \leq 2 \zeta(2) + \sum_{k > 2} \zeta_5(d_k)$. Since $\zeta_5(2) < .52$ and $\min(\zeta_5(d),\zetao(d)) < .4$ for $d > 2$, we have either $\sum \kappa_i \leq 2 \zeta(2) + 2 \zeta_5(2) + .4 < 2.94$ or $d_k =2$ for all $k > 2$. If $v(x_1) + v(x_2) = 3$, then $\kappa_1 + \kappa_2 \leq \zeta(2) + \zeta_2(2) = 1.4$ and $\sum \kappa_i \leq \kappa_1 + \kappa_2 + \sum_{k > 2} \zeta_4(d_k)$. Since $\zeta_4(2) < .54$ and $\min(\zeta_4(d), \zetao(d)) < .41$ when $d > 2$, either $\sum \kappa_i < 2.9$ or $d_k = 2$ for all $k > 2$. Finally, if $v(x_1) + v(x_2) = 4$, then $\kappa_1 + \kappa_2 \leq \zeta(2) + \zeta_3(2) < 1.32$. Considering that $\zeta_3(2) < .57$ and $\min (\zeta_3(d), \zetao(d)) < .44$ for $d > 2$, either $\sum \kappa_i < 2.9$ or $d_k = 2$ for all $k > 2$. Since $\sum \kappa_i \geq S$ we conclude in every case that $d_k = 2$ for all $k > 2$. This completes the argument that if $v(x_i) + v(x_j) \leq 4$ for some $i \neq j$, then $d_k = 2$ whenever $k \neq i, j$.
Reverting to the ordering of $x_i$, so that $d_5$ is the largest value of $d_i$, the previous paragraph implies that if $d_5 > 2$, then $v(x_i) + v(x_j) \geq 5$ for every pair of distinct $i, j < 5$. In that case, $\sum_{i < 5} \kappa_i \leq \max(\zeta(2) + 3 \zeta_4(2), \zeta_2(2) + 3 \zeta_3(2)) < 2.4$, and $\kappa_5 \leq \zetao(d_5) \leq .5$, whence $\sum \kappa_i < S$. We conclude that $d_i = 2$ for all $i$. From Lemma~\ref{translation.lemma} with $\ue = (2,2,2,1,1)$ it follows that $v(x_i) + v(x_j) \geq 7$ whenever $i \neq j$, whence $\kappa_i \leq \max(\{\zeta_a(2) + 4 \zeta_{7-a}(2) \ : \ a = 1,2,3\}) < 2.8 < S$. This completes the argument that $r \neq 5$.
\begin{step} \label{step3} $r = 3$. \end{step}
Suppose $r = 4$. Since $v(x) + v(x') + v(x'') \geq 14$ for every set of $3$ generators $\{x,x',x''\}$ it follows that $v(x) \geq 5$ for at least two of the four generators, so $\kappa_i \leq \zeta_5(d_i)$ for at least two values of $i$.
We claim that $A(\ud) \leq 3$. Suppose $A(\ud) > 3$. Then $\sum 1/d_i < 1$. The ordering assumption on $d_i$ implies that $d_2 > 2$ and $d_4 > 4$, so $\kappa_i \leq .5$ for $i > 1$ and $\kappa_4 \leq .375$. If $d_1 > 2$, then $\sum \kappa_i \leq 1.875$, which is not the case, so $d_1 = 2$. It follows that $d_3 > 4$, since otherwise $1/d_1 + 1/d_2 + 1/d_3 \geq 1$. This implies that $\kappa_1 + \kappa_2 + \kappa_3 \leq 1.625$. Therefore $\kappa_4 > .3$, so $d_4 \leq 6$ and $A(\ud) \leq A(2,4,6,6) < 3$, a contradiction. This establishes the claim, and we conclude that $S \geq 1.9698$.
Set $w = v(x_1) + v(x_2)$. Then, by Lemma~\ref{LemmaX}, $\kappa_3 \leq \zeta^w(d_3)$, $\kappa_4 \leq \zeta^w(d_4)$, and $d_3 \geq 14/w$. If $w \leq 3$, then $\kappa_1 + \kappa_2 \leq 1.5$, $d_i \geq 5$, and $\kappa_i < \zeta^3(d_i) < .201$ for $i > 2$. If $w = 4$, then $\kappa_1 + \kappa_2 \leq \zeta(2) + \zeta_3(2) < 1.32$, $d_i \geq 4$, and $\kappa_i < \zeta^4(d_i) < .26$ for $i > 2$. If $w = 5$, then $\kappa_1 + \kappa_2 \leq \zeta(2) + \zeta_4(2) < 1.282$, $d_i \geq 3$, and $\kappa_i < \zeta^5(d_i) < .335$ for $i > 2$. If $w = 6$, then $\kappa_1 + \kappa_2 \leq \zeta(2) + \zeta_5(2) < 1.266$, $d_i \geq 3$, and $\kappa_i < \zeta^6(d_i) < .336$ for $i > 2$. In each case, $\sum \kappa_i \leq 1.96 < S$. This implies that $v(x_1) + v(x_2) \geq 7$. More generally, $v(x_i) + v(x_j) \geq 7$ whenever $i \neq j$.
Suppose $d_3 > 2$ and set $v = v(x_1)$. We claim that $v = 1$. If $v = 2$, then $\kappa_i \leq \zeta_5(d_i)$ for all $i > 1$. Since $\zetao(d) < \zeta_5(4) < .5 $ when $d > 4$ and $\zetao_k(3) \leq \zetao_k(4)$ for all $k$ it follows that $\Sigma \leq \zeta_2(2) + \zeta_5(2) + 2 \zetao_5(4) < 1.93$. Similarly, if $v = 3$, then $\Sigma \leq \zeta_3(2) + \zeta_4(2) + 2 \zetao_4(4) < 1.91$. If $v = 4$, then $\Sigma \leq 2\zeta_4(2) + \zetao_3(4) + \zetao_4(4) < 1.91$. If $v = 5$, then $\Sigma \leq 2\zeta_5(2) + \zetao_2(4) + \zetao_5(4) < 1.93$. If $v \geq 6$, then $\kappa_1 + \kappa_2 \leq 2 \zeta_6(2) < 1.02$ and $\kappa_3 + \kappa_4 \leq \max_{t = 1,2,3}(\zetao_t(4) + \zetao_{7-t}(4)) < .9$. In all cases, $\Sigma < S$.
Therefore $v(x_1) = 1$ and $v(x_i) \geq 6$ for $i > 1$. We have $\kappa_1 + \kappa_2 \leq \zeta(2) + \zeta_6(2) < 1.26$. Also, $\kappa_i \leq \zetao_6(d_i)$ when $i > 2$. If $d > 4$, then $\zetao_6(d) < .34$. Therefore $d_3 \leq 4$ and $\kappa_3 < .39$. From Lemma~\ref{translation.lemma} with $\ue = (1,2,d_3,2)$ we have $2d_3 + d_3 v(x_4^2) \geq 28$, whence $v(x_4^2) \geq 28/d_3 -2 \geq 5$. Therefore $\kappa_4 \leq \zetao_{6,5}(d_4)$. If $d_4 \geq 4$, then $\kappa_4 < .3$ and $\Sigma < 1.95$. Therefore $d_4 = 3$, whence $d_3 = 3$, and $\kappa_i \leq \zeta_6(3) < .35$ for $i = 3$ or $4$. Once again, $\Sigma < S$. This shows that $d_3 = 2$.
We have $A(\ud) < 2.5$, so $S > 1.9748$. As before, set $v = v(x_1)$. From Lemma~\ref{LemmaX4}, $\kappa_4 \leq \zeta^{2v}(d_4)$ and $d_4 \geq 7/v$. If $v = 1$, then $\kappa_1 + \kappa_2 + \kappa_3 \leq \zeta(2) + 2\zeta_6(2) < 1.766$ and $\kappa_4 \leq \zeta^2(d_4) < .144$ because $d_4 \geq 7$. If $v = 2$, then $\kappa_1 + \kappa_2 + \kappa_3 \leq \zeta_2(2) + 2\zeta_5(2) < 1.657$ and $\kappa_4 \leq \zeta^4(d_4) < .255$ because $d_4 \geq 4$. If $v = 3$, then $\kappa_1 + \kappa_2 + \kappa_3 \leq \zeta_3(2) + 2\zeta_4(2) = 1.625$ and $\kappa_4 \leq \zeta^6(d_4) < .336$ because $d_4 \geq 3$. This shows that $\Sigma < S$ when $v \leq 3$. Therefore $v \geq 4$ and $\kappa_1 + \kappa_2 + \kappa_3 \leq 3 \zeta_4(2) < 1.594$. From Lemma~\ref{translation.lemma} with $\ue = (2,2,2,1)$ we have $v(x_4) \geq 7$, so $\kappa_4 \leq \zeta_7(d_4) < .379$ because $d_4 > 2$. In this case as well, $\sum \kappa_i < S$. This shows that $r < 4$.
\begin{step} \label{leq4} $v(x_i) \geq 4$ for all $i$. \end{step}
Since $A(\ud) < r = 3$, $S > .9698$. Set $v = v(x_1)$. We apply Lemma~\ref{LemmaX} once again to bound $v$ from below. If $v = 1, 2$, or $3$, then $\kappa_1 \leq \zeta_v(2) \leq .75, \ .625, \ .563$, respectively. For $i > 1$, $\kappa_i \leq \zeta^v(d_i)$ where $d_i \geq 14, 7, 5$ in the respective cases. Using Step~\ref{step1} and inspection, we have $\kappa_i < .08, .15, .201$ in the respective cases. It follows that $\sum \kappa_i < S$ whenever $v(x_1) < 4$. Therefore $v(x_1) \geq 4$. More generally, since the argument that established this does not use the ordering assumption on $x_i$, it follows that $v(x_i) \geq 4$ for all $i$.
\begin{step} \label{step5} $d_1 = 2$. \end{step}
Assume that $d_1 > 2$. It follows from Step~\ref{step1} that $d_1 \leq 6$, so $A(\ud) < 2.84$ and $S > .9714$. Since $v(x_1) \geq 4$, we have $\kappa_1 \leq \zetao_4(d_1)$. It follows from Step~\ref{step1} and inspection that $\kappa_i < .41$ for all $i$.
If $v(x_1) = 4$, then $d_i \geq 4$, $i = 2, 3$ and $\kappa_i \leq \zeta^4(d_i) < .255$, which implies that $\sum \kappa_i < S$. Therefore $v(x_1) \geq 5$, and, similarly, $v(x_i) \geq 5$ for all $i$. Thus $\kappa_i \leq \zetao_5(d_i)$ for all $i$. In particular, $\kappa_i < .3907$ for all $i$.
Suppose $d_1 > 4$. Since $\zetao_5(d) < .27$ when $d> 4$, $d \neq 6$ and $\zetao_5(6) < .35$, it follows that $d_i = 6$ for all $i$. Lemma~\ref{zeta5*} implies that $v(x_i^2) \geq 3$ for at least two choices of $i$, so $\sum \kappa_i \leq \zetao_5(6) + 2 \zetao_{5,3}(6)< .95$. Therefore $d_1 \leq 4$.
Suppose $d_1 = 4$. Then $d_2 \leq 6$ since otherwise $\kappa_i \leq \zetao_5(d_i) < .27$ for $i = 2, 3$ and $\Sigma < \zetao_5(4) + 2 \cdot .27 < S$. It follows from Step~\ref{step1} that $d_3 \leq 12$ as otherwise $\Sigma < S$. This implies that $A(\ud) \leq A(4,6,12) = 2.5$, so $S \geq .9748$. Also, Lemma~\ref{zeta5*} implies that $v(x_1^2) + v(x_2^2) \geq 3$. We claim that $d_3 \leq 8$. If $d_2 = 5$ or $6$, then $\kappa_1 + \kappa_2 < \zetao_5(4) + \zetao_5(6) < .7345$, so $\zetao_5(d_3) \geq \kappa_3 > .24$. It follows from inspection that $d_3 \leq 8$ in this case. If $d_2 = 4$, then $\kappa_1 + \kappa_2 \leq \zetao_5(4) + \zetao_{5,2}(4) < .7188$, so $\zetao_5(d_3) > .25$ and $d_3 \leq 8$ in this case as well.
From Lemma~\ref{zeta5*}.\ref{part1} we have $v(x_1^2) + v(x_2^2) \geq 4$ and $v(x_1^2) + v(x_3^2) \geq 5$. If $v(x_1^2) = 1$, then $v(x_2^2) \geq 3$ and $v(x_3^2) \geq 4$. so $\kappa_2 \leq \zetao_{5,3}(d_2) < .3021$, $\kappa_3 \leq \zetao_{5,4}(d_3) < .2813$, and $\sum \kappa_i < .974 < S$. If $v(x_1^2) = 2$, then $\kappa_1 \leq \zetao_{5,2}(4) < .3282$, $\kappa_2 \leq \zetao_{5,2}(d_2) < .3438$, and $\kappa_3 \leq \zetao_{5,3}(d_3) < .3021$, whence $\sum \kappa_1 < .9741 < S$. We conclude that $v(x_1^2) \geq 3$, so that $\kappa_1 \leq \zeta_{5,3}(4) < .3$. Without loss, if $d_i = 4$, $i = 2, 3$, then $\kappa_i \leq \zetao_{5,3}(4) < .3$. If $d_i > 4$ for some $i$, then $\kappa_i \leq \zetao_5(d_i) < .35$. Since $v(x_2^2) + v(x_3^2) \geq 7$ by Lemma~\ref{zeta5*}, we have either $v(x_2^2) \geq 4$ or $v(x_3^2) \geq 4$, whence $\kappa_i \leq \zetao_{5,4}(d_i) < .3$ for some $i > 1$. It follows that $\Sigma < .95 < S$, so we conclude that $d_1 \neq 4$.
We may therefore suppose $d_1 = 3$, so that $\kappa_1 \leq \zetao_5(3) \leq .3542$. By Lemma~\ref{Scott3}, $\kappa_i \leq \zetao_5(d_i)$ for $i = 2,3$. As in the argument when $d_1 = 4$, it follows that $d_2 \leq 6$. By Lemma~\ref{translation.lemma} with $\ue = (1,1,1)$, we have $v(x_i) \geq 7$ for two choices of $i$. If $d_2 = 6$, then $\kappa_1 + \kappa_2 \leq \max(\zetao_5(3)+\zetao_7(6), \zetao_7(3)+\zetao_5(6)) < .6902$. It follows from inspection of $\zetao_5$ values that $d_3 = 6$. Since $v(x_2^2) + v(x_3^2) \geq 10$ by Lemma~\ref{zeta5*}.\ref{part1} we have $\sum \kappa_i \leq \zetao_5(3) + \zetao_5(6) + \zetao_{5,5}(6) < .3542 + .3438 + .2709 < .97 < S$.
If $d_2 = 5$, then $\kappa_2 \leq .225$ and $\kappa_3 \leq \zetao_5(d_3) < .344$, so $\Sigma < S$.
If $d_2 = 4$, then $\kappa_1 + \kappa_2 \leq \max(\zetao_5(3) + \zetao_7(4), \zeta_7(3) + \zeta_5(4) ) < .7332$. By Lemma~\ref{zeta5*}.\ref{part3}, $\kappa_3 \leq \zetao_{5,3}(d)$. It follows that $\zetao_{5,3}(d) > .24$, so $d_3 = 4$ or $6$ by inspection. If $d_3 = 4$, then $\kappa_2 \leq \zetao_{5,3}(4) < .3$ by the same result, and $\sum \kappa_i < \zetao_5(3) + 2 \zetao_{5,3}(4) < S$. Therefore $d_3 = 6$. If $v(x_2^2) \geq 3$, then $\sum \kappa_i \leq \zetao_5(3) + \zetao_{5,3}(4) + \zetao_{5,3}(6) < S$. If $v(x_2^2) = 2$, then $v(x_3^2) \geq 8$ by Lemma~\ref{zeta5*} and $\sum \kappa_i < \zetao_5(3) + \zetao_{5,2}(4) + \zetao_{5,5}(6) < S$, so we may assume that $v(x_2^2) = 1$. From Lemma~\ref{translation.lemma} with $\ue = (3,2,3)$ we have $4v(x_3^3) + 6 \geq 28$ whence $v(x_3^3) \geq 6$ and $\kappa_3 < \zetao_{5,5,6}(6) < .2 < S - \kappa_1 - \kappa_2$. This shows that $d_2 \neq 4$.
If $d_2 = 3$ then $\kappa_2 \leq \zetao_7(3) \leq .3386$ and, by Lemma~\ref{zeta5*}.\ref{part2},
$\kappa_3 \leq \zetao_{5,5}(d_3) \leq .2735$, so $\Sigma \leq .97 < S$.
\begin{step} \label{step6} $d_2 \leq 4$ \end{step}
By Lemma~\ref{S.bounds} and the previous step, $S \geq .9748$. Assume that $d_2 > 4$. By Step~\ref{leq4}, $v(x_1) \geq 4$. If $v(x_1) = 4$, then $\kappa_1 \leq \zeta_4(2) < .532$, and $\kappa_i \leq \zetao_{10,6,2}(d_i)$ by Lemma~\ref{LemmaX}. By inspection, $\kappa_i < .22$ for $i \geq 2$. This implies that $\Sigma < S$. We conclude that $v(x_1) > 4$.
We have $\kappa_1 \leq \zeta_5(2) < .5157$. If $d_i > 8$ and $d_i \neq 12$, then $\kappa_i \leq \zetao(d_i) < .2$. If $d_i = 12$, then $\kappa_i \leq \zetao_7(12) < .232$. It follows that either $d_2 \leq 8$ or $d_2 = d_3 = 12$. In the latter case, Lemma~\ref{translation.lemma} with $\ue = (2,3,3)$ shows that $v(x_2^3) + v(x_3^3) \geq 7$, so $v(x_i^3) \geq 4$ for some $i > 1$, and $\kappa_i \leq \zetao_{7,1,4}(12) < .21$. This implies that $\Sigma < S$. We conclude that $d_2 \leq 8$. From Lemma~\ref{translation.lemma} with $\ue = (2,d_2,2)$ we have $v(x_3^2) \geq 2 \cdot 14/d_2 > 3$. Consequently, $\kappa_3 \leq \zetao_{7,4}(d_3)$. Suppose $d_2 = 8$. Then $\kappa_2 \leq \zetao_7(8) < .254$. If $d_2 > 12$, then $\kappa_3 < .19$ by Step~\ref{step1} and $\Sigma < S$, so $d_2 \leq 12$. By Lemma~\ref{translation.lemma} with $\ue = (2,2,12)$, $v(x_2^2) > 2$, so $\kappa_2 \leq \zetao_{7,3}(8) < .223$. Since $\zetao_{7,4}(12) < .222$, we conclude that $\kappa_2 + \kappa_3 < .446 < S - \kappa_1$, a contradiction. Therefore $d_2 < 8$. Since $\zetao_7(7) < .15$, it is evident that $d_2 \neq 7$.
Suppose $d_2 = 6$. Then $A(\ud) < 2.34$ and $S > .9764$, so $\kappa_2 + \kappa_3 \geq S- \kappa_1 > .4607$. Set $w =v(x_2^2)$. Then $w$ is necessarily even because $x_2^2$ has order $3$. If $w = 2$, then $\kappa_2 \leq \zetao_7(6) < .336$. By Lemma~\ref{LemmaY}, $d_{3} \geq 14$ and $\kappa_3 \leq \zeta^1(d_3)$. By Step~\ref{step1} and inspection of the values of $\zeta^1(d)$ for $14 \leq d \leq 30$ we have $\kappa_3 < .08$, so $\Sigma < S$ in this case. If $w= 4$, then $\kappa_2 \leq \zetao_{7,4}(6) < .2735$. By Lemma~\ref{LemmaY}, $d_3 \geq 7$ and $\kappa_3 \leq \zeta^2(d_3)$. Observing that $\zeta^2(d) < .1431$ for $7 \leq d \leq 28$, we conclude from Step~\ref{step1} that $\Sigma < S$ in this case as well. If $w = 6$, then $\kappa_2 \leq \zetao_{7,6}(6) < .2579$. We have $d_3 \geq d_2 = 6$, and, by Lemma~\ref{LemmaY}, $\kappa_3 \leq \zeta^3(d_3)$. Since $\zeta^3(d) < .18$ for $6 \leq d \leq 12$ we conclude from Step~\ref{step1} that $\kappa_3 < .18$, whence, once again, $\Sigma < S$. It follows that $w \geq 8$, so $\kappa_2 \leq \zetao_{8,8}(6) < .2527$. From Lemma~\ref{translation.lemma} with $\ud = (2,6,2)$ we have $v(x_3^2) \geq 5$, so $\kappa_3 \leq \zetao_{7,5}(d_3)$. If $d_3 > 6$ and $d_3 \neq 12$, then $\zetao_{7,5}(d_3) < .2$ and $\Sigma < S$. Therefore either $d_3 = 6$ or $d_3 = 12$. Recall that, by Lemma~\ref{translation.lemma} with $\ue = (2,3,3)$, $v(x_2^3) + v(x_3^3) \geq 7$. If $v(x_2^3) = 1$, then $\kappa_3 \leq \zetao_{7,5,6}(d_3) < .2$, and $\Sigma < S$. Therefore $v(x_2^3) \geq 2$, so $\kappa_2 < \zetao_{8,8,2}(6) < .211$. If $d_3 = 6 = d_2$, then we may assume that $\kappa_3 \leq \kappa_2$, whence $\kappa_2 + \kappa_3 < .43$. If $d_3 = 12$, then $\kappa_3 \leq \zetao_{7,5}(12) < .217$ and $\kappa_2 + \kappa_3 < .43$. In either case, $\Sigma < S$. Therefore $d_2 \neq 6$.
Suppose $d_2 = 5$. Then $\kappa_2 < .2063$, so $\kappa_3 \geq S - \kappa_1 - \kappa_2 > .25$. We have $\kappa_3 < \zetao_{7,6}(d_3)$ by Lemma~\ref{translation.lemma} with $\ue = (2,5,2)$. It follows from Step~\ref{step1} and inspection that $d_3 = 6$. From Lemma~\ref{translation.lemma} with $\ue = (2,5,3)$ we have $v(x_3^3) \geq 2$, so $\kappa_3 < \zetao_{7,6,2}(d_3) < .22$, a contradiction.
\begin{step} \label{step7} If $d_2 = 4$, then $n = 16$, $\ud = (2,4,5)$, $v(x_1) = 4$, $v(x_2) = 12$, and $v(x_3) = 16$. \end{step}
Suppose $d_2 = 4$. Then $A < 2.25$ and $S > .9773$. Also, $\kappa_2 \leq \zetao_7(4) < .379$.
Assume that $v(x_1) = 4$. then $\kappa_1 \leq \zeta_4(2) < .532$. By Lemma~\ref{LemmaX}, $d_3 > 3$ and $\kappa_2 \leq \zeta^4(d_3)$, so $\kappa_2 < .255$ by inspection and Step~\ref{step1}. From Lemma~\ref{translation.lemma} with $\ue = (1,4,4)$ we have $v(x_3^4) \geq 2n - 4 \cdot 4 \geq 12$, so $v(x_3) \geq 12$ and $v(x_3^2) \geq 12$ as well. By Lemma~\ref{24d} we have $v(x_3^3) \geq 4$. Therefore $\kappa_3 \leq \zetao_{12,12,4,12}(d_3)$. If $d_3 > 5$ then $\kappa_3 < .178$ by Step~\ref{step1} and inspection. Therefore $d_3 = 5$. We have $n \leq d_2 v(x_1) \leq 16$ and $v(x_i) \geq n - 4 \geq 10$, $i = 2,3$. Since $2$ has multiplicative order $4$ modulo $5$, we also have
$4 | v(x_3)$, so $v(x_3) = 12$ or $16$. If $v(x_3) = 12$, then $v(x_2) \geq 2n - v(x_1) - v(x_3) = 16$. However, $v(x_2) \leq 3n/4$ because $x_2$ is an element of order $4$ acting in characteristic $2$. These inequalities are not compatible with the condition $n \leq 16$. We conclude that $v(x_3) = 16$, $n= 16$, and $v(x_2) = 12$.
We may therefore assume that $v(x_1) > 4$. Then $\kappa_1 \leq \zeta_5(2) < .5157$. Set $w = v(x_2^2)$. Assume that $w \leq 2$. Then, by Lemma~\ref{LemmaY}, $d_3 \geq 14$, and $\kappa_3 \leq \zeta^1(d_3)$. So $\kappa_3 < .08 < S - \kappa_1 - \kappa_2$. Therefore $w > 2$. If $w = 3$ or $4$, then $\kappa_2 \leq \zetao_{7,3}(4) < .2852$, $d_3 \geq 7$, and $\kappa_3 \leq \zeta^2(d_3)$, so $\kappa_3 < .144$ by inspection and Step~\ref{step1}. Once again, $\Sigma < S$. If $w = 5$ or $6$, then $\kappa_2 \leq \zetao_{7,5}(4) < .2618$, and $\kappa_3 \leq \zeta^3(d_3)$. If $d_3 \geq 6$, then $\kappa_3 <.19$ and $\Sigma < S$, so $d_3 = 5$. By Lemma~\ref{LemmaY}, $w = 6$. Thus, $\kappa_2 \leq \zetao_{7,6}(4) < .2579$ and $\kappa_3 \leq \zeta^3(5) < .2004$, so $\Sigma < S$. We conclude that $w = v(x_2^2) \geq 7$, so $\kappa_2 \leq \zetao_{7,7}(4) < .2559$. By Lemma~\ref{24d}, $\kappa_3 \leq \zetao_{7,7,4}(d_3)$. If $d_3 > 5$, then $\kappa_3 < .2$ by Step~\ref{step1} and inspection, so $\Sigma < S$. If $d_3 = 5$, then $S = .9793$, and $\kappa_3 \leq .2063$, so once again $\Sigma < S$. This completes the argument that $d_3 \neq 4$.
\begin{step} \label{step8} If $d_2 = 3$ then $\ud = (2,3,7)$. \end{step}
It suffices to assume that $d_2 = 3$ and $d_3 > 7$. We have $A(\ud) < 2.17$ and $S > .9781$. Also, $v(x_2)$ is even because $x_2$ is an element of order $3$ acting over $\F_2$. In particular, $v(x_2) \geq 8$ and $\kappa_2 < .33595$. We have $\kappa_3 \leq \zetao_{10,10,7,5,3}(d_3)$ by Lemma~\ref{23d}. By inspection, $\kappa_3 \leq .132$. If $v(x_1) \geq 6$, then $\kappa_1 < .50782$ and $\Sigma < S$, so $v(x_1) = 5$ by Lemma~\ref{Scott3}. We have $\kappa_1 \leq \zeta_5(2) < .5157$.
It follows that $v(x_2) \geq n-5 \geq 9$, whence $v(x_2) \geq 10$, and $\kappa_2 \leq \zeta_{10}(3) < .334$. We have $\kappa_1 + \kappa_2 < .8497$.
By inspection, if $d > 7$ and $d \neq 8$ or $12$, then $\zetao_{10,10,7,5,3}(d) < .114$. It follows that $d_3 = 8$ or $12$. If $d_3 = 8$, then $A(\ud) < 2.05$, so $S > .9793$ and $\Sigma \leq \zeta_5(2) + \zeta_{10}(3) + \zetao_{10,10,7,5}(8) < .9793 < S$. We conclude that $d_3 = 12$, whence $A(\ud) < 2.09$ and $S > .9789$. Since $x_3^4$ has order $3$, $v(x_3^4)$ must be even, and $v(x_3^4) \geq 6$. If $v(x_2) = 10$, then $v(x_3) \geq 2n-v(x_1) - v(x_2) \geq 13$, so $\kappa_3 \leq \zetao_{13,10,7,6}(12)$, and $\sum \kappa_i \leq \zeta_5(2) + \zeta_{10}(3) + \zetao_{13,10,7,6}(12)$. If $v(x_2) > 10$, then $v(x_2) \geq 12$ and $\sum \kappa_i \leq \zeta_5(2) + \zeta_{10}(3) + \zetao_{10,10,7,6}(12)$. In either case, $\Sigma < S$, a contradiction.
\begin{step} Conclusion \end{step}
By Steps~\ref{step3}, \ref{step5}, \ref{step6}, \ref{step7}, and \ref{step8}, it suffices to show that if $\ud = (2,3,7)$ then $14 \leq n \leq 21$.
Assume that $\ud = (2,3,7)$. By Lemma~\ref{n.bounds}, $n \geq 14$. If $n > 21$, then $v(x_1) \geq 8$, $v(x_2) \geq 11$, and $v(x_3) \geq 11$, so $\sum \kappa_i \leq \zeta_8(2) + \zeta_{11}(3) + \zeta_{11}(7) < .9795 < S$. \eop
Theorem~\ref{basic.result} now follows from Propositions~\ref{large.p}--\ref{2prop}.
Note that for $p = 2,3$, or $5$, further information about values of $v(y)$ for certain elements $y$ is recorded in Propositions~\ref{p=5.prop}, \ref{p=3.prop}, and~\ref{2prop}.
\section{Proof of Theorem~\ref{grassmann.exceptions}}
Retaining the notation of \ref{notation.section}, assume that $\Omega$ is a primitive point action for $G$ with $\order{\Omega} \geq 10^4$ and that $x \in G$.
\subsection{Linear and Symplectic Groups} \begin{prop} \label{linear} If $\Omega$ consists of all points in the $L$ action or $Sp$ action, then $f(x) - q^{-v(x)} < 1/100$. \end{prop}
\pf We have $N = (q^n-1)/(q-1)$, so $q^{n-1} < N < 2q^{n-1} \leq q^n$.
Suppose $x$ is a linear transformation. Then the fixed points of $x$ are contained in the union of its eigenspaces, the largest of which has dimension $n-v$. We claim $f(x) - q^{-v(x)} < q^{-n/2} < 1/100$. It suffices to establish the first inequality.
If $v \leq n/2$, then the fixed points of $x$ lying outside the largest eigenspace are contained in a space of dimension $v$. This implies that $f(x) \leq \displaystyle \frac{q^{n-v} -1}{q-1} + \frac{q^v-1}{q-1}$, so $$ \begin{array}{rcl} \displaystyle \frac{F(x)}{N} - q^{-v} & \leq & \displaystyle \frac{q^{n-v} - 1}{q^n - 1} + \frac{q^v-1}{q^n-1} - q^{-v} \\ & < & q^{-(n-v)} \leq q^{-n/2}. \end{array}$$
If $v = \frac{n+1}{2}$, then the fixed points of $x$ lying outside the largest eigenspace are contained in the union of two nontrivial spaces having total dimension $n-v=(n+1)/2$. For fixed $m$, the largest value of $q^{a} + q^{m-a}$ for $a$ in $\{1,2,\ldots,m-1\}$ is $q^{m-1} + q$. Therefore $\displaystyle F(x) \leq \frac{q^{n-v}-1}{q^n-1} + \frac{q^{(n-1)/2}-1}{q-1} + 1$, so $$\frac{F(x)}N - q^{-v} < \frac{q^{(n-1)/2} - 1}{q^n -1} + \frac{q-1}{q^n-1} < q^{-n/2}.$$
If $v \geq n/2 + 1$, then $x$ has at most $q-1$ eigenspaces, each of which has dimension at most $n/2 - 1$, so $\displaystyle F(x) \leq (q-1) \frac{q^{n/2-1}-1}{q-1}$ and $$\frac{F(x)}{N} \leq (q-1)\left(\frac{q^{n/2-1} - 1}{q^n-1}\right) < q^{-n/2}.$$ This completes the analysis for $x$ a linear transformation.
Now suppose $x$ is not a linear transformation. Then $x$ induces a field automorphism because graph automorphisms do not act on $\Omega$. Let $d$ be the order of $x$ modulo InnDiag. Then $\displaystyle F(x) \leq \frac{q^{n/d}-1}{q^{1/d} -1}$, so $f(x) > .01$ implies that $$q^{n(d-1)/d} < \frac{q^n-1}{q^{n/d} -1} < 100 \frac{q-1}{q^{1/d} - 1} = 100 q^{(d-1)/d}\left( \frac{1- q^{-1}}{1 - q^{-1/d}}\right).$$ Since $q^{-1/d} \leq 1/2$, we have $q^{n(d-1)/d} < 200 q^{(d-1)/d}(1-q^{-1}) < 200q^{(d-1)/d}$. It follows that $q^{(n-1)(d-1)/d} < 200$, so $200^{d/(d-1)} > q^{n-1}$.
By the first line of this argument, $2q^{n-1} > N > 10000$. Therefore $q^{n-1} > 5000 > 200^{3/2}$, whence $\frac{d}{d-1} > \frac32$, and $d = 2$.
If $x$ is not a standard field automorphism, then $F(x) \leq \displaystyle \frac{q^{n/2-1}-1}{q^{1/2}-1} + 1$, so $$ \begin{array}{rcl} .01 < f(x) & \leq & (q^{1/2}+1)\left(\frac{q^{n/2-1}-1}{q^n-1}\right) + \frac1N \\ & < & \frac32q^{1/2}\cdot q^{-(n/2 + 1)} + .0001 \end{array}$$ This implies that $q^{n+1} < \displaystyle \left(\frac1{.0066}\right)^2 < 160^2$.
On the other hand, we have $F(x) > .01 N > 100$, so $\displaystyle \frac{q^{n/2-1} - 1}{q^{1/2} - 1} + 1 > 100$. It follows from this that $q^{n-2} > 99^2$, whence $q^3 < (160/99)^2$, which is impossible. Therefore $x$ must be a standard field automorphism.
We have $f(x) = \frac{q^{1/2} + 1}{q^{n/2} + 1}$ and $v_q(x) = n/2$. If $f(x) - q^{-v_q(x)} > .01$, then $q^{-(n-1)/2} > .01$, whence $q^{n-1} < 10000$. On the other hand, $q^{n-1} \cdot \displaystyle \frac{q}{q-1} > \frac{q^n-1}{q-1} = N > 10000$. That is, $$q^{n-1} < 10000 < \frac{q^n}{q-1}.$$ Since $n > 2$, the first inequality implies that $q < 100$. Since $q$ is both a perfect square and a prime power, it follow easily by inspection that these two inequalities cannot both hold. \eop
\begin{prop} \label{Sphypprop} If $\Omega$ consists of hyperplanes of type $\delta$ in the $Sp$ action, then $f(x) < q^{-v(x)} + 1/100$. \end{prop}
\pf We have $N = \frac12(q^n + \delta q^{n/2})$. Since $q^n$ is an even power of $2$ and $2^{14}+2^7 < 20000$, we have $q^n \geq 2^{16}$.
If $x$ is a field automorphism, then $F(x) \leq q^{n/2}$ in either action, so $f(x) \leq 2(q^{n/2}-1)^{-1} < .01$.
If $x$ is in InnDiag, then $F(x) \leq \frac12(q^{n-v}+ q^{n/2})$, so $F(x) - q^{-v(x)}N < q^{n/2}$, and $f(x) - q^{-v} < .01$, as before. \eop
\subsection{Actions of Unitary and Orthogonal groups} We record here properties of orthogonal and unitary actions that will be used in the analysis.
\newcommand\AAA{P} \newcommand\BBB{S} \newcommand\cB{{\cal B}} \begin{fact} \label{point.count.lemma} Let $W$ be an orthogonal or hermitian space of dimension $m$ over $\F_q$, and let $\pi(W)$ be the number of points of a given type in $W$. If $\rad W$, the totally singular radical of $W$, has dimension $r$, then $$\AAA(m) - \BBB(m+r) \leq \pi(W) \leq \AAA(m) + \BBB(m+r)$$ where $\AAA(m)$ and $\BBB(m)$ are as given below. $$ \begin{array}{ccc} {\rm{type}} & \AAA(m) & \BBB(m) \\ \\ U,\bs & \displaystyle \frac{q^{m-1/2} - 1}{q-1} &
\displaystyle \frac{q^{m/2 - 1/2}}{q^{1/2} + 1} \\ \\ U,\bn & \displaystyle \frac{q^{m-1/2}}{q^{1/2} + 1} &
\displaystyle \frac{q^{m/2 - 1/2}}{q^{1/2} + 1} \\ \\ O,\bs & \displaystyle \frac{q^{m-1} - 1}{q-1} &
\displaystyle q^{m/2 - 1} \\ \\ O,\bn\ (\ q \mbox{\rm{ even }}) & \displaystyle q^{m-1} &
\displaystyle q^{m/2 - 1} \\ \\ O,\bn\ (\ q \mbox{\rm{ odd }}) & \displaystyle \frac{1}{2} q^{m-1} &
\displaystyle \frac{1}{2} q^{m/2 - 1/2} \\ \end{array} $$ In particular, $N > q^{n-2}$. \end{fact}
\pf When $r = 0$, this follows immediately from Table~\ref{point.table} for all cases except odd-dimensional orthogonal spaces in even characteristic, in which case $\pi(W) = \AAA(m)$. The general case follows since $\pi(W) = \displaystyle \frac{q^r - 1}{q-1} + q^r \pi (W/R)$ for singular points and $\pi(W) = q^r \pi (W/R)$ for nonsingular points.
\eop
\begin{fact} Assume that $V$ is an even-dimensional unitary space or orthogonal space of type $+$. For $k \leq n/2 -1$, set $F_k = \AAA(n-k) + \BBB(n)$. Set $F_{n/2} = 2\left( \frac{q^{n/2} - 1}{ q - 1 }\right)$. If $q = q_0^2$, set $F^\ast = \frac{q_0^{n} - 1}{q_0 - 1}$. \begin{enumerate} \item If $x$ is linear and $v(x) = k$, $k \leq n/2$, then $F(x) \leq F_k$. \item If $x$ is linear and $v(x) \geq n/2$, then $F(x) \leq F_{n/2}$. \item If $x$ is semilinear then $F(x) \leq F^\ast$. \end{enumerate} \end{fact}
\pf This is a straightforward consequence of the previous statement. \eop
\begin{fact} \label{geneigenspacefact} Suppose $x$ preserves a non-degenerate sesquilinear or bilinear form on $V$, $X_\lambda = \ker (X-\lambda I)^n$, and $X_\mu = \ker (X-\mu I)^n$. If $\lambda \bar{\mu} \neq 1$ then $X_\mu \subseteq X_\lambda^\perp$. \end{fact}
\pf Argue by induction on $k + l$ that if $k$ and $l$ are positive integers, $v \in \ker (X- \lambda I )^k$, and $w \in \ker (X- \mu I )^l$ then $\langle v , w \rangle = 0$. \eop
\subsection{Unitary and Orthogonal Groups}
To complete the proof of Theorem~\ref{grassmann.exceptions} we assume that $V$ admits a nondegenerate orthogonal or unitary form, and that the action of $G$ is on the points of type $t$ in $V$.
To estimate $f(x) - q^{-v(x)}$ we bound $F(x)$ from above and $N$ from below. For a subspace $\U$ of $V$, let $\zzeta(\U) = \zzeta_t(\U)$ be the number of points of type $t$ in $\U$. It is apparent that $F(x) = \sum \pi(E_\lambda)$ where $\{E_\lambda\}$ is the collection of eigenspaces for $x$.
\begin{lemma} \label{Lemma2} If $x$ acts linearly on $V$, then either \begin{enumerate} \item $f(x) - q^{-v(x)} < 1/100$ or \item \label{halfdimension.exception} $V$ has even dimension, the action is on singular points, and some eigenspace for the action of $x$ on $V$ is a totally singular subspace of dimension $\dim V/2$. \end{enumerate} \end{lemma}
\pf We have $F(x) = \sum \pi(E_\mu)$ where the sum is over the eigenspaces $E_\mu$ for the action of (some pull-back of) $x$ in the group of linear transformations of $V$.
Suppose $v = v(x) \leq n/2$, and let $\lambda$ be the principal eigenvalue. Then $\dim E_\lambda = n - v$. Let $X_\lambda$ be the corresponding generalized eigenspace, that is $X_\lambda = \ker (x-\lambda I)^n$, and set $w = \codim_V X_\lambda$. Then $w \leq v$.
If $X_\lambda$ is totally singular, then $\dim X_\lambda \leq n/2$, and it follows that $X_\lambda = E_\lambda$, so the second alternative holds. We may therefore suppose that $X_\lambda$ is not totally singular.
It follows from {\bf \ref{geneigenspacefact}\/} that $\lambda \bar\lambda = 1$ and that $E_\mu \subseteq X_\lambda^\perp$ whenever $\mu \neq \lambda$.
This implies that $$F(x) \leq \pi(E_\lambda) + \pi( X_\lambda^\perp).$$
Setting $r = \dim \rad(E_\lambda)$, we have $r \leq \codim_{X_\lambda}(E_\lambda) = v - w$. So $\dim X_\lambda^\perp = w \leq v-r$. By {\bf \ref{point.count.lemma}\/}, $F(x) \leq \AAA(n-v) + \BBB(n-v+r) + \AAA(v-r) + \BBB(v-r)$ because $X_\lambda^\perp$ is non-degenerate.
Since $N \geq \AAA(n) - \BBB(n)$ and $\AAA(n-v) \leq q^{-v}\AAA(n)$, it follows that
$$ \begin{array}{ccl} F(x) - q^{-v}N & \leq &
\BBB(n-v+r) + \AAA(v-r) + \BBB(v-r) + q^{-v} \BBB(n) \\
& = & \BBB(n-v+r) + \AAA(v-r) + \BBB(v-r) + \BBB(n-2v). \end{array} $$
because $\BBB(n) = K q^{n/2}$ where $K$ is independent of $n$.
Set $D(x) = F(x) - q^{-v} N$. We claim that $D(x) < (q+1) \BBB(n-2) + 2$.
We have shown that $D(x) \leq \phi(v,r)$ where $\phi(v,r) = \BBB(n-v+r) + \AAA(v-r) + \BBB(v-r) + \BBB(n-2v)$. By elementary calculus, $\phi$ attains its maximum on the region $\{ (v,r) \ : \ 1 \leq v \leq n/2, 0 \leq r \leq v\}$ at $(1,1)$. Therefore $D(x) \leq \phi(1,1) = \BBB(n) + \BBB(n-2) + \AAA(0) + \BBB(0) < (q+1) \BBB(n-2) + 2$ since $\AAA(0) < 1$ and $\BBB(0) < 1$.
\newcommand\DDD{D} Set $\DDD = (q+1) \BBB(n-2) + 2$. It suffices to show that if $N \geq 10000$ then $\DDD/N < 1/100$.
In all cases, $N > q^{n-2}$ by {\bf \ref{point.count.lemma}\/}. When $V$ is unitary, $\BBB(n-2) = q^{(n-3)/2}/(q^{1/2}+1)$, and
it is easy to see that $\DDD < q^{(n-2)/2}$. So $\DDD^2 < N$, which implies that $\DDD/N < 1/100$.
We may therefore assume that $V$ is orthogonal. If either the action is on singular points or $q$ is even, then $\BBB(n-2) = q^{n/2 -2}$,
and $\DDD < a^{n/2-1} (1 + q^{-1} + 2/(q^{n/2-1}) ) < 8q^{(n-2)/2}/5$. Therefore, $\DDD/N < \frac85 q^{- (n-2)/2}$, and $q^{(n-2)/2} < 160$. This implies that $n \leq 16$ and $q \leq 11$ since $n \geq 6$. Among the pairs $(n,q)$ of such values, the only ones for which both $q^{(n-2)/2} < 160$ and $N > 10000$ are $(6,11), (7, 7) , (8,5) , ( 11, 3)$, and $(16,2)$.
In the nonsingular case when $q$ is odd, $\BBB(n-2) = \frac12 q^{(n-3)/2}$, so $\DDD \leq q^{(n-1)/2}(q+1)/2q + 2 < q^{(n-1)/2}$. In this case, $N \geq \frac12 q^{n-1}(1 - q^{-(n-1)/2}) > \frac49 q^{(n-1)}$. Therefore, $\DDD/N < \frac94 q^{- (n-1)/2}$, and $q^{(n-1)/2} < 225$. This implies that $n \leq 10$ and $q \leq 7$. By inspection, the only pairs $( n , q )$ for which both $q^{(n-2)/2} < 160$ and $N > 10000$ are $(6,8), (8,4) , ( 16, 2 )$. A straightforward calculation shows that $f(x) - q^{-v} < 1/100$ in these cases.
A straightforward calculation shows that $f(x) - q^{-v} < 1/100$ in these cases. This shows that the result holds when
$v \leq n/2$.
If $v \geq n/2 + 1$, then every eigenspace has dimension at most $n/2 -1$, and there are at most $q-1$ eigenspaces. Therefore $F(x) \leq (q-1) \left( \displaystyle \frac{q^{n/2-1} -1}{q-1}\right) < q^{n/2 -1} < \sqrt{N}$. Since $N \geq 10000$ this implies that $F(x)/N < 1/100$.
This leaves the case $v = (n+1)/2$, where $n$ is necessarily odd.
Every eigenspace has dimension at most $(n-1)/2$,
so $F(x) \leq 2(q^{(n-1)/2} - 1)/(q-1) + 1 \leq F$ where $F = q^{(n-1)/2}$. A short calculation, described below, shows that the conclusion holds in this case.
Suppose $V$ is unitary and set $q_0 = q^{1/2}$. Then $N =
\displaystyle \frac{q^{n-1/2} - 1}{q-1} - \frac{q^{(n-1)/2}}{q_0 + 1}$ in the singular case, and $N = \displaystyle \frac{q^{n-1/2} + q^{(n-1)/2}}{q_0 + 1}$ in the nonsingular case. By computation, $F/N < 1/100$ when $(n,q_0) = (3,5), (5,3)$, or $(9,2)$. Since $F/N$ is a decreasing function of both $q$ and $n$, it follows that $n = 3, 5$, or $7$. Furthermore, $q_0 \leq 4$ when $n = 3$ and $q_0 = 2$ when $n = 5$ or $7$. By inspection, $N < 10000$ in these cases.
In the orthogonal case, $q$ is necessarily odd because $n$ is odd. We have $N = \displaystyle \frac{q^{n-1} - 1}2$ in the singular case, and $N = \displaystyle \frac{q^{n-1} \pm q^{(n-1)/2}}{2}$ in the nonsingular case. By computation, $F/N < 1/100$ when $(n,q) = (7,7)$ or $(11,3)$. Since $F/N$ is a decreasing function of both $q$ and $n$, it follows that $n = 7$, or $9$. Furthermore, $q \leq 5$ when $n = 7$ and $q = 3$ when $n = 9$. By inspection, $N < 10000$ in these cases.
This completes the proof of Lemma~\ref{Lemma2}. \eop
\begin{lemma} \label{Lemma3} If $x$ acts semilinearly on $V$ then either \begin{enumerate} \item $f(x) - q^{-v(x)} < 1/100$ or \item \label{field.exception} The dimension $n$ of $V$ is even, and $x$ has a totally singular eigenspace of dimension $n$ over $F_{q^{1/2}}$. \end{enumerate} \end{lemma}
\pf Assume that $x$ acts semilinearly on $V$ with $f(x) \geq 1/100$. Let $d$ be the order of $x$ mod $PGL(V)$. We claim that $d = 2$.
Suppose $d > 2$ and set $q_1 = q^{1/d}$. Then the points fixed by $x$ must lie in an $n$-dimensional space over $GF(q_1)$, so $F(x) \leq \displaystyle \psi(d) = \frac{q^{n/d} - 1}{q^{1/d} - 1}$.
If $V$ is orthogonal, and the action is on singular points, then $\psi(d) < 1/100$ when $(d,q_1,n) = (3,2,8), (4,2,6), (3,3,6)$, or $(3,3,7)$. If the action is on nonsingular points, then Then $\psi(d) < 1/100$ when $(d,q_1,n) = (3,2,6), (3,3,6)$, or $(3,3,7)$. For a given parity of $n$ and a given parity of $q$ the ratio $\psi(d)$ is a decreasing function of $d,n$, and $q$. We have $N < 10000$ when $V$ is an orthogonal space of dimension $6$ over $F_8$, so $d=2$ in the orthogonal case.
In the unitary case, $q$ is necessarily a square. We have $F(x)/N < 1/100$ when $(d,q_1,n) = (3,2^2,3), (3,2^2,4), (4,2,3)$, or $(4,2,4)$, and the claim holds for the unitary case as well.
We have $d = 2$. Let $E$ be the primary eigenspace for $x$. Then $E$ is an $\F_{q_0}$ space of dimension at most $n$ where $q_0 = q^{1/2}$. If $\dim E \leq n-1$, then $F(x) < (q_0^{n-1}-1)/(q_0-1) + 1$, and a short computation shows that $F(x)/N < 1/100$ whenever $N > 10000$. Therefore $E$ has dimension $n$, and $F(x) \leq \psi(2)$.
We may assume that $\psi(2) - q_0^{-n}N \geq N/100$, and in particular, that $\psi(2) > N/100$.
Then $F(x) \leq \displaystyle \frac{q_0^n-1}{q_0 - 1}$, and the conditions $F(x)/N \geq 1/100$, $N \geq 10000$ imply that
$(n,q_0)$ is on one of the following lists.
Unitary groups, nonsingular action:
$(8,2), (9,2), (4,5)$; \\
singular action: $(8,2), (9,2), (6,3), (5,4), (4,7), (4,8), (4,9)$. \\
Orthogonal groups,
nonsingular action: $(8,2), (6,3)$; \\
singular action: $(10,2), (7,3), (6,4)$.
For $U_9(2^2)$, we have $\psi(2) \leq N/100 + q^{-n/2} N $.
For all other cases, the upper bounds in {\bf \ref{point.count.lemma}\/} imply that $F(x) < N/100 - q^{-n/2} N$ whenever $E$ is not totally singular. \eop
\begin{lemma} \label{Lemma4} If $f(x) - q^{-v} \geq 1/100$ for some $x \in G^\sharp$ then the action is on singular points and one of the following is true. \begin{enumerate} \item \label{u.alternative} $V$ is unitary and $(n,q_0) \in \{ (4 , 7 ) , ( 4, 8 ) , ( 4, 9 ), ( 6 , 3 ) , ( 8 , 2 ) \}$. \item \label{o.alternative} $V$ is orthogonal of $+$ type and $(n,q) \in \{ (6,11) , ( 6, 13 ) , ( 6, 16 ), ( 8 , 5 ) , ( 10 , 4 ) \}$. \end{enumerate} \end{lemma}
\pf The two previous lemmas show that it suffices to assume
the action is on the singular points of an even-dimensional space $V$ and $V$ contains a totally singular subspace of dimension $\dim V/2$.
Suppose first that $V$ is a unitary space of dimension $2m$ over $\F_{q}$ where $m \geq 2$ and $q = q_0^2$. Then $N = \displaystyle \frac{ (q_0^{2m} - 1 ) (q_0 ^ {2m - 1} + 1)}{q_0^2 - 1}$.
If $x$ has a totally singular eigenspace of dimension $m$, then the fixed points of $x$ are contained in the union of two subspaces of $V$ each of dimension $m$, so $F(x) \leq \displaystyle 2 \left( \frac{q_0^{2m} - 1 }{ q_0^2 - 1} \right)$. Otherwise, $x$ has an eigenspace of dimension $2m$ over $\F_{q_0}$, and $F(x) \leq \displaystyle \frac{q_0^{2m} - 1}{q_0 - 1}$.
In either case, $f(x) \leq \displaystyle \frac{{q_0}+1}{q_0^{2m-1} + 1 }$. By assumption, $f(x) \geq 1/100$. Therefore $q_0^{2m-2} < 100\left(\displaystyle \frac{{q_0}+1}{{q_0}} \right)\leq 150$. Since $2^8 > 150$, it follows that $2m -2 < 8$. Therefore $m \leq 4$. By inspection, one of the following holds:
$m = 2$, ${q_0} \leq 9$;
$m = 3$, ${q_0} \leq 3$; or
$m = 4$, ${q_0} = 2$.
By further inspection, $N < 10^4$ when $m = 2$, ${q_0}\leq 5$ and when $m = 3$, ${q_0} = 2$. One of the conditions in \ref{u.alternative} must therefore hold.
Now suppose $V$ is an orthogonal $+$ space of dimension $2m$ over $\F_q$, $m \geq 3$. Then $N = \displaystyle \frac{ ( q^m - 1) ( q^{m-1} + 1 )}{ q - 1 }$.
Suppose $q = 2$. Then $x$ has a single eigenspace and $F(x) \leq q^m - 1$, so $f(x) \leq \displaystyle \frac{1}{q^{m-1} + 1}$. Since $N > 10000$, we have $m \geq 8$. Therefore $f(x) < 1/100$. We may therefore assume that $q \geq 3$.
Suppose $x$ fixes a totally singular subspace of dimension $m$. Then the fixed points of $x$ are contained in the union of two subspaces of $V$ each of dimension $m$, so $F(x) \leq \displaystyle 2 \left( \frac{q^{m} - 1 }{ q - 1} \right)$.
We have $f(x) \leq \displaystyle \frac{2}{q^{m-1} + 1}$. The assumption that $f(x) > 1/100$ implies that $q^{m-1} < 200$. Since $q \geq 3$ and $3^5 > 200$, it follows that $m \leq 5$ and that one of the following holds:
$m = 3$, $q \leq 13$;
$m = 4$, $q \leq 5$; or
$m = 5$, $q = 3$.
By inspection, $N < 10^4$ when $m = 3$, $q \leq 9$, when $m = 4$, $q \leq 4$, and when $m = 5$, $q = 3$, so one of the following must hold: $2m = 6$ and $q = 11$ or $13$; $2m = 8$ and $q = 5$.
If $x$ fixes a subspace of dimension $2m$ over $\F_{q^{1/2}}$, then $F(x) \leq \displaystyle \frac{q^{m} - 1}{q^{1/2} - 1}$. Therefore $f(x) \leq \displaystyle \frac{q^{1/2} + 1}{q^{m-1} + 1} $.
The condition $f(x) \geq 1/100$ implies that $q^{m-1} + 1 \leq 100 (q_0 + 1)$, so $q_0^{2m-2} \leq 100(q_0 + 1) \leq 150 q_0$, and $q_0^{2m-3} \leq 150$.
We have $m \leq 5$ because $2^8 > 150$, and one of the following holds:
$m = 3$, $q_0 \leq 5$;
$m = 4$, $q_0 = 2$; or
$m = 5$, $q_0 = 2$.
By inspection, $N < 10^4$ for $m = 3$, $q_0 \leq 3$ and for $m = 4$, $q_0 = 2$. Since $\frac{5+1}{25^2 + 1} < 1/100$, the case $m = 3$, $q_0 = 5$ does not satisfy the hypotheses. This leaves the cases $2m = 6$, $q= 4^2$ and $2m = 10$, $q = 2^2$. \eop
\begin{lemma} \label{newLemma5} If one of the conclusions of Lemma~\ref{Lemma4} holds then $g(\ux) > 2$ whenever $\ux$ is a normalized generating tuple for $G$. \end{lemma}
\pf We consider the cases in turn. We assume throughout that $N > 10^4$, that $\ux$ is a normalized generating tuple for $G$, with signature $\ud$, and that $g(\ux) \leq 2$.
By Theorem~\ref{basic.result}, it suffices to assume that there is an element $y$ involved in $\ux$ which violates Grassmann Condition $1/100$.
\begin{stp} \label{two.cases} For some $i$, $\langle x_i \rangle$ contains an element $y$ such that one of the following is true. \begin{enumerate} \item $y$ fixes two totally singular subspaces of dimension $n/2$. \item $y$ is a semilinear map on $V$, $y$ has order $2$, and $y$ fixes a subspace of dimension $n$ over $F_{q^{1/2}}$. \end{enumerate} \end{stp}
\begin{stp} \label{ruleout237} $\ux$ does not have signature $(2,3,7)$. \end{stp}
\pf If $\ux$ has signature $(2,3,7)$, then every element of $G$ must act linearly on $V$. By Step~\ref{two.cases}, $\ux$ must involve an element $y$ which has two totally singular eigenspaces of dimension $n/2$. No such element can violate Grassmann Condition $1/100$ when $G$ is of type $U_4(q_0^2)$, $U_6(3^2)$, $O_6(16)$, or $O_{10}(4)$. When $G$ is of type $U_8(2^2)$ no element can have two distinct totally singular eigenspaces. In all other cases, the element of order $7$ can have at most one eigenvalue. When $q = 11$ or $5$, the element of order $3$ can have at most one eigenvalue as well. A short computation using fixed point estimates shows that $g(\ux) > 2$ in all cases. \eop
\begin{stp} $G$ is not of type $U_4(q^2)$. \end{stp}
\pf $N = (q^2 + 1 ) ( q^3 + 1)$, and, for all $x \in G^\sharp$, we have $F(x) \leq F$ where $F = (q+1)(q^2 + 1)$. The Riemann-Hurwitz Formula implies that $A(\ud) \leq (2N + 2 )/(N - F)$. Since $\ud \neq (2,3,7)$, we must have $q=7$, $\ud = (2,3,8)$. In this case, $v(x_3^2) > 1$, and $x_3^2$ must act linearly on $V$, so $F(x_3) \leq F(x_3^2) \leq 2(q^2 + 1)$. By computation, $g(\ux) > 2$. \eop
\begin{stp} \label{O6step} $G$ is not of type $O_6^+(q)$. \end{stp}
\pf We have $N = (q^3-1)(q^2+1)/(q-1)$. Set $F_1 = (q^4-1)/(q-1) + q^2$, $F_2 = (q^3-1)/(q-1) + q^2$, and $F_3 = 2(q^3-1)/(q-1)$. When $q = 16$, set $q_0 = 4$ and $F^\ast = (q^3-1)/(q_0-1)$.
Then $F(x) \leq F_1$ for all $x \in G^\sharp$, and the Riemann-Hurwitz Formula implies that $A(\ud) \leq (2N + 2)/(N- F_1)$. It follows that $\ud$ is one of the following:
$(2,3,d)$;
$(2,4,d)$, $d \leq 29$;
$(2,5,d)$, $d \leq 11$;
$(2,6,d)$, $d \leq 8$;
$(2,7,7)$;
$(3,3,d)$, $d \leq 8$;
$(3,4,4)$; or
$(2,2,2,3)$.
By inspection, if $\cB$ is the set of all elements in $G^\sharp$ for which $F(x) > \max(F_2,F^\ast)$ then
$\sum \displaystyle \frac{|\{ \cB \cap \langle x_i \rangle \}|}{d_i} < 1$. Set $F' = \max(F_2, F_3, F^\ast)$.
The Riemann-Hurwitz Formula now implies that $A(\ud) \leq (2N + 2 + 1(F_1-F'))/(N- F')$, whence $\ud$ is one of the following:
$(2,3,d)$, $d \leq 19$;
$(2,4,d)$, $d \leq 7$;
$(2,5,5)$; or
$(3,4,4)$.
Inspecting this list it follows that
$\sum \displaystyle \frac{|\{ \cB \cap \langle x_i \rangle \}|}{d_i} \leq 1/2$, so $A(\ud) \leq (2N + 2 + \frac12(F_1-F'))/(N- F')$, which further reduces the possible signatures. Further iterations of this procedure show that $\ux$ must have signature $(2,3,7)$, which was already ruled out by Step~\ref{ruleout237}.
\eop
\begin{stp} \label{U6step} $G$ is not of type $U_6(3^2)$. \end{stp}
\pf In this case, using $N = (q_0^6-1)(q_0^5 + 1) / ( q - 1 )$, $F_1 = (q_0^9 -1)/ ( q - 1 ) + q_0^5/(q_0 + 1)$, $F_2 = (q_0^7 -1)/ ( q - 1 ) + q_0^5/(q_0 + 1)$, $F_3 = 2(q^3 - 1 )/( q - 1 )$, and $F^\ast = (q_0^6 - 1) / ( q_0 - 1)$, a short modification of the analysis in the previous step again reduces to the case $\ud = (2,3,7)$, which was treated earlier. \eop
\begin{stp} $G$ is not of type $O_8^+(5)$. \end{stp}
\pf The argument of the previous two steps shows that either $\ud = (2, 3, d)$ for some $d$ or $\ud \in \{ (2, 4, \leq 8), (2, 5, \leq 6), ( 3, 3, \leq 5) \}$.
In the former situation, the contribution of elements having $v(y) = 1$ is less than $2/3$, and it follows that $d < 200$, whence the contribution is less than $5/12$. Continuing in this way shows that no tuple $\ux$ can have $g(\ux) \leq 2$.
In the remaining cases, bounding the contributions from elements with $v(y) \leq 2$ leads to the same conclusion. \eop
\begin{stp} \label{U8step} $G$ is not of type $U_8(2^2)$. \end{stp}
\pf The argument of the previous steps shows that either $\ud = (2, 3, d)$ or $(2, 4, d)$ for some $d$ or $\ud \in \{ (2, 5, \leq 17), (2, 6, \leq 10), (2, 7, \leq 8), ( 3, 3, \leq 10) , ( 3, 4, \leq 5) , ( 2, 2, 2, 3 ) \}$.
If $\beta_1$ is the contribution to $A(\ud)$ from elements having $v(y) = 1$ and $\beta_2$ is the contribution from elements with $v(y) \leq 2$, then the Riemann-Hurwitz Formula implies that $A(\ud) \leq (2N + 2 - \beta_1(F - F') - \beta_2(F' - F'') ) / (N - F'')$, where $F(x) \leq F$ for all $x \in G^\sharp$, $F(x) \leq F'$ for all $x$ with $v(x) > 1$, and $F(x) \leq F''$ for all $x$ with $v(x) > 2$.
Using this criterion eliminates the individual cases other than $(2,3,d)$, $(2,4,d)$.
Using estimates for indexes, this reduces to $(2,3, \leq 19)$, or $(2,4, \leq 9)$.
In the $(2,3,d)$ case, we have $\Ind(x_2) \geq \frac23(N - 2 (q_0^4-1)(q_0^3+1)/(q-1) )$ because $v(x_2) \geq 4$ and the eigenspaces for $x_2$, an element of order $3$, must be nondegenerate. Bounding $v(x_3^k)$, and hence $F(x_3^k)$, for $k = 1,2,3,4$, shows that $g(\ux) > 2 $ for for all choices of $d$.
In the $(2,4,d)$ case, consideration of the subcases $v(x_2^2) = 1$, $v(x_2^2) > 1$ leads to the same conclusion. \eop
\begin{stp} $G$ is not of type $O_{10}^+(4)$. \end{stp}
\pf The argument in Step~\ref{O6step} shows that either $\ud = (2, 3, d)$ or $(2, 4, d)$ for some $d$ or $\ud \in \{ (2, 5, \leq 12), (2, 6, \leq 9), (2, 7, 7), ( 3, 3, \leq 9) , ( 3, 4, \leq 5) , ( 2, 2, 2, 3 ) \}$.
Its extension in Step~\ref{U8step} reduces to the earlier treated case $\ud = (2,3,7)$.
\eop
Combining Lemmas~\ref{Lemma4} and \ref{newLemma5} we have the following result.
\begin{prop} \label{OUprop} If $G$ is unitary or orthogonal and $G$ violates Grassmann Condition $1/100$ then $g(\ux) > 2$ for every normalized generating tuple for $G$. \end{prop}
Propositions~\ref{linear}, \ref{Sphypprop}, and \ref{OUprop} establish Theorem~\ref{grassmann.exceptions}.
\section{Proof of Theorem~\ref{touch.up}}
We assume here that $\ux$ and $V$ satisfy one of the conditions listed in Table~\ref{pnd.table}. Suppose $\Omega$ is a primitive $G$-set of [projective] points in $V$ with $\order{\Omega} \geq 10000$.
That is, one of the following is true where $n_p = \dim_{\F_p}(V)$.
\begin{enumerate}
\item $\ux$ has signature $(2,3,7)$ and one of the following holds.
\begin{enumerate}
\item $p = 11$ and $n_p = 5$ or $6$.
\item $p = 7$ and $n_p = 6$.
\item $p = 5$ and $n_p = 7,8$, or $9$.
\item $p = 3$ and $n_p = 12$.
\item $p = 2$ and $14 \leq n_p \leq 21$.
\end{enumerate}
\item $\ux$ has signature $(2,3,8)$, $p = 3$, and $n_p = 10$.
\item $\ux$ has signature $(2,4,5)$, $p = 2$, and $n_p = 16$
Furthermore $v_p(x_1) = 4$, $v_p(x_2) = 12$,
and $v_p(x_3) = 16$.
\end{enumerate}
Then $V$ is an $n_q$-dimensional $\F_q$-module where $q^{n_q} = p^n$. and $n_q$ and $q$ satisfy the conditions listed for point actions.
\begin{fact} The number $CP(X,n,q,t)$ of $t$-points in a classical $n$-space of type X over $GF(q)$ is given in Table~\ref{point.table}. \end{fact}
\pf See \cite{FM1}. \eop
We calculate a lower bound for $g(\ux)$ in each of the cases using the following lemma.
\begin{lemma} \begin{enumerate} \item If $\ud = (2,3,7)$, then $v(x_1) \geq n/3$, $v(x_2) \geq n/2$, and $v(x_3) \geq n/2$. \item If $\ud = (2,3,8)$ then $v(x_1) \geq n/3$, $v(x_3) \geq n/2$, $v(x_2) \geq n/2$, $v(x_2^2) \geq n/2$, and $v(x_2^4) \geq n/5$. \item If $\ud = (2,4,5)$, then $v(x_1) \geq n/4$, $v(x_2) \geq n/2$, $v(x_2^2) \geq n/4$, and $v(x_3) \geq n/2$. \item The number of $t$-points in an $n$-space with radical of dimension $r$ of type $X$ over $\F_q$ is $(q^r-1)/(q-1) + q^r CP(X,n-r,q,t)$ for singular points and $q^r CP(X,n-r,q,t)$ for non-singular points. \item \label{fifth.statement} Assume that $q$ is even. Let $G = O(2m+1,q) \cong Sp(2m,q)$ act on the $2m+1$-dimensional orthogonal space $V$, where $V$ has a $1$-dimensional radical $R$. If $x$ is a linear transformation in $G$ then $x$ fixes at most $q^m(q^{m-v(x)}+1)/2$ complements to $R$ of each type. \item If $W$ is a space of codimension $v$ in the non-degenerate space $V$ then $\dim \rad W \leq v$. \item \label{seventh.statement} Let $\Fix_2(x)$ be the number of fixed points of $x$ lying outside its principal eigenspace. Set $v = v(x)$. Then \begin{enumerate} \item \label{seventh.1} If $(o(x),q-1) = 1$ then $\Fix_2(x)=0$. \item \label{seventh.2} $\Fix_2(x) = 0$ in case of type $S$. \item \label{seventh.3} If $2v \leq n$ then $\Fix_2(x)$ is bounded by the number of type $t$ points in some $v$-dimensional space. \item \label{seventh.4} If $(o(x),q-1) = d_0$ and every $n-v$-dimensional space contains at most $M$ points then $\Fix_2(x) \leq (d_0-1)M$. \end{enumerate} \item \label{eighth} If $\Fix(x^j) \leq F_j$ for all positive powers of $x$, then
$$\Ind(x) \geq \frac{d-1}{d}N - \frac1d \left( \sum_{k|d,k<d} \phi(\frac{d}{k})F_k \right)$$ \item \label{ninth} If $\Ind x_i \geq H_i$ for all $i$ then $g(\ux) \geq \frac12\sum H_i -N + 1$. \end{enumerate} \end{lemma}
\pf The first three statements follow from Lemma~\ref{translation.lemma}.
The fourth statement is a straightforward count of points in $R \oplus W$ where $R$ is totally singular of dimension $r$ and $W$ is non-degenerate.
Statement \ref{fifth.statement} follows from a straightforward calculation, as in the proof of Proposition 8.1 of \cite{FM1}.
The next statement is clear because $ \rad W \subseteq W^\perp$.
To prove \ref{seventh.statement}, note that the principal eigenspace of $x$ has dimension $n-v$, and every fixed point of $x$ lying outside the principal eigenspace must lie in an eigenspace of dimension at most $n-v$.
All eigenvalues of $x$ must have order dividing both $o(x)$ and $q-1$, so there are at most $d_0 = (o(x),q-1)$ eigenvalues in toto. Statements \ref{seventh.1} and \ref{seventh.4} now follow immediately.
In type $S$ only the eigenvalue $\lambda = 1$ corresponds to fixed points, so statement \ref{seventh.2} holds.
The total dimension of all secondary eigenspaces is at most $v$, and all secondary fixed points of $x$ lie in the direct sum of such subspaces. Statement \ref{seventh.3} follows.
Statements \ref{eighth} and \ref{ninth} follow easily from the Cauchy-Frobenius and Riemann-Hurwitz Formulas, respectively. \eop
\begin{center} \begin{table} \caption{Number of $t$-points in classical $n$-space of type $X$ over $\F_q$.} \begin{tabular}{cccc} \label{point.table} $X$ & condition & $t$ & $CP(X,n,q,t)$ \\ \\ $L$ & & & $\displaystyle \frac{q^n-1}{q-1}$ \\ \\ $O^\eps$ & $n = 2m$ & singular & $\displaystyle\frac{(q^m-\eps1)(q^{m-1}+\eps1)}{q-1}$ \\ \\ $O^\eps$ & $n = 2m$ & $\delta$ & $\displaystyle\frac{(2,q)}2 {(q^m-\eps1)q^{m-1}} $ \\ \\ $O^\eps$ & $n = 2m+1$ & singular & $\displaystyle\frac{q^{2m}-1}{q-1}$ \\ \\ $O^\eps$ & $n = 2m+1$ & $\delta$ & $\displaystyle\frac{q^m(q^{m}-\eps\delta)}{2}$ \\ \\ $U$ & $q = q_0^2$ & singular & $\displaystyle \frac{(q_0^{n} - (-1)^{n})(q_0^{n-1} + (-1)^{n})}{q-1}$ \\ \\ $U$ & $q = q_0^2$ & non-singular & $\displaystyle \frac{(q_0^{n}-(-1)^{n})q_0^{n-1}}{q_0 + 1}$ \\ \\ $S$ & $n = 2m,q$ even & $\eps$ hyperplane & $\displaystyle \frac{q^{m}(q^{m}+\eps1)}{2}$ \\ \end{tabular} \end{table} \end{center}
In all cases except $L_{14}(2)$ acting on the points in its natural module and $U_8(2^2)$ acting on singular points the lower bound is larger than 2.
However, in those cases, we use the following additional facts: \begin{enumerate} \item If $x$ has order $7$
and acts as a linear transformation over $\F_2$ or $\F_4$ then $x$ has a single eigenspace and $3|v(x)$. \item If $x$ has order $3$
and acts as a linear transformation over $\F_2$ then $x$ has a single eigenspace and $2|v(x)$. \end{enumerate}
Using these additional facts, it is easy to establish the following lemma and complete the proof of Theorem~\ref{touch.up}. \begin{lemma} If $\ud = (2,3,7)$ and the action is either $L_{14}(2)$ on points or $U_{8}(2^2)$ on singular points, then the genus is at least $20$. \end{lemma}
\pf Suppose $G = L_{14}(2)$. Then $x_i$ has only one eigenspace for $i = 1, 2, 3$,
$2 | v(x_2)$, and $3 | v(x_3)$. It follows that $v_1 \geq 5$, $v_2 \geq 8$, and $v_3 \geq 9$. Furthermore, $\Ind(x_1) \geq \frac12(2^{14}-2^9) = 7936$, $\Ind(x_2) \geq \frac23(2^{14}-2^6) = 10880$, and $\Ind(x_3) \geq \frac67(2^{14}-2^5) = 14016$. This implies that $g(\ux) > 30$.
Suppose $G = U_8(2^2)$. Then $x_1$ and $x_3$ have at most one eigenspace, and $3 | v(x_3)$. We have $v_1 \geq 3$, $v_2 \geq 4$, and $v_3 = 6$, and it follows that $g(\ux) > 2$. \eop
\end{document} | arXiv |
\begin{definition}[Definition:Separable Space]
A topological space $T = \struct {S, \tau}$ is '''separable''' {{iff}} there exists a countable subset of $S$ which is everywhere dense in $T$.
\end{definition} | ProofWiki |
Bolsinov A. V., Borisov A. V., Mamaev I. S.
In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.
The paper is devoted to the bifurcation analysis and the Conley index in Hamiltonian dynamical systems. We discuss the phenomenon of appearance (disappearance) of equilibrium points under the change of the Morse index of a critical point of a Hamiltonian. As an application of these techniques we find new relative equilibria in the problem of the motion of three point vortices of equal intensity in a circular domain.
The problem of Hamiltonization of nonholonomic systems, both integrable and non-integrable, is considered. This question is important in the qualitative analysis of such systems and it enables one to determine possible dynamical effects. The first part of the paper is devoted to representing integrable systems in a conformally Hamiltonian form. In the second part, the existence of a conformally Hamiltonian representation in a neighborhood of a periodic solution is proved for an arbitrary (including integrable) system preserving an invariant measure. Throughout the paper, general constructions are illustrated by examples in nonholonomic mechanics.
Bolsinov A. V., Oshemkov A. A.
A Hamiltonian system on a Poisson manifold $M$ is called integrable if it possesses sufficiently many commuting first integrals $f_1, \ldots f_s$ which are functionally independent on $M$ almost everywhere. We study the structure of the singular set $K$ where the differentials $df_1, \ldots, df_s$ become linearly dependent and show that in the case of bi-Hamiltonian systems this structure is closely related to the properties of the corresponding pencil of compatible Poisson brackets. The main goal of the paper is to illustrate this relationship and to show that the bi-Hamiltonian approach can be extremely effective in the study of singularities of integrable systems, especially in the case of many degrees of freedom when using other methods leads to serious computational problems. Since in many examples the underlying bi-Hamiltonian structure has a natural algebraic interpretation, the technology developed in this paper allows one to reformulate analytic and topological questions related to the dynamics of a given system into pure algebraic language, which leads to simple and natural answers.
The work introduces a naive description of dynamics of point vortices on a plane in terms of variables of distances and areas which generate Lie–Poisson structure. Using this approach a qualitative description of dynamics of point vortices on a plane and a sphere is obtained in the works [14,15]. In this paper we consider more formal constructions of the general problem of n vortices on a plane and a sphere. The developed methods of algebraization are also applied to the classical problem of the reduction in the three-body problem.
Bolsinov A. V., Dullin H. R.
Using two classical integrable problems, we demonstrate some methods of a new theory of orbital classification for integrable Hamiltonian systems with two degrees of freedom. We show that the Liouville foliations (i.e., decompositions of the phase space into Liouville tori) of the two systems under consideration are diffeomorphic. Moreover, these systems are orbitally topologically equivalent, but this equivalence cannot be made smooth. | CommonCrawl |
Show that a scale mixtures of normals is a power exponential
I'm trying to show that a scale mixture of normals yields a Laplace distribution. I've gotten to the point where I have $\int N(0,\tau)\times Ga(\tau\:;\:1,\frac{\lambda^{2}}{2}) \:d\tau$ should equal a Laplace. It's unclear to me how to solve the integral.
Alternatively, I've seen a derivation of the scale of mixture normals as a power exponential here, with certain parameter values yielding the Laplace. However, to be honest it's hard to follow. Can someone please help decipher this derivation or provide a clue as to how to solve the integral?
self-study mixture gaussian-mixture laplace-distribution
ilanmanilanman
$\begingroup$ I found the integral above (with Gamma, not Inverse Gamma) in this presentation: newton.ac.uk/files/seminar/20061214161517004-150400.pdf (page 6)Are they formulating the problem incorrectly? $\endgroup$ – ilanman Oct 5 '15 at 16:45
$\begingroup$ Ok great. However, the calculation of this integral still eludes me (it's been a while since I integrated anything...) $\endgroup$ – ilanman Oct 5 '15 at 18:38
The marginal distribution of $\beta$ associated with $$\beta|\tau\sim\mathcal{N}(0,\tau)\quad\tau\sim\mathcal{E}(\lambda^2/2)$$ [with the convention that $\tau$ is the variance] has density \begin{align*}\mathfrak{t}(\beta) &=\int_0^\infty \tau^{-1/2}\varphi(\beta/\sqrt{\tau})\frac{\lambda^2}{2}\exp\{-\lambda^2\tau/2\}\text{d}\tau\\ &=\frac{\lambda^2 e^{-\lambda|\beta|}}{2\sqrt{2\pi}}\int_0^\infty \tau^{-1/2}\exp\left\{-\frac{1}{2}\left(\frac{|\beta|}{\sqrt{\tau}}-\lambda\sqrt{\tau}\right)^2\right\}\text{d}\tau\\ \end{align*} [where the $\lambda|\beta|$ appears by creating a perfect square in the exponential]. This suggests the change of variable $\nu=\sqrt{\tau}$ and leads to $$\mathfrak{t}(\beta) = \frac{\lambda^2 e^{-\lambda|\beta|}}{2\sqrt{2\pi}}\int_0^\infty \exp\left\{-\frac{1}{2}\left(\frac{|\beta|}{\nu}-\lambda\nu\right)^2\right\}\text{d}\nu$$ [since $\tau^{-1/2}\text{d}\tau=2\text{d}\nu$]. This further suggests the change of variable $$\zeta=\frac{|\beta|}{\nu}-\lambda\nu$$ with its inverse $$\nu=\left\{-\zeta+\sqrt{\zeta^2+4\lambda|\beta|} \right\}\big/2\lambda$$ [obtained by solving a second degree polynomial equation] and the Jacobian $$\frac{\text{d}\nu}{\text{d}\zeta}=\left\{-1+\frac{\zeta}{\sqrt{\zeta^2+4\lambda|\beta|}} \right\}\big/2\lambda$$ which is always negative. Hence \begin{align*}\mathfrak{t}(\beta)&=\frac{\lambda^2 e^{-\lambda|\beta|}}{4\lambda\sqrt{2\pi}}\int_{-\infty}^\infty \exp\left\{-\frac{\zeta^2}{2}\right\}\left\{1-\frac{\zeta}{\sqrt{\zeta^2+4\lambda|\beta|}} \right\}\text{d}\zeta\\ &=\frac{\lambda e^{-\lambda|\beta|}}{2}\int_{-\infty}^\infty \left\{1-\frac{\zeta}{\sqrt{\zeta^2+4\lambda|\beta|}} \right\}\varphi(\zeta)\text{d}\zeta\\ &=\frac{\lambda e^{-\lambda|\beta|}}{2}\left\{1-\int_{-\infty}^\infty\frac{\zeta\varphi(\zeta)}{\sqrt{\zeta^2+4\lambda|\beta|}}\text{d}\zeta\right\}=\frac{\lambda e^{-\lambda|\beta|}}{2}\end{align*} [since the integrand is an odd function of $\zeta$ in the last integral]. This establishes [without complex calculus] that the marginal distribution of $\beta$ is indeed a Laplace or double-exponential distribution.
Xi'anXi'an
I've always found the direct integration in this case to be a complicated integral. The Moment Generating Function (MGF) approach works too.
The MGF: $\beta | \tau \sim N(0, \tau)$ and then
$M_{\beta|\tau}(t)=e^{\frac{\tau t^2}{2}}.$ Now to get the MGF of $\beta$ marginally, take the expectation with respect to $\tau$.
$$\mathbb{E}(M_{\beta|\tau}(t)) = \int_0^\infty e^{\frac{\tau t^2}{2}} \frac{\lambda^2}{2}e^{-\tau \frac{\lambda^2}{2}}d\tau =\int_0^\infty \frac{\lambda^2}{2} e^{-\tau \left(-\frac{t^2}{2} +\frac{\lambda^2}{2}\right)} d\tau =\frac{\lambda^2/2}{\lambda^2/2 - t^2/2} =\frac{1}{1 - \frac{t^2}{\lambda^2}}, $$ Now you can recognize this last function as the MGF of a Laplace (double exponential) distribution.
Lucas RobertsLucas Roberts
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\begin{document}
\begin{frontmatter}
\title{Dispersion forces in macroscopic quantum electrodynamics}
\author{Stefan Yoshi Buhmann\corauthref{cor}\thanksref{thanks}}, \ead{[email protected]} \author{Dirk-Gunnar Welsch\thanksref{thanks}} \address{Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit\"{a}t Jena, Max-Wien-Platz 1, 07743 Jena, Germany} \ead[url]{www.tpi.uni-jena.de/tpi/qophysics/qo.html} \corauth[cor]{Corresponding author.} \thanks[thanks]{This work was supported by the Deutsche Forschungsgemeinschaft.}
\begin{abstract} The description of dispersion forces within the framework of macroscopic quantum electrodynamics in linear, dispersing and absorbing media combines the benefits of approaches based on normal-mode techniques of standard quantum electrodynamics and methods based on linear-response theory in a natural way. It renders generally valid expressions for both the forces between bodies and the forces on atoms in the presence of bodies while showing very clearly the intimate relation between the different types of dispersion forces. By considering examples, the influence of various factors like form, size, electric and magnetic properties, or intervening media on the forces is addressed. Since the approach based on macroscopic quantum electrodynamics does not only apply to equilibrium systems, it can be used to investigate dynamical effects such as the temporal evolution of forces on arbitrarily excited atoms. \end{abstract}
\begin{keyword} dispersion force \sep macroscopic quantum electrodynamics \sep Casimir effect \sep van der Waals force \sep atom--surface interaction \sep intermolecular potential \sep atomic polarizability \sep magneto-electric medium \sep multilayer structure \sep spontaneous decay \sep weak atom--field coupling \sep strong atom--field coupling \sep Rabi oscillations \PACS 12.20.-m \sep 42.50.Ct
\sep 34.50.Dy
\sep 42.50.Nn
\end{keyword}
\end{frontmatter}
\tableofcontents
\section{Introduction} \label{sec1}
Dispersion forces originate from the electromagnetic interaction between electrically neutral objects which do not carry permanent electric and magnetic moments. They have been of increasing interest because of their important impact on many areas of science. In particular, the extremely miniaturized components in nanotechnology can be strongly affected by dispersion forces. Recent progress in experimental techniques has led to accurate measurements of dispersion forces which have confirmed some of the theoretical predictions while posing new questions at the same time.
\subsection{Dispersion forces} \label{sec1.1}
The prediction of dispersion forces is one of the most prominent achievements of quantum electrodynamics (QED) where they can be regarded as being a consequence of quantum ground-state fluctuations. In order to understand how quantum fluctuations lead to dispersion forces, it may be helpful to first recall the corresponding classical situation. According to classical electrodynamics, electrically neutral, unpolarized material objects will not interact with each other, even if they are polarizable. An interaction can only occur if (i) at least one of the objects is polarized or (ii) an electromagnetic field is applied to at least one of the objects. In the former case the object's polarization will give rise to an electromagnetic field which can induce a polarization of the other polarizable object(s); in the latter case the applied field induces a polarization of the object which in turn gives rise to an electromagnetic field acting on the other object(s). Both cases result in polarized objects interacting with each other via an electromagnetic field, the interaction and the resulting attractive forces between them being a consequence of the departure from the classical ground state---unpolarized objects and vanishing electromagnetic field.
In QED, the state that most closely corresponds to the classical ground state is given by the material objects being in their (unpolarized) quantum ground states and the electromagnetic field being in its vacuum state, such that both the electromagnetic field and the polarization of all objects vanish on the quantum average. At first glance, one could hence expect the absence of any interaction between the objects. However, the Heisenberg uncertainty principle necessarily implies the existence of ground state fluctuations, i.e., both (i) fluctuating polarizations of the objects and (ii) a fluctuating electromagnetic field will always be present. These fluctuations give rise to an interaction between the objects---a purely quantum effect which is manifested in the dispersion forces acting on them. At finite temperatures, additional thermal fluctuations come into play.
Thus, dispersion forces---also known as Casimir or van der Waals forces\footnote{Often, the term Casimir force is used to denote dispersion forces on a macroscopic level whereas dispersion forces on a microscopic level are referred to as van der Waals forces.}---are ever-present long-range forces between atoms and/or macroscopic bodies, i.e., they exist even if the interacting objects are electrically neutral and do not carry electric or magnetic moments. Naturally, dispersion forces have many important consequences. On a microscopic level, they influence, e.g., the properties of weakly bound molecules \cite{0529,0531}. A prominent macroscopic signature of dispersion forces is the well-known correction to the equation of state of an ideal gas, leading to the more general van der Waals equation.\footnote{In fact, it was in this context that the existence of dispersion forces was first predicted, for a historical review, see Ref.~\cite{0487}.} But dispersion forces also influence the macroscopic properties of liquids and solids such as the anomalies of water \cite{0532}, the magnetic, thermal and optical properties of solid oxygen \cite{0544} or the melting behavior of weakly bound crystals \cite{0530}.
The influence of dispersion forces becomes even more pronounced in the presence of interfaces between different phases and/or media. Atom--surface dispersion interactions drive the adsorption of inert gas atoms to solid surfaces \cite{0117,0225,0505}, influence the wetting properties of liquids on such surfaces \cite{0225,0534,0533} and lead to the phenomenon of capillarity \cite{0539}. The mutual dispersion attractions of colloidal particles suspended in a liquid \cite{0509} influence the stability of such suspensions \cite{0540,0541}; unless sufficiently balanced by repulsive forces, they lead to a clustering of the particles, commonly known as flocculation \cite{0543,0542}.\vspace*{-1ex}
The above mentioned relevance of dispersion forces to material sciences and physical chemistry being rather obvious, it is perhaps more surprising to note that they also play a role in astrophysics and biology. Thus, dispersion forces initiate the preplanetary dust aggregation leading to the formation of planets around a star \cite{0508}. Furthermore, they are needed for an understanding of the interaction of molecules with cell membranes \cite{0359,0188} and of cell adhesion driven by mutual cell-membrane interactions \cite{0359,0495}. Recently, dispersion forces have been found to be responsible for the remarkable abilities of some gecko \cite{0112} and spider species \cite{0111} to climb smooth, dry surfaces.\vspace*{-1ex}
\subsection{Experimental observations}\vspace*{-1ex} \label{sec1.2}
A force between two macroscopic bodies can most easily be measured in a quasi-static way where the bodies are brought close together and the force to be measured is compensated by a force of known magnitude. In the first use of this idea for measuring dispersion forces, the compensating force was provided by a Hookean spring and the distance of the bodies was measured by means of optical interferometry \cite{0568}. In this way, forces between two dielectric \cite{0568,0569,0520,0567,0565} and metal plates \cite{0566,0570,0571,0572} were investigated. Difficulties in aligning the plates were overcome by using alternative setups of a plate interacting with a spherical lens \cite{0565,0581,0576,0560}, two interacting spheres \cite{0559}, and crossed cylinders \cite{0559,0561}. Magnetic forces generated by electric currents were also used to compensate dispersion forces where via feedback, the force values could be inferred from current measurements \cite{0517,0558,0585,0573}. The typical outcome of all these experiments was the observation of attractive forces which follow $1/z^4$ and $1/z^3$ power laws for the plate--plate and plate--sphere\footnote{Note that the plate--sphere separation was much smaller than the sphere radius.} geometries, respectively, with $z$ denoting the object--object separation (see Ref.~\cite{0556} for a review). However, owing to the rather low accuracy---the main limitations being the presence of electrostatic forces due to residual charges, the roughness of the samples, the lack of accurate position control and the low resolution of the actual force measurements---the early results remained controversial. This is best illustrated by the fact that differing power laws \cite{0568,0569} and even signs \cite{0566} were found.
Substantial progress was made by using a torsion-balance scheme \cite{0575} where very smooth bodies, piezoelectric devices for position control, and capacitive force detection, were used to measure the force between a metal sphere and plate with high accuracy, thereby confirming an attractive force proportional to $1/z^3$. This experiment has been followed by a number of experiments which have profited by recent developments in nanotechnology (for an overview, see Refs.~\cite{0378,0135}). By attaching a microsphere to the cantilever of an atomic force microscope (playing the role of the spring) and monitoring the dispersion-force induced bending of the cantilever via deflection of an optical beam, the force between the sphere and a nearby surface was measured to obtain very accurate results for various metals \cite{0574,0553,0555,0551,0547} and/or dielectrics \cite{0588,0512}, including the influence of finite conductivity \cite{0574,0553,0551} as well as surface roughness \cite{0555,0547}. It was further demonstrated that one-dimensional periodic surface corrugations can lead to a sinusoidally varying tangential force in addition to the attractive normal Casimir force \cite{0547,0591,0590}. A similar experimental setup was used to measure the force between two dielectric cylinders \cite{0548} and spheres \cite{0546}. Dispersion forces have also been measured by means of a micromachined torsion-balance scheme where a small plate suspended by two thin rods rotates in response to the Casimir force exerted by a nearby microsphere and this tilting is monitored by capacitive measurements. The scheme was used to study the force between dissimilar metals \cite{0545,0516} and to demonstrate its dependence on the thickness of the interacting objects \cite{0587,0586}. Changing the objects' reflectivity in the visible region by means of hydrogen deposition was found to have no observable influence on the force \cite{0586,0580}, indicating that it should depend on the frequency-dependent body properties in some integral form.
Dispersion forces can also be measured in a dynamical setup, based on the idea that any interaction will affect the relative motion of two objects. In the first experimental realization of this idea, a spherical lens was mounted on a loud speaker and periodically driven, thereby inducing---by means of the Casimir force---a similar motion of a nearby plate mounted on a microphone \cite{0564,0583}. Detection of the amplitude of the induced oscillations led to accurate force measurements. This idea has been applied in modern experiments to infer the Casimir force from the periodic motion of an object such as a microsphere oscillating at the tip of an atomic force microscope cantilever while interacting with a surface \cite{0557,0579} or an oscillating micromachined torsion balance interacting with a rigidly mounted sphere \cite{0545,0516,0550,0401}. Dynamical measurements of this kind can also provide high-precision results for atom--body dispersion interactions, which was demonstrated by observing the change of the oscillatory motion of a single excited ion trapped in a standing electromagnetic wave \cite{0148} and of cold gases of (ground-state) atoms confined in a magnetic trap \cite{0404} or an optical lattice \cite{0403}, induced by their dispersion interaction with a nearby surface. In the latter experiment, a temperature dependence of dispersion forces was observed.
Controlled dynamical measurements of dispersion forces on atoms have only become feasible recently due to the availability of efficient techniques to cool and trap single atoms. In the early experiments, scattering techniques were employed which are of course much simpler to implement. Atom--atom interactions have been observed by scattering a beam of ground-state atoms with known narrow velocity profile off a second beam of atoms with thermal velocity distribution \cite{0592,0584,0589} or a stationary target gas \cite{0582,0594,0595,0598} where an attractive $1/z^7$ force, was found. Scattering techniques have also been employed to probe the interactions of atoms with anisotropic molecules \cite{0597,0599,0600} and even the interaction of excited atoms with ground-state atoms \cite{0596} where in the latter case a strong enhancement of the force, was observed. Evidence of atom--body forces was first found by observing the deflection of a beam of ground-state atoms or molecules passing near the surface of a metal or dielectric cylinder, the results suggesting an attractive $1/z^4$ force \cite{0137,0138,0139,0140}.\footnote{In the experiments, the minimum atom--surface separation was so small that to a good approximation, the cylinder surfaces can be regarded as planar.} In a similar scheme, the deflection of atoms passing between two metal plates was monitored by observing the atom flux losses due to the sticking of atoms to the plates. In doing so, a strong enhancement of the force on excited atoms was observed \cite{0141}. It was further found \cite{0143,0142} that the distance dependence of the ground-state force changes from a $1/z^4$ power law for small atom--surface separations (non-retarded regime) to the more rapidly decreasing $1/z^5$ power law as soon as the separations exceed the relevant transition wavelengths of the atoms and the bodies (retarded regime\footnote{Note that the above mentioned measurements of Casimir forces between macroscopic bodies typically operate in the retarded regime.}).
It has turned out that introducing a controllable compensating force is also useful in the context of atom-scattering experiments; this is the central idea of the evanescent-wave mirror: A laser beam is incident on the surface of a dielectric from the inside at a sufficiently shallow angle, such that total reflection leads to an exponentially decaying electric field at its exterior. An atom placed in the vicinity of the body will interact with this evanescent field, leading to an optical potential. If the laser frequency is larger than the relevant atomic transition frequency (blue detuning), then this potential is repulsive; thus creating the required compensating force which can be controlled by varying the laser frequency and intensity. In this way, dispersion forces on ground-state atoms can be measured by monitoring the reflection of the atoms incident on evanescent-wave mirrors \cite{0172,0164,0294}. Alternatively, compensating forces can be provided by the magnetic fields created by magnetic films, the strength being controlled by varying the film thickness \cite{0354}.
Effects due to the wave nature of the atomic motion become relevant for small values of the (center-of-mass) momentum such that the atomic de Broglie wavelength becomes sufficiently large. In this case quantum reflection of an atom from the potential associated with the atom--body force may occur \cite{0076}. Quantum reflection of ground-state \cite{0444,0167,0402,0449} and excited atoms \cite{0177,0450} incident on the surface of dielectric bodies was observed in various experiments where a detailed measurement of the atom--surface dispersion potential, was achieved by recording the reflectivities at different normal velocities. Another prominent wave phenomenon that can be exploited for the measurement of atom--body potentials is the diffraction of an atomic wave incident on a transmission grating forming a periodic array of parallel slits. When passing the slits (which may be regarded as small planar cavities), the atomic matter wave acquires a phase shift due to the dispersion potential which affects the interference pattern forming behind the slits. By comparing the experimental observations with theoretical simulations, the interaction of ground-state \cite{0160,0452,0453,0454} as well as excited atoms \cite{0162,0161,0455} with dielectrics in the non-retarded regime has been measured.\vspace*{-1ex}
Spectroscopic measurements provide a powerful indirect method for studying atom--body dispersion interactions. Here, the fact is exploited that the dispersion potential of an atom can be identified with the position-dependent shift of the respective atomic energy level \cite{0030}. The resulting shifts of the atomic transition frequencies can be observed by spectroscopic means. As the shifts are usually much more noticeable for excited levels, this approach yields good estimates of the dispersion potentials of atoms in excited energy eigenstates. This was demonstrated in experiments measuring dispersion potentials of atoms inside planar \cite{0144,0456,0145} and spherical metal cavities \cite{0146,0340}, near a dielectric half space \cite{0457,0458}, and of an ion near a metal plate \cite{0147}. In this context, selective reflection spectroscopy of atomic gases has proven to be a particularly powerful method \cite{0153}. It is based on the fact that the reflection of a laser beam incident on a gas cell is modified due to the laser-induced polarization of the gas atoms which in turn is strongly influenced by the dispersion interaction of the atoms with the walls of the cell. By comparing measured reflectivity spectra with theoretically computed ones, very accurate information on the non-retarded dispersion interaction of atoms with dielectric plates \cite{0151,0152,0154,0180,0459,0156} was obtained, including the potentials of atoms in very short-lived excited energy eigenstates which are difficult to study by scattering methods. Note that the dispersion interaction with metal plates is much more difficult to observe via selective reflection spectroscopy \cite{0460}. As a major achievement, the method has shown that the dispersion forces on excited atoms can be repulsive \cite{0157,0159}.\vspace*{-1ex}
\subsection{Applications}\vspace*{-1ex} \label{sec1.3}
Taking advantage of the substantially improved sensitivity of dispersion-force measurements, comparison of the experimental results with theoretical predictions can nowadays even be used to place constraints on other short-scale interactions of fundamental interest, such as non-standard gravitational forces \cite{0378,0516,0404,0554,0552,0549,0538}. In addition, dispersion forces have become of increasing importance in applied science such as nanotechnology and related fields. While providing a powerful tool for surface control, e.g., in near-field scanning microscopy \cite{0114,0115}, they can also be a disturbing factor whose influence will become more and more pronounced with proceeding miniaturization. In particular, they can lead to an undesired and permanent sticking of (small) objects to surfaces \cite{0578,0593,0577}. A similarly disturbing effect is observed when atom traps are operated near surfaces where dispersion forces can diminish the depth of magneto-optical traps, thereby imposing limits upon the near-surface operation of such traps \cite{0116,0194}. Traps based on evanescent waves \cite{0173,0461,0176,0466,0386} necessarily operate in the near-surface regime so that dispersion forces automatically come into play. The influence of dispersion forces also needs to be taken into account when constructing evanescent-wave based elements for atom guiding \cite{0175,0465,0387,0195,0463}.
Dispersion forces are indispensable in atom optics \cite{0118} where mirrors and beam splitters for atomic matter waves, have been constructed based on the dispersion interactions of atoms with flat surfaces and transmission gratings, respectively. Transmission gratings can be used to realize Mach--Zehnder-type interferometers for atoms \cite{0453}. Flat quantum reflective mirrors provide a focussing mechanism when dispersion and gravitational forces are combined with in an appropriate way \cite{0467}. In addition, by locally enhancing the reflectivity of the mirrors via a Fresnel reflection structure \cite{0468}, reflection holograms for atomic matter waves can be realized \cite{0178}. The efficiency of atomic mirrors can also be enhanced by using evanescent-wave mirrors which can even operate quantum-state selectively \cite{0174}. As recently reported, the quantum reflection of ultracold gases at dielectric surfaces gives rise to interesting phenomena, such as the excitation of solitons and vortex structures \cite{0384}.
Further impact on the application of dispersion forces has been made by the recent proposal \cite{0477} and subsequent fabrication \cite{0479} of materials with tailored magneto-electric properties, also known as metamaterials.\footnote{For the current state-of-art of metamaterial fabrication, see, e.g., Refs.~\cite{0482,0484,0486,0485,0483}.} Metamaterials displaying simultaneous negative permittivity and permeability in some frequency range, allow for the existence of traveling electromagnetic waves whose electric-field, magnetic-field and wave vector, form a left-handed triad \cite{0476}\footnote{For this reason materials with these properties are commonly referred to as left-handed materials.}, leading to a number of unusual effects. It is yet an open question whether left-handed properties can lead to interesting phenomena in the context of dispersion forces and to what extent metamaterials can be exploited to tailor the shape and sign of these forces. An interesting behavior of dispersion forces may also occur in conjunction with soft-magnetic alloys, such as permalloy or Mumetal \cite{0694,0635}. After heating and rapid cooling (a process called annealing), these materials are in a state of extremely high permeability; values of more than $5\times 10^4$ have been reported for Mumetal \cite{0609}.
\subsection{Theoretical approaches} \label{sec1.4}
As already mentioned, dispersion forces arise from quantum zero-point fluctuations, namely the fluctuating charge and current distributions of the interacting objects and the vacuum fluctuations of the (transverse) electromagnetic field. If the separation of the objects is smaller than the wavelengths of the relevant field fluctuations, then the latter can be disregarded, allowing for a simplified treatment of dispersion forces. In this non-retarded regime, dispersion forces are dominated by the Coulomb interaction of fluctuating charge distributions. In particular, the Coulomb interaction between two atoms may within a leading-order multipole expansion be regarded as the interaction of two electric dipoles $\hat{\vect{d}}$ and $\hat{\vect{d}}'$,
\begin{equation} \label{1.1} \hat{V}=\frac{\hat{\vect{d}}\!\cdot\!\hat{\vect{d}}'
-3\hat{d}_z\hat{d}'_z}{4\pi\varepsilon_0z^3}\,. \end{equation}
This approach was first used by London in conjunction with leading-order perturbation theory to derive the potential energy of two isotropic ground-state atoms to be
\begin{equation} \label{1.2} U(z)=-\frac{C}{z^6}\,,\qquad C=\frac{1}{24\pi^2\varepsilon_0^2}
\sum_{kk'}\frac{\left|\langle 0|\hat{\vect{d}}|k\rangle\right|^2
\left|\langle 0'|\hat{\vect{d}}'|k'\rangle\right|^2}
{E_k+E_{k'}-(E_0+E_{0'})} \end{equation}
with $|k^{(\prime)}\rangle$ and $E_{k^{(\prime)}}$ denoting the eigenstates and -energies of the unperturbed atoms \cite{0374}. The London potential implies an attractive force proportional to $1/z^7$. The idea of deriving dispersion forces from dipole--dipole interactions by means of perturbation theory was later applied to three- \cite{0084,0085,0086,0515,0510,0507,0506}, four- \cite{0511} and $N$-atom interaction potentials \cite{0494,0119,0120,0501}. Lennard-Jones showed \cite{0022} that the interaction of an atom with a perfectly conducting plate can be treated on an equal footing, by using the image-charge method. Considering the dipole--dipole interaction of the atom with its own image in the plate (instead of a second atom) within first-order perturbation theory, he found a potential
\begin{equation} \label{1.3}
U(z)=-\frac{\langle 0|\hat{\vect{d}}^2|0\rangle}
{48\pi\varepsilon_0z^3}\,, \end{equation}
implying an attractive $1/z^4$ force. The influence of fluctuating magnetic dipoles \cite{0499,0496,0121}, electric quadrupoles \cite{0514} and higher multipoles \cite{0513,0237} as well as permanent electric \cite{0374,0515,0497,0502} and magnetic dipoles \cite{0497} on the atom--atom interaction has also been discussed. In this way, it was found that the force between a magnetizable and a polarizable atom is repulsive and proportional to $1/z^5$, in contrast to the attractive $1/z^7$ force between two polarizable atoms. Furthermore, studying the interaction of atoms prepared in excited energy eigenstates showed that the contributions to the force which arise from real, resonant transitions can be attractive or repulsive \cite{0515,0496,0497} (for further reading regarding the atom--atom interaction, refer to Refs.~\cite{0693,0261,0608}). Similar extensions have been accomplished regarding the interaction of atoms with (planar) bodies. Quadrupole \cite{0288} and higher-order multipole atomic moments \cite{0282,0289,0412} were included in the interaction of ground-state atoms with perfectly conducting plates and extensions of the image-charge method to the interaction of ground-state \cite{0029,0277} and atoms in excited energy eigenstates \cite{0347} with planar dielectric bodies were given.
The method can be further improved by describing the atom and the body on an equal footing in terms of their charge densities and expressing the resulting interaction potential in terms of electrostatic linear response functions\footnote{In contrast to the quantities appearing in earlier attempts to treat conductors in a more realistic way \cite{0025,0023,0026}, the two response functions are directly accessible to measurements.} of the two systems. This was first demonstrated for a ground-state atom interacting with a realistic electric\footnote{The term electric is used where no explicit distinction is made between metals (conductors) and dielectrics (insulators). Likewise, the notion magneto-electric is used to refer to metals or dielectrics possessing non-trivial magnetic properties.} half space exhibiting non-local properties \cite{0027}. The approach was demonstrated to lead to a finite value of the interaction potential in the limit $z\to 0$ \cite{0282,0375,0269,0281,0273}.\footnote{For the dispersion interaction between two dielectric half spaces, this is shown in Refs.~\cite{0650,0649}.} For sufficiently large values of $z$, the potential for an atom in front of a half space can be given by an asymptotic power series in $1/z$ \cite{0027,0286,0276,0341,0348}
\begin{equation} \label{1.4} U(z)=-\frac{\hbar}{16\pi^2\varepsilon_0z^3}
\int_0^\infty\mathrm{d}\xi\,\alpha(\mathrm{i}\xi)\,
\frac{\varepsilon(\mathrm{i}\xi)-1}{\varepsilon(\mathrm{i}\xi)+1}
+O(1/z^4) \end{equation}
[$\alpha(\omega)$, dipole polarizability of the atom; $\varepsilon(\omega)$, local permittivity of the half space] where the leading-order term corresponds to an attractive $1/z^4$ force and coincides with the perfect conductor result (\ref{1.3}) in the limit of infinite permittivity. Next-order corrections are due to the atomic quadrupole polarizability on the one hand \cite{0273} and the leading-order non-local dielectric response on the other hand \cite{0027}. The response-function approach has been used to study the interaction of various ground-state objects with half spaces of different kinds, such as the forces on an ion \cite{0298} and a permanently polarized atom \cite{0272,0283} in front of a metal half space, an anisotropic molecule in front of an electric half space \cite{0290} as well as the interaction of two atoms in front of a metal \cite{0361,0518} and an electric half space \cite{0036}. Extensions include the interaction of an atom in an excited energy eigenstate with an electric \cite{0285} and a birefringent dielectric half space \cite{0028}; non-perturbative effects \cite{0287}; effects due to a constant external magnetic field \cite{0295}; and the interaction of single ground-state atoms/molecules with bodies of various shapes where perfectly conducting \cite{0110}, non-local metallic \cite{0270,0060} and electric spheres \cite{0110}, non-local metallic \cite{0040,0397} and electric cylinders \cite{0284}, perfectly conducting planar \cite{0070} and non-local metallic spherical cavities \cite{0393,0396}, have been considered.\vspace*{-1ex}
The interaction of two macroscopic bodies $B$ and $B'$ was first treated by pairwise summation over the microscopic London potentials~(\ref{1.2}) between the atoms constituting the bodies \cite{0642,0641},\vspace*{-1ex}
\begin{equation} \label{1.4b} U(z)=-\sum_{\vect{r}\in B}\sum_{\vect{r}'\in B'}
\frac{C}{|\vect{r}-\vect{r}'|^6}\,,\vspace*{-1ex} \end{equation}
yielding an attractive $1/z^3$ force between two dielectric half spaces \cite{0642,0656}. Though applicable to bodies of various shapes (cf. also Refs.~\cite{0375,0322}), the method could only yield approximate results due to the restriction to two-atom interactions. By modeling the body atoms by harmonic oscillators, the interaction energy of the bodies could be shown to be a sum of all possible many-atom interaction potentials \cite{0642,0656,0342}. Applications to the interaction of two half spaces \cite{0120} and two spheres \cite{0343} were studied. Microscopic calculations of the dispersion interaction between bodies were soon realized to be very cumbersome, in particular for more involved geometries. In an alternative approach based on macroscopic electrostatics, the interaction energy can be derived from the eigenmodes of the electrostatic Coulomb potential which are subject to the boundary conditions imposed by the surfaces of discontinuity \cite{0640,0654} (for an overview, cf. Ref.~\cite{0135}). The method was used to calculate Casimir forces between electric spheres \cite{0654}; electric spherical cavities \cite{0504}; metal half spaces exhibiting non-local properties \cite{0341}; rough electric half spaces \cite{0630,0631}; and electrolytic half spaces separated by a dielectric \cite{0651}.\vspace*{-1ex}
Even though electrostatic methods have been developed into a sophisticated theory covering various aspects of dispersion forces, they can only render approximate results valid in the non-retarded limit where the object separations are sufficiently small so that the influence of the transverse electromagnetic field can be disregarded. This was first demonstrated by Casimir and Polder \cite{0030,0373}. Using a normal-mode expansion of the quantized electromagnetic field inside a planar cavity bounded by perfectly conducting plates, they showed that the force between the plates can be derived from the total zero-point energy of the modes\vspace*{-1ex}
\begin{equation} \label{1.5} E=\sum_k{\textstyle\frac{1}{2}}\hbar\omega_k. \end{equation}
The difficulty that this energy is divergent was overcome by subtracting the respective diverging energy corresponding to infinite plate separation, the finite result implying a force per unit area
\begin{equation} \label{1.6} \bar{F}=\frac{\pi^2\hbar c}{240}\,\frac{1}{z^4}\,. \end{equation}
In a similar way, they obtained the force on an atom near one of such plates and the force between two atoms in free space from the ground-state energy of the respective system in leading-order perturbation theory. They recovered the results of the non-retarded limit, Eqs.~(\ref{1.2}) and (\ref{1.3}), and found that in the retarded limit the atom--atom and atom--plate potentials are given by
\begin{equation} \label{1.7} U(z)=-\frac{23\hbar c\alpha(0)\alpha'(0)}
{64\pi^3\varepsilon_0^2z^7} \end{equation}
and
\begin{equation} \label{1.8} U(z)=-\frac{3\hbar c\alpha(0)}{32\pi^2\varepsilon_0z^4}\,, \end{equation}
respectively, which correspond to attractive $1/z^8$ and $1/z^5$ forces that decrease more rapidly than the ones in the non-retarded limit. Casimir and Polder had thus developed a unified theory to describe dispersion interactions over a large range of distances.
Normal-mode techniques have since been widely used to study dispersion interactions. The two-atom interaction has been confirmed in various ways \cite{0011,0521,0325,0498,0047,0055,0054,0056,0320,0051,0323,0007, 0067,0201,0200}, inter alia by basing the calculations on the multipolar-coupling scheme \cite{0011,0325,0007} in place of the minimal-coupling scheme originally used by Casimir and Polder, and relativistic corrections have been considered \cite{0189}. Extensions include the interaction of three \cite{0047,0067,0088,0106,0202,0109} or more atoms \cite{0090,0091}, the influence of higher-order multipole moments \cite{0490} and permanent dipole moments \cite{0492} on the two-atom force, the interaction between anisotropically polarizable atoms \cite{0107} and that between a polarizable and a magnetizable atom \cite{0089,0095,0094,0097,0537,0104,0096} (for an overview, cf.~Ref.~\cite{0620}). In particular, it was found that in the retarded limit the force between a polarizable and a magnetizable atom is repulsive as in the non-retarded limit, but follows the same $1/z^8$ power law as that between two polarizable atoms. Furthermore, the interaction of atoms in excited energy eigenstates \cite{0527,0526,0493,0099,0098,0333,0522} and the influence of external conditions such as finite temperature \cite{0104,0103,0376,0105,0101}, applied electromagnetic fields \cite{0105}, or additional bodies \cite{0107,0367,0309,0679} on the atom--atom interaction have been studied. In particular, when the interatomic separation exceeds the thermal wavelength, the force decreases more slowly ($\sim 1/z^7$) than in the zero-temperature limit. Similarly, the Casimir--Polder result for the atom--plate interaction has been confirmed \cite{0047,0055,0054,0056,0051,0095,0321}, atoms that carry permanent electric dipole moments \cite{0327} or are magnetizable \cite{0293} have been considered and the influence of finite temperature \cite{0376,0034} as well as force fluctuations \cite{0072} has been studied. In close analogy to the atom--atom interaction, it was found that the interaction between a magnetizable atom and a perfectly conducting plate is repulsive and that the force decreases more slowly ($\sim 1/z^4$) than in the zero-temperature limit as soon as the atom--plate separation exceeds the thermal wavelength. In contrast to the atom--atom interaction, the atom--plate potential for an atom in an excited energy eigenstate was found to show an oscillatory behavior in the retarded limit \cite{0320,0367,0061,0062}, thereby making the effect of the transverse electromagnetic field more explicit. In addition, atoms interacting with bodies of different shapes and materials have been considered, such as: Perfectly conducting planar \cite{0327,0292,0278,0301,0063,0315} and parabolic cavities \cite{0390,0389}; metal \cite{0296}, electric \cite{0053,0031,0308,0331} and magneto-electric half spaces \cite{0330}; electric planar \cite{0032,0033} and spherical cavities \cite{0441}.\vspace*{-1.3ex}
Needless to say that the pioneering work of Casimir and Polder on dispersion forces has also stimulated further studies of the problem of body--body interactions (for reviews see Refs.~\cite{0378,0136}). Apart from confirming and interpreting the original results \cite{0068,0131,0746}, normal-mode techniques have been employed to include effects that arise from finite temperatures \cite{0632}, surface roughness \cite{0747,0748,0749}, the presence of a dielectric medium between the plates \cite{0668} and even virtual electron-positron pairs \cite{0602,0611} (where the latter were found to be negligibly small). As in the case of atom--body interactions, various other geometries and materials have been considered such as: Two electric \cite{0666,0652,0667,0197}, dielectric \cite{0688}, locally \cite{0626,0625,0645} and non-locally responding metal plates \cite{0682}; two plates that are polarizable and magnetizable \cite{0122,0123,0125,0659,0126,0129,0127}; the faces of a perfectly conducting rectangular cavity \cite{0629,0622}; two electric multilayer stacks \cite{0658}; a perfectly conducting plate and cylinder \cite{0750}; two electric spheres \cite{0377}; a perfectly conducting plate and a small electric sphere \cite{0071}; a sphere and a surrounding spherical cavity \cite{0643}. The results qualitatively resemble the findings for the atom--atom and atom--body interactions. In particular, retardation was found to lead to a stronger asymptotic decrease of the forces which is softened due to thermal effects as soon as the separations exceed the thermal wavelengths; and the force between a polarizable object and a magnetizable one was found to be repulsive. Perhaps a more surprising result is the fact that two birefringent plates may exert a non-vanishing dispersion torque on each other \cite{0639,0684}. Moreover, the problem of Casimir energies of single bodies\footnote{The Casimir energy of a single body can be defined as the geometry-dependent part of the total electromagnetic energy where the notion geometry-dependent part is subject to ambiguities, cf.~the discussion below. For further reading on the Casimir energy of a single body, cf. Ref.~\cite{0136}.} has been addressed, motivated by a conjecture made by Casimir \cite{0692}, according to which an attractive Casimir energy of an electron (modeled as a small perfectly conducting sphere) should be able to counterbalance the repulsive self-energy of the electron charge and thus explain its stability.\footnote{For a further a discussion of this idea, cf. Ref.~\cite{0487}.} However, the energy of a perfectly conducting sphere was found to be repulsive \cite{0519,0648}, with similar findings for a weakly dielectric sphere \cite{0525,0627,0624,0615}. On the contrary, the Casimir energy of a weakly dielectric cylinder was found to be attractive, in agreement with expectations \cite{0627,0623}. The physical significance of Casimir energies of single objects is yet unclear; in particular it was shown by pairwise summation over microscopic dispersion energies that dispersion energies of macroscopic objects are in fact dominated by the always attractive volume and surface energies and may hence never be observed \cite{0204}. In standard calculations of Casimir energies, these volume and surface energies are either not considered from the very beginning or discarded during regularization procedures \cite{0615}.\vspace*{-1ex}
Normal-mode techniques have proved to be a powerful tool for studying dispersion forces (cf.~also Refs.~\cite{0487,0636}). Nevertheless, some principal limitations of the approach have become apparent recently, in particular in view of the new challenges in connection with recent improvements on the experimental side. So, normal-mode calculations can become extremely cumbersome when applied to object geometries relevant to practice or when a realistic description of the electromagnetic properties of the interacting objects is required. The limitations are also illustrated by the controversy regarding the low-temperature behavior of dispersion forces on bodies (for a recent account of the debate, see Ref.~\cite{0681} and references therein). The answer to this question requires detailed knowledge of the complicated interplay of positional, thermal and spectral factors. To see this, one has to bear in mind that, in general, a large range of frequencies contributes to the forces where the relative influence of different frequency intervals is determined by the object--object separation, temperature and the frequency dependence of the object properties. As a consequence, approximations such as long-/short-range, high-/low-temperature or perfect-reflectivity limits become intrinsically intertwined. A typical material property relevant to dispersion forces is the permittivity which is a complex function of frequency, with the real and the imaginary part being responsible for dispersion and absorption, respectively. In particular, absorption which introduces additional noise into a system, inhibits the application of normal-mode expansion on a macroscopic level. This point was first taken into account by Lifshitz in his calculation of the dispersion force between two electric half spaces at finite temperature \cite{0057,0264} where he derived the force from the average of the stress tensor of the fluctuating electromagnetic field at the surfaces of the half spaces, with the source of the field being the fluctuating noise current within the dielectric matter. The required average was obtained by noting that the current fluctuations are linked to the imaginary part of the permittivity via the fluctuation--dissipation theorem. In this way, Lifshitz could express the force per unit area in terms of the permittivities $\varepsilon(\omega)$, $\varepsilon'(\omega)$ of the two half spaces where in particular, in the non-retarded (zero-temperature) limit the force per unit area, was obtained to be\vspace*{-1ex}
\begin{equation} \label{1.9} \bar{F}=\frac{\hbar}{8\pi^2z^3}\int_0^\infty\mathrm{d}\xi\,
\frac{\varepsilon(\mathrm{i}\xi)-1}{\varepsilon(\mathrm{i}\xi)+1}\,
\frac{\varepsilon'(\mathrm{i}\xi)-1}{\varepsilon'(\mathrm{i}\xi)+1}\,. \end{equation}
The Lifshitz theory has been applied and extended by a number of authors (for an overview, see Refs.~\cite{0378,0136}), who studied the influence of different frequency ranges \cite{0644,0132,0676,0690}, effects of finite temperatures \cite{0646,0691,0689,0683} and surface roughness \cite{0607}, and other planar structures such as electrolytic half spaces separated by a dielectric \cite{0657}, magneto-electric half spaces \cite{0134}, metal plates of finite thickness \cite{0612}, metal half spaces exhibiting non-local properties \cite{0689,0680} and electric multilayer systems \cite{0655} (for further reading, cf., e.g., Ref.~\cite{0322}). A typical approximation for treating small deviations from planar structures (like a sphere that is sufficiently close by a plate \cite{0625}) is the proximity force approximation where it is assumed that the interaction of two objects with gently curved surfaces can be obtained by simply integrating the (Lifshitz) force per unit area along the surfaces \cite{0601}.\footnote{Recently, validity limits for the proximity force approximation in the case of perfectly conducting objects have been discussed on the basis of numerical calculations \cite{0695}.} While the debate regarding the temperature dependence of the force between realistic metal half spaces still seems unsettled, general consensus is reached that inclusion or neglect of material absorption (i.e., use of a Drude-type or a plasma-type permittivity) leads to the disagreeing results \cite{0681}. It is worth noting that the forces in a planar structure can be reexpressed in terms of (frequency-dependent) reflection coefficients directly accessible from experiments. This formulation of the theory has been applied to metal \cite{0628,0621,0742} and electric half spaces \cite{0678}, metal half spaces with non-local properties \cite{0604}, electric multilayer stacks \cite{0741,0664} and, in some approximation, to rough perfectly conducting \cite{0677,0674} and metal half spaces \cite{0675,0672} where the surface roughness can give rise to a tangential force component \cite{0673} and a torque \cite{0743}.
Lifshitz's idea of expressing dispersion forces in terms of response functions is of course not restricted to planar systems, but can be extended to arbitrary geometries. This can be achieved by expressing the results obtained by normal-mode expansion in terms of the Green tensor of the (classical) electromagnetic field \cite{0616,0528,0124}. Alternatively, the Green tensor which contains all the necessary information on the shape and the relevant electromagnetic properties of the objects, can be introduced by applying path-integral techniques \cite{0050,0069} or employing the fluctuation--dissipation theorem \cite{0665}. The theory has been used to study the forces between two perfectly conducting plates \cite{0616,0637,0744}, a perfectly conducting plate and a perfectly permeable plate \cite{0124}, two dielectric half spaces \cite{0688,0687}, two electric plates \cite{0665} and, in some approximation, two perfectly conducting spheres \cite{0124,0637}, a perfectly conducting sphere and a perfectly conducting plate \cite{0637,0744}; the force on a rectangular piston \cite{0745}; and the force and the torque between weakly dielectric objects of arbitrary shapes \cite{0528}. In particular, it was shown that the force between two mirror-symmetric electric objects is always attractive \cite{0503}. Casimir energies of a perfectly conducting sphere \cite{0616}, a dielectric sphere \cite{0069}, a magneto-dielectric sphere \cite{0614} and a perfectly conducting cylinder \cite{0617} have also been studied in this way.
The concept of linear-response theory has also been widely used to study dispersion forces on atoms. In particular, it can be shown that the (position-dependent part of the) interaction energy between a ground-state atom and the (body-assisted) electromagnetic vacuum in leading-order perturbation theory can be expressed in terms of the linear response functions of the two systems,
\begin{equation} \label{1.10} U(\vect{r})=\frac{\hbar\mu_0}{2\pi}
\int_{0}^{\infty}\mathrm{d}\xi\,\xi^2 \alpha(\mathrm{i}\xi)
\,\mathrm{Tr}\ten{G}{^{(1)}}(\vect{r},\vect{r},\mathrm{i}\xi), \end{equation}
i.e., the atomic polarizability $\alpha(\omega)$ on the one hand and the scattering Green tensor $\ten{G}{^{(1)}}(\vect{r},\vect{r},\omega)$ of the electromagnetic field on the other, thus rendering a general expression for the force on an atom in the presence of arbitrary bodies \cite{0035,0039,0275,0041}\footnote{Note that the method is the natural extension of the approach based on the electrostatic response function which is now replaced by the response function for the complete electromagnetic field including its transverse part.} (for an alternative, semiclassical approach based on finding the eigenenergies of the classical electromagnetic field interacting with a harmonic-oscillator atom, see Ref.~\cite{0375}). The method which can easily be extended to thermal fields \cite{0037,0046,0394}, also applies to the dispersion interaction of two ground-state atoms in free space \cite{0035,0037} or in the presence of bodies \cite{0036} (cf. also Refs.~\cite{0375,0092}). However, it cannot directly be applied to atoms in excited energy eigenstates where it is necessary to again start from the leading-order interaction energy and only express the field contribution in terms of the respective response function \cite{0235,0042}. Linear response theory has been used to study the dispersion interaction of a single ground-state atom with a multitude of bodies such as: Perfectly conducting plates \cite{0035,0039,0041,0037}; dielectric \cite{0077}, electric \cite{0039,0041,0046,0653} and magneto-electric half spaces \cite{0043}; metal half spaces exhibiting non-local properties \cite{0275,0400} and/or surface roughness \cite{0275,0044} or being covered by a thin overlayer \cite{0280}; perfectly conducting \cite{0346} and dielectric spheres \cite{0069,0077,0349,0372}; dielectric cylinders \cite{0077,0395,0316,0442}; perfectly conducting and electric planar cavities \cite{0314}; dielectric spherical \cite{0313}, cylindrical \cite{0316} and perfectly conducting wedge-shaped cavities \cite{0069,0372}. Moreover, the force on an atom in an excited energy eigenstate in front of a perfectly conducting \cite{0235,0042}, dielectric \cite{0042} and birefringent dielectric half space \cite{0045} has been considered; and the interaction of two ground-state atoms embedded in a non-locally responding electrolyte \cite{0351}, placed near a perfectly conducting \cite{0093} and a electric half space \cite{0653,0523} or inside a perfectly conducting \cite{0092,0093} and dielectric planar cavity \cite{0108} has been studied.
Finally, relations between microscopic and macroscopic dispersion forces have been established whose validity is no longer restricted to the non-retarded limit. Modeling macroscopic bodies as collections of harmonic-oscillator atoms interacting with the electromagnetic field and calculating the total energy of the interacting system, both the the force on a single ground-state atom in the presence of a dielectric half space and the force between two dielectric half spaces were derived from microscopic atom--atom interactions \cite{0048} where the former result was later extended beyond the harmonic-oscillator model \cite{0087}. A harmonic-oscillator model of atoms with the atoms being coupled to a heat bath, was used to derive the force between absorbing dielectric half spaces, confirming the result of Lifshitz theory \cite{0203}. The microscopic-model calculations show that only in the limit of weakly polarizable bodies, i.e., small values of the susceptibilities, a pairwise sum over two-atom forces is sufficient to obtain the total force, stressing once more the importance of many-atom interactions in the context of body-assisted dispersion forces. Pairwise summation of two-atom interactions can be used to obtain an approximate description of the interaction between intricately shaped objects, e.g., bodies with rough surfaces \cite{0638},\footnote{For bodies with small deviations from the planar geometry, the result of pairwise summation can be improved by introducing a correction factor obtained from Lifshitz theory \cite{0606,0603}.} or of atom--body/body--body interactions involving excited atoms and/or bodies \cite{0333,0522,0443}. Conversely, from well-known formulas for the body--body interaction, formulas for the atom--body interaction \cite{0057,0264,0050,0069,0305,0388,0399,0392,0391,0670,0669} as well as the atom--atom interaction \cite{0101,0057,0264,0392,0669,0102} can be obtained in the limit of the respective susceptibilities being asymptotically small.
As we have seen, various concepts have been developed to describe dispersion forces---an overview over the different scenarios which have been studied theoretically, is given in App.~\ref{app1} in tabular form. These concepts, to some extent, are based on different basic assumptions and hence impose different limitations upon the applicability. The QED concepts based on normal-mode expansion of the quantized electromagnetic field typically suffer from the fact that when macroscopic bodies come into play, these bodies should be regarded as non-absorbing and hence also non-dispersing. To overcome this difficulty, arguments from other theories, such as the fluctua\-tion--dissipation theorem of statistical physics, must be borrowed. On the contrary, the concepts based on linear response theory abandon an explicit field quantization and employ the fluctuation--dissipation theorem from the beginning. However, the applicability of methods that make use of the fluctuation--dissipation theorem by some means or other is limited to equilibrium systems ---a disadvantage when dynamical aspects of excited atoms are to be considered. All concepts have in common that macroscopic bodies are typically described in terms of macroscopic electrodynamics, i.e., boundary conditions at surfaces of discontinuity and/or constitutive relations.
In this article we show that by following the formalism of macroscopic QED in media (as developed, e.g., in Refs.~\cite{0003,0002,0605}) from the very beginning, one can obtain a unified approach to dispersion forces which does not only combine the benefits of normal-mode QED and linear-response theory in a natural way, but also accentuates the common origin of and intimate relations between the different types of forces and extends the range of application. In particular, the approach can be used to study dispersion forces for a wide class of different scenarios, including many of those listed in the above overview which have originally been studied by means of a variety of different methods.
The further contents of the article are organized as follows. In Sec.~\ref{sec2}, the main features of the quantization of the electromagnetic field in linear, dispersing and absorbing media and the interaction of the medium-assisted field with atoms is outlined, with special emphasis on magneto-electric media described in terms of spatially varying permittivities and permeabilities which are complex functions of frequency. On this basis, in Sec.~\ref{sec3}, very general formulas for the force on a macroscopic body due to its interaction with other macroscopic bodies are presented which are valid for arbitrarily shaped bodies as all the relevant properties of the bodies are fully expressed in terms of the Green tensor of the associated macroscopic Maxwell equations. Both Casimir stress and Casimir force are introduced, and a very general relation to many-atom van der Waals forces is established. In particular, it is shown that both the force on a single ground-state atom interacting with a body and the force between two ground-state atoms, can be obtained as limiting cases of the general formulas. In Sec.~\ref{sec4}, forces on individual atoms interacting with the body-assisted electromagnetic field are studied in more detail, with special emphasis on explicitly solving the corresponding quantum-mechanical interaction problem. It is demonstrated how the force on one or two ground-state atoms in the presence of magneto-electric bodies can be calculated by leading-order perturbation theory, the results agreeing with those obtained in Sec.~\ref{sec3}. A number of examples is studied where it is shown that dispersion forces are often given by simple asymptotic power laws in the retarded and non-retarded limits. The force on a single atom initially prepared in an arbitrary excited quantum state is calculated by solving the atom--field dynamics, leading to explicitly time-dependent results. Some concluding are given in Sec.~\ref{sec5}.
\section{Elements of QED in linearly responding media} \label{sec2}
It is well known that the properties of the electromagnetic field in media can significantly differ from those observed in free space, and hence, the interaction of the field with atoms can strongly be influenced by the presence of media. In classical electrodynamics, linear media are commonly described in terms of phenomenologically introduced macroscopic electric and magnetic susceptibilities (or permittivities and permeabilities, respectively) available from measurable data. This concept which can be transferred to quantum electrodynamics, has the benefit of being universally valid because it uses only very general physical properties, without the need for specific microscopic matter models and involved ab initio calculations.
\subsection{The medium-assisted electromagnetic field} \label{sec2.1}
The medium-assisted electromagnetic field in the absence of free charges or currents obeys the macroscopic Maxwell equations which in the Fourier domain read
\begin{align} \label{2.1} &\bm{\nabla}\!\cdot\!\underline{\hat{\vect{B}}}(\vect{r},\omega)=0,
\\[.5ex] \label{2.3} &\bm{\nabla}\!\times\!\underline{\hat{\vect{E}}}(\vect{r},\omega)
-\mathrm{i}\omega\underline{\hat{\vect{B}}}(\vect{r},\omega)=\vect{0},
\\[.5ex ] \label{2.2b} &\varepsilon_0\bm{\nabla}\!\cdot\!
\underline{\hat{\vect{E}}}(\vect{r},\omega)
=\underline{\hat{\rho}}_\mathrm{in}(\vect{r},\omega),\\[.5ex] \label{2.4b} &\kappa_0
\bm{\nabla}\!\times\!\underline{\hat{\vect{B}}}(\vect{r},\omega)
+\mathrm{i}\omega\varepsilon_0
\underline{\hat{\vect{E}}}(\vect{r},\omega)
=\underline{\hat{\vect{j}}}_\mathrm{in}(\vect{r},\omega) \end{align}
($\kappa_0$ $\!=$ $\!\mu_0^{-1}$) where the internal charge and current densities of the magneto-electric media $\underline{\hat{\rho}}_\mathrm{in}(\vect{r},\omega)$ and $\underline{\hat{\vect{j}}}_\mathrm{in}(\vect{r},\omega)$ are the sources for the electric field $\underline{\hat{\vect{E}}}(\vect{r},\omega)$ and the induction field $\hat{\underline{\vect{B}}}(\vect{r},\omega)$. Note that the picture-independent Fourier components $\hat{\underline{O}}(\vect{r},\omega)$ of an operator field $\hat{O}(\vect{r})$ are defined according to
\begin{equation} \label{2.0}
\hat{O}(\vect{r})=\int_0^{\infty}\mathrm{d}\omega\,
\hat{\underline{O}}(\vect{r},\omega)+\mathrm{H.c.} \end{equation}
so that $\underline{\hat{O}}(\vect{r},\omega,t)$ $\!=$ $\!\mathrm{e}^{-\mathrm{i}\omega(t-t')} \underline{\hat{O}}(\vect{r},\omega,t')$ in the Heisenberg picture. Since the internal charge and current densities are subject to the continuity equation
\begin{equation} \label{2.0-1} -\mathrm{i}\omega\hat{\underline{\rho}}_\mathrm{in}(\vect{r},\omega)
+\bm{\nabla}\!\cdot\!
\hat{\underline{\vect{j}}}_\mathrm{in}(\vect{r},\omega)
=0, \end{equation}
they may be related to polarization and magnetization fields $\hat{\underline{\vect{P}}}(\vect{r},\omega)$ and $\hat{\underline{\vect{M}}}(\vect{r},\omega)$ as follows:
\begin{align} \label{2.0-2} \hat{\underline{\rho}}_\mathrm{in}(\vect{r},\omega)
=&-\bm{\nabla}\!\cdot\!
\hat{\underline{\vect{P}}}(\vect{r},\omega),\\[.5ex] \label{2.0-3}
\hat{\underline{\vect{j}}}_\mathrm{in}(\vect{r},\omega)
=&-\mathrm{i}\omega\hat{\underline{\vect{P}}}(\vect{r},\omega)
+\bm{\nabla}\!\times\!
\hat{\underline{\vect{M}}}(\vect{r},\omega). \end{align}
Upon introducing the displacement field
\begin{equation} \label{2.5} \hat{\underline{\vect{D}}}(\vect{r},\omega) = \varepsilon_0\hat{\underline{\vect{E}}}(\vect{r},\omega)
+\hat{\underline{\vect{P}}}(\vect{r},\omega) \end{equation}
and the magnetic field
\begin{equation} \label{2.6} \hat{\underline{\vect{H}}}(\vect{r},\omega) = \kappa_0\hat{\underline{\vect{B}}}(\vect{r},\omega)
-\hat{\underline{\vect{M}}}(\vect{r},\omega), \end{equation}
the inhomogeneous Maxwell equations (\ref{2.2b}) and (\ref{2.4b}) can hence be written in the familiar equivalent form
\begin{align} \label{2.2} &\bm{\nabla}\!\cdot\!\underline{\hat{\vect{D}}}(\vect{r},\omega)=0,
\\[.5ex] \label{2.4} &\bm{\nabla}\!\times\!\underline{\hat{\vect{H}}}(\vect{r},\omega)
+\mathrm{i}\omega\underline{\hat{\vect{D}}}(\vect{r},\omega) = \vect{0} \end{align}
where the source terms associated with the internal charge and current densities are now contained in the displacement and magnetic fields.
In particular in the case of linearly and locally responding magneto-electric media, Eqs.~(\ref{2.5}) and (\ref{2.6}) take the form
\begin{align} \label{2.7} &\hat{\underline{\vect{P}}}(\vect{r},\omega) =\varepsilon_0[\varepsilon(\vect{r},\omega)-1]
\hat{\underline{\vect{E}}}(\vect{r},\omega)
+\hat{\underline{\vect{P}}}_\mathrm{N}(\vect{r},\omega),\\[.5ex] \label{2.8} &\hat{\underline{\vect{M}}}(\vect{r},\omega) =\kappa_0[1-\kappa(\vect{r},\omega)]
\hat{\underline{\vect{B}}}(\vect{r},\omega)
+\hat{\underline{\vect{M}}}_\mathrm{N}(\vect{r},\omega) \end{align}
[$\kappa(\vect{r},\omega)$ $\!=$ $\!\mu^{-1}(\vect{r},\omega)$] with $\varepsilon(\vect{r},\omega)$ and $\mu(\vect{r},\omega)$ being the (relative) electric permittivity and magnetic permeability of the media, respectively. Causality implies that $\varepsilon(\vect{r},\omega)$ and $\mu(\vect{r},\omega)$ which vary with space in general, are complex-valued functions of frequency with the Kramers--Kronig relations being satisfied \cite{0001}.\footnote{Note that both metals and dielectrics can be described in terms of their permittivity with the main difference being that the permittivity of a dielectric is analytic in the whole upper half of the complex frequency plane whereas that of a metal is commonly assumed to exhibit a simple pole at $\omega$ $\!=$ $\!0$ \cite{0001}.} According to the fluctuation--dissipation theorem, $\hat{\underline{\vect{P}}}_\mathrm{N}(\vect{r},\omega)$ and $\hat{\underline{\vect{M}}}_\mathrm{N}(\vect{r},\omega)$ are the (linear) noise polarization and magnetization, respectively, associated with the (linear) absorption described by the imaginary parts of $\varepsilon(\vect{r},\omega)$ \mbox{[$\mathrm{Im}\,\varepsilon(\vect{r}, \omega)$ $\!>$ $\!0$]} and $\mu(\vect{r},\omega)$ \mbox{[$\mathrm{Im}\,\mu(\vect{r},\omega)$ $\!>$ $\!0$]}. For simplicity, in Eqs.~(\ref{2.7}) and (\ref{2.8}) the material is assumed to be isotropic.\footnote{The theory can be extended to arbitrary media, by starting from the general linear response relation between the current density and the electric field. Formulas in this article which do not explicitly refer to material properties (but solely via the Green tensor) are valid for arbitrary linear media \cite{1019}.}
Substituting Eqs.~(\ref{2.5}), (\ref{2.6}), (\ref{2.7}) and (\ref{2.8}) into Eq.~(\ref{2.4}) and making use of Eq.~(\ref{2.3}), one can verify that the electric field obeys the inhomogeneous Helmholtz equation
\begin{equation} \label{2.9} \biggl[\bm{\nabla}\!\times\!\kappa(\vect{r},\omega)\bm{\nabla}\!\times\!
\,-\,\frac{\omega^2}{c^2}\varepsilon(\vect{r},\omega)\biggr]
\underline{\hat{\vect{E}}}(\vect{r},\omega)
=\mathrm{i}\omega\mu_0
\underline{\hat{\vect{j}}}_\mathrm{N}(\vect{r},\omega) \end{equation}
where the source term is determined by the noise current density
\begin{equation} \label{2.10} \hat{\underline{\vect{j}}}_\mathrm{N}(\vect{r},\omega)
=-\mathrm{i}\omega\hat{\underline{\vect{P}}}_\mathrm{N}(\vect{r},\omega)
+\bm{\nabla}\!\times\!
\hat{\underline{\vect{M}}}_\mathrm{N}(\vect{r},\omega). \end{equation}
Note that noise current density and noise charge density
\begin{equation} \label{2.11} \hat{\underline{\rho}}_\mathrm{N}(\vect{r},\omega)
=-\bm{\nabla}\!\cdot\!
\hat{\underline{\vect{P}}}_\mathrm{N}(\vect{r},\omega) \end{equation}
fulfill the continuity equation
\begin{equation} \label{2.12} -\mathrm{i}\omega\hat{\underline{\rho}}_\mathrm{N}(\vect{r},\omega)
+\bm{\nabla}\!\cdot\!
\hat{\underline{\vect{j}}}_\mathrm{N}(\vect{r},\omega)
=0 \end{equation}
[recall Eqs.~(\ref{2.0-1})--(\ref{2.0-3})]. The solution to Eq.~(\ref{2.9}) can be given in the form
\begin{equation} \label{2.15} \hat{\underline{\vect{E}}}(\vect{r},\omega)
=\mathrm{i}\omega\mu_0 \int \mathrm{d}^3 r'\,
\ten{G}(\vect{r},\vect{r}',\omega)
\!\cdot\!\hat{\underline{\vect{j}}}_\mathrm{N}(\vect{r}',\omega) \end{equation}
which, according to Eq.~(\ref{2.3}), implies that
\begin{equation} \label{2.15-1} \hat{\underline{\vect{B}}}(\vect{r},\omega)
=\mu_0\int\mathrm{d}^3 r'\,\bm{\nabla}\!\times\!
\ten{G}(\vect{r},\vect{r}',\omega)
\!\cdot\!\hat{\underline{\vect{j}}}_\mathrm{N}(\vect{r}',\omega). \end{equation}
Here, $\ten{G}(\vect{r},\vect{r}',\omega)$ is the (classical) Green tensor which is defined by the equation
\begin{equation} \label{2.13} \biggl[\bm{\nabla}\!\times\!\kappa(\vect{r},\omega)\bm{\nabla}\!\times\!
\,-\,\frac{\omega^2}{c^2}\,\varepsilon(\vect{r},\omega)\biggr]
\ten{G}(\vect{r},\vect{r}',\omega)=\delta(\vect{r}-\vect{r}')\ten{I} \end{equation}
($\ten{I}$: unit tensor) together with the boundary condition
\begin{equation} \label{2.14} \ten{G}(\vect{r},\vect{r}',\omega)\to \ten{0} \quad\mbox{for
}\quad |\vect{r}-\vect{r}'|\to \infty. \end{equation}
It should be pointed out that the Green tensor is uniquely defined by Eqs.~(\ref{2.13}) and (\ref{2.14}) provided that the strict inequalities $\mathrm{Im}\,\varepsilon(\vect{r},\omega)$ $\!>$ $\!0$ and \mbox{$\mathrm{Im}\,\mu(\vect{r},\omega)$ $\!>$ $\!0$} hold. Note that it is an analytic function of $\omega$ in the upper complex half plane and and has the following useful properties ($\cten{A}{}_{ij}^\mathsf{T}$ $\!=$ $\!\cten{A}_{ji}$):
\begin{gather} \label{2.17} \ten{G}^{\ast}(\vect{r},\vect{r}',\omega)
=\ten{G}(\vect{r},\vect{r}',-\omega^{\ast}), \\[.5ex] \label{2.18} \ten{G}(\vect{r},\vect{r}',\omega)
=\ten{G}^\mathsf{T}(\vect{r}',\vect{r},\omega), \end{gather}
\begin{multline} \label{2.19} \int\!\mathrm{d}^3 s\,\Bigl\{
-\mathrm{Im}\,\kappa(\vect{s},\omega)
\bigl[{\bm{\nabla}}_{\!\vect{s}}\!\times\!
\ten{G}(\vect{s},\vect{r},\omega)\bigr]^\mathsf{T}\!\cdot\!
\bigl[{\bm{\nabla}}_{\!\vect{s}}\!\times\!
\ten{G}^\ast(\vect{s},\vect{r}',\omega)\bigr] \\[.5ex]
+\frac{\omega^2}{c^2}\,
\mathrm{Im}\,\varepsilon(\vect{s},\omega)
\,\ten{G}(\vect{r},\vect{s},\omega)
\!\cdot\!\ten{G}^\ast(\vect{s},\vect{r}',\omega)\Bigr\}
=\mathrm{Im}\ten{G}(\vect{r},\vect{r}',\omega). \end{multline}
Noise polarization and magnetization and hence noise current density, can be related to dynamical variables $\hat{\vect{f}}_\lambda(\vect{r},\omega)$ and $\hat{\vect{f}}_\lambda^\dagger(\vect{r},\omega)$ \mbox{($\lambda$ $\!\in$ $\!\{{e},{m}\}$)} of the system which consists of the electromagnetic field and the magneto-electric matter, including the dissipative system responsible for absorption, as follows \cite{0003,0002}:
\begin{align} \label{2.22} &\hat{\underline{\vect{P}}}_\mathrm{N}(\vect{r},\omega) =\mathrm{i}\sqrt{\frac{\hbar\varepsilon_0}{\pi}
\mathrm{Im}\,\varepsilon(\vect{r},\omega)}
\,\hat{\vect{f}}_{e}(\vect{r},\omega),\\[.5ex] \label{2.23} &\hat{\underline{\vect{M}}}_\mathrm{N}(\vect{r},\omega) =\sqrt{-\frac{\hbar\kappa_0}{\pi}
\mathrm{Im}\,\kappa(\vect{r},\omega)}
\,\hat{\vect{f}}_{m}(\vect{r},\omega)
=\sqrt{\frac{\hbar}{\pi\mu_0}\,
\frac{\mathrm{Im}\,\mu(\vect{r},\omega)}
{|\mu(\vect{r},\omega)|^2}}
\,\hat{\vect{f}}_{m}(\vect{r},\omega) \end{align}
with the $\hat{\vect{f}}_\lambda(\vect{r},\omega)$ and $\hat{\vect{f}}_\lambda^\dagger(\vect{r},\omega)$ being attributed to the collective Bosonic excitations of the system,
\begin{align} \label{2.20} &\left[\hat{f}_{\lambda i}(\vect{r},\omega),
\hat{f}_{\lambda'i'}^\dagger(\vect{r}',\omega')\right] = \delta_{\lambda\lambda'}
\delta_{ii'}\delta(\vect{r}-\vect{r}')
\delta(\omega-\omega'),\\[.5ex] \label{2.21} &\left[\hat{f}_{\lambda i}(\vect{r},\omega),
\hat{f}_{\lambda'i'}(\vect{r}',\omega')\right] = 0. \end{align}
By substituting Eqs.~(\ref{2.22}) and (\ref{2.23}) into Eq.~(\ref{2.15}), on recalling Eq.~(\ref{2.10}), we may express the medium-assisted electric field in terms of the dynamical variables $\hat{\vect{f}}_\lambda(\vect{r},\omega)$ and $\hat{\vect{f}}_\lambda^\dagger(\vect{r},\omega)$ to obtain
\begin{equation} \label{2.24}
\underline{\hat{\vect{E}}}(\vect{r},\omega)
=\sum_{\lambda={e},{m}}
\int\mathrm{d}^3r'\,
\ten{G}_\lambda(\vect{r},\vect{r}',\omega)
\!\cdot\!\hat{\vect{f}}_\lambda(\vect{r}',\omega) \end{equation}
where
\begin{align} \label{2.25} &\ten{G}_{e}(\vect{r},\vect{r}',\omega)
=\mathrm{i}\,\frac{\omega^2}{c^2}
\sqrt{\frac{\hbar}{\pi\varepsilon_0}\,
\mathrm{Im}\,\varepsilon(\vect{r}',\omega)}\;
\ten{G}(\vect{r},\vect{r}',\omega),\\[.5ex] \label{2.30} &\ten{G}_{m}(\vect{r},\vect{r}',\omega)
=\mathrm{i}\,\frac{\omega}{c}
\sqrt{-\frac{\hbar}{\pi\varepsilon_0}\,
\mathrm{Im}\,\kappa(\vect{r}',\omega)}\,
\left[\bm{\nabla}'
\!\!\times\!\ten{G}(\vect{r}',\vect{r},\omega)
\right]^\mathsf{T}, \end{align}
so that, according to Eq.~(\ref{2.0}),
\begin{align} \label{2.24-1}
\hat{\vect{E}}(\vect{r}) &=\int_0^\infty\mathrm{d}\omega\,
\underline{\hat{\vect{E}}}(\vect{r},\omega)
+\mathrm{H.c.}\nonumber\\[.5ex] &=\sum_{\lambda={e},{m}}\int\mathrm{d}^3r'
\int_0^\infty\mathrm{d}\omega\,
\ten{G}_\lambda(\vect{r},\vect{r}',\omega)
\!\cdot\!\hat{\vect{f}}_\lambda(\vect{r}',\omega) +\mathrm{H.c.}. \end{align}
Note that the relation (\ref{2.19}) can be written in the more compact form
\begin{equation} \label{2.30b} \sum_{\lambda={e},{m}}\int\mathrm{d}^3s\,
\ten{G}_\lambda(\vect{r},\vect{s},\omega)\!\cdot\!
\ten{G}^{\ast\mathsf{T}}_\lambda(\vect{r}',\vect{s},\omega)
=\frac{\hbar\mu_0}{\pi}\,\omega^2\mathrm{Im}
\ten{G}(\vect{r},\vect{r}',\omega). \end{equation}
By starting from Eq.~(\ref{2.24}) and making use of the Maxwell equations in the Fourier domain, Eqs.~(\ref{2.3}) and (\ref{2.4}) together with Eqs.~(\ref{2.5}), (\ref{2.6}), (\ref{2.7}), (\ref{2.8}), (\ref{2.22}) and (\ref{2.23}), the other electromagnetic-field quantities such as $\hat{\vect{B}}(\vect{r})$, $\hat{\vect{D}}(\vect{r})$ and $\hat{\vect{H}}(\vect{r})$, can be expressed in terms of the dynamical variables $\hat{\vect{f}}_\lambda(\vect{r},\omega)$ and $\hat{\vect{f}}_\lambda^\dagger(\vect{r},\omega)$ in a straightforward way. In particular, one derives
\begin{equation} \label{2.31}
\underline{\hat{\vect{B}}}(\vect{r},\omega)
=\frac{1}{\mathrm{i}\omega}\sum_{\lambda={e},{m}}
\int\mathrm{d}^3r'\,
\bm{\nabla}\!\times\!\ten{G}_\lambda(\vect{r},\vect{r}',\omega)
\!\cdot\!\hat{\vect{f}}_\lambda(\vect{r}',\omega), \end{equation}
and hence,
\begin{align} \label{2.31-1}
\hat{\vect{B}}(\vect{r}) &=\int_0^\infty\mathrm{d}\omega\,
\underline{\hat{\vect{B}}}(\vect{r},\omega)
+\mathrm{H.c.}\nonumber\\[.5ex] &=\sum_{\lambda={e},{m}}\int\mathrm{d}^3r'
\int_0^\infty\frac{\mathrm{d}\omega}{\mathrm{i}\omega}\,
\bm{\nabla}\!\times\!\ten{G}_\lambda(\vect{r},\vect{r}',\omega)
\!\cdot\!\hat{\vect{f}}_\lambda(\vect{r}',\omega)
+\mathrm{H.c.} \end{align}
In view of the treatment of the interaction of the medium-assisted electromagnetic field with atoms, it may be expedient to express the electric and induction fields in terms of potentials,
\begin{align} \label{2.45} &\hat{\vect{E}}(\vect{r}) = -\bm{\nabla}\hat{\varphi}(\vect{r}) -\dot{\hat{\vect{A}}}(\vect{r}),
\\[.5ex] \label{2.46} &\hat{\vect{B}}(\vect{r}) = \bm{\nabla}\!\times\!\hat{\vect{A}}(\vect{r}). \end{align}
In Coulomb gauge, $\bm{\nabla}\!\cdot\!\hat{\vect{A}}(\vect{r})$ $\!=$ $\!0$, the first and second terms on the r.h.s. of Eq.~(\ref{2.45}) are equal to the longitudinal ($\parallel$) and transverse ($\perp$) parts of the electric field, respectively. {F}rom Eq.~(\ref{2.24-1}) it then follows that $\bm{\nabla}\hat{\varphi}(\vect{r})$ and $\hat{\vect{A}}(\vect{r})$ can be expressed in terms of the dynamical variables $\hat{\vect{f}}_\lambda(\vect{r},\omega)$ and $\hat{\vect{f}}_\lambda^\dagger(\vect{r},\omega)$ as
\begin{align} \label{2.49} &\bm{\nabla}
\hat{\varphi}(\vect{r})
=-\hat{\vect{E}}{}^\parallel(\vect{r})
=-\sum_{\lambda={e},{m}}\int\mathrm{d}^3r'
\int_0^\infty \mathrm{d}\omega\,
{}^\parallel\ten{G}_\lambda(\vect{r},\vect{r}',\omega)
\!\cdot\!\hat{\vect{f}}_\lambda(\vect{r}',\omega)
+ \mathrm{H.c.}, \\[.5ex] \label{2.50} & \hat{\vect{A}}(\vect{r})
=\sum_{\lambda={e},{m}}\int\mathrm{d}^3r'
\int_0^\infty\frac{\mathrm{d}\omega}{\mathrm{i}\omega}\,
{}^\perp\ten{G}_\lambda(\vect{r},\vect{r}',\omega)
\!\cdot\!\hat{\vect{f}}_\lambda(\vect{r}',\omega)
+ \mathrm{H.c.} \end{align}
Note that the longitudinal (transverse) part of a vector field is given by
\begin{equation} \label{2.47} \vect{F}^{\parallel(\perp)}(\vect{r})
=\int\mathrm{d}^3 r'\,
\bm{\delta}^{\parallel(\perp)}(\vect{r}-\vect{r}')
\!\cdot\!\vect{F}(\vect{r}') \end{equation}
where
\begin{equation} \label{2.48} \bm{\delta}^\parallel(\vect{r})
=-\bm{\nabla}\tprod\bm{\nabla}\left(\frac{1}{4\pi r}\right), \quad\bm{\delta}^\perp(\vect{r})
=\delta(\vect{r})\ten{I}-\bm{\delta}^\parallel(\vect{r}) \end{equation}
(and for a tensor field accordingly).
The commutation relations for the electromagnetic fields can be deduced from the commutation relations for the dynamical variables $\hat{\vect{f}}_\lambda(\vect{r},\omega)$ and $\hat{\vect{f}}_\lambda^\dagger(\vect{r},\omega)$ as given by Eqs.~(\ref{2.20}) and (\ref{2.21}). In particular, it can be shown \cite{0003,0002} that the electric and induction fields obey the well-known (equal-time) commutation relations
\begin{align} \label{2.35} &\left[\hat{E}_i(\vect{r}),\hat{B}_{i'}(\vect{r}')\right]
=-\mathrm{i}\hbar\varepsilon_0^{-1}\epsilon_{ii'k}\partial_k
\delta(\vect{r}-\vect{r}'),\\[.5ex] \label{2.34} &\left[\hat{E}_i(\vect{r}),\hat{E}_{i'}(\vect{r}')\right]=0
=\left[\hat{B}_i(\vect{r}),\hat{B}_{i'}(\vect{r}')\right]. \end{align}
Introducing the canonical momentum field associated with the (transverse) vector potential,
\begin{equation} \label{2.51b} \hat{\vect{\Pi}}(\vect{r})
=-\varepsilon_0\hat{\vect{E}}{^\perp}(\vect{r}), \end{equation}
one can also prove that the (equal-time) commutation relations
\begin{align} \label{2.53} & \left[\hat{A}_i(\vect{r}),\hat{\Pi}_{i'}(\vect{r}')\right]
=\mathrm{i}\hbar\delta^\perp_{ii'}(\vect{r}-\vect{r}'), \\[.5ex] \label{2.52} & \left[\hat{A}_i(\vect{r}),\hat{A}_{i'}(\vect{r}')\right]=0
=\left[\hat{\Pi}_i(\vect{r}),\hat{\Pi}_{i'}(\vect{r}')\right] \end{align}
are fulfilled.
It is an almost trivial consequence of the quantization scheme that upon choosing the Hamiltonian of the combined system to be
\begin{equation} \label{2.39} \hat{H}_\mathrm{mf} =\sum_{\lambda={e},{m}}\int\mathrm{d}^3r \int_0^\infty
\mathrm{d}\omega\,\hbar\omega\,
\hat{\vect{f}}_{\lambda}^\dagger(\vect{r},\omega)
\!\cdot\!\hat{\vect{f}}_{\lambda}(\vect{r},\omega), \end{equation}
the Heisenberg equation of motion
\begin{equation} \label{2.40} \dot{\hat{O}}=\frac{\mathrm{i}}{\hbar}
\bigl[\hat{H}_\mathrm{mf},\hat{O}\bigr] \end{equation}
generates the correct Maxwell equations in the time domain,
\begin{align} &\bm{\nabla}\!\times\!\hat{\vect{E}}(\vect{r})
+\dot{\hat{\vect{B}}}(\vect{r})=\vect{0},\\[.5ex] \label{2.44} &\bm{\nabla}\!\times\!\hat{\vect{H}}(\vect{r})
-\dot{\hat{\vect{D}}}(\vect{r})=\vect{0}. \end{align}
Note that the Maxwell equations $\bm{\nabla}\!\cdot\!\hat{\vect{B}}(\vect{r})$ $\!=$ $\!0$ and $\bm{\nabla}\!\cdot\!\hat{\vect{D}}(\vect{r})$ $\!=$ $\!0$ are fulfilled by construction.
The Hilbert space can be spanned by Fock states obtained in the usual way by repeated application of the creation operators
$\hat{\vect{f}}_\lambda^\dagger(\vect{r},\omega)$ on the ground state $|\{0\}\rangle$ which is defined by
\begin{equation}
\label{2.36} \hat{\vect{f}}_\lambda(\vect{r},\omega)|\{0\}\rangle
=\vect{0}\ \quad\forall\ \lambda,\vect{r},\omega. \end{equation}
Note that the ground state refers to the combined system of the electromagnetic field and the medium. In particular, from Eq.~(\ref{2.24}) together with the commutation relations (\ref{2.20}) and (\ref{2.21}) one derives, on using the integral relation (\ref{2.30b}),
\begin{equation}
\label{2.36-1} \langle\{0\}|\underline{\hat{E}}_i(\vect{r},\omega)
\underline{\hat{E}}^\dagger_{i'}(\vect{r}',\omega')|\{0\}\rangle = \pi^{-1}\hbar\mu_0\omega^2 \mathrm{Im}\, \cten{G}_{ii'}(\vect{r},\vect{r}',\omega)\delta(\omega-\omega'), \end{equation}
which reveals that the ground-state fluctuations of the electric field are determined by the imaginary part of the Green tensor---in full agreement with the fluctuation--dissipation theorem.
It should be stressed that $\mathrm{Im}\,\varepsilon(\vect{r},\omega)$ $\!>$ $\!0$ and $\mathrm{Im}\,\mu(\vect{r},\omega)$ $\!>$ $\!0$ are assumed to hold everywhere. Even in almost empty regions or regions where absorption is very small and can be neglected in practice, the imaginary parts of the permittivity and permeability must not be set equal to zero in the integrands of expressions of the type (\ref{2.24-1}). To allow for empty-space regions, the limits \mbox{$\mathrm{Im}\,\varepsilon(\vect{r},\omega)$ $\!\to$ $\!0$} and $\mathrm{Im}\,\mu(\vect{r},\omega)$ $\!\to$ $\!0$ may be performed \emph{a posteriori}, i.e., after taking the desired expectation values and having carried out all spatial integrals. In this sense, the theory provides the quantized electromagnetic field in the presence of an arbitrary arrangement of linear, causal magneto-electric bodies characterized by their permittivities and permeabilities where $\mathrm{Im}\,\varepsilon(\vect{r},\omega)$ $\!\ge$ $\!0$ and \mbox{$\mathrm{Im}\,\mu(\vect{r},\omega)$ $\!\ge$ $\!0$}.
As outlined above, quantization of the electromagnetic field in the presence of dispersing and absorbing magneto-electric bodies can be performed by starting from the macroscopic Maxwell equations including noise terms associated with absorption, expressing the electromagnetic field in terms of these noise terms and relating them to Bosonic dynamical variables in an appropriate way (cf. also Refs.~\cite{0716,0717,0718}). Alternatively, absorption can be accounted for by expressing the electromagnetic field in terms of auxiliary fields with the dynamics of the auxiliary fields being such that the Maxwell equations are fulfilled \cite{0710}. It has been shown that the two approaches are equivalent \cite{0711}. It is worth noting that the quantization scheme is in full agreement with the results of (quasi-)microscopic models of dielectric matter where the polarization is modeled by harmonic-oscillator fields and damping is accounted for by introducing a bath of additional harmonic oscillators \cite{1000}. After a Fano diagonalization \cite{0252} of the total Hamiltonian of the system (which consists of the electromagnetic field, the polarization and the bath), an expression of the form~(\ref{2.39}) is obtained. A model of this type was first developed for homogeneous dielectrics \cite{1000} and later extended to inhomogeneous dielectric bodies \cite{0713,1001,0714}, including bodies exhibiting non-local properties \cite{0715}. In particular in the latter case, the differential equation (\ref{2.13}) for the Green tensor obviously changes to an integro-differential equation and Eqs.~(\ref{2.22}) and (\ref{2.23}) must be modified accordingly.
\subsection{Atom--field interaction} \label{sec2.2}
Let us consider a system of nonrelativistic particles of masses $m_\alpha$ and charges $q_\alpha$ which form an atomic system, e.g., an atom or a molecule, interacting with the medium-assisted electromagnetic field. The Hamiltonian governing the dynamics of the atomic system (briefly referred to as atom in the following) in the absence of the medium-assisted electromagnetic field is commonly given in the form
\begin{equation} \label{2.55} \hat{H}_\mathrm{at}
=\sum_{\alpha}\frac{\hat{\vect{p}}_{\alpha}^2}{2m_{\alpha}}
+{\textstyle\frac{1}{2}}\int\mathrm{d}^3r\,
\hat{\rho}_\mathrm{at}(\vect{r})
\hat{\varphi}_\mathrm{at}(\vect{r}) \end{equation}
where $\hat{\rho}_\mathrm{at}(\vect{r})$ and $\hat{\varphi}_\mathrm{at}(\vect{r})$, respectively, are the charge density and the scalar potential which are attributed to the atom,
\begin{align} \label{2.56} &\hat{\rho}_\mathrm{at}(\vect{r})
=\sum_{\alpha}q_{\alpha}\delta(\vect{r}-\hat{\vect{r}}_{\alpha}),
\\[.5ex] \label{2.57} &\hat{\varphi}_\mathrm{at}(\vect{r})
=\int\mathrm{d}^3r'
\frac{\hat{\rho}_\mathrm{at}(\vect{r}')}
{4\pi\varepsilon_0|\vect{r}-\vect{r}'|}
=\sum_{\alpha}\frac{q_{\alpha}}
{4\pi\varepsilon_0|\vect{r}-\hat{\vect{r}}_{\alpha}|}\,, \end{align}
and the standard commutation relations
\begin{align} \label{2.54} &\left[\hat{r}_{\alpha i},\hat{p}_{\alpha'i'}\right]
=\mathrm{i}\hbar\delta_{\alpha\alpha'}\delta_{ii'}, \\[.5ex] \label{2.54-1} &\left[\hat{r}_{\alpha i},\hat{r}_{\alpha'i'}\right]
=0=\left[\hat{p}_{\alpha i},\hat{p}_{\alpha'i'}\right] \end{align}
hold. Obviously, $\hat{\varphi}_\mathrm{at}(\vect{r})$ and $\hat{\rho}_\mathrm{at}(\vect{r})$ obey the Poisson equation
\begin{equation} \label{2.57b} \varepsilon_0\Delta\hat{\varphi}_\mathrm{at}(\vect{r})
=-\hat{\rho}_\mathrm{at}(\vect{r}), \end{equation}
and the continuity equation
\begin{equation} \label{2.57-1} \dot{\hat{\rho}}_\mathrm{at}(\vect{r}) +\bm{\nabla}\!\cdot\!\hat{\vect{j}}_\mathrm{at}(\vect{r})=0 \end{equation}
is fulfilled where the atomic current density $\hat{\vect{j}}_\mathrm{at}(\vect{r})$ reads
\begin{equation} \label{2.63a} \hat{\vect{j}}_\mathrm{at}(\vect{r})
={\textstyle\frac{1}{2}}\sum_\alpha
q_\alpha\left[\dot{\hat{\vect{r}}}_\alpha
\delta(\vect{r}-\hat{\vect{r}}_\alpha)
+\delta(\vect{r}-\hat{\vect{r}}_\alpha)
\dot{\hat{\vect{r}}}_\alpha\right]. \end{equation}
It may be useful \cite{0005,0008} to introduce center-of-mass and relative coordinates
\begin{equation} \label{2.58} \hat{\vect{r}}_{A}=\sum_\alpha\frac{m_\alpha}{m_{A}}
\,\hat{\vect{r}}_\alpha,\qquad \hat{\overline{\vect{r}}}_\alpha=\hat{\vect{r}}_\alpha
-\hat{\vect{r}}_{A} \end{equation}
($m_{A}=\sum_\alpha m_\alpha$) with the associated momenta being
\begin{equation} \label{2.59} \hat{\vect{p}}_{A}=\sum_\alpha\hat{\vect{p}}_\alpha, \qquad \hat{\overline{\vect{p}}}_\alpha=\hat{\vect{p}}_\alpha
-\frac{m_\alpha}{m_{A}}\,\hat{\vect{p}}_{A}. \end{equation}
Combining Eqs.~(\ref{2.55}) and (\ref{2.59}), the atomic Hamiltonian may be written in the form
\begin{align} \label{2.63} \hat{H}_\mathrm{at}
=&\;\frac{\hat{\vect{p}}_{A}^2}{2m_{A}}
+\sum_{\alpha}
\frac{\hat{\overline{\vect{p}}}{}_{\alpha}^2}{2m_{\alpha}}
+{\textstyle\frac{1}{2}}\int\mathrm{d}^3r\,
\hat{\rho}_\mathrm{at}(\vect{r})
\hat{\varphi}_\mathrm{at}(\vect{r})
\nonumber\\[.5ex]
=&\;\frac{\hat{\vect{p}}_{A}^2}{2m_{A}}
+\sum_n E_n |n\rangle\langle n| \end{align}
where $E_n$ and $|n\rangle$ are the eigenenergies and eigenstates of the internal Hamiltonian. {F}rom the commutation relations (\ref{2.54}) and (\ref{2.54-1}) it then follows that the non-vanishing commutators of the new variables are
\begin{align} \label{2.60} &\left[\hat{r}_{{A}i},\hat{p}_{{A}i'}\right]
=\mathrm{i}\hbar\delta_{ii'},\\[.5ex] \label{2.61} & \left[\hat{\overline{r}}_{\alpha i},
\hat{\overline{p}}_{\alpha'i'}\right]
=\mathrm{i}\hbar\delta_{ii'}
\left(\delta_{\alpha\alpha'}
-\frac{m_{\alpha'}}{m_{A}}\right). \end{align}
In particular, when $m_{\alpha'}/m_{A}$ $\!\ll$ $\!1$, then
\begin{equation} \label{2.61-1}
\left[\hat{\overline{r}}_{\alpha i},
\hat{\overline{p}}_{\alpha'i'}\right]
\simeq \mathrm{i}\hbar\delta_{ii'}\delta_{\alpha\alpha'}. \end{equation}
Further atomic quantities that will be of interest are the atomic polarization $\hat{\vect{P}}_\mathrm{at}(\vect{r})$ and magnetization $\hat{\vect{M}}_\mathrm{at}(\vect{r})$ \cite{0006},
\begin{align} \label{2.64} &\hspace{-1ex}\hat{\vect{P}}_\mathrm{at}(\vect{r})
=\sum_\alpha q_\alpha
\hat{\overline{\vect{r}}}_\alpha\int _0^1\mathrm{d}\sigma\,
\delta\bigl(\vect{r}-\hat{\vect{r}}_{A}
-\sigma\hat{\overline{\vect{r}}}_\alpha\bigr),\\[.5ex] \label{2.65} &\hspace{-1ex}\hat{\vect{M}}_\mathrm{at}(\vect{r})
={\textstyle\frac{1}{2}}\sum_\alpha
q_\alpha\!\int _0^1\!\!\mathrm{d}\sigma\,\sigma
\left[\delta\bigl(\vect{r}\!-\!\hat{\vect{r}}_{A}
\!-\!\sigma\hat{\overline{\vect{r}}}_\alpha\bigr)
\hat{\overline{\vect{r}}}_\alpha\!\times\!
\dot{\hat{\overline{\vect{r}}}}_\alpha
-\dot{\hat{\overline{\vect{r}}}}_\alpha\!\times\!
\hat{\overline{\vect{r}}}_\alpha
\delta\bigl(\vect{r}\!-\!\hat{\vect{r}}_{A}
\!-\!\sigma\hat{\overline{\vect{r}}}_\alpha\bigr)\right]\!; \end{align}
it can be shown that for neutral atoms, the atomic charge and current densities are related to the atomic polarization and magnetization according to
\begin{equation} \label{2.66c}
\hat{\rho}_\mathrm{at}(\vect{r})
=-\bm{\nabla}\!\cdot\!\hat{\vect{P}}_\mathrm{at}(\vect{r}) \end{equation}
and
\begin{equation} \label{2.66d} \hat{\vect{j}}_\mathrm{at}(\vect{r})
=\dot{\hat{\vect{P}}}_\mathrm{at}(\vect{r})
+\bm{\nabla}\!\times\!\hat{\vect{M}}_\mathrm{at}(\vect{r})
+\hat{\vect{j}}_\mathrm{r}(\vect{r}) \end{equation}
where
\begin{equation} \label{2.66d2} \hat{\vect{j}}_\mathrm{r}(\vect{r})
={\textstyle\frac{1}{2}}\bm{\nabla}\!\times\!\left[
\hat{\vect{P}}_\mathrm{at}(\vect{r})\!\times\!
\dot{\hat{\vect{r}}}_{A}
-\dot{\hat{\vect{r}}}_{A}
\!\times\!\hat{\vect{P}}_\mathrm{at}(\vect{r})\right] \end{equation}
which is due to the center-of-mass motion, is known as the R\"{o}ntgen current density \cite{0007,0005}. Note that Eqs.~(\ref{2.57b}) and (\ref{2.66c}) imply
\begin{equation} \label{2.66e} \varepsilon_0\bm{\nabla}\hat{\varphi}_\mathrm{at}(\vect{r})
= \hat{\vect{P}}_\mathrm{at}^\parallel(\vect{r}). \end{equation}
Expanding the delta functions in Eqs.~(\ref{2.64}) and (\ref{2.65}) in powers of the relative coordinates $\hat{\overline{\vect{r}}}_\alpha$, we see that the leading-order terms are the electric and magnetic dipole densities associated with the atom,
\begin{align} \label{2.64-1} &\hat{\vect{P}}_\mathrm{at}(\vect{r}) = \hat{\vect{d}}\delta(\vect{r}-\hat{\vect{r}}_{A}),\\[.5ex] \label{2.65-1} &\hat{\vect{M}}_\mathrm{at}(\vect{r}) = \hat{\vect{m}}\delta(\vect{r}-\hat{\vect{r}}_{A}) \end{align}
where the electric and magnetic atomic dipole moments read
\begin{align} \label{2.66} &\hat{\vect{d}}
=\sum_\alpha q_\alpha\hat{\overline{\vect{r}}}_\alpha
=\sum_\alpha q_\alpha\hat{\vect{r}}_\alpha,\\[0.5ex] \label{2.66b} &\hat{\vect{m}}
={\textstyle\frac{1}{2}}\sum_\alpha q_\alpha
\hat{\overline{\vect{r}}}_\alpha\!\times\!
\dot{\hat{\overline{\vect{r}}}}_\alpha. \end{align}
Note that the second equality in Eq.~(\ref{2.66}) only holds for neutral atoms. Using the atomic Hamiltonian (\ref{2.63}) together with the commutation relation (\ref{2.61}) and the definition (\ref{2.59}), one can easily verify the useful relation
\begin{equation} \label{2.66f} \sum_\alpha\frac{q_\alpha}{m_\alpha}\,
\langle m|\hat{\overline{\vect{p}}}_\alpha|n\rangle
=\mathrm{i}\omega_{mn}\vect{d}_{mn} \end{equation}
[$\omega_{mn}$ $\!=$ $\!(E_m$ $\!-$ $\!E_n)/\hbar$, $\vect{d}_{mn}$
$\!=$ $\!\langle m|\hat{\vect{d}}|n\rangle$] which in turn implies the well-known sum rule
\begin{equation} \label{2.66i} \frac{1}{2\hbar}\sum_m\omega_{mn}
(\vect{d}_{nm}\tprod\vect{d}_{mn}
+\vect{d}_{mn}\tprod\vect{d}_{nm})
=\sum_\alpha\frac{q_\alpha^2}{2m_\alpha}\,\ten{I}. \end{equation}
\subsubsection{Minimal coupling} \label{sec2.2.1}
Having established the Hamiltonians of the medium-assisted field and the atom, we next consider the atom--field interaction. According to the minimal coupling scheme (cf.,~e.g., Ref.~\cite{0007}), this may be done by making the replacement $\hat{\vect{p}}_\alpha \mapsto \hat{\vect{p}}_\alpha$ $\!-$ $\!q_\alpha\hat{\vect{A}}(\hat{\vect{r}}_\alpha)$ in the atomic Hamiltonian (\ref{2.55}), summing the Hamiltonians of the medium-assisted field and the atom and adding the Coulomb interaction of the atom with the medium-assisted field, leading to \cite{0003,0008}
\begin{align} \label{2.67} \hat{H} =&\,\sum_{\lambda={e},{m}}\int\mathrm{d}^3r
\int_0^\infty\mathrm{d}\omega\,
\hbar\omega\,\hat{\vect{f}}_\lambda^{\dagger}(\vect{r},\omega)
\!\cdot\!\hat{\vect{f}}_\lambda(\vect{r},\omega)
+{\textstyle\frac{1}{2}}\sum_\alpha m_\alpha^{-1}
\left[\hat{\vect{p}}_\alpha
-q_{\alpha}\hat{\vect{A}}(\hat{\vect{r}}_\alpha)\right]^2
\nonumber\\[.5ex] &\,+{\textstyle\frac{1}{2}}\int\mathrm{d}^3r\,
\hat{\rho}_\mathrm{at}(\vect{r})
\hat{\varphi}_\mathrm{at}(\vect{r})
+\int\mathrm{d}^3r\,\hat{\rho}_\mathrm{at}(\vect{r})
\hat{\varphi}(\vect{r})\nonumber\\[.5ex] =&\;\hat{H}_\mathrm{mf}+\hat{H}_\mathrm{at}+\hat{H}_\mathrm{int} \end{align}
where $\hat{\varphi}(\vect{r})$ and $\hat{\vect{A}}(\vect{r})$ must be thought of as being expressed in terms of the dynamical variables $\hat{\vect{f}}_\lambda(\vect{r},\omega)$ and $\hat{\vect{f}}_\lambda^\dagger(\vect{r},\omega)$, according to Eqs.~(\ref{2.49}) and (\ref{2.50}). Hence, the atom--field interaction energy reads
\begin{equation} \label{2.68} \hat{H}_\mathrm{int}=
\sum_\alpha q_\alpha\hat{\varphi}(\hat{\vect{r}}_\alpha)
-\sum_\alpha\frac{q_\alpha}{m_\alpha}\,
\hat{\vect{p}}_\alpha\!\cdot\!
\hat{\vect{A}}(\hat{\vect{r}}_\alpha)
+\sum_\alpha\frac{q_\alpha^2}{2m_\alpha}\,
\hat{\vect{A}}^2(\hat{\vect{r}}_\alpha). \end{equation}
Note that the scalar product of $\hat{\vect{p}}_\alpha$ and $\hat{\vect{A}}(\hat{\vect{r}}_\alpha)$ commutes in the Coulomb gauge used.
The total electromagnetic field in the presence of the atom reads
\begin{alignat}{4} \label{2.69} & \hat{\bm{\mathcal{E}}}(\vect{r})
=\hat{\vect{E}}(\vect{r})
-\bm{\nabla}\hat{\varphi}_\mathrm{at}(\vect{r}), &\qquad&\hat{\bm{\mathcal{B}}}(\vect{r})
=\hat{\vect{B}}(\vect{r}),\\[.5ex] \label{2.70} &\hat{\bm{\mathcal{D}}}(\vect{r}) =\hat{\vect{D}}(\vect{r})
-\varepsilon_0\bm{\nabla}\hat{\varphi}_\mathrm{at}(\vect{r}), &\qquad&\hat{\bm{\mathcal{H}}}(\vect{r})
=\hat{\vect{H}}(\vect{r}). \end{alignat}
Obviously, $\hat{\bm{\mathcal{B}}}(\vect{r})$ and $\hat{\bm{\mathcal{D}}}(\vect{r})$ obey the Maxwell equations
\begin{align} \label{2.71} &\bm{\nabla}\!\cdot\!\hat{\bm{\mathcal{B}}}(\vect{r}) =0,\\[.5ex] \label{2.72} &\bm{\nabla}\!\cdot\!\hat{\bm{\mathcal{D}}}(\vect{r})
=\hat{\rho}_\mathrm{at}(\vect{r}), \end{align}
and it is a straightforward calculation \cite{0003,0008} to verify that the Hamiltonian (\ref{2.67}) generates the remaining two Maxwell equations
\begin{align} \label{2.73} &\bm{\nabla}\!\times\!\hat{\bm{\mathcal{E}}}(\vect{r})
+\dot{\hat{\bm{\mathcal{B}}}}(\vect{r})
=\vect{0},\\[.5ex] \label{2.74} &\bm{\nabla}\!\times\!\hat{\bm{\mathcal{H}}}(\vect{r})
-\dot{\hat{\bm{\mathcal{D}}}}(\vect{r})
=\hat{\vect{j}}_\mathrm{at}(\vect{r}) \end{align}
and the Newton equations of motion for the charged particles,
\begin{equation} \label{2.76} m_\alpha\ddot{\hat{\vect{r}}}_\alpha = q_\alpha\hat{\bm{\mathcal{E}}}(\vect{r}_\alpha)
+{\textstyle\frac{1}{2}}q_\alpha
\left[\dot{\hat{\vect{r}}}_\alpha
\!\times\!\hat{\bm{\mathcal{B}}}(\vect{r}_\alpha)
-\hat{\bm{\mathcal{B}}}(\vect{r}_\alpha)
\!\times\!\dot{\hat{\vect{r}}}_\alpha\right] \end{equation}
where
\begin{equation} \label{2.75} \dot{\hat{\vect{r}}}_\alpha = m_\alpha^{-1}
\left[\hat{\vect{p}}_\alpha
-q_\alpha\hat{\vect{A}}(\hat{\vect{r}}_\alpha)\right]\!. \end{equation}
In many cases of practical interest one may assume that the atom is small compared to the wavelength of the relevant electromagnetic field. It is hence useful to employ center-of-mass and relative coordinates [Eqs.~(\ref{2.58}) and (\ref{2.59})] and apply the long-wavelength approximation by performing a leading-order expansion of the interaction Hamiltonian (\ref{2.68}) in terms of the relative particle coordinates $\hat{\overline{\vect{r}}}_\alpha$. Considering a neutral atom and recalling Eq.~(\ref{2.49}), one finds
\begin{equation} \label{2.77} \hat{H}_\mathrm{int}
=-\hat{\vect{d}}\!\cdot\!\hat{\vect{E}}{}^\parallel(\hat{\vect{r}}_{A})
-\sum_{\alpha}\frac{q_\alpha}{m_\alpha}\,
\hat{\overline{\vect{p}}}_\alpha\!\cdot\!
\hat{\vect{A}}(\hat{\vect{r}}_{A})
+\sum_\alpha\frac{q_\alpha^2}{2m_\alpha}
\,\hat{\vect{A}}^2(\hat{\vect{r}}_{A}). \end{equation}
Note that the last term on the r.h.s. of Eq.~(\ref{2.77}) is independent of the relative particle coordinates and hence does not act on the internal state of the atom. When considering processes caused by strong resonant transitions between different internal states of the atom, it may therefore be neglected.
\subsubsection{Multipolar coupling} \label{sec2.2.2}
An equivalent description of the atom--field interaction that is widely used is based on the multipolar-coupling Hamiltonian.\footnote{For an extension of the formulas given below to the case of more than one atoms, see Ref.~\cite{0009}.} For a neutral atom, the transition from the minimal-coupling Hamiltonian (\ref{2.67}) to the multipolar-coupling Hamiltonian is a canonical transformation of the dynamical variables, corresponding to a unitary transformation with the transformation operator being given by
\begin{equation} \label{2.79} \hat{U}=\exp\left[\frac{\mathrm{i}}{\hbar}\int\mathrm{d}^3r\,
\hat{\vect{P}}_\mathrm{at}(\vect{r})\!\cdot\!
\hat{\vect{A}}(\vect{r})\right] \end{equation}
where $\hat{\vect{A}}(\vect{r})$ and $\hat{\vect{P}}_\mathrm{at}(\vect{r})$ are defined by Eqs.~(\ref{2.50}) and (\ref{2.64}), respectively. This transformation is commonly known as the Power--Zienau--Woolley transformation \cite{0013,0014}; obviously it does not change $\hat{\vect{r}}_\alpha$,
\begin{equation} \label{2.79-1} \hat{\vect{r}}_\alpha' = \hat{U}\hat{\vect{r}}_\alpha\hat{U}^\dagger = \hat{\vect{r}}_\alpha \end{equation}
and a straightforward calculation yields \cite{0003,0008}
\begin{equation} \label{2.79-2} \hat{\vect{p}}'_\alpha =\hat{U}\hat{\vect{p}}_\alpha\hat{U}^\dagger = \hat{\vect{p}}_\alpha
-q_\alpha\hat{\vect{A}}(\hat{\vect{r}}_\alpha)
-\int\mathrm{d}^3 r\,\hat{\bm{\Xi}}_\alpha(\vect{r})
\!\times\!\hat{\vect{B}}(\vect{r}) \end{equation}
and
\begin{equation} \label{2.81} \hat{\vect{f}}_\lambda'(\vect{r},\omega) =\hat{U}\hat{\vect{f}}_\lambda(\vect{r},\omega)\hat{U}^\dagger
=\hat{\vect{f}}_\lambda(\vect{r},\omega)
+\frac{1}{\hbar\omega}\int\mathrm{d}^3 r'\,
\hat{\vect{P}}_\mathrm{at}^\perp(\vect{r}')
\!\cdot\!\ten{G}_\lambda^\ast(\vect{r}',\vect{r},\omega) \end{equation}
where
\begin{multline} \label{2.86} \hat{\bm{\Xi}}_\alpha(\vect{r}) = q_\alpha\hat{\overline{\vect{r}}}_\alpha
\int _0^1\mathrm{d}\sigma\, \sigma
\delta\bigl(\vect{r}-\hat{\vect{r}}_{A}
-\sigma\hat{\overline{\vect{r}}}_\alpha\bigr)\\[.5ex] -\frac{m_\alpha}{m_{A}}
\sum_\beta q_\beta
\hat{\overline{\vect{r}}}_\beta
\int _0^1\mathrm{d}\sigma\, \sigma
\delta\bigl(\vect{r}-\hat{\vect{r}}_{A}
-\sigma\hat{\overline{\vect{r}}}_\beta\bigr)
+\frac{m_\alpha}{m_{A}}\,\hat{\vect{P}}_\mathrm{at}(\vect{r}). \end{multline}
Now we may express the minimal-coupling Hamiltonian~(\ref{2.67}) in terms of the transformed variables to obtain the multipolar-coupling Hamiltonian in the form
\begin{multline} \label{2.82} \hat{H}=\sum_{\lambda=e,m}\int\mathrm{d}^3r
\int_0^\infty\mathrm{d}\omega\,\hbar\omega
\hat{\vect{f}}_\lambda^{\prime\dagger}(\vect{r},\omega)
\!\cdot\!\hat{\vect{f}}_\lambda'(\vect{r},\omega)
+\frac{1}{2\varepsilon_0}\int\mathrm{d}^3r\,
\hat{\vect{P}}^{\prime 2}_\mathrm{at}(\vect{r})\\[.5ex]
+\sum_\alpha\frac{1}{2 m_\alpha}
\left[\hat{\vect{p}}'_\alpha
+\int\mathrm{d}^3 r\,\hat{\bm{\Xi}}'_\alpha(\vect{r})
\!\times\!\hat{\vect{B}}'(\vect{r})\right]^2
-\int\mathrm{d}^3r\,\hat{\vect{P}}'_\mathrm{at}(\vect{r})
\!\cdot\!\hat{\vect{E}}'(\vect{r}). \end{multline}
Here, $\hat{\vect{E}}'(\vect{r})$ and $\hat{\vect{B}}'(\vect{r})$, respectively, are given by Eqs.~(\ref{2.24-1}) and (\ref{2.31-1}) with $\hat{\vect{f}}_\lambda(\vect{r},\omega)$ [$\hat{\vect{f}}^\dagger_\lambda(\vect{r},\omega)$] being replaced with $\hat{\vect{f}}'_\lambda(\vect{r},\omega)$ [$\hat{\vect{f}}^{\prime\dagger}_\lambda(\vect{r},\omega)$]. Note that $\hat{\vect{r}}'_\alpha$ $\!=$ $\!\hat{\vect{r}}_\alpha$, \mbox{$\hat{\overline{\vect{r}}}{}'_\alpha$ $\!=$ $\!\hat{\overline{\vect{r}}}_\alpha$}, $\hat{\overline{\vect{r}}}{}'_{A}$ $\!=$ $\!\hat{\overline{\vect{r}}}_{A}$, $\hat{\vect{P}}'_\mathrm{at}(\vect{r})$ $\!=$ $\!\hat{\vect{P}}_\mathrm{at}(\vect{r})$, $\hat{\vect{\bm{\Xi}}}'_\alpha(\vect{r})$ $\!=$ $\!\hat{\vect{\bm{\Xi}}}_\alpha(\vect{r})$, $\hat{\vect{B}}'(\vect{r})$ $\!=$ $\!\hat{\vect{B}}(\vect{r})$, but
\begin{equation} \label{2.93} \hat{\vect{E}}'(\vect{r})=\hat{\vect{E}}(\vect{r})
+\varepsilon_0^{-1}
\hat{\vect{P}}^{\perp}_\mathrm{at}(\vect{r}) \end{equation}
which means that the transformed (medium-assisted) electric field $\hat{\vect{E}}'(\vect{r})$ has the physical meaning of a displacement field, in contrast to $\hat{\vect{E}}(\vect{r})$ which has the physical meaning of an electric field.
Hamiltonian (\ref{2.82}) can be decomposed into three parts,
\begin{equation} \label{2.82-1} \hat{H}=\hat{H}_\mathrm{mf'}+\hat{H}_\mathrm{at'}
+\hat{H}_\mathrm{int'} \end{equation}
where $\hat{H}_\mathrm{mf'}$ is given by Eq.~(\ref{2.39}) with the primed variables in place of the unprimed ones,
\begin{equation} \label{2.83} \hat{H}_\mathrm{mf'} =
\sum_{\lambda={e},{m}}\int\mathrm{d}^3r \int_0^\infty
\mathrm{d}\omega\,\hbar\omega\,
\hat{\vect{f}}_{\lambda}^{\prime\dagger}(\vect{r},\omega)
\!\cdot\!\hat{\vect{f}}'_{\lambda}(\vect{r},\omega), \end{equation}
$\hat{H}_\mathrm{at'}$ is the atomic Hamiltonian,
\begin{align} \label{2.84} \hat{H}_\mathrm{at'} &= \frac{\hat{\vect{p}}_{A}^{\prime 2}}{2m_{A}} + \sum_{\alpha} \frac{\hat{\overline{\vect{p}}}{}_{\alpha}^{\prime 2}}{2m_{\alpha}}
+\frac{1}{2\varepsilon_0}\int\mathrm{d}^3r\, \hat{\vect{P}}^{\prime 2}_\mathrm{at}(\vect{r}) \nonumber\\[.5ex] &= \frac{\hat{\vect{p}}_{A}^{\prime 2}}{2m_{A}}
+\sum_n E'_n |n'\rangle\langle n'| \end{align}
and $\hat{H}_\mathrm{int'}$ is the coupling term,
\begin{multline} \label{2.85} \hat{H}_\mathrm{int'}=
-\int\mathrm{d}^3r\,\hat{\vect{P}}'_\mathrm{at}(\vect{r})
\!\cdot\!\hat{\vect{E}}'(\vect{r})
-\int\mathrm{d}^3r\,\hat{\widetilde{\vect{M}}}{}'_\mathrm{at}(\vect{r})
\!\cdot\!\hat{\vect{B}}'(\vect{r}) \\[.5ex]
+\sum_\alpha\frac{1}{2 m_\alpha}
\left[\int\mathrm{d}^3 r\,\hat{\bm{\Xi}}'_\alpha(\vect{r})
\!\times\!\hat{\vect{B}}'(\vect{r})\right]^2
+ \frac{1}{m_{A}}\int\mathrm{d}^3 r\,
\hat{\vect{p}}'_{A}\!\cdot\!
\hat{\vect{P}}'_\mathrm{at}(\vect{r})
\!\times\!\hat{\vect{B}}'(\vect{r}) \end{multline}
where
\begin{equation} \label{2.85b} \hat{\widetilde{\vect{M}}}{}'_\mathrm{at}(\vect{r}) =\sum_\alpha \frac{q_\alpha}{2m_\alpha}\int_0^1\mathrm{d}\sigma\,\sigma
\left[\delta\bigl(\vect{r}\!-\!\hat{\vect{r}}'_{A}
\!-\!\sigma\hat{\overline{\vect{r}}}{}'_\alpha\bigr)
\hat{\overline{\vect{r}}}{}'_\alpha\!\times\!
\hat{\overline{\vect{p}}}{}'_\alpha
-\hat{\overline{\vect{p}}}{}'_\alpha\!\times\!
\hat{\overline{\vect{r}}}{}'_\alpha
\delta\bigl(\vect{r}\!-\!\hat{\vect{r}}'_{A}
\!-\!\sigma\hat{\overline{\vect{r}}}{}'_\alpha\bigr)\right]. \end{equation}
Note that in contrast to the physical magnetization $\hat{\vect{M}}_\mathrm{at}(\vect{r})$ [Eq.~(\ref{2.65})], $\hat{\widetilde{\vect{M}}}_\mathrm{at}(\vect{r})$ is defined in terms of the canonically conjugated momenta rather than the velocities, as is required in a canonical formalism. The Hamiltonian (\ref{2.82}) implies the relation
\begin{equation} \label{2.92} m_\alpha\dot{\hat{\vect{r}}}'_\alpha
=\hat{\vect{p}}'_\alpha
+\int\mathrm{d}^3r\,\hat{\bm{\Xi}}'_\alpha(\vect{r})
\!\times\!\hat{\vect{B}}'(\vect{r}) \end{equation}
and it is not difficult to see [recall Eqs.~(\ref{2.75}) and (\ref{2.79-2})] that $m_\alpha\dot{\hat{\vect{r}}}'_\alpha$ $\!=$ $\!m_\alpha\dot{\hat{\vect{r}}}_\alpha$. It should be pointed out that the eigenenergies $E'_n$ of the internal Hamiltonian in Eq.~(\ref{2.84}) may be different from the corresponding ones of the internal Hamiltonian in Eq.~(\ref{2.63}), because of the additional term contained in
\begin{equation} \label{2.84-1} \frac{1}{2\varepsilon_0}\int\mathrm{d}^3r\, \hat{\vect{P}}^{\prime 2}_\mathrm{at}(\vect{r}) = {\textstyle\frac{1}{2}}\int\mathrm{d}{r}\, \hat{\rho}'_\mathrm{at}(\vect{r})\varphi'_\mathrm{at}(\vect{r}) + \frac{1}{2\varepsilon_0}\int\mathrm{d}^3r \left[\hat{\vect{P}}^{\prime\perp}_\mathrm{at}(\vect{r})\right]^2. \end{equation}
Accordingly, the eigenstates of the two internal Hamiltonians are not related to each other via the unitary transformation $\hat{U}$ [Eq.~(\ref{2.79})] in general.
One of the advantages of the multipolar coupling scheme is the fact that it allows for a systematic expansion in terms of the electric and magnetic multipole moments of the atom. In particular, in the long-wavelength approximation, by retaining only the leading-order terms in the relative coordinates $\hat{\overline{\vect{r}}}{}'_\alpha$, the interaction energy (\ref{2.85}) reads
\begin{align} \label{2.88} \hat{H}_\mathrm{int'}=&\;
-\hat{\vect{d}}'\!\cdot\!
\hat{\vect{E}}'(\hat{\vect{r}}'_{A})
-\hat{\widetilde{\vect{m}}}{}'\!\cdot\!
\hat{\vect{B}}'(\hat{\vect{r}}'_{A})
+\sum_\alpha\frac{q_\alpha^2}{8m_\alpha}
\left[\hat{\bar{\vect{r}}}{}'_\alpha\!\times\!
\hat{\vect{B}}'(\hat{\vect{r}}'_{A})\right]^2
\nonumber\\[.5ex] &\;+\frac{3}{8m_{A}}\left[\hat{\vect{d}}'\!\times\!
\hat{\vect{B}}'(\hat{\vect{r}}_{A})\right]^2
+\frac{1}{m_{A}}\,\hat{\vect{p}}'_{A}
\!\cdot\!\hat{\vect{d}}'\!\times\!
\hat{\vect{B}}'(\hat{\vect{r}}'_{A}) \end{align}
where
\begin{equation} \label{2.89} \hat{\widetilde{\vect{m}}}{}'
=\sum_\alpha\frac{q_\alpha}{2m_\alpha}\,
\hat{\overline{\vect{r}}}{}'_\alpha\!\times\!
\hat{\overline{\vect{p}}}{}'_\alpha. \end{equation}
Note that, in contrast to $\hat{\vect{m}}$ [Eq.~(\ref{2.66b})], $\hat{\widetilde{\vect{m}}}$ is defined in terms of the canonical momenta. The first two terms on the r.h.s. of Eq.~(\ref{2.88}) represent electric and magnetic dipole interactions, respectively; the next two terms describe the (generalized) diamagnetic interaction; and the last term is the R\"{o}ntgen interaction due to the center-of-mass motion. For non-magnetic atoms, Eq.~(\ref{2.89}) reduces to the interaction Hamiltonian in electric-dipole approximation,
\begin{equation} \label{2.90} \hat{H}_\mathrm{int'}=
-\hat{\vect{d}}'\!\cdot\!
\hat{\vect{E}}'(\hat{\vect{r}}'_{A})
+\frac{\hat{\vect{p}}'_{A}}{m_{A}}
\!\cdot\!\hat{\vect{d}}'\!\times\!
\hat{\vect{B}}'(\hat{\vect{r}}'_{A}) \end{equation}
which in cases where the influence of the center-of-mass motion on the atom--field interaction does not need to be taken into account, reduces to
\begin{equation} \label{2.90-1} \hat{H}_\mathrm{int'}= -\hat{\vect{d}}'\!\cdot\!
\hat{\vect{E}}'(\hat{\vect{r}}'_{A}). \end{equation}
\section{Forces on bodies} \label{sec3}
Electromagnetic forces are Lorentz forces. As known, the total Lorentz force $\hat{\vect{F}}_\mathrm{L}$ acting on the matter contained in a volume $V$ is given by
\begin{equation} \label{3.4} \hat{\vect{F}}_\mathrm{L}
=\int_{V}\mathrm{d}^3r\,\left[
\hat{\rho}(\vect{r})\hat{\vect{E}}(\vect{r})
+\hat{\vect{j}}(\vect{r})\!\times\!\hat{\vect{B}}(\vect{r})\right]\!. \end{equation}
Here, the electromagnetic field acts on the the total charge and current densities $\hat{\rho}(\vect{r})$ and $\hat{\vect{j}}(\vect{r})$, respectively, which in general include the internal charge and current densities $\hat{\rho}_\mathrm{in}(\vect{r})$ and $\hat{\vect{j}}_\mathrm{in}(\vect{r})$, respectively, which are attributed to a medium [Eqs.~(\ref{2.0-2}) and (\ref{2.0-3})] as well as those due to the presence of additional sources, such as $\hat{\rho}_\mathrm{at}(\vect{r})$ and $\hat{\vect{j}}_\mathrm{at}(\vect{r})$ [Eqs.~(\ref{2.56}) and (\ref{2.63a})]. With the help of the Maxwell equations [as given by Eqs.~(\ref{2.1})--(\ref{2.4b}) with $\hat{\rho}_\mathrm{in}(\vect{r})$ $\!\mapsto$ $\hat{\rho}(\vect{r})$, \mbox{$\hat{\vect{j}}_\mathrm{in}(\vect{r})$ $\!\mapsto$ $\!\hat{\vect{j}}(\vect{r})$}] one easily finds
\begin{equation} \label{3.2} \hat{\rho}(\vect{r})\hat{\vect{E}}(\vect{r})
+\hat{\vect{j}}(\vect{r})\!\times\!\hat{\vect{B}}(\vect{r}) =\bm{\nabla}\!\cdot\!\hat{\ten{T}}(\vect{r})
-\varepsilon_{0}\frac{\partial}{\partial t}
\left[\hat{\vect{E}}(\vect{r})\!\times\!\hat{\vect{B}}(\vect{r})\right] \end{equation}
so that
\begin{equation} \label{3.5} \hat{\vect{F}}_\mathrm{L} =\int_{\partial V}\mathrm{d}\vect{a}\!\cdot\!\hat{\ten{T}}(\vect{r})
-\varepsilon_0\,\frac{\mathrm{d}}{\mathrm{d} t}
\int_V \mathrm{d}^3r\,\hat{\vect{E}}(\vect{r})
\!\times\!\hat{\vect{B}}(\vect{r}) \end{equation}
where the Maxwell stress tensor
\begin{equation} \label{3.3} \hat{\ten{T}}(\vect{r}) =\varepsilon_0\hat{\vect{E}}(\vect{r})
\tprod\hat{\vect{E}}(\vect{r})
+\mu_0^{-1}\hat{\vect{B}}(\vect{r})
\tprod\hat{\vect{B}}(\vect{r})
-\textstyle{\frac{1}{2}}\left[\varepsilon_0
\hat{\vect{E}}^2(\vect{r})
+\mu_0^{-1}\hat{\vect{B}}^2(\vect{r})\right]\ten{I} \end{equation}
has been introduced. In particular, if the volume integral in the second term on the r.h.s. of Eq.~(\ref{3.5}) does not depend on time, then the total force reduces to the surface integral
\begin{equation} \label{3.6} \hat{\vect{F}}_\mathrm{L}
=\int_{\partial V}\mathrm{d}\hat{\vect{F}}_\mathrm{L} \end{equation}
where
\begin{equation} \label{3.7} \mathrm{d}\hat{\vect{F}}_\mathrm{L}
=\mathrm{d}\vect{a}\!\cdot\! \hat{\ten{T}}(\vect{r})
=\hat{\ten{T}}(\vect{r})\!\cdot\!\mathrm{d}\vect{a} \end{equation}
may be regarded as the infinitesimal force element acting on an infinitesimal surface element $\mathrm{d}\vect{a}$. Note that a constant term in the stress tensor does not contribute to the integral in Eq.~(\ref{3.6}) and can therefore be omitted.
If the Minkowski stress tensor
\begin{multline} \label{3.8} \hat{\ten{T}}{}^\mathrm{(M)}(\vect{r}) =\hat{\vect{D}}(\vect{r})\tprod\hat{\vect{E}}(\vect{r})
+\hat{\vect{H}}(\vect{r})\tprod\hat{\vect{B}}(\vect{r})
-\textstyle{\frac{1}{2}}\left[
\hat{\vect{D}}(\vect{r})\!\cdot\!\hat{\vect{E}}(\vect{r})
+\hat{\vect{H}}(\vect{r})\!\cdot\!\hat{\vect{B}}(\vect{r})
\right]\ten{I} \\[.5ex] =\hat{\ten{T}}(\vect{r})
+\hat{\vect{P}}(\vect{r})\tprod\hat{\vect{E}}(\vect{r})
-\hat{\vect{M}}(\vect{r})\tprod\hat{\vect{B}}(\vect{r})
-\textstyle{\frac{1}{2}}\left[
\hat{\vect{P}}(\vect{r})\!\cdot\!\hat{\vect{E}}(\vect{r})
-\hat{\vect{M}}(\vect{r})\!\cdot\!\hat{\vect{B}}(\vect{r})\right]\ten{I} \end{multline}
(which agrees with Abraham's stress tensor \cite{1003}) is used in Eq.~(\ref{3.6}) [together with Eq.~(\ref{3.7})] instead of the Maxwell stress tensor $\hat{\ten{T}}(\vect{r})$ to calculate the force, one finds
\begin{multline} \label{3.9} \mathrm{d}\hat{\vect{F}}^\mathrm{(M)} =\mathrm{d}\vect{a}\!\cdot\!\hat{\ten{T}}{}^\mathrm{(M)}(\vect{r})
=\mathrm{d}\hat{\vect{F}}_\mathrm{L}\\[.5ex] +\mathrm{d}\vect{a}\!\cdot\!\left\{
\hat{\vect{P}}(\vect{r})\tprod\hat{\vect{E}}(\vect{r})
-\hat{\vect{M}}(\vect{r})\tprod\hat{\vect{B}}(\vect{r})
-\textstyle{\frac{1}{2}}\left[
\hat{\vect{P}}(\vect{r})\!\cdot\!\hat{\vect{E}}(\vect{r})
-\hat{\vect{M}}(\vect{r})\!\cdot\!\hat{\vect{B}}(\vect{r})\right]
\ten{I}\right\}\!, \end{multline}
and it is seen that in general
\begin{equation} \label{3.10} \mathrm{d}\hat{\vect{F}}_\mathrm{L}
\neq\mathrm{d}\vect{a}\!\cdot\!\hat{\ten{T}}{}^\mathrm{(M)}(\vect{r}). \end{equation}
That is to say, the use of the Minkowski stress tensor is expected not to yield the Lorentz force, in general. Indeed, a careful analysis and interpretation of classical electromagnetic force experiments \cite{1010,1011,1007,1006,1008,1009,1005,1012} shows that the (energy--momentum four-tensor associated with the) Lorentz force passes the theoretical and experimental tests and qualifies for a correct description of the energy--momentum properties of the electromagnetic field in macroscopic electrodynamics \cite{1004} (also see Secs.~\ref{sec3.1} and \ref{sec3.2}).
In classical electrodynamics, electrically neutral material bodies at zero temperature which do not carry a permanent polarization and/or magnetization are not subject to a Lorentz force in the absence of external electromagnetic fields. As already noted in Sec.~\ref{sec1.1}, the situation changes in quantum electrodynamics, since the ground-state fluctuations of the body-assisted electromagnetic field and the body's polarization/magnetization charge and current densities can give rise to a non-vanishing ground-state expectation value of the Lorentz force---the Casimir force \cite{0198}
\begin{equation} \label{3.10-2}
\vect{F}=\int_{V}\mathrm{d}^3r\,\left\{\langle\{0\}|\left[
\hat{\rho}(\vect{r})\hat{\vect{E}}(\vect{r}')
+\hat{\vect{j}}(\vect{r})\!\times\!\hat{\vect{B}}(\vect{r}')\right]
|\{0\}\rangle\right\}_{\vect{r'}\to\vect{r}}. \end{equation}
Here, the coincidence limit $\vect{r'}\to\vect{r}$ must be performed in such a way that unphysical (divergent) self-force contributions are discarded after the vacuum expectation value has been calculated for $\vect{r}'$ $\neq$ $\!\vect{r}$, an explicit prescription will be given below Eq.~(\ref{3.19}).
To calculate the Casimir force, let us consider linear media that locally respond to the electromagnetic field and can be characterized by a spatially varying complex permittivity $\varepsilon(\vect{r},\omega)$ and a spatially varying complex permeability $\mu(\vect{r},\omega)$. Following Sec.~\ref{sec2.1}, we may write the medium-assisted electric and induction fields in the form of Eqs.~(\ref{2.24-1}) and (\ref{2.31-1}). Provided that the volume of interest $V$ does not contain any additional charges or currents, the charge and current densities that are subject to the Lorentz force~(\ref{3.4}) are the internal ones, $\hat{\rho}(\vect{r})$ $\!=$ $\!\hat{\rho}_\mathrm{in}(\vect{r})$ and $\hat{\vect{j}}(\vect{r})$ $\!=$ $\!\hat{\vect{j}}_\mathrm{in}(\vect{r})$ [recall Eqs.~(\ref{2.0-2}) and (\ref{2.0-3})]. Making use of Eqs.~(\ref{2.7}) and (\ref{2.8}) together with Eqs.~(\ref{2.11})--(\ref{2.13}), one can easily see that
\begin{equation} \label{3.15} \hat{\underline{\rho}}(\vect{r},\omega)
=-\varepsilon_{0}
\bm{\nabla}\!\cdot\!\left\{[\varepsilon(\vect{r},\omega)-1]
\hat{\underline{\vect{E}}}(\vect{r},\omega)\right\}
+(\mathrm{i}\omega)^{-1}\bm{\nabla}\!\cdot\!
\hat{\underline{\vect{j}}}_\mathrm{N}(\vect{r},\omega) \end{equation}
and
\begin{align} \label{3.16} \hat{\underline{\vect{j}}}(\vect{r},\omega) =&\;-\mathrm{i}\omega \varepsilon_{0}
[\varepsilon(\vect{r},\omega)-1]
\hat{\underline{\vect{E}}}(\vect{r},\omega)
\nonumber\\[.5ex] &\;+\,\bm{\nabla}\!\times\!\left\{\kappa_{0}
[1-\kappa(\vect{r},\omega)]
\hat{\underline{\vect{B}}}(\vect{r},\omega)\right\}
+\hat{\underline{\vect{j}}}_\mathrm{N}(\vect{r},\omega). \end{align}
Taking into account that $\hat{\underline{\vect{E}}}(\vect{r},\omega)$ and $\hat{\underline{\vect{B}}}(\vect{r},\omega)$ can be given in the forms (\ref{2.15}) and (\ref{2.31}), respectively, and that the Green tensor $\ten{G}(\vect{r},\vect{r'},\omega)$ obeys the differential equation (\ref{2.13}), one may perform Eqs.~(\ref{3.15}) and (\ref{3.16}) to obtain
\begin{align} \label{3.17} &\hat{\underline{\rho}}(\vect{r},\omega)
=\frac{\mathrm{i}\omega}{c^2}\int\mathrm{d}^3r'\,
\bm{\nabla}\!\cdot\!\ten{G}(\vect{r},\vect{r'},\omega)\!\cdot\!
\hat{\underline{\vect{j}}}_\mathrm{N}(\vect{r}',\omega),\\[.5ex] \label{3.18} &\hat{\underline{\vect{j}}}(\vect{r},\omega)
=\int \mathrm{d}^3r'\biggl[\bm{\nabla}\!\times\!\bm{\nabla}\!\times\!\,
-\frac{\omega^2}{c^2}\biggr]
\ten{G}(\vect{r},\vect{r'},\omega)\!\cdot\!
\hat{\underline{\vect{j}}}_\mathrm{N}(\vect{r}',\omega). \end{align}
In this way, the fields $\hat{\underline{\rho}}(\vect{r},\omega)$, $\hat{\underline{\vect{j}}}(\vect{r},\omega)$, $\hat{\underline{\vect{E}}}(\vect{r},\omega)$ [Eq.~(\ref{2.15})] and $\hat{\underline{\vect{B}}}(\vect{r},\omega)$ [Eq.~(\ref{2.15-1})] are expressed in terms of the noise current density $\hat{\underline{\vect{j}}}_\mathrm{N}(\vect{r},\omega)$. Making use of Eq.~(\ref{2.10}) together with Eqs.~(\ref{2.22}) and (\ref{2.23}) and recalling the commutation relations (\ref{2.20}) and (\ref{2.21}), one can easily calculate the ground-state correlation function
$\langle\{0\}|\underline{\hat{\vect{j}}}_\mathrm{N}(\vect{r},\omega) \tprod \underline{\hat{\vect{j}}}_\mathrm{N}^\dagger(\vect{r}',\omega')
|\{0\}\rangle$ which can then be used, on recalling Eqs.~(\ref{2.15}), (\ref{2.15-1}), (\ref{3.17}) and (\ref{3.18}), to calculate all the correlation functions relevant to the Casimir force, as given by Eq.~(\ref{3.10-2}). The result is \cite{0663}
\begin{align} \label{3.19} \vect{F} =&\;
\frac{\hbar}{\pi}\int_{V}\mathrm{d}^3r\int_{0}^\infty\mathrm{d}\omega\,
\biggl(\frac{\omega^2}{c^2}\bm{\nabla}\!\cdot\!
\mathrm{Im}\ten{G}(\vect{r},\vect{r'},\omega)
\nonumber\\[.5ex] &\hspace{19ex}+\mathrm{Tr}\biggl\{\ten{I}\!\times\!
\biggl[\bm{\nabla}\!\times\!\bm{\nabla}\!\times\!\,
-\frac{\omega^2}{c^2}\biggr]
\mathrm{Im}\ten{G}(\vect{r},\vect{r'},\omega)\!\times\!
\overleftarrow{\bm{\nabla}}'\biggr\}
\biggr)_{\vect{r'}\to\vect{r}}\nonumber\\[.5ex] =&\;-\frac{\hbar}{\pi}\int_{V}\mathrm{d}^3r\int_{0}^\infty\mathrm{d}\xi\,
\biggl(\frac{\xi^2}{c^2}\bm{\nabla}\!\cdot\!
\ten{G}(\vect{r},\vect{r'},\mathrm{i}\xi)
\nonumber\\[.5ex] &\hspace{17ex} -\mathrm{Tr}\biggl\{\ten{I}\!\times\! \biggl[\bm{\nabla}\!\times\!\bm{\nabla}\!\times\!\,
+\frac{\xi^2}{c^2}\biggr]
\ten{G}(\vect{r},\vect{r'},\mathrm{i}\xi)\!\times\!
\overleftarrow{\bm{\nabla}}'\biggr\}
\biggr)
_{\vect{r'}\to\vect{r}} \end{align}
[$(\mathrm{Tr}\ten{T})_j$ $\!=$ $\!\cten{T}_{ljl}$, $\overleftarrow{\bm{\nabla}}$ introduces differentiation to the left] where it is now apparent that in the coincidence limit $\vect{r'}\to\vect{r}$ the Green tensor has to be replaced with its scattering part at each space point. In particular, when the material in the space region $V$ is homogeneous, then the Green tensor therein can be globally decomposed into a bulk part $\ten{G}^{(0)}(\vect{r},\vect{r'},\omega)$ and a scattering part $\ten{G}^{(1)}(\vect{r},\vect{r'},\omega)$,
\begin{equation} \label{3.22} \ten{G}(\vect{r},\vect{r'},\omega)
=\ten{G}^{(0)}(\vect{r},\vect{r'},\omega)
+\ten{G}^{(1)}(\vect{r},\vect{r'},\omega)\qquad(\vect{r}\in V). \end{equation}
In this case, the coincidence limit $\vect{r'}\to\vect{r}$ simply means that the Green tensor $\ten{G}(\vect{r},\vect{r'},\omega)$ can be globally replaced by its well-behaved scattering part $\ten{G}^{(1)}(\vect{r},\vect{r'},\omega)$.
According to Eqs.~(\ref{3.3})--(\ref{3.7}), the Casimir force can be equivalently rewritten as a surface integral over a stress tensor \cite{0198},
\begin{equation} \label{3.23} \vect{F}=\int_{\partial V}\mathrm{d}\vect{a}
\!\cdot\!\ten{T}(\vect{r},\vect{r}')_{\vect{r'}\to\vect{r}} \end{equation}
where
\begin{multline} \label{3.24} \ten{T}(\vect{r},\vect{r}')
=\langle\{0\}|\bigl\{\varepsilon_0
\hat{\vect{E}}(\vect{r})\tprod\hat{\vect{E}}(\vect{r'})
+\mu_0^{-1}\hat{\vect{B}}(\vect{r})\tprod\hat{\vect{B}}(\vect{r'})
\\[.5ex] -\textstyle{\frac{1}{2}}\bigl[\varepsilon_0
\hat{\vect{E}}(\vect{r})\!\cdot\!\hat{\vect{E}}(\vect{r'})
+\mu_0^{-1}\hat{\vect{B}}(\vect{r})\!\cdot\!\hat{\vect{B}}(\vect{r'})
\bigr]\ten{I}\bigr\}|\{0\}\rangle \end{multline}
which leads to
\begin{equation} \label{3.25} \ten{T}(\vect{r},\vect{r}')
=\ten{S}(\vect{r},\vect{r}')
-{\textstyle\frac{1}{2}}
\bigl[\mathrm{Tr}\ten{S}(\vect{r},\vect{r}')\bigr]\ten{I} \end{equation}
with
\begin{align} \label{3.26} \ten{S}(\vect{r},\vect{r}') =&\;\frac{\hbar}{\pi}\int_{0}^{\infty} \mathrm{d}\omega
\biggl[\frac{\omega^2}{c^2}\,
\mathrm{Im}\ten{G}(\vect{r},\vect{r}',\omega)
-\bm{\nabla}\!\times\!
\mathrm{Im}\ten{G}(\vect{r},\vect{r}',\omega)
\!\times\!\overleftarrow{\bm{\nabla}}'\biggr]\nonumber\\[.5ex] =&\,-\frac{\hbar}{\pi}\int_{0}^{\infty} \mathrm{d}\xi
\biggl[\frac{\xi^2}{c^2}\,\ten{G}(\vect{r},\vect{r}',\mathrm{i}\xi)
+\bm{\nabla}\!\times\!\ten{G}(\vect{r},\vect{r}',\mathrm{i}\xi)
\!\times\!\overleftarrow{\bm{\nabla}}'\biggr]. \end{align}
Both Eq.~(\ref{3.19}) and Eq.~(\ref{3.23}) [together with Eqs.~(\ref{3.25}) and (\ref{3.26})] are valid for arbitrary bodies that linearly respond to the electromagnetic field, since the force is fully determined by the Green tensor of the classical, macroscopic Maxwell equations with the material properties entering the force formulas only via the Green tensor. Moreover, Eqs.~(\ref{3.19}) and (\ref{3.26}) reveal that the force is proportional to $\hbar$ and hence represents a pure quantum effect. The results can be generalized to finite temperatures $T$ in a straightforward way, by averaging in Eqs.~(\ref{3.10-2}) and (\ref{3.24}) over the thermal state instead of the vacuum state. As a consequence, the r.h.s. of the first equalities of Eqs.~(\ref{3.19}) and (\ref{3.26}) are modified according to \cite{0198}
\begin{equation} \label{3.26-2} \int_{0}^{\infty}\mathrm{d}\omega\,\ldots\
\mapsto\ \int_{0}^{\infty}\mathrm{d}\omega\,
\coth{\left(\frac{\hbar\omega}{2k_\mathrm{B}T}\right)}\ldots, \end{equation}
($k_\mathrm{B}$, Boltzmann constant) so that the final forms of these equations change as
\begin{equation} \label{3.26-3} \frac{\hbar}{\pi}\int_{0}^{\infty}\mathrm{d}\xi\,f(\mathrm{i}\xi)\
\mapsto\ 2k_\mathrm{B}T\sum_{n=0}^\infty
\bigl(1-{\textstyle\frac{1}{2}}\delta_{n0}\bigr)f(\mathrm{i}\xi_n) \end{equation}
with
\begin{equation} \label{3.26-4} \xi_n=\frac{2\pi k_\mathrm{B}T}{\hbar}\,n \end{equation}
being the Matsubara frequencies.
\subsection{Casimir stress in planar structures} \label{sec3.1}
Let us apply the theory to a planar magneto-electric structure defined according to
\begin{equation} \label{3.27} \varepsilon(\vect{r},\omega)
=\begin{cases}
\varepsilon_{-}(z,\omega)&\quad z<0,\\
\varepsilon(\omega)&\quad 0<z<d,\\
\varepsilon_{+}(z,\omega)&\quad z>d,
\end{cases} \end{equation}
\begin{equation} \label{3.28} \mu(\vect{r},\omega)
=\begin{cases}
\mu_{-}(z,\omega)&\quad z<0,\\
\mu(\omega) & \quad 0<z<d,\\
\mu_{+}(z,\omega)&\quad z>d
\end{cases} \end{equation}
and restrict our attention to the zero-temperature limit. To determine the Casimir stress in the interspace $0$ $\!<$ $\!z$ $\!<$ $\!d$, we need the scattering part of the Green tensor in Eq.~(\ref{3.26}) for both spatial arguments within the interspace ($0\!<\!z\!=\!z'\!<\!d$). Since the component $\vect{q}$ of the wave vector parallel to the interfaces is conserved and the polarizations $\sigma$ $\!=$ $\!s,p$ decouple, the required Green tensor (as given in App.~\ref{appA}) can be expressed in terms of reflection coefficients $r_{\sigma\pm}$ $\!=$
$\!r_{\sigma\pm}(\omega,q)$ ($q$ $\!=$ $\!|\vect{q}|$) referring to reflection of waves at the right ($+$) and left ($-$) wall, respectively, as seen from the interspace. Explicit (recurrence) expressions for the reflection coefficients are available if the walls are multi-slab magneto-electrics, cf. Eqs.~(\ref{A.7}) and (\ref{A.8}).\footnote{For continuous wall profiles, Riccati-type equations have to be solved \cite{0217}.} In the simplest case of two homogeneous, semi-infinite half spaces, the coefficients $r_{\sigma\pm}$ reduce to the well-known Fresnel amplitudes, Eq.~(\ref{A.10}).
In order to determine the Casimir force, it is clear for symmetry reasons that one requires the $z$ component $\cten{T}_{\!zz}(\vect{r})$ $\!\equiv$ $\cten{T}_{\!zz}(\vect{r},\vect{r})_{\vect{r}'\to\vect{r}}$ of the stress tensor in the interspace \mbox{$0$ $\!<$ $\!z$ $\!<$ $\!d$} which, upon using the Green tensor from App.~\ref{appA}, can be given in the form \cite{0198}
\begin{equation} \label{3.29} \cten{T}_{\!zz}(\vect{r}) =-\frac{\hbar}{8\pi^2}\int_{0}^{\infty}\mathrm{d}\xi
\int_{0}^{\infty}\mathrm{d} q\,\frac{q}{b}\,\mu(\mathrm{i}\xi)
g(z,\mathrm{i}\xi,q) \end{equation}
where the function $g(z,\xi,q)$ is defined by
\begin{align} \label{3.30} g(z,\mathrm{i}\xi,q) =&\;-2\left[b^2 (1+n^{-2})+q^2(1-n^{-2})\right]
\mathrm{e} ^{-2bd}\,r_{s+}r_{s-}D_s^{-1}\nonumber\\[.5ex] &\;-2\left[b^2 (1+n^{-2})-q^2 (1-n^{-2})\right]
\mathrm{e}^{-2bd}\,r_{p+}r_{p-}D_p^{-1}\nonumber\\[.5ex] &\;+(b^2-q^2)(1-n^{-2})\left[\mathrm{e}^{-2b z}r_{s-}+
\mathrm{e}^{-2b(d-z)}r_{s+}\right]D_s^{-1}\nonumber\\[.5ex] &\;-(b^2-q^2)(1-n^{-2})\left[\mathrm{e} ^{-2bz}r_{p-}+
\mathrm{e}^{-2b(d-z)}r_{p+}\right]D_p^{-1} \end{align}
with
\begin{gather} \label{3.31} n=n(\mathrm{i}\xi)
=\sqrt{\varepsilon(\mathrm{i}\xi)\mu(\mathrm{i}\xi)}\,,\\[.5ex] \label{3.32} b=b(\mathrm{i}\xi,q)
=\sqrt{n^2(\mathrm{i}\xi)\frac{\xi^2}{c^2}+q^2}\,,\\[.5ex] \label{3.33} D_\sigma=D_\sigma(\mathrm{i}\xi,q)
=1-r_{\sigma +} r_{\sigma -} \mathrm{e}^{-2bd}. \end{gather}
According to Eq.~(\ref{3.23}), Eq.~(\ref{3.29}) [together with Eqs.~(\ref{3.30})--(\ref{3.33})] gives the force per unit area between two arbitrary planar multilayer stacks of (locally responding) dispersing and absorbing magneto-electric material where the interspace between them may contain an additional magneto-electric medium. It is worth noting that many specific planar systems that can be addressed by means of Eq.~(\ref{3.29}) have been studied by using alternative methods: The most prominent example is the case of two electric half spaces separated by vacuum, as first considered by Lifshitz \cite{0057,0264} and later readdressed \cite{0644,0132,0676,0690,0628,0678}, inter alia based on normal-mode QED \cite{0652,0667,0197,0626}. Extended scenarios range from electric half spaces separated by an electric medium (as studied by means of electrostatic theory \cite{0650,0649,0640,0651}, normal-mode QED \cite{0668} and an extended Lifshitz theory \cite{0657}) over electric plates of finite thickness (as addressed on the basis of the Lifshitz theory \cite{0612} and linear-response theory \cite{0665}), electric multilayer stacks (as treated by generalizing the results of the Lifshitz theory \cite{0655,0741,0664} as well as normal-mode QED \cite{0658}) to magneto-electric half spaces (as studied by means of Lifshitz theory \cite{0124} as well as normal-mode QED \cite{0122,0123,0125,0659,0126,0134}). For purely electric multilayer systems, various effects not being taken into account by Eq.~(\ref{3.29}), have also been addressed in a number of works, such as the influence on the force of finite temperature \cite{0666,0688,0625,0645,0682,0125,0126,0057,0264,0646,0691, 0689,0683,0621,0742,0616}, surface roughness \cite{0747,0748,0749,0630,0631,0607,0677,0674,0675,0672,0673} and non-locally responding materials \cite{0341,0682,0689,0680}. As outlined above, finite temperature can be easily included in Eq.~(\ref{3.29}) by applying Eqs.~(\ref{3.26-3}) and (\ref{3.26-4}) \cite{0198}, whereas surface roughness as well as non-local material response can be taken into account by returning to the more general formula (\ref{3.23}) [together with Eqs.~(\ref{3.25}) and (\ref{3.26})] and specifying the Green tensor appropriately.
To further (numerically) evaluate Eq.~(\ref{3.29}), knowledge of the $\xi$- and $q$-depen\-dence of the reflection coefficients which depend on the respective planar system (see, e.g., the examples studied in Refs.~\cite{0670,0669,0647,0660,0133}), is required. Let us here restrict our attention to the limit of perfectly conducting surfaces, i.e., $r_{p\pm}$ $\!=$ $\!-r_{s\pm}$ $\!=$ $\!1$ and assume that the wall separation $d$ is sufficiently large, so that the permittivity and the permeability of the medium in the interspace can be replaced by their static values $\varepsilon$ $\!\equiv$ $\varepsilon(0)$ and $\mu$ $\!\equiv$ $\mu(0)$. It is then not difficult to calculate the simplified integrals in Eq.~(\ref{3.29}) analytically to obtain the attractive Casimir force per unit area, $\bar{F}$ $\!=$ $\!\cten{T}_{\!zz}(d)$ [recall Eq.~(\ref{3.23})] which acts between two perfectly reflecting plates as\footnote{Note that the terms proportional to $\mathrm{e}^{-2b(d-z)}$ in Eq.~(\ref{3.30}) give rise to divergent integrals in Eq.~(\ref{3.29}) in the limit \mbox{$z$ $\!\to$ $\!d$} which obviously results from the assumptions of frequency-independent response of the plates and the intervening medium and infinite lateral extension of the plates. Since in a realistic system these contributions are canceled by similar contributions occurring at the backs of the plates \cite{0198}, they can be discarded.}
\begin{equation} \label{3.34} \bar{F}
=\frac{\pi^2\hbar c}{240}\,\sqrt{\frac{\mu}{\varepsilon}}
\biggl(\frac{2}{3}+\frac{1}{3\varepsilon\mu}\biggr)
\frac{1}{d^4}\,. \end{equation}
In particular, when the interspace is empty ($\varepsilon$ $\!=$ $\!1$, $\mu$ $\!=$ $\!1$), then Eq.~(\ref{3.34}) reduces to
\begin{equation} \label{3.35} \bar{F}=\frac{\pi^2\hbar c}{240}\,\frac{1}{d^4}\,, \end{equation}
in agreement with the famous result~(\ref{1.6}) first obtained by Casimir on the basis of a normal-mode expansion \cite{0373} and subsequently rederived \cite{0131,0632}, inter alia based on classical orbits \cite{0124,0637} or dimensional arguments \cite{0068}.
In the same approximation, the force that acts on a perfectly conducting plate in a planar cavity bounded by perfectly conducting walls and filled with a magneto-electric medium obviously reads
\begin{equation} \label{3.36} \bar{F}=\frac{\pi^2\hbar c}{240}\,\sqrt{\frac{\mu}{\varepsilon}}\,
\biggl(\frac{2}{3}+\frac{1}{3\varepsilon\mu}\biggr)
\biggl(\frac{1}{d_\mathrm{r}^4}-\frac{1}{d_\mathrm{l}^4}\biggr) \end{equation}
where $d_\mathrm{l}$ ($d_\mathrm{r}$) is the left (right) plate--wall separation. On the contrary, use of the Minkowski stress tensor (\ref{3.8}) leads, for $\mu$ $\!=$ $\!1$, to \cite{0647}
\begin{equation} \label{3.37} \bar{F}^\mathrm{(M)} =\frac{\pi^2\hbar c }{240}\,\frac{1}{\sqrt{\varepsilon}} \biggl(\frac{1}{d_\mathrm{r}^{4}} -\frac{1}{d_\mathrm{l}^{4}}\biggr). \end{equation}
Comparing the two results for $\mu$ $\!=$ $\!1$, we see that
$|\bar{F}|$ $\!\leq$ $|\bar{F}^\mathrm{(M)}|$. Introduction of a (polarizable) medium into the interspace between the plate and the cavity walls is obviously associated with some screening, thereby reducing the force acting on the plate. Since the internal charges and currents of the interspace medium are fully included only in the Lorentz force [recall Eqs.~(\ref{3.4})--(\ref{3.7})], the force based on the Minkowski stress tensor or an equivalent quantity underestimates the screening effect and is hence larger than the Lorentz force in general.
\subsection{Macro- and micro-objects} \label{sec3.2}
Let us return to the general formula (\ref{3.19}) and consider the Casimir force acting on electric matter of susceptibility $\chi(\vect{r},\omega)$ $\!=$ $\!\varepsilon(\vect{r},\omega)\!-\!1$ in some particular space region $V$ in the presence of arbitrary linearly responding bodies (outside~$V$) in more detail. If $\overline{\ten{G}}(\vect{r},\vect{r}',\omega)$ and $\ten{G}(\vect{r},\vect{r}',\omega)$, respectively, denote the Green tensors in the absence and presence of the electric matter in $V$ with both of them taking into account the bodies in the remaining space, the differential equation (\ref{2.13}) for $\ten{G}(\vect{r},\vect{r}',\omega)$ can be converted into the Dyson-type integral equation
\begin{equation} \label{3.38} \ten{G}(\vect{r},\vect{r}',\omega) =\overline{\ten{G}}(\vect{r},\vect{r}',\omega)
+\frac{\omega^2}{c^2}\int_V \mathrm{d}^3s\,\chi(\vect{s},\omega)
\overline{\ten{G}}(\vect{r},\vect{s},\omega)\!\cdot\!
\ten{G}(\vect{s},\vect{r}',\omega) \end{equation}
where, for $\vect{r}$ $\!\in$ $\!V$, the Green tensor $\overline{\ten{G}}(\vect{r},\vect{r}',\omega)$ satisfies the same differential equation as the free-space Green tensor,
\begin{equation} \label{3.39} \left[\bm{\nabla}\!\times\!\bm{\nabla}\!\times\!\,
-\frac{\omega^2}{c^2}\right]
\overline{\ten{G}}(\vect{r},\vect{r}',\omega)
=\delta(\vect{r}-\vect{r}')\ten{I}, \end{equation}
from which it follows that
\begin{equation} \label{3.40} \frac{\omega^2}{c^2}\,\bm{\nabla}\!\cdot\!
\overline{\ten{G}}(\vect{r},\vect{r}',\omega) =-\bm{\nabla}\delta(\vect{r}-\vect{r}') \end{equation}
for $\vect{r}$ $\!\in$ $\!V$. Equations~(\ref{3.38}) and (\ref{3.40}) imply ($\vect{r}\in V$, $\omega$ real)
\begin{equation} \label{3.41} \bm{\nabla}\!\cdot\!\mathrm{Im}\ten{G}(\vect{r},\vect{r}',\omega)
=-\bm{\nabla}\!\cdot\!\mathrm{Im}[
\chi(\vect{r},\omega)\ten{G}(\vect{r},\vect{r}',\omega)]. \end{equation}
In a similar way, one finds that ($\vect{r}\in V$, $\omega$ real)
\begin{equation} \label{3.43} \left[\bm{\nabla}\!\times\!\bm{\nabla}\!\times\!\,
-\frac{\omega^2}{c^2}\right]
\mathrm{Im}\ten{G}(\vect{r},\vect{r'},\omega)\!\times\!
\overleftarrow{\bm{\nabla}}' =\frac{\omega^2}{c^2}\,\mathrm{Im}[ \chi(\vect{r},\omega)\ten{G}(\vect{r},\vect{r}',\omega)] \!\times\!\overleftarrow{\bm{\nabla}}'. \end{equation}
Substituting Eqs.~(\ref{3.41}) and (\ref{3.43}) into Eq.~(\ref{3.19}) one can then show that the Casimir force acting on an electric body of volume $V_\mathrm{I}$ which is an inner part of a larger electric body (occupying volume $V$) reads \cite{0663}
\begin{multline} \label{3.44} \vect{F}=-\frac{\hbar}{2\pi}\int_{0}^{\infty}\mathrm{d}\xi\,
\frac{\xi^2}{c^2}
\Bigl\{\int_{V_\mathrm{I}}\mathrm{d}^3r\,\chi(\vect{r},\mathrm{i}\xi)\bm{\nabla}
\mathrm{Tr}[\ten{G}(\vect{r},\vect{r}',\mathrm{i}\xi)]_{\vect{r'}\to\vect{r}}
\\[.5ex] -2\int_{\partial V_\mathrm{I}}\mathrm{d}\vect{a}\!\cdot\!
\chi(\vect{r},\mathrm{i}\xi)[\ten{G}(\vect{r},\vect{r}',\mathrm{i}\xi)
]_{\vect{r'}\to\vect{r}}\Bigr\}. \end{multline}
In particular, in the case of an isolated body, i.e., when the region $V\supset V_\mathrm{I}$ is empty apart from the electric matter contained in $V_\mathrm{I}$, then the surface integral can be dropped, hence
\begin{equation} \label{3.45} \vect{F}=-\frac{\hbar}{2\pi}\int_{V_\mathrm{I}}\mathrm{d}^3r
\int_{0}^{\infty}\mathrm{d}\xi\,
\frac{\xi^2}{c^2}\,\chi(\vect{r},\mathrm{i}\xi)\bm{\nabla}
\mathrm{Tr}[\ten{G}(\vect{r},\vect{r}',\mathrm{i}\xi)]_{\vect{r'}\to\vect{r}}. \end{equation}
Let us briefly compare Eq.~(\ref{3.44}) with the equation obtained on the basis of the Minkowski stress tensor,
\begin{equation} \label{3.44-1} \vect{F}^{\mathrm{(M)}}=\frac{\hbar}{2\pi}\int_{V_\mathrm{I}}\mathrm{d}^3r
\int_{0}^{\infty}\mathrm{d}\xi\,\frac{\xi^2}{c^2}
\left[\bm{\nabla}\chi(\vect{r},\mathrm{i}\xi)\right]\,
\mathrm{Tr}[\ten{G}(\vect{r},\vect{r},\mathrm{i}\xi)]_{\vect{r}'\to\vect{r}}. \end{equation}
It differs from Eq.~(\ref{3.44}) by a surface integral, in general \cite{0663}. Hence, the two force formulas agree in the case of an isolated body where the surface integrals do not contribute to the force and both equations reduce to Eq.~(\ref{3.45}). In contrast to Eq.~(\ref{3.44}), application of Eq.~(\ref{3.44-1}) to any inner, homogeneous part of a body leads to the paradoxical result that the force identically vanishes, because of $\bm{\nabla}\chi(\vect{r},\mathrm{i}\xi)$ $\!=$ $\!0$. In other words, the only atoms that are subject to a force are those at the surface of the body. On the contrary, it is known that the van der Waals forces on all atoms of a body contribute to the Casimir force (Sec.~\ref{sec3.2.1}).
We have seen that within the framework of macroscopic QED, Casimir forces on linearly responding bodies can be expressed in terms of the respective Green tensor of the Maxwell equations for the body-assisted electromagnetic field. Hence, the main problem to be solved in practice is the determination of the Green tensors for the specific systems of interest. Since closed formulas for Green tensors are only available for highly symmetric systems (see, e.g., Ref.~\cite{0217}), approximative and numerical methods are required. For example, one can start from an appropriately chosen Green tensor as zeroth-order approximation to the exact one and perform a Born expansion of the exact Green tensor by iteratively solving the corresponding Dyson-type equation. In particular, iteratively solving Eq.~(\ref{3.38}) yields the Born series
\begin{multline} \label{3.46} \ten{G}(\vect{r},\vect{r}',\omega)
=\overline{\ten{G}}(\vect{r},\vect{r}',\omega)
+\sum_{K=1}^\infty\Bigl(\frac{\omega}{c}\Bigr)^{2K}
\Biggl[\prod_{J=1}^K\int_V\mathrm{d}^3s_J\,
\chi(\vect{s}_J,\omega)\Biggr]\\[.5ex]
\times\overline{\ten{G}}(\vect{r},\vect{s}_1,\omega)\!\cdot\!
\overline{\ten{G}}(\vect{s}_1,\vect{s}_2,\omega)\cdots
\overline{\ten{G}}(\vect{s}_K,\vect{r}',\omega). \end{multline}
\subsubsection{Weakly polarizable bodies, micro-objects and atoms} \label{sec3.2.1}
The force formulas (\ref{3.44}) and (\ref{3.45}) which follow from macroscopic QED without involved microscopic considerations, do not only apply to electric macro-objects but also to micro-objects. Moreover they also allow for studying the limiting case of individual atoms and determining in this way even the dispersion forces with which bodies act on atoms and atoms act on each other in the presence of bodies. To see this, let us consider dielectric bodies which may be typically thought of as consisting of distinguishable (electrically neutral but polarizable) micro-constituents (again briefly referred to as atoms), so that the Clausius--Mossotti relation \cite{0001,1018}
\begin{align} \label{3.47} \chi(\vect{r},\omega) &=\varepsilon_{0}^{-1}\eta(\vect{r})\alpha(\omega)
[1-\eta(\vect{r})\alpha(\omega)/(3\varepsilon_{0})]^{-1}
\nonumber\\[.5ex] &=\varepsilon_{0}^{-1} \eta(\vect{r})\alpha(\omega)\,
[1+\chi(\vect{r},\omega)/3] \end{align}
may be assumed to be valid where $\alpha(\omega)$ is the atomic polarizability and $\eta(\vect{r})$ their number density.\footnote{Note that Eq.~(\ref{3.47}) is consistent with the requirement that both $\alpha(\omega)$ and $\chi(\vect{r},\omega)$ be Fourier transforms of response functions iff $\eta(\vect{r})\alpha(0)/(3\varepsilon_{0})$ $\!<$ $\!1$.} Let $V_\mathrm{I}$ be the volume of an isolated dielectric body of susceptibility $\chi(\vect{r},\omega)$. The Born series (\ref{3.46}) and the Clausius--Mossotti relation (\ref{3.47}) imply that when the body is sufficiently small and/or weakly polarizable, then the force, as given by Eq.~(\ref{3.45}), is essentially determined by the leading-order term proportional to $\chi(\vect{r},\omega)$ $\!\simeq$ $\!\varepsilon_{0}^{-1}\eta(\vect{r})\alpha(\omega)$, so that Eq.~(\ref{3.45}) approximates to\footnote{For a small and/or weakly polarizable body that is an inner part of a larger body, see Refs.~\cite{0392,0663,0661}.}
\begin{equation} \label{3.48} \vect{F}=-\frac{\hbar\mu_0}{2\pi}\int_{V_\mathrm{I}}\mathrm{d}^3r\,
\eta(\vect{r})
\int_{0}^{\infty} \mathrm{d}\xi\,\xi^2\alpha(\mathrm{i}\xi)
\bm{\nabla}\,\mathrm{Tr}
\overline{\ten{G}}{^{(1)}}(\vect{r},\vect{r},\mathrm{i}\xi) \end{equation}
where, according to the decomposition
\begin{equation} \label{3.49} \overline{\ten{G}}(\vect{r},\vect{r'},\omega)
=\overline{\ten{G}}{^{(0)}}(\vect{r},\vect{r'},\omega)
+\overline{\ten{G}}{^{(1)}}(\vect{r},\vect{r'},\omega)
\quad(\vect{r}\in V) \end{equation}
[cf.~Eq.~(\ref{3.22})], $\overline{\ten{G}}{^{(1)}}(\vect{r},\vect{r}',\omega)$ is simply the scattering part of the Green tensor $\overline{\ten{G}}(\vect{r},\vect{r}',\omega)$ of the system without the dielectric body under consideration.
It can be easily seen that Eq.~(\ref{3.48}) may be rewritten as \cite{0663}
\begin{equation} \label{3.50} \vect{F}
=\int_{V_\mathrm{I}}\mathrm{d}^3r\,\eta(\vect{r})\vect{F}(\vect{r}) \end{equation}
where
\begin{equation} \label{3.51} \vect{F}(\vect{r})=-\bm{\nabla}U(\vect{r}) \end{equation}
with
\begin{equation} \label{3.52} U(\vect{r})=\frac{\hbar\mu_0}{2\pi}
\int_{0}^{\infty}\mathrm{d}\xi\,\xi^2 \alpha(\mathrm{i}\xi)
\,\mathrm{Tr}\overline{\ten{G}}{^{(1)}}(\vect{r},\vect{r},\mathrm{i}\xi) \end{equation}
being nothing but the van der Waals potential of a single ground-state atom of polarizability $\alpha(\omega)$ at position $\vect{r}$ in the presence of arbitrary linearly responding bodies at zero temperature (Sec.~\ref{sec4}). Note that in the limiting case when $V_\mathrm{I}\to 0$ and $\eta\to\infty$ but $\eta V_\mathrm{I}$ $\!=$ $\!1$, such that $V_\mathrm{I}$ covers a single atom at position $\vect{r}_{A}$, then $\vect{F}$ reduces to
$\vect{F}(\vect{r}_{A})$---the force acting on a single atom. Note that a relation of the kind~(\ref{3.50}) was already used by Lifshitz to deduce the dispersion force between a single atom and an electric half space from that between a dielectric and an electric half space \cite{0057,0264}.
Equation (\ref{3.50}) reveals that the force acting on a weakly polarizable dielectric body is the sum of the forces acting on all body atoms due to their interaction with other bodies giving rise to the Green tensor $\overline{\ten{G}}(\vect{r},\vect{r}',\omega)$. Let us consider in more detail the interaction of a weakly polarizable body with a second isolated dielectric body of volume $V'_\mathrm{I}$ which is also weakly polarizable [$\chi'(\vect{r},\omega)$ $\!\simeq$ $\!\varepsilon_{0}^{-1}\eta'(\vect{r})\alpha'(\omega)$]. Denoting the Green tensor associated with all remaining bodies except for the two under consideration by $\widetilde{\ten{G}}(\vect{r},\vect{r}',\omega)$, expanding $\overline{\ten{G}}(\vect{r},\vect{r}',\omega)$, by starting from $\widetilde{\ten{G}}(\vect{r},\vect{r}',\omega)$ in the Born series [i.e., using Eq.~(\ref{3.46}) \mbox{with $\ten{G}$ $\!\mapsto$ $\!\overline{\ten{G}}$}, $\overline{\ten{G}}$ $\!\mapsto$ $\!\widetilde{\ten{G}}$ and $V$ $\!\mapsto$ $\!V'_\mathrm{I}$] and again omitting terms of higher than linear order in the susceptibility, Eq.~(\ref{3.48}) leads to \cite{0663}
\begin{equation} \label{3.54} \vect{F}=\int_{V_\mathrm{I}}\mathrm{d}^3r\,\eta(\vect{r})
\int_{V'_\mathrm{I}}\mathrm{d}^3r'\,\eta'(\vect{r})
\vect{F}(\vect{r},\vect{r}') \end{equation}
where
\begin{equation} \label{3.55} \vect{F}(\vect{r},\vect{r}') = - \bm{\nabla}U(\vect{r},\vect{r}'), \end{equation}
is the force with which an atom of polarizability $\alpha'(\omega)$ at position $\vect{r}'$ acts on an atom of polarizability $\alpha(\omega)$ at position $\vect{r}$ with
\begin{equation} \label{3.56} U(\vect{r},\vect{r}')=-\frac{\hbar\mu_0^2}{2\pi}
\int_{0}^{\infty}\mathrm{d}\xi\,\xi^4\alpha(\mathrm{i}\xi)\alpha'(\mathrm{i}\xi)
\mathrm{Tr}\bigl[\widetilde{\ten{G}}(\vect{r},\vect{r}',\mathrm{i}\xi)
\!\cdot\!\widetilde{\ten{G}}(\vect{r}',\vect{r},\mathrm{i}\xi)\bigr] \end{equation}
being the two-atom van der Waals potential \cite{0009,0113}. Equation~(\ref{3.54}) clearly shows that the Casimir force is a volume force and not a surface force as could be suggested on the basis of the Minkowski stress tensor. According to Eq.~(\ref{3.54}), the Casimir force between weakly polarizable dielectric bodies is the sum of all two-atom van der Waals forces between the body atoms---a result which was already obtained by Lifshitz for the special case of two dielectric half spaces \cite{0057,0264}. In fact, such a relation formed the basis of early calculations of dispersion forces between bodies \cite{0642,0641} and it is still used for treating bodies exhibiting surface roughness \cite{0638} or containing excited media \cite{0333,0522}.
The force between a polarizable atom and a magnetizable one can be obtained in a similar way \cite{0491}. For this purpose, we consider the interaction of polarizable atoms contained in the first, weakly polarizable body (volume $V_\mathrm{I}$) with a second, weakly magnetizable body of volume $V'_\mathrm{I}$ and magnetic susceptibility $\zeta(\vect{r},\omega)$ $\!=$ $\!\mu(\vect{r},\omega)\!-\!1$ $\!=$ $\!\mu_0\eta'(\vect{r})\beta'(\omega)$ where $\beta'(\omega)$ denotes the magnetizability of the atoms contained in $V'_\mathrm{I}$. Again expanding the Green tensor $\overline{\ten{G}}(\vect{r},\vect{r}',\omega)$ associated with all the bodies except for the first one by starting from the Green tensor $\widetilde{\ten{G}}(\vect{r},\vect{r}',\omega)$ (associated with all the bodies except for the two under consideration) and retaining only the linear order in $\zeta(\vect{r},\omega)$, one obtains
\begin{equation} \label{3.67} \overline{\ten{G}}(\vect{r},\vect{r}',\omega)
=\widetilde{\ten{G}}(\vect{r},\vect{r}',\omega)
-\int_{V'_\mathrm{I}}\mathrm{d}^3s\,\zeta(\vect{s},\omega)
\left[\widetilde{\ten{G}}(\vect{r},\vect{s},\omega)\!\times\!
\overleftarrow{\bm{\nabla}}_{\!\vect{s}}\right]\!\cdot\!
\bm{\nabla}_{\!\vect{s}}\!\times\!
\widetilde{\ten{G}}(\vect{s},\vect{r}',\omega). \end{equation}
Upon substitution of Eq.~(\ref{3.67}), the force on the first body (\ref{3.48}) can again be written as a sum over two-atom forces, Eqs.~(\ref{3.54}) and (\ref{3.55}) where the potential of a polarizable atom interacting with a magnetizable one reads \cite{0491}
\begin{multline} \label{3.68} U(\vect{r},\vect{r}')=-\frac{\hbar\mu_0^2}{2\pi}
\int_0^\infty\mathrm{d}\xi\,\xi^2
\alpha(\mathrm{i}\xi)\beta'(\mathrm{i}\xi)\\[.5ex]
\times\mathrm{Tr}\!\left\{\left[
\widetilde{\ten{G}}(\vect{r},\vect{s},\mathrm{i}\xi)
\!\times\!\overleftarrow{\bm{\nabla}}_{\!\vect{r}}\right]
\!\cdot\!\bm{\nabla}_{\!\vect{s}}\!\times\!
\widetilde{\ten{G}} (\vect{s},\vect{r},\mathrm{i}\xi)
\right\}_{\vect{s}=\vect{r}'}. \end{multline}
\subsubsection{Many-atom van der Waals interactions} \label{sec3.2.2}
In general, not only two-atom interactions but all many-atom interactions must be taken into account to obtain exact dispersion forces involving macroscopic bodies. To illustrate this point, let us return to Eq.~(\ref{3.52}) and consider the interaction of a single atom with a dielectric body (volume $V$), whose susceptibility $\chi(\vect{r},\omega)$ is given by the Clausius--Mossotti relation (\ref{3.47}). Recall from Sec.~\ref{sec3.2.1} that $\overline{\ten{G}}(\vect{r},\vect{r}',\omega)$ denotes the Green tensor of the whole arrangement of bodies and $\widetilde{\ten{G}}(\vect{r},\vect{r}',\omega)$ is the Green tensor of all (background) bodies except for the one under consideration. Substituting the Born series (\ref{3.46}) [\mbox{$\ten{G}$ $\!\mapsto$ $\!\overline{\ten{G}}$}, $\!\overline{\ten{G}}$ $\!\mapsto$ $\!\widetilde{\ten{G}}$] into Eq.~(\ref{3.52}), one can write the atom--body potential in the form
\begin{equation} \label{3.59} U(\vect{r})=\sum_{K=0}^\infty U_K(\vect{r}) \end{equation}
where
\begin{equation} \label{3.60} U_0(\vect{r})
=\frac{\hbar\mu_0}{2\pi}\int_0^\infty\mathrm{d}\xi\,\xi^2\alpha(\mathrm{i}\xi)
\mathrm{Tr}\widetilde{\ten{G}}^{(1)}(\vect{r},\vect{r},\mathrm{i}\xi) \end{equation}
is the atomic potential due to the background bodies\footnote{Note that $U_0(\vect{r})$ $\!=$ $\!0$ for free-space background, i.e., if there are no further bodies.} and ($K$ $\!\ge$ $\!1$)
\begin{multline} \label{3.61} U_K(\vect{r})=\frac{(-1)^K\hbar\mu_0}{2\pi c^{2K}}
\Biggl[\prod_{J=1}^K\int_V\mathrm{d}^3s_J\,\chi(\vect{s}_J,\mathrm{i}\xi)\Biggr]
\int_0^{\infty}\mathrm{d} \xi\,\xi^{2K+2}\alpha(\mathrm{i}\xi) \\[.5ex]
\times\mathrm{Tr}\bigl[
\widetilde{\ten{G}}(\vect{r},\vect{s}_1,\mathrm{i}\xi)\!\cdot\!
\widetilde{\ten{G}}(\vect{s}_1,\vect{s}_2,\mathrm{i}\xi)\cdots
\widetilde{\ten{G}}(\vect{s}_K,\vect{r},\mathrm{i}\xi)\bigr] \end{multline}
is the contribution to the potential that is of $K$th order in the susceptibility of the dielectric body. To further treat the sum on the right-hand side of Eq.~(\ref{3.61}), the Green tensors therein are decomposed into singular and regular parts according to
\begin{equation} \label{3.62} \widetilde{\ten{G}}(\vect{r},\vect{r}',\omega)
=-\frac{1}{3}\Bigl(\frac{c}{\omega}\Bigr)^2
\delta(\vect{r}-\vect{r}')\ten{I}
+\widetilde{\ten{G}}{}'(\vect{r},\vect{r}',\omega) \end{equation}
and use is made of the Clausius--Mossotti relation (\ref{3.47}). A somewhat lengthy calculation then leads to the result that \cite{0113,0020}
\begin{equation} \label{3.63} U(\vect{r})
=U_0(\vect{r})+\sum_{K=1}^\infty\frac{1}{K!}
\Biggl[\prod_{J=1}^K\int\mathrm{d}^3s_J\,\eta(\vect{s}_J)\Biggr]
U(\vect{r},\vect{s}_1,\ldots,\vect{s}_K) \end{equation}
where
\begin{multline} \label{3.65} U(\vect{r}_1,\vect{r}_2,\ldots,\vect{r}_N)
=\frac{(-1)^{N-1}\hbar\mu_0^N}{(1+\delta_{2N})\pi}
\int_0^\infty\mathrm{d}\xi\,\xi^{2N}
\alpha_1(\mathrm{i}\xi)\ldots\alpha_N(\mathrm{i}\xi)\\[.5ex] \times\mathcal{S}\,\mathrm{Tr}\!\bigl[
\widetilde{\ten{G}}{'}(\vect{r}_1,\vect{r}_2,\mathrm{i}\xi)\!\cdot\!
\widetilde{\ten{G}}{'}(\vect{r}_2,\vect{r}_3,\mathrm{i}\xi)\cdots
\widetilde{\ten{G}}{'}(\vect{r}_N,\vect{r}_1,\mathrm{i}\xi)\bigr] \end{multline}
is the $N$-atom van der Waals potential in the presence of arbitrary linearly responding (background) bodies at zero temperature and hence generalizes the free-space result given in Refs.~\cite{0090,0091}. In Eq.~(\ref{3.65}), the symbol $\mathcal{S}$ introduces the symmetrization prescription
\begin{equation} \label{3.66} \mathcal{S}f(\vect{r}_1,\vect{r}_2,\ldots,\vect{r}_N) = \frac{1}{(2-\delta_{2N})N}\sum_{\Pi\in P(N)}
f(\vect{r}_{\Pi(1)},\vect{r}_{\Pi(2)},\ldots,\vect{r}_{\Pi(N)}) \end{equation}
where $P(N)$ denotes the permutation group of the numbers $1,2,\ldots,N$. Note that $\widetilde{\ten{G}}{'}(\vect{r},\vect{r}',\omega)$ $\!=$ $\!\widetilde{\ten{G}}(\vect{r},\vect{r}',\omega)$ for $\vect{r}$ $\!\neq$ $\!\vect{r}'$. Clearly, for $N$ $\!=$ $\!2$, Eq.~(\ref{3.65}) reduces to the two-atom potential already given, Eq.~(\ref{3.56}).
Equation~(\ref{3.63}) reveals that under very general conditions the interaction of a single ground-state atom with a dielectric body of Clausius--Mossotti susceptibility may be regarded as being due to all many-atom interactions of the atom in question with the body atoms. A relation of this type was first derived for the special case of a homogeneous half space filled with harmonic-oscillator atoms \cite{0119,0120,0048} and later extended to homogeneous dielectric bodies of arbitrary shapes with vacuum background by means of the Ewald--Oseen extinction theorem \cite{0087}. Note that in close analogy to Eq.~(\ref{3.63}), the dispersion force between two dielectric bodies is due to all many-atom interactions of atoms in the first body with atoms in the second one; this was explicitly verified for the cases of two homogeneous half spaces \cite{0120,0048} and spheres \cite{0343} and was also shown for two homogeneous bodies of arbitrary shapes \cite{0342}.
\section{Forces on atoms} \label{sec4}
As demonstrated in Secs.~\ref{sec3.2.1} and \ref{sec3.2.2}, forces on individual ground-state atoms in the presence of linearly responding bodies can be deduced from the forces between dielectric bodies of Clausius--Mossotti type in the limiting case of the bodies being weakly polarizable. Alternatively, these forces can be derived by explicitly studying the interaction of atoms with the body-assisted electromagnetic field according to Sec.~\ref{sec2.2}. This approach to dispersion forces on atoms allows for studying the influence of the internal atomic dynamics on the forces; in particular, excited atoms can also be considered.
\subsection{Ground-state atoms} \label{sec4.1}
Atoms initially prepared in their ground state will remain in this state provided that the body-assisted field is also initially prepared in the ground state. In addition, the atom--field interaction in this case will involve only virtual, off-resonant transitions of the atoms and the body-assisted field. Consequently, dispersion forces on ground-state atoms may adequately be described within the framework of time-independent perturbation theory.
\subsubsection{Single-atom force} \label{sec4.1.1}
Following the idea of Casimir and Polder \cite{0030}, one can derive the force on a single atom at zero temperature from the shift $\Delta E$ of the system's ground-state energy $E$ for given center-of-mass position of the atom which arises from the atom--field coupling. The potential $U(\vect{r}_{A})$, whose negative gradient gives the sought van der Waals force, is the position-dependent part of this energy shift,\footnote{The position-independent part $\Delta^{(0)}E$ is a contribution to the Lamb shift in free space; for a discussion of the Lamb shift within the multipolar coupling scheme, see,~e.g., Ref.~\cite{0013}.}
\begin{equation} \label{4.1} \Delta E=\Delta^{(0)}E+U(\vect{r}_{A}). \end{equation}
It can be seen that the effective center-of-mass Hamiltonian
\begin{equation} \label{4.2} \hat{H}_\mathrm{eff}
=\frac{\hat{\vect{p}}_{A}^2}{2m_{A}}+U(\hat{\vect{r}}_{A}) \end{equation}
leads to the equation of motion
\begin{equation} \label{4.3} \vect{F}(\hat{\vect{r}}_{A}) =m_{A}\ddot{\hat{\vect{r}}}_{A}
=-\frac{1}{\hbar^2}\bigl[\hat{H}_\mathrm{eff},
\bigl[\hat{H}_\mathrm{eff},m_{A}\hat{\vect{r}}_{A}\bigr]\bigr]
=-\bm{\nabla}_{\!\!{A}}U(\hat{\vect{r}}_{A}). \end{equation}
Making use of the interaction Hamiltonian (\ref{2.90-1}), one obtains the ground-state energy shift for a non-magnetic atom in second-order (i.e., leading-order) perturbation theory\footnote{In the following, the multipolar coupling scheme will be employed and the primes discriminating the respective atomic and field variables from the minimal coupling ones will be dropped, for notational convenience. Equation~(\ref{2.90-1}) can be employed, because the second term in Eq.~(\ref{2.90}) gives rise to a contribution of order $v/c$ ($v$, center-of-mass speed) \cite{0008} which can be neglected.} as
\begin{equation} \label{4.4} \Delta E
=\sum_k\sum_{\lambda=e,m}\int\mathrm{d}^3r\,
\mathcal{P}\!\int_0^\infty\mathrm{d}\omega\,
\frac{\bigl|\langle 0|\langle\{0\}|
-\!\hat{\vect{d}}\!\cdot\!\hat{\vect{E}}(\vect{r}_{A})
|\bm{1}_\lambda(\vect{r},\omega)\rangle
|k\rangle\bigr|^2}{-\hbar(\omega_{k0}+\omega)} \end{equation}
[$\mathcal{P}$, principal part;
$|\vect{1}_\lambda(\vect{r},\omega)\rangle$ $\!=$
$\!\hat{\vect{f}}^\dagger_\lambda(\vect{r},\omega)|\{0\}\rangle$]. Using Eqs.~(\ref{2.20})--(\ref{2.24}), (\ref{2.24-1}) and (\ref{2.30b}), one derives
\begin{equation} \label{4.5} \Delta E
=-\frac{\mu_0}{\pi}\sum_k
\mathcal{P}\int_0^\infty
\frac{\mathrm{d}\omega}{\omega_{k0}+\omega}\,\omega^2
\vect{d}_{0k}\!\cdot\!\mathrm{Im}\,\ten{G}
(\vect{r}_{A},\vect{r}_{A},\omega)\!\cdot\!\vect{d}_{k0} \end{equation}
where $\ten{G}(\vect{r},\vect{r}',\omega)$ is the Green tensor of the body configuration considered. By discarding the position-independent contribution $\Delta E^{(0)}$ associated with $\ten{G}^{(0)}(\vect{r}_{A},\vect{r}_{A},\omega)$ [recall Eqs.~(\ref{3.22}) and (\ref{4.1})] which may be thought of as being already included in the unperturbed energy, the van der Waals potential can be written in the form \cite{0008,0012,0018}
\begin{equation} \label{4.6-1} U(\vect{r}_{A})=\frac{\hbar\mu_0}{2\pi}
\int_0^\infty\mathrm{d}\xi\,\xi^2\mathrm{Tr}\bigl[\bm{\alpha}(\mathrm{i}\xi)\!\cdot\!
\ten{G}^{(1)}(\vect{r}_{A},\vect{r}_{A},\mathrm{i}\xi)\bigr] \end{equation}
where
\begin{equation} \label{4.7-1} \bm{\alpha}(\omega)
=\lim_{\epsilon\to 0}\frac{2}{\hbar}\sum_k
\frac{\omega_{k0}\vect{d}_{0k}\tprod\vect{d}_{k0}}
{\omega_{k0}^2-\omega^2-\mathrm{i} \omega\epsilon} \end{equation}
is the ground-state polarizability of the atom. For isotropic atoms, it simplifies to
\begin{equation} \label{4.7} \bm{\alpha}(\omega)=\alpha(\omega)\ten{I}
=\lim_{\epsilon\to 0}\frac{2}{3\hbar}\sum_k
\frac{\omega_{k0}|\vect{d}_{0k}|^2}
{\omega_{k0}^2-\omega^2-\mathrm{i} \omega\epsilon}\,\ten{I}, \end{equation}
so the potential simplifies to
\begin{equation} \label{4.6} U(\vect{r}_{A})=\frac{\hbar\mu_0}{2\pi}
\int_0^\infty\mathrm{d}\xi\,\xi^2\alpha(\mathrm{i}\xi)\,\mathrm{Tr}
\ten{G}^{(1)}(\vect{r}_{A},\vect{r}_{A},\mathrm{i}\xi). \end{equation}
The perturbative result hence agrees with what has been inferred from the force on weakly polarizable bodies and renders an explicit expression for the polarizability. Note that the scattering Green tensor $\ten{G}^{(1)}$ in Eq.~(\ref{4.6}) has exactly the same meaning as $\overline{\ten{G}}$ in Eq.~(\ref{3.52}). {F}rom Eq.~(\ref{4.6}) together with Eq.~(\ref{4.7}) it can be seen that the potential can be given in the equivalent form
\begin{equation} \label{4.8} U(\vect{r}_{A})
=-\frac{\hbar\mu_0}{2\pi}\int_0^\infty\mathrm{d}\omega\,\omega^2
\mathrm{Im}\bigl[\alpha(\omega)\mathrm{Tr}
\ten{G}^{(1)}(\vect{r}_{A},\vect{r}_{A},\omega)\bigr] \end{equation}
which allows for a simple physical interpretation of the force as being due to correlation of the fluctuating electromagnetic field with the corresponding induced electric dipole of the atom plus the correlation of the fluctuating electric dipole with its induced electric field \cite{0046}. It should be pointed out that an analogous treatment based on the minimal-coupling Hamiltonian (\ref{2.77}) leads to the formally same result \cite{0008} where of course the unperturbed eigenstates and energies occurring in the polarizability (\ref{4.7-1}) are now determined by the atomic Hamiltonian (\ref{2.55}) in place of (\ref{2.84}). Needless to say that both results are approximations to the same Hamiltonian of the total system. Bearing in mind that the ground-state energy shift is entirely due to virtual, off-resonant transitions, it is crucial to retain the $\vect{A}^2$~term in the minimal coupling scheme which contributes to the ground-state energy shift already in first-order perturbation theory.
Equation~(\ref{4.6-1}) gives the potential of a single ground-state atom in the presence of an arbitrary arrangement of linearly responding bodies at zero temperature in terms of the polarizability of the atom and the Green tensor of the body-assisted electromagnetic field. A relation of this kind was first derived for arbitrary electric bodies on the basis of linear-response theory \cite{0035,0039,0041}, recall Eq.~(\ref{1.10}); alternatively, it was obtained from a QED path-integral approach \cite{0050} and semiclassical considerations \cite{0375}. A perturbative derivation based on macroscopic QED very similar to that presented here is given in Refs.~\cite{0186,0439,0187}. The linear-response approach has also been applied to magneto-electric bodies \cite{0043} and finite temperatures \cite{0037,0046,0394}. Note that in close analogy to the case of the Casimir force, the single-atom potential at finite temperature can be obtained from the zero-temperature result~(\ref{4.6-1}) or (\ref{4.8}) by making the replacements~(\ref{3.26-3}) or (\ref{3.26-2}), respectively.
\subsubsection{Two-atom force} \label{sec4.1.2}
The interaction potential of two polarizable ground-state atoms in the presence of linearly responding bodies giving rise to the Green tensor $\ten{G}(\vect{r},\vect{r}',\omega)$ can also be obtained by means of time-independent perturbation theory. The leading contribution is now of fourth order in the atom--field interaction and a somewhat lengthy calculation yields \cite{0009,0113}
\begin{multline} \label{4.9-1} U(\vect{r}_{A},\vect{r}_{B})
=-\frac{\hbar\mu_0^2}{2\pi}\int_0^\infty\mathrm{d}\xi\,\xi^4\\[.5ex] \times\mathrm{Tr}[\bm{\alpha}_{A}(\mathrm{i}\xi)\!\cdot\!
\ten{G}(\vect{r}_{A},\vect{r}_{B},\mathrm{i}\xi)
\!\cdot\!\bm{\alpha}_{B}(\mathrm{i}\xi)
\!\cdot\!\ten{G}(\vect{r}_{B},\vect{r}_{A},\mathrm{i}\xi)] \end{multline}
which for isotropic atoms reduces to
\begin{multline} \label{4.9} U(\vect{r}_{A},\vect{r}_{B})
=-\frac{\hbar\mu_0^2}{2\pi}\int_0^\infty\mathrm{d}\xi\,
\xi^4\alpha_{A}(\mathrm{i}\xi)\alpha_{B}(\mathrm{i}\xi)\\[.5ex] \times\mathrm{Tr}[\ten{G}(\vect{r}_{A},\vect{r}_{B},\mathrm{i}\xi)
\!\cdot\!\ten{G}(\vect{r}_{B},\vect{r}_{A},\mathrm{i}\xi)]. \end{multline}
Note that Eq.~(\ref{4.9}) agrees with Eq.~(\ref{3.56}) [$\widetilde{\ten{G}}{'}(\vect{r}_1,\vect{r}_2,\omega)$ $\!\mapsto$ $\!\ten{G}(\vect{r}_{A},\vect{r}_{B},\omega)$ for \mbox{$\vect{r}_1$ $\!\neq$ $\!\vect{r}_2$}]. The total force acting on atom ${A}$ (${B}$) can be obtained by supplementing the single-atom force $\vect{F}(\vect{r}_{{A}({B})})$ with the two-atom force
\begin{equation} \label{4.10} \vect{F}(\vect{r}_{{A}({B})},\vect{r}_{{B}({A})})
=-\bm{\nabla}_{\!\!{A}({B})}U(\vect{r}_{A},\vect{r}_{B}) \end{equation}
where in general $\vect{F}(\vect{r}_{A},\vect{r}_{B})$ $\!\neq$ $-\vect{F}(\vect{r}_{B},\vect{r}_{A})$, due to the presence of the bodies.
Equations~(\ref{4.9-1})--(\ref{4.10}) form a general basis for calculating two-atom potentials in the presence of linearly responding bodies at zero temperature. An equation of the type~(\ref{4.9}) was first derived for the case of electric bodies by means of linear-response theory \cite{0036}. Derivations based on semiclassical models of harmonic-oscillator atoms interacting in the presence of perfectly conducting \cite{0092} and electric bodies~\cite{0375} were given and extended to allow for finite temperatures \cite{0093}. Equation (\ref{4.9}) has been used to study the interaction of two atoms embedded in an electrolyte \cite{0351}, situated near perfectly conducting \cite{0093}, non-local metallic \cite{0518}, electric \cite{0036,0653,0523} and magneto-electric half spaces \cite{0009,0491}, placed inside perfectly conducting \cite{0092,0093} and electric planar cavities \cite{0108}.
Let us consider the simplest case of two atoms in free space where the single-atom force identically vanishes and the two-atom potential (\ref{4.9}) can be calculated by using the free-space Green tensor $\ten{G}(\vect{r},\vect{r}',\omega)$ $\!=$ $\!\ten{G}_\mathrm{free}(\vect{r},\vect{r}',\omega)$
(App.~\ref{appA}), leading to ($r$ $\!=$ $|\vect{r}_{A}$ $\!-$
$\!\vect{r}_{B}|$)
\begin{equation} \label{4.11} U(\vect{r}_{A},\vect{r}_{B})
=-\frac{\hbar}{32\pi^3\varepsilon_0^2r^6}
\int_0^\infty\mathrm{d}\xi\,
\alpha_A(\mathrm{i}\xi)\alpha_B(\mathrm{i}\xi)g(\xi r/c) \end{equation}
where
\begin{equation} \label{4.12} g(x)=2\mathrm{e}^{-2x}\bigl(3+6x+5x^2+2x^3+x^4\bigr), \end{equation}
which shows that the interaction between two polarizable ground-state atoms is always attractive. Equations~(\ref{4.11}) and (\ref{4.12}) are in agreement with the famous result of Casimir and Polder \cite{0030}, whose derivation was based on fourth-order perturbation theory and normal-mode QED. The problem has been reconsidered many times, inter alia within the frameworks of normal-mode QED \cite{0011,0521,0498,0056,0320,0051,0201,0200} and linear-response theory \cite{0035}; and it has even become a common textbook example \cite{0325,0323,0007}.
In the retarded limit where $r$ $\!\gg$ $\!c/\omega_{\mathrm{min}}$ ($\omega_{\mathrm{min}}$ denoting the minimum of the relevant resonance frequencies of atoms ${A}$ and ${B}$), the function $g(\xi r/c)$ effectively limits the $\xi$~integral in Eq.~(\ref{4.11}) to a range where $\alpha_{{A}({B})}(\mathrm{i}\xi)$ $\!\simeq$ $\!\alpha_{{A}({B})}(0)$, so the potential approaches
\begin{equation} \label{4.13} U(\vect{r}_{A},\vect{r}_{B}) =-\frac{23\hbar c\alpha_{A}(0)\alpha_{B}(0)}
{64\pi^3\varepsilon_0^2r^7}\,, \end{equation}
as already pointed out in Ref.~\cite{0030}, cf. Eq.~(\ref{1.7}). In the non-retarded limit where \mbox{$r$ $\!\ll$ $\!c/\omega_\mathrm{max}$} ($\omega_{\mathrm{max}}$ denoting the maximum of the relevant resonance frequencies of atoms $A$ and $B$), the integral is limited by the polarizabilities $\alpha_{{A}({B})}(\mathrm{i}\xi)$ to a range where $g(\mathrm{i}\xi)$ $\!\simeq$ $g(0)$ $\!=$ $\!6$, leading to
\begin{equation} \label{4.14} U(\vect{r}_{A},\vect{r}_{B})=-\frac{3\hbar}{16\pi^3\varepsilon_0^2r^6}
\int_0^\infty\mathrm{d}\xi\,\alpha_{A}(\mathrm{i}\xi)\alpha_{B}(\mathrm{i}\xi). \end{equation}
Upon recalling Eq.~(\ref{4.7}), one may easily verify that this non-retarded asymptote is nothing but the well-known London potential~(\ref{1.2}) which was originally obtained from a perturbative treatment of the Coulomb interaction \cite{0374} (cf.~Refs.~\cite{0510,0507} for similar derivations).
It is illustrative to compare the potential between two polarizable atoms with the potential between a polarizable atom $A$ of polarizability $\alpha_{A}(\omega)$ and a magnetizable atom $B$ of magnetizability $\beta_{B}(\omega)$ which, according to Eq.~(\ref{3.68}), is given by
\begin{multline} \label{4.15} U(\vect{r}_{A},\vect{r}_{B})
=-\frac{\hbar\mu_0^2}{2\pi}
\int_0^\infty\mathrm{d}\xi\,\xi^2
\alpha_{A}(\mathrm{i}\xi)\beta_{B}(\mathrm{i}\xi)\\[.5ex]
\times\mathrm{Tr}\bigl\{\bigl[
\ten{G}(\vect{r}_{A},\vect{r},\mathrm{i}\xi)
\!\times\!\overleftarrow{\bm{\nabla}}\bigr]
\!\cdot\!\bm{\nabla}\!\times\!
\ten{G} (\vect{r},\vect{r}_{A},\mathrm{i}\xi)
\bigr\}_{\vect{r}=\vect{r}_{B}}. \end{multline}
When the two atoms are in free space so that $\ten{G}(\vect{r},\vect{r}',\omega)$ $\!=$ $\!\ten{G}_\mathrm{free}(\vect{r},\vect{r}',\omega)$, then Eq.~(\ref{4.15}) reads
\begin{equation} \label{4.16} U(\vect{r}_{A},\vect{r}_{B}) = \frac{\hbar\mu_0^2}{32\pi^3r^4}
\int_0^{\infty}\mathrm{d}\xi\,\xi^2
\alpha_{A}(\mathrm{i}\xi)\beta_{B}(\mathrm{i}\xi) h(\xi r/c) \end{equation}
where
\begin{equation} \label{4.17}
h(x) = 2\mathrm{e}^{-2x}\bigl(1+2x+x^2\bigr) \end{equation}
which is in agreement with results found on the basis of normal-mode QED \cite{0094,0096}. In contrast to the attractive interaction between two polarizable atoms, the interaction between a polarizable and a magnetizable atom is always repulsive, as can be easily seen from Eq.~(\ref{4.16}) together with Eq.~(\ref{4.17}). In particular in the retarded and non-retarded limits, respectively, Eq.~(\ref{4.16}) reduces to
\begin{equation} \label{4.19} U(\vect{r}_{A},\vect{r}_{B})
=\frac{7\hbar c\mu_0\alpha_{A}(0)\beta_{B}(0)}
{64\pi^3\varepsilon_0 r^7} \end{equation}
and
\begin{equation} \label{4.20} U(\vect{r}_{A},\vect{r}_{B}) =\frac{\hbar\mu_0^2}{16\pi^3r^4}
\int_0^\infty\mathrm{d}\xi\,\xi^2
\alpha_{A}(\mathrm{i}\xi)\beta_{B}(\mathrm{i}\xi) \end{equation}
which was already given in Refs.~\cite{0089,0095} and \cite{0499,0121}.
Comparing Eqs.~(\ref{4.13}) and (\ref{4.19}), we see that in the retarded limit the absolute value of the force between two polarizable atoms and that between a polarizable and a magnetizable atom follow the same $1/r^8$~power law with the strength being weaker in the latter case by a factor $7/23$. Comparison of Eqs.~(\ref{4.14}) and (\ref{4.20}) shows that in the non-retarded limit the absolute value of the force between a polarizable and a magnetizable atom which follows a $1/r^5$~power law, is more weakly diverging than that between two polarizable atoms which follows a $1/r^7$~power law.
\subsubsection{Atom in a planar structure} \label{sec4.1.3}
Let us return to Eq.~(\ref{4.6-1}) for the potential of a single ground-state atom in the presence of an arbitrary arrangement of linearly responding bodies at zero temperature. It has been used to calculate the potential for a variety of particular geometries, including different planar structures (see below) as well as perfectly conducting \cite{0110,0346}, non-local metallic \cite{0270,0060}, dielectric \cite{0110,0069,0077,0349,0372,0017} and magneto-electric spheres \cite{0113}; dielectric \cite{0284,0077,0395,0316} and non-local metallic cylinders \cite{0040,0397}; magneto-dielectric rings \cite{0020}; electric cylindrical shells \cite{0186,0439,0187}; electric \cite{0313} and non-local metallic spherical cavities \cite{0396}; electric cylindrical cavities \cite{0316} and perfectly conducting wedge-shaped cavities \cite{0069,0372}.
To illustrate the theory, let us consider an atom placed in a free-space region between two planar magneto-electric walls, as defined by Eqs.~(\ref{3.27}) and (\ref{3.28}) with $\varepsilon(\omega)$ $\!=$ $\!\mu(\omega)$ $\!=$ $\!1$. Inserting the Green tensor for this system (App.~\ref{appA}) into Eq.~(\ref{4.6}) leads to the single-atom potential \cite{0012,0019}
\begin{multline} \label{4.21} U(z_{A}) =\frac{\hbar\mu_0}{8\pi^2}
\int_0^\infty\mathrm{d}\xi\,\xi^2\alpha(\mathrm{i}\xi)
\int_0^\infty\mathrm{d} q\,\frac{q}{b}\biggl\{
\mathrm{e}^{-2bz_{A}}\biggl[\frac{r_{s-}}{D_s}
-\biggl(1+2\,\frac{q^2c^2}{\xi^2}\biggr)
\frac{r_{p-}}{D_p}\biggr]\\[.5ex] +\,\mathrm{e}^{-2b(d-z_{A})}
\biggl[\frac{r_{s+}}{D_s}-\biggl(1+2\,\frac{q^2c^2}{\xi^2}\biggr)
\frac{r_{p+}}{D_p}\biggr]\biggr\} \end{multline}
where $z_{A}$ is the separation of the atom from the left wall, $d$ is the separation of the two walls, $b$ and $D_\sigma$ are given by Eqs.~(\ref{3.32}) and (\ref{3.33}), respectively, with $n(\mathrm{i}\xi)$ $\!=$ $\!1$, and $r_{\sigma\pm}$ $\!=$ $r_{\sigma\pm}(\xi,q)$ are again the reflection coefficients associated with the left/right walls. If the atom is placed near a single wall, say the right wall is missing, Eq.~(\ref{4.21}) reduces to ($r_{\sigma}$ $\!\equiv$ $\!r_{\sigma -}$)
\begin{equation} \label{4.22} U(z_{A})=\frac{\hbar\mu_0}{8\pi^2}
\int_0^\infty\mathrm{d}\xi\,\xi^2
\alpha(\mathrm{i}\xi)\int_0^\infty\mathrm{d} q\,
\frac{q}{b}\,\mathrm{e}^{-2bz_{A}}\biggl[r_s
-\biggl(1+2\frac{q^2c^2}{\xi^2}\biggr)r_p\biggr]. \end{equation}
\paragraph{Perfectly reflecting plate} \label{sec4.1.3.1}
Consider first an atom placed near a perfectly reflecting electric (i.e., perfectly conducting) plate, $r_p$ $\!=$ $\!-r_s$ $\!=$ $\!1$. By changing the integration variable in Eq.~(\ref{4.22}) from $q$ to $b$ [Eq.~(\ref{3.32})], the resulting integral can be performed to obtain
\begin{equation} \label{4.23} U(z_{A})=-\frac{\hbar}{16\pi^2\varepsilon_0z_{A}^3}
\int_0^\infty\mathrm{d}\xi\,\alpha(\mathrm{i}\xi)
\,\mathrm{e}^{-2\xi z_{A}/c}
\Biggl[1+2\biggl(\frac{\xi z_{A}}{c}\biggr)
+2\biggl(\frac{\xi z_{A}}{c}\biggr)^2\Biggr], \end{equation}
in agreement with the result found by Casimir and Polder on the basis of normal-mode QED \cite{0030} (cf.~also Refs.~\cite{0047,0056}) which has been reproduced by means of linear-response theory \cite{0035,0039,0041} and dynamical image-dipole treatments \cite{0321}. In the retarded limit, $z_{A}$ $\!\gg$ $\!c/\omega_{\mathrm{min}}$, the exponential $\mathrm{e}^{-2\xi z_{A}/c}$ effectively limits the $\xi$~integral to the region where $\alpha(\mathrm{i}\xi)$ $\!\simeq$ $\alpha(0)$, so the potential approaches
\begin{equation} \label{4.24} U(z_A)=-\frac{3\hbar c\alpha(0)}{32\pi^2\varepsilon_0z_A^4}\,, \end{equation}
as already noted in Refs.~\cite{0030,0325,0054}, cf. Eq.~(\ref{1.8}). In the non-retarded limit, $z_{A}$ $\!\ll$ $\!c/\omega_\mathrm{max}$, the polarizability $\alpha(\mathrm{i}\xi)$ restricts the integration to the region where $\xi z_{A}/c$ $\!\simeq$ $\!0$, leading to
\begin{equation} \label{4.25}
U(z_A)=-\frac{1}{48\pi\varepsilon_0z_A^3}\sum_k|\vect{d}_{0k}|^2
=-\frac{\langle 0|\hat{\vect{d}}^2|0\rangle}
{48\pi\varepsilon_0z_A^3}\,, \end{equation}
in agreement with the result~(\ref{1.3}) obtained by Lennard-Jones on the basis of an electrostatic calculation \cite{0022} [recall Eq.~(\ref{4.7})].
In contrast, the potential of an atom in front of an infinitely permeable plate is repulsive, as can be seen by setting $r_p$ $\!=$ $\!-r_s$ $\!=$ $\!-1$ in Eq.~(\ref{4.22}),
\begin{equation} \label{4.26} U(z_{A})=\frac{\hbar}{16\pi^2\varepsilon_0z_{A}^3}
\int_0^\infty\mathrm{d}\xi\,\alpha(\mathrm{i}\xi)
\mathrm{e}^{-2\xi z_{A}/c}
\Biggl[1+2\biggl(\frac{\xi z_A}{c}\biggr)
+2\biggl(\frac{\xi z_A}{c}\biggr)^2\Biggr]\,. \end{equation}
It approaches
\begin{equation} \label{4.27} U(z_A)=\frac{3\hbar c\alpha(0)}{32\pi^2\varepsilon_0z_A^4} \end{equation}
in the retarded limit (cf. also Ref.~\cite{0330}) and
\begin{equation} \label{4.28} U(z_A)=
\frac{\langle 0|\hat{\vect{d}}^2|0\rangle} {48\pi\varepsilon_0z_A^3} \end{equation}
in the non-retarded limit.
The different signs of the non-retarded potentials (\ref{4.25}) and (\ref{4.28}) in the cases of a perfectly reflecting electric and a perfectly reflecting magnetic plate, respectively, can be understood from an image-dipole model \cite{0022}. The non-retarded potential can be regarded as being due to the Coulomb interaction of an electric dipole $\hat{\vect{d}}$ $\!=$ $(\hat{d}_x,\hat{d}_y,\hat{d}_z)$ situated at distance $z_{A}$ from the plate with its image $\hat{\vect{d}}'$ $\!=$ $(-\hat{d}_x,-\hat{d}_y,\hat{d}_z)$ in the plate [Fig~\ref{fig1}(a)].
\begin{figure}
\caption{ The image dipole construction for an (a) electric (b) magnetic dipole in front of a perfectly reflecting electric plate, is shown. }
\label{fig1}
\end{figure}
The average interaction energy of the dipole and its image hence reads \cite{0001}\footnote{The factor $1/2$ in Eq.~(\ref{4.29}) accounts for the fact that the second dipole is induced by the first one.}
\begin{equation} \label{4.29} U(z_{A})
=\frac{1}{2}\,\frac{\langle 0|\hat{\vect{d}}\!\cdot\!\hat{\vect{d}}'
-3\hat{d}_z\hat{d'_z}|0\rangle}
{4\pi\varepsilon_0(2z_{A})^3}
=-\frac{\langle 0|\hat{\vect{d}}^2+\hat{d}_z^2|0\rangle}
{64\pi\varepsilon_0z_{A}^3}
=-\frac{\langle 0|\hat{\vect{d}}^2|0\rangle}
{48\pi\varepsilon_0z_{A}^3} \end{equation}
[$\langle 0|\hat{d}_x^2|0\rangle$ $\!=$
$\!\langle 0|\hat{d}_y^2|0\rangle$ $\!=$
$\!\langle 0|\hat{d}_z^2|0\rangle$ $\!=$
$(1/3)\langle 0|\hat{\vect{d}}^2|0\rangle$], in agreement with Eq.~(\ref{4.25}).
Since the interaction of a polarizable atom with an infinitely permeable plate is equivalent to the interaction of a magnetizable atom with a perfectly reflecting electric plate by virtue of the duality of electric and magnetic fields, a magnetic dipole $\hat{\vect{m}}$ $\!=$ $(\hat{m}_x,\hat{m}_y,\hat{m}_z)$ in front of a perfectly reflecting electric plate can be considered. A magnetic dipole behaves like a pseudo-vector under reflection, so its image is given by $\hat{\vect{m}}'$ $\!=$ $(\hat{m}_x,\hat{m}_y,-\hat{m}_z)$ [Fig~\ref{fig1}(b)]. The interaction energy of the magnetic dipole and its image reads
\begin{equation} \label{4.30} U(z_{A})
=\frac{1}{2}\,\frac{\langle 0|\hat{\vect{m}}\!\cdot\!\hat{\vect{m}}'
-3\hat{m}_z\hat{m'_z}|0\rangle}
{4\pi\varepsilon_0(2z_{A})^3}
=\frac{\langle 0|\hat{\vect{m}}^2|0\rangle}
{48\pi\varepsilon_0z_{A}^3} \end{equation}
which by means of a duality transformation is equivalent to Eq.~(\ref{4.28}). The different signs of the potentials (\ref{4.25}) and (\ref{4.28}) can thus be understood from the different reflection behavior of electric and magnetic dipoles.
\paragraph{Magneto-electric half space} \label{sec4.1.3.2}
To be more realistic, let us next consider an atom in front of a semi-infinite half space of given $\varepsilon(\omega)$ and $\mu(\omega)$. Upon substitution of the Fresnel reflection coefficients~(\ref{A.10}), Eq.~(\ref{4.22}) takes the form \cite{0012,0019}
\begin{multline} \label{4.31} U(z_{A}) =\frac{\hbar\mu_0}{8\pi^2}
\int_0^\infty\mathrm{d}\xi\,\xi^2\alpha(\mathrm{i}\xi)
\int_0^\infty\mathrm{d} q\,\frac{q}{b}\,\mathrm{e}^{-2bz_{A}}
\biggl[\frac{\mu(\mathrm{i}\xi)b-b_1}
{\mu(\mathrm{i}\xi)b+b_1}\\[.5ex] -\biggl(1+2\frac{q^2c^2}{\xi^2}\biggr)
\frac{\varepsilon(\mathrm{i}\xi)b-b_1}
{\varepsilon(\mathrm{i}\xi)b+b_1}\biggr] \end{multline}
with $b_1$ $\!\equiv$ $\!b^1_-$ defined as in Eq.~(\ref{A.9}), in agreement with the result found by means of linear-response theory \cite{0043}. From Eq.~(\ref{4.31}), the results obtained by means of normal-mode QED \cite{0051,0031,0330} and linear-response theory \cite{0036,0039,0041} for an electric half space can also be recovered.
One can show that in the retarded limit $z_A$ $\!\gg$ $\!c/\omega_\mathrm{min}$ (with $\omega_\mathrm{min}$ being the minimum of all relevant atom and medium resonance frequencies) the potential takes the asymptotic form \cite{0012,0019}\footnote{Obviously this limit does not apply for metals where the condition can never be fulfilled. Similarly, Eqs.~(\ref{4.33}), (\ref{4.39}) and (\ref{4.40}) only hold for dielectrics.}
\begin{multline} \label{4.32} U(z_{A})=-\frac{3\hbar c\alpha(0)}{64\pi^2\varepsilon_0z_{A}^4}
\int_{1}^\infty\mathrm{d} v\,
\biggl[\Bigl(\frac{2}{v^2}-\frac{1}{v^4}\Bigr)
\frac{\varepsilon(0)v-\sqrt{\varepsilon(0)\mu(0)-1+v^2}}
{\varepsilon(0)v+\sqrt{\varepsilon(0)\mu(0)-1+v^2 }}\\[.5ex] -\,\frac{1}{v^4}\,
\frac{\mu(0)v-\sqrt{\varepsilon(0)\mu(0)-1+v^2}}
{\mu(0)v+\sqrt{\varepsilon(0)\mu(0)-1+v^2}}\,\biggr] \end{multline}
which can be attractive or repulsive, depending on the strengths of the competing magnetic and electric properties of the half space. Figure~\ref{fig2} shows the borderline between attractive and repulsive potentials in the $\varepsilon(0)\mu(0)$-plane. In particular, repulsion occurs iff $\mu(0)\!-\!1$ $\!>$ $\!3.29[\varepsilon(0)\!-\!1]$ or $\mu(0)$ $\!>$ $\!5.11\varepsilon(0)$ for weak and strong magneto-dielectric properties, respectively.
\begin{figure}
\caption{ Borderline of attractive and repulsive retarded potentials of a ground-state atom in front of a magneto-dielectric half space. }
\label{fig2}
\end{figure}
In the non-retarded limit, $n(0)z_A$ $\!\ll$ $\!c/\omega_\mathrm{max}$ [$\omega_\mathrm{max}$, maximum of all relevant atom and medium resonance frequencies; $n(0)$ $\!=$ $\!\sqrt{\varepsilon(0)\mu(0)}$], the situation is more complex, because electric and magnetic medium properties give rise to potentials with different asymptotic power laws. In particular, the potential approaches
\begin{equation} \label{4.33} U(z_{A})=-\frac{\hbar}{16\pi^2\varepsilon_0z_{A}^3}
\int_0^\infty\mathrm{d}\xi\,\alpha(\mathrm{i}\xi)\,
\frac{\varepsilon(\mathrm{i}\xi)-1}{\varepsilon(\mathrm{i}\xi)+1} \end{equation}
in the case of a purely dielectric half space---in agreement with the result~(\ref{1.4}) found by considering the Coulomb interaction and using image-dipole methods \cite{0277} or linear-response theory \cite{0027}---and
\begin{equation} \label{4.34} U(z_{A})=\frac{\hbar}{32\pi^2\varepsilon_0z_{A}}
\int_0^\infty\mathrm{d}\xi\,\biggl(\frac{\xi}{c}\biggr)^2\alpha(\mathrm{i}\xi)\,
\frac{[\mu(\mathrm{i}\xi)-1][\mu(\mathrm{i}\xi)+3]}{\mu(\mathrm{i}\xi)+1} \end{equation}
\begin{figure}\label{fig3}
\end{figure}
in the case of a purely magnetic one \cite{0012,0019}. Thus for a magneto-dielectric half space the attractive $1/z_{A}^3$ potential associated with the polarizability of the half space will always dominate the repulsive $1/z_{A}$ potential related to its magnetizability. It should be noted that the non-retarded limit is in general incompatible with the limit of perfect reflectivity considered in Sec.~\ref{sec4.1.3.1}. So, Eq.~(\ref{4.34}) does not approach Eq.~(\ref{4.28}) as $\mu(\mathrm{i}\xi)$ tends to infinity. On the contrary, Eq.~(\ref{4.33}) converges to Eq.~(\ref{4.25}) as $\varepsilon(\mathrm{i}\xi)$ tends to infinity~\cite{0296}.
Combining the observations from the retarded and non-retarded limits, one may thus expect a potential barrier, provided that the permeability of the half space is sufficiently large \cite{0012,0019}. This is illustrated in Fig.~\ref{fig3} where the potential of a two-level atom as a function of its distance from the half space, is shown for various values of the (static) permeability. In the figure, the permittivity and permeability of the half space have been assumed to be of Drude--Lorentz type,
\begin{equation} \label{4.35} \varepsilon(\omega)=1+\frac{\omega_{\mathrm{P}e}^2}
{\omega^2_{{\mathrm{T}e}}-\omega^2-\mathrm{i}\omega\gamma_e}\,, \quad \mu(\omega)=1+\frac{\omega_{\mathrm{P}m}^2}
{\omega^2_{\mathrm{T}m}-\omega^2-\mathrm{i}\omega\gamma_m}\,. \end{equation}
{F}rom the figure it is seen that with increasing value of $\mu(0)$, a potential barrier begins to form at intermediate distances, as expected. It is shifted to smaller distances and increases in height as the value of $\mu(0)$ is further increased.
\paragraph{Plate of finite thickness} \label{sec4.1.3.3}
Consider now an atom in front of magneto-electric plate of finite thickness $d$. Evaluating the relevant reflection coefficients (\ref{A.7}) and (\ref{A.8}) ($d$ $\!\equiv$ $\!d_-^1$), one finds that Eq.~(\ref{4.22}) takes the form \cite{0012,0019}
\begin{multline} \label{4.37} U(z_{A}) =\frac{\hbar\mu_0}{8\pi^2}
\int_0^\infty\mathrm{d}\xi\,\xi^2\alpha(\mathrm{i}\xi)
\int_0^\infty\mathrm{d} q\,\frac{q}{b}\,\mathrm{e}^{-2bz_{A}}\\[.5ex]
\times\biggl\{\frac{[\mu^2(\mathrm{i}\xi)b^2-b_1^2]\tanh(b_1d)}
{2\mu(\mathrm{i}\xi)b b_1+[\mu^2(\mathrm{i}\xi)b^2+b_1^2]\tanh(b_1d)}\\[.5ex] -\,\biggl(1+2\,\frac{q^2c^2}{\xi^2}\biggr)
\frac{[\varepsilon^2(\mathrm{i}\xi)b^2-b_1^2]\tanh(b_1d)}
{2\varepsilon(\mathrm{i}\xi)b b_1+
[\varepsilon^2(\mathrm{i}\xi)b^2+b_1^2]\tanh(b_1d)}
\biggr\} \end{multline}
which reduces to the result in Ref.~\cite{0032} in the special case of an electric plate. For an asymptotically thick plate, $d$ $\!\gg$ $\!z_A$, the exponential factor restricts the integral in Eq.~(\ref{4.37}) to a region where \mbox{$b_1d$ $\!\sim$ $\!d/(2z_A)$ $\!\gg$ $\!1$}. One may hence make the approximation $\tanh(b_1d)$ $\!\simeq$ $\!1$, leading back to Eq.~(\ref{4.31}) which demonstrates that the semi-infinite half space is a good model provided that $d$ $\!\gg$ $\!z_A$.
\begin{figure}
\caption{ The potential of a ground-state two-level atom in front of a magneto-dielectric plate, Eq.~(\ref{4.37}), is shown as a function of the atom--plate separation for different values of the plate thickness $d$ [$\mu(0)$ $\!=$ $\!5$; whereas all other parameters are the same as in Fig.~\ref{fig2}]. }
\label{fig4}
\end{figure}
On the contrary, in the limit of an asymptotically thin plate, $n(0)d$ $\!\ll$ $\!z_A$, the approximation $b_1d$ $\!\ll$ $\!1$ results in \cite{0012,0019}
\begin{multline} \label{4.38} U(z_{A}) =\frac{\hbar\mu_0d}{8\pi^2}
\int_0^\infty\mathrm{d}\xi\,\xi^2\alpha(\mathrm{i}\xi)
\int_0^\infty\mathrm{d} q\,\frac{q}{b}\,\mathrm{e}^{-2bz_{A}}
\biggl[\frac{\mu^2(\mathrm{i}\xi)b^2-b_1^2}{2\mu(\mathrm{i}\xi)b}\\[.5ex] -\,\biggl(1+2\frac{q^2c^2}{\xi^2}\biggr)
\frac{\varepsilon^2(\mathrm{i}\xi)b^2-b_1^2}
{2\varepsilon(\mathrm{i}\xi)b}\biggr] \end{multline}
which in the retarded limit approaches
\begin{equation} \label{4.39} U(z_{A})=-\frac{\hbar c\alpha(0)d}
{160\pi^2\varepsilon_0z_{A}^5}\,
\biggl[\frac{14\varepsilon^2(0)-9}{\varepsilon(0)}
-\frac{6\mu^2(0)-1}{\mu(0)}\biggr]. \end{equation}
In the non-retarded limit one can again distinguish between a purely dielectric plate and a purely magnetic plate. Equation~(\ref{4.38}) approaches
\begin{equation} \label{4.40} U(z_{A})=-\frac{3\hbar d}{64\pi^2\varepsilon_0z_{A}^4}
\int_0^\infty\mathrm{d}\xi\,\alpha(\mathrm{i}\xi)\,
\frac{\varepsilon^2(\mathrm{i}\xi)-1}{\varepsilon(\mathrm{i}\xi)} \end{equation}
in the former case and
\begin{equation} \label{4.41} U(z_{A})=\frac{\hbar d}{64\pi^2\varepsilon_0z_{A}^2}
\int_0^\infty\mathrm{d}\xi\,\biggl(\frac{\xi}{c}\biggr)^2\alpha(\mathrm{i}\xi)\,
\frac{[\mu(\mathrm{i}\xi)-1][3\mu(\mathrm{i}\xi)+1]}{\mu(\mathrm{i}\xi)} \end{equation}
in the latter case. Comparing Eqs.~(\ref{4.39})--(\ref{4.41}) with Eqs.~(\ref{4.32})--(\ref{4.34}), we see that the power laws change from $z_{A}^{-n}$ to $z_{A}^{-(n+1)}$ when the plate thickness changes from being infinitely large to being infinitely small.
If the permeability is sufficiently big, a magneto-dielectric plate of finite thickness features a potential wall \cite{0012,0019}, as illustrated in Fig.~\ref{fig4} for a two-level atom, with the permittivity and permeability of the plate being again given by Eq.~(\ref{4.35}). It is seen that for a thin plate the barrier is very low. It raises with increasing thickness of the plate, reaches a maximal height for some intermediate thickness and then lowers slowly towards the asymptotic half space value as the thickness is further increased.
\paragraph{Planar cavity} \label{sec4.1.3.4}
When an atom is placed between two magneto-electric plates, then the competing effects of attractive and repulsive interaction with the two plates can result in the formation of a trapping potential \cite{0012,0019}. Let us model a magneto-electric planar cavity by two identical half spaces of permittivity $\varepsilon(\omega)$ and permeability $\mu(\omega)$ which are separated by a distance $d$.\footnote{Purely electric planar cavities have been modeled with various degrees of detail, e.g., by two perfectly conducting plates \cite{0070,0321,0034,0278,0301,0063,0315,0314}, two electric half spaces \cite{0314}, or even two electric plates of finite thickness \cite{0032,0033}.} Substitution of the Fresnel reflection coefficients (\ref{A.10}) into Eq.~(\ref{4.21}) yields the potential of an atom placed within a cavity bounded by the half spaces:
\begin{multline} \label{4.42} U(z_{A})=\frac{\hbar\mu_0}{8\pi^2}\int_0^\infty\mathrm{d}\xi\,\xi^2
\alpha(\mathrm{i}\xi)\int_0^\infty\mathrm{d} q\,\frac{q}{b}
\bigl[\mathrm{e}^{-2bz_{A}}+\mathrm{e}^{-2b(d-z_{A})}\bigr]
\biggl[\frac{1}{D_s}\,\frac{\mu(\mathrm{i}\xi)b-b_1}{\mu(\mathrm{i}\xi)b+b_1}
\\[.5ex] -\,\biggl(1+2\frac{q^2c^2}{\xi^2}\biggr)\frac{1}{D_p}\,
\frac{\varepsilon(\mathrm{i}\xi)b-b_1}
{\varepsilon(\mathrm{i}\xi)b+b_1}\biggr]. \end{multline}
\begin{figure}
\caption{ The potential of a ground-state two-level atom placed between two (a) magneto-dielectric (all parameters as in Fig.~\ref{fig3}), (b) purely dielectric [$\mu(\omega)$ $\!=$ $\!1$, other parameters as in (a)], and (c) purely magnetic [$\varepsilon(\omega)$ $\!=$ $\!1$, other parameters as in (a)] half spaces of separation $d$ $\!=$ $\!15c/\omega_{10}$, Eq.~(\ref{4.42}), is shown as a function of the position of the atom. }
\label{fig5}
\end{figure}
As expected, the potential is in general not the sum of the potentials associated with the left and right plates separately, as a comparison of Eq.~(\ref{4.42}) with Eq.~(\ref{4.31}) shows. Clearly, the difference is due to the effect of multiple reflection between the two plates which gives rise to the denominators $D_\sigma$,
\begin{equation} \label{4.43} \frac{1}{D_\sigma}=\sum_{n=0}^\infty\big(r_{\sigma -}
\mathrm{e}^{-bd}r_{\sigma +} \mathrm{e}^{-bd}\big)^n, \end{equation}
recall Eq.~(\ref{3.33}).
The formation of a potential well is illustrated in Fig.~\ref{fig5} where the potentials of an atom placed between purely dielectric and purely magnetic plates, are also shown. It is seen that the attractive (repulsive) potentials associated with each of two purely dielectric (magnetic) plates combine to an infinite potential wall (well) at the center of the cavity, while a potential well of finite depth can be realized within the cavity in the case of two magneto-dielectric plates of sufficiently large permeability.
\subsubsection{Asymptotic power laws} \label{sec4.1.4}
As we have seen, the dispersive interaction of polarizable/magnetizable objects in their ground states can often be given in terms of simple asymptotic power laws in the retarded and non-retarded limits. Typical examples are given in Tab.~\ref{tab1} where the asymptotic power laws for the dispersion force on an atom interacting with another atom [Eqs.~(\ref{4.13}), (\ref{4.14}), (\ref{4.19}) and (\ref{4.20})], a small sphere \cite{0113}, thin ring \cite{0020}, a thin plate [Eqs.~(\ref{4.39})--(\ref{4.41})] and a semi-infinite half space [Eqs.~(\ref{4.32})--(\ref{4.34})], and for the force per unit area between two half spaces \cite{0134,0133}, are shown.
\begin{table} \begin{center}
\begin{tabular}{|c||c|c|c|c|} \hline
Distance $\rightarrow$
&\multicolumn{2}{c|}{Retarded}&\multicolumn{2}{c|}{Nonretarded} \\ \hline
Polarizability $\rightarrow$
&$\mathrm{p}\leftrightarrow \mathrm{p}$
&$\mathrm{p}\leftrightarrow \mathrm{m}$
&$\mathrm{p}\leftrightarrow \mathrm{p}$
&$\mathrm{p}\leftrightarrow \mathrm{m}$\\ \hline\hline
\hspace*{1ex}(a)\hspace*{4.4cm}\begin{picture}(1,1)
\put(-117,-17){\includegraphics[width=4cm]{smallfigure1.eps}}
\end{picture}
&\parbox{5ex}{$$-\frac{1}{r^8}$$}
&\parbox{5ex}{$$+\frac{1}{r^8}$$}
&\parbox{5ex}{$$-\frac{1}{r^7}$$}
&\parbox{5ex}{$$+\frac{1}{r^5}$$}\\ \hline
\hspace*{1ex}(b)\hspace*{4.4cm}\begin{picture}(1,1)
\put(-117,-17){\includegraphics[width=4cm]{smallfigure2.eps}}
\end{picture}
&\parbox{5ex}{$$-\frac{1}{r_{A}^8}$$}
&\parbox{5ex}{$$+\frac{1}{r_{A}^8}$$}
&\parbox{5ex}{$$-\frac{1}{r_{A}^7}$$}
&\parbox{5ex}{$$+\frac{1}{r_{A}^5}$$}\\ \hline
\hspace*{1ex}(c)\hspace*{4.4cm}\begin{picture}(1,1)
\put(-117,-17){\includegraphics[width=4cm]{smallfigure3.eps}}
\end{picture}
&\parbox{5ex}{$$-\frac{1}{\rho_{A}^8}$$}
&\parbox{5ex}{$$+\frac{1}{\rho_{A}^8}$$}
&\parbox{5ex}{$$-\frac{1}{\rho_{A}^7}$$}
&\parbox{5ex}{$$+\frac{1}{\rho_{A}^5}$$}\\ \hline
\hspace*{1ex}(d)\hspace*{4.4cm}\begin{picture}(1,1)
\put(-117,-17){\includegraphics[width=4cm]{smallfigure4.eps}}
\end{picture}
&\parbox{5ex}{$$-\frac{1}{z_{A}^6}$$}
&\parbox{5ex}{$$+\frac{1}{z_{A}^6}$$}
&\parbox{5ex}{$$-\frac{1}{z_{A}^5}$$}
&\parbox{5ex}{$$+\frac{1}{z_{A}^3}$$}\\ \hline
\hspace*{1ex}(e)\hspace*{4.4cm}\begin{picture}(1,1)
\put(-117,-17){\includegraphics[width=4cm]{smallfigure5.eps}}
\end{picture}
&\parbox{5ex}{$$-\frac{1}{z_{A}^5}$$}
&\parbox{5ex}{$$+\frac{1}{z_{A}^5}$$}
&\parbox{5ex}{$$-\frac{1}{z_{A}^4}$$}
&\parbox{5ex}{$$+\frac{1}{z_{A}^2}$$}\\ \hline
\hspace*{1ex}(f)\hspace*{4.4cm}\begin{picture}(1,1)
\put(-117,-17){\includegraphics[width=4cm]{smallfigure6.eps}}
\end{picture}
&\parbox{5ex}{$$-\frac{1}{z^4}$$}
&\parbox{5ex}{$$+\frac{1}{z^4}$$}
&\parbox{5ex}{$$-\frac{1}{z^3}$$}
&\parbox{5ex}{$$+\frac{1}{z}$$}\\ \hline \end{tabular} \end{center} \vspace*{2ex} \caption{ Asymptotic power laws for the forces between (a) two atoms, (b) an atom and a small sphere, (c) an atom and a thin ring, (d) an atom an a thin plate, (e) an atom and a half space and (f) for the force per unit area between two half spaces. In the table heading, $\mathrm{p}$ stands for a polarizable object and $\mathrm{m}$ for a magnetizable one. The signs $+$ and $-$ denote repulsive and attractive forces, respectively.} \label{tab1} \end{table}
Comparing the dispersion forces between objects of different shapes and sizes, it is seen that the signs are always the same, while the leading inverse powers are the same or changed by some global power when moving from one row of the table to another. This can be understood by assuming that the leading inverse power is determined by the contribution to the force which results from the two-atom interaction [row (a)] by pairwise summation. Obviously, integration of two-atom forces over the (finite) volumes of a small sphere (b) or a thin ring (c) does not change the respective power law, while integration over an infinite volume lowers the leading inverse power according to the number of infinite dimensions. So, the leading inverse powers are lowered by two and three for the interaction of an atom with a thin plate of infinite lateral extension (d) and a half space (e), respectively. The power laws for the force between two half spaces (f) can then be obtained from the atom--half-space force (e) by integrating over the three infinite dimensions where integration over $z$ lowers the leading inverse powers by one while the trivial integrations over $x$ and $y$ yield an infinite force, i.e., a finite force per unit area. It follows from the table that many-atom interactions do not change the leading power laws resulting from the summation of pairwise interactions, but only modify the proportionality factors.
All dispersion forces in Tab.~\ref{tab1} are seen to be attractive for two polarizable objects and repulsive for a polarizable object interacting with a magnetizable one. It can further be noted that in the retarded limit the forces decrease more rapidly with increasing distance than might be expected from considering only the non-retarded limit. Finally, the table shows that the retarded dispersion forces between polarizable/polarizable and polarizable/magnetizable objects follow the same power laws, while in the non-retarded limit the forces between polarizable and magnetizable objects are weaker than those between polarizable objects by two powers in the object separation. This can be understood by regarding the forces as being due to the electromagnetic field created by the first object interacting with the second. While the electric and magnetic far fields created by an oscillating electric dipole display the same distance dependence, the electric near field (which can interact with a second polarizable object) is stronger than the magnetic near field (which interacts with a second magnetizable object) by one power in the object separation (giving rise to a difference of two powers in second-order perturbation theory).
\subsection{Excited atoms} \label{sec4.2}
In a first attempt, dispersion forces on atoms in excited energy eigenstates can also be derived from perturbative energy shifts. A straightforward generalization of the calculation outlined in Sec.~\ref{sec4.1.1} to an atom prepared in an arbitrary energy eigenstate $|m\rangle$ yields the potential \cite{0008,0012,0018}
\begin{equation} \label{4.15x} U_m(\vect{r}_{A})=U_m^\mathrm{or}(\vect{r}_{A})
+U_m^\mathrm{r}(\vect{r}_{A}) \end{equation}
where
\begin{equation} \label{4.16x} U_m^\mathrm{or}(\vect{r}_{A})=\frac{\hbar\mu_0}{2\pi}
\int_0^\infty\mathrm{d}\xi\,\xi^2 \mathrm{Tr}\bigl[\bm{\alpha}_m(\mathrm{i}\xi)
\!\cdot\!\ten{G}^{(1)}(\vect{r}_{A},\vect{r}_{A},\mathrm{i}\xi)
\bigr] \end{equation}
and
\begin{equation} \label{4.17x} U_m^\mathrm{r}(\vect{r}_{A})
=-\mu_0\sum_k \Theta(\omega_{mk})\omega_{mk}^2
\vect{d}_{mk}\!\cdot\!\mathrm{Re}\,
\ten{G}^{(1)}(\vect{r}_{A},\vect{r}_{A},\omega_{mk})
\!\cdot\!\vect{d}_{km} \end{equation}
[$\Theta(z)$, unit step function] are the off-resonant and resonant contributions to the potential, respectively, and
\begin{align} \label{4.18x} \bm{\alpha}_m(\omega) =\lim_{\epsilon\to 0}\frac{1}{\hbar}\sum_k\biggl[
\frac{\vect{d}_{mk}\tprod\vect{d}_{km}}
{\omega_{km}-\omega-\mathrm{i}\epsilon}
+\frac{\vect{d}_{km}\tprod\vect{d}_{mk}}
{\omega_{km}+\omega+\mathrm{i}\epsilon}
\biggr] \end{align}
is the atomic polarizability tensor. Equation~(\ref{4.15x}) [together with Eqs.~(\ref{4.16x}) and (\ref{4.17x})] obviously reduces to the ground-state result (\ref{4.6-1}) in the special case $m$ $\!=$ $\!0$. Note that the resonant contribution vanishes in the ground state; it is only present for an excited atom that can undergo real transitions. Equations~(\ref{4.15x})--(\ref{4.17x}) which can also be obtained by means of linear-response theory \cite{0235,0042}, have been used to calculate the potential of an excited atom near a perfectly conducting \cite{0235,0042}, dielectric \cite{0042}, birefringent dielectric half space \cite{0045} and an electric cylinder \cite{0442}.
The above mentioned approaches are time-independent and essentially perturbative and inspection of Eqs.~(\ref{4.15x})--(\ref{4.18x}) reveals that the application of (static) perturbative methods to excited atoms is problematic in several respects. Firstly, the potential is determined by quantities that are attributed to the unperturbed atomic transitions which do not take into account the effect of line broadening, whereas in practice finite line widths are observed which are known to strongly affect resonant transitions. Secondly, the potential and hence also the force remains constant in time; this is not very realistic for excited atoms which undergo spontaneous decay with the allowed (dipole-) transitions being the same as those entering the potential. And thirdly, perturbation theory does not apply to the case of strong atom--field coupling. These problems can be overcome by a dynamical approach to the calculation of forces acting on excited atoms.
\subsubsection{Dynamical approach} \label{sec4.2.1}
Instead of deriving the dispersion force from an energy shift by some means or other, we return to the origin of the force by starting from the Lorentz force acting on an atom and calculating its expectation value for a given initial state. In particular, when the electromagnetic field is initially in its ground state, then this expression yields the sought dispersion force which is genuinely time-dependent for atoms initially prepared in excited states.
Summing the physical momenta $m_\alpha\dot{\hat{\vect{r}}}_\alpha$ of the particles constituting the atom [as given by Eq.~(\ref{2.92})], one obtains for the atom as a whole
\begin{equation} \label{4.45} m_{A}\dot{\hat{\vect{r}}}_{A} =\hat{\vect{p}}_{A}
+\int\mathrm{d}^3 r\,\hat{\vect{P}}_\mathrm{at}(\vect{r})
\!\times\!\hat{\vect{B}}(\vect{r}). \end{equation}
Hence, the center-of-mass motion is governed by the Newton equation
\begin{equation} \label{4.46} m_{A}\ddot{\hat{\vect{r}}}_{A}
=\hat{\vect{F}}_\mathrm{L} \end{equation}
where, according to Eq.~(\ref{4.45}), the Lorentz force is given by
\begin{equation} \label{4.47} \hat{\vect{F}}_\mathrm{L}
=\frac{\mathrm{i}}{\hbar}\bigl[\hat{H},
\hat{\vect{p}}_{A}\bigr]
+\frac{\mathrm{d}}{\mathrm{d} t}
\int\mathrm{d}^3 r\,
\hat{\vect{P}}_\mathrm{at}(\vect{r})
\!\times\!\hat{\vect{B}}(\vect{r}) \end{equation}
with $\hat{\vect{P}}_\mathrm{at}(\vect{r})$ from Eq.~(\ref{2.64}). The first term in Eq.~(\ref{4.47}) can be further evaluated by recalling Eq.~(\ref{2.82}) and using the commutation relations (\ref{2.60}). By making use of the identity $\bm{\nabla}_{\!\!{A}}\tprod\hat{\vect{P}}_\mathrm{at}(\vect{r})$ $\!=$ $-\bm{\nabla}\tprod\hat{\vect{P}}_\mathrm{at}(\vect{r})$ [recall Eq.~(\ref{2.64})], one can show that
\begin{equation} \label{4.48}
\frac{\mathrm{i}}{\hbar}\biggl[
\frac{1}{2\varepsilon_0}\int\mathrm{d}^3r\,
\hat{\vect{P}}^2_\mathrm{at}(\vect{r}),
\hat{\vect{p}}_{A}\biggr]
=\frac{1}{2\varepsilon_0}\int\mathrm{d}^3r\,
\bm{\nabla}\hat{\vect{P}}_\mathrm{at}^2(\vect{r})
=\vect{0} \end{equation}
and by recalling Eq.~(\ref{2.86}) and using the definitions~(\ref{2.64}) and (\ref{2.65}), one derives
\begin{multline} \label{4.49} \frac{\mathrm{i}}{\hbar}\biggl[
\sum_\alpha\frac{1}{2 m_\alpha}
\Bigl\{\hat{\vect{p}}_\alpha
+\int\mathrm{d}^3 r\,\hat{\bm{\Xi}}_\alpha(\vect{r})
\!\times\!\hat{\vect{B}}(\vect{r})\Bigr\}^2,
\hat{\vect{p}}_{A}\biggr]\\[.5ex] =\bm{\nabla}_{\!\!{A}}\Bigl\{
\int\mathrm{d}^3r\,\bigl[\hat{\vect{M}}_\mathrm{at}(\vect{r})
+\hat{\vect{P}}_\mathrm{at}(\vect{r})\!\times\!
\dot{\hat{\vect{r}}}_{A}\bigr]
\!\cdot\!\hat{\vect{B}}(\vect{r})\Bigr\}. \end{multline}
Equations~(\ref{4.48}) and (\ref{4.49}) then imply that Eq.~(\ref{4.47}) can be written as
\begin{align} \label{4.50} \hat{\vect{F}}_\mathrm{L} =&\,\bm{\nabla}_{\!\!{A}}\Bigl\{\int\mathrm{d}^3r\,
\hat{\vect{P}}_\mathrm{at}(\vect{r})\!\cdot\!\hat{\vect{E}}(\vect{r})
+\int\mathrm{d}^3r\,\bigl[\hat{\vect{M}}_\mathrm{at}(\vect{r})
+\hat{\vect{P}}_\mathrm{at}(\vect{r})\!\times\!
\dot{\hat{\vect{r}}}_{A}\bigr]
\!\cdot\!\hat{\vect{B}}(\vect{r})\Bigr\}
\nonumber\\[.5ex] &\,+\frac{\mathrm{d}}{\mathrm{d} t}\int\mathrm{d}^3r\,
\hat{\vect{P}}_\mathrm{at}(\vect{r})\!\times\!\hat{\vect{B}}(\vect{r}). \end{align}
It should be mentioned that by using Eqs.~(\ref{2.66c})--(\ref{2.66d2}) together with the Maxwell equations (\ref{2.71}) and (\ref{2.73}), this equation can be given in the equivalent form
\begin{equation} \label{4.51} \hat{\vect{F}}_\mathrm{L} =\int\mathrm{d}^3r\bigl[\hat{\rho}_\mathrm{at}(\vect{r})
\hat{\vect{E}}(\vect{r})
+\hat{\vect{j}}_\mathrm{at}(\vect{r})
\!\times\!\hat{\vect{B}}(\vect{r})\bigr] \end{equation}
which corresponds to the Eq.~(\ref{3.4}) used in Sec.~\ref{sec3} as a starting point for calculating dispersion forces on bodies.\footnote{Note that the field created by the atom only gives rise to internal forces, so that one may equivalently write Eq.~(\ref{4.51}) with the total fields $\hat{\bm{\mathcal{E}}}(\vect{r})$ and $\hat{\bm{\mathcal{B}}}(\vect{r})$ [Eq.~(\ref{2.69})] instead of $\hat{\vect{E}}(\vect{r})$ and $\hat{\vect{B}}(\vect{r})$.}
{F}rom Eq.~(\ref{4.50}), the Lorentz force in long-wavelength approximation can be obtained by performing a leading-order expansion in the relative particle coordinates $\hat{\overline{\vect{r}}}_\alpha$, resulting in
\begin{equation} \label{4.52} \hat{\vect{F}}_\mathrm{L} =\bm{\nabla}_{\!\!{A}}\bigl[
\hat{\vect{d}}\!\cdot\!\hat{\vect{E}}(\hat{\vect{r}}_{A})
+\hat{\vect{m}}\!\cdot\!\hat{\vect{B}}(\hat{\vect{r}}_{A})
+\hat{\vect{d}}\!\times\!\dot{\hat{\vect{r}}}_{A}
\!\cdot\!\hat{\vect{B}}(\hat{\vect{r}}_{A})\bigr]
+\frac{\mathrm{d}}{\mathrm{d} t}\bigl[\hat{\vect{d}}\!\times\!
\hat{\vect{B}}(\hat{\vect{r}}_{A})\bigr], \end{equation}
[recall Eqs.~(\ref{2.66}) and (\ref{2.66b})] where
\begin{align} \label{4.53} \frac{\mathrm{d}}{\mathrm{d} t}\bigl[\hat{\vect{d}}\!\times\!
\hat{\vect{B}}(\hat{\vect{r}}_{A})\bigr] =&\;\frac{\mathrm{i}}{\hbar}\bigl[\hat{H},\hat{\vect{d}}\!\times\!
\hat{\vect{B}}(\hat{\vect{r}}_{A})\bigr]
=\dot{\hat{\vect{d}}}\!\times\!
\hat{\vect{B}}(\hat{\vect{r}}_{A})
+\hat{\vect{d}}\!\times\!\dot{\hat{\vect{B}}}(\vect{r})
\big|_{\vect{r}=\hat{\vect{r}}_{A}}\nonumber\\[.5ex] &+{\textstyle\frac{1}{2}}\hat{\vect{d}}\!\times\!
\bigl[\bigl(\dot{\hat{\vect{r}}}_{A}
\!\cdot\!\bm{\nabla}_{\!\!{A}}\bigr)
\hat{\vect{B}}(\hat{\vect{r}}_{A})
+\hat{\vect{B}}(\hat{\vect{r}}_{A})
\bigl(\overleftarrow{\bm{\nabla}}_{\!\!{A}}\!\cdot\!
\dot{\hat{\vect{r}}}_{A}\bigr)\bigr]. \end{align}
Discarding all terms proportional to $\dot{\hat{\vect{r}}}_{A}$ (which are of the order $v/c$ and thus negligible for nonrelativistic center-of-mass motion), as well as the contribution from the magnetic interactions, Eq.~(\ref{4.52}) reduces to
\begin{equation} \label{4.54} \hat{\vect{F}}_\mathrm{L} =\left\{\bm{\nabla}
\left[\hat{\vect{d}}\!\cdot\!\hat{\vect{E}}(\vect{r})\right]
+\frac{\mathrm{d}}{\mathrm{d} t}\left[\hat{\vect{d}}\!\times\!
\hat{\vect{B}}(\vect{r})\right]
\right\}_{\vect{r}=\hat{\vect{r}}_{A}}, \end{equation}
whose expectation value
\begin{equation} \label{4.54-1} \vect{F} =\langle\hat{\vect{F}}_\mathrm{L}\rangle \end{equation}
quite generally provides a basis for calculating electromagnetic forces on non-magnetic atoms, including dispersion forces. Needless to say that Eq.~(\ref{4.54}) is valid regardless of the state the atom and the body-assisted field are prepared in.
At this point we recall that, according to Eqs.~(\ref{2.24-1}) and (\ref{2.31-1}), the electric and the magnetic induction fields are expressed in terms of the dynamical variables $\hat{\vect{f}}_\lambda(\vect{r},\omega)$ and $\hat{\vect{f}}_\lambda^\dagger(\vect{r},\omega)$. It is not difficult to prove that in electric-dipole approximation, $\hat{\vect{f}}_\lambda(\vect{r},\omega)$ obeys the Heisenberg equation of motion
\begin{equation} \label{4.55} \dot{\hat{\vect{f}}}_\lambda(\vect{r},\omega) =\frac{\mathrm{i}}{\hbar}\bigl[\hat{H},
\hat{\vect{f}}_\lambda(\vect{r},\omega)\bigr]
=-\mathrm{i}\omega\hat{\vect{f}}_\lambda(\vect{r},\omega)
+\frac{\mathrm{i}}{\hbar}\,\hat{\vect{d}}\!\cdot\! \ten{G}_\lambda^\ast(\hat{\vect{r}}_{A},\vect{r},\omega) \end{equation}
[recall the Hamiltonian (\ref{2.82-1}) together with Eqs.~(\ref{2.83}), (\ref{2.84}) and (\ref{2.90-1})], whose formal solution reads
\begin{equation} \label{4.56} \hat{\vect{f}}_\lambda(\vect{r},\omega,t) =\hat{\vect{f}}_{\lambda\mathrm{free}}(\vect{r},\omega,t)
+\hat{\vect{f}}_{\lambda\mathrm{source}}(\vect{r},\omega,t) \end{equation}
where
\begin{equation} \label{4.57}
\hat{\vect{f}}_{\lambda\mathrm{free}}(\vect{r},\omega,t)
=\mathrm{e}^{-\mathrm{i}\omega (t-t_0)}
\hat{\vect{f}}_\lambda(\vect{r},\omega) \end{equation}
and
\begin{equation} \label{4.58} \hat{\vect{f}}_{\lambda\mathrm{source}}(\vect{r},\omega,t) =\frac{\mathrm{i}}{\hbar}\int_{t_0}^t\mathrm{d}\tau\,\mathrm{e}^{-\mathrm{i}\omega(t-\tau)}
\hat{\vect{d}}(\tau)\!\cdot\!\ten{G}_\lambda^\ast
[\hat{\vect{r}}_{A}(\tau),\vect{r},\omega] \end{equation}
($t_0$, initial time), respectively, determine the free-field parts $\underline{\hat{\vect{E}}}_\mathrm{free}(\vect{r},\omega,t)$ and $\underline{\hat{\vect{B}}}_\mathrm{free}(\vect{r},\omega,t)$ and the source-field parts $\underline{\hat{\vect{E}}}_\mathrm{source}(\vect{r},\omega,t)$ and $\underline{\hat{\vect{B}}}_\mathrm{source}(\vect{r},\omega,t)$ of the electric and the induction field in the $\omega$ domain. Substitution of Eqs.~(\ref{4.56})--(\ref{4.58}) together with Eqs.~(\ref{2.24-1}) and (\ref{2.31-1}) into Eq.~(\ref{4.54-1}) together with Eq.~(\ref{4.54}) and use of Eq.~(\ref{2.30b}) leads to the following expression for the mean force \cite{0008,0018}:
\begin{equation} \label{4.59} \vect{F}(t)=\vect{F}_\mathrm{free}(t)+\vect{F}_\mathrm{source}(t) \end{equation}
with
\begin{multline} \label{4.60} \vect{F}_\mathrm{free}(t) = \int_0^\infty\mathrm{d}\omega
\biggl\{\bm{\nabla}
\bigl\langle\hat{\vect{d}}(t)\!\cdot\!
\underline{\hat{\vect{E}}}_\mathrm{free}(\vect{r},\omega,t)
\bigr\rangle\\[.5ex] +\frac{\mathrm{d}}{\mathrm{d} t}\bigl[\bigl\langle\hat{\vect{d}}(t)\!\times\!
\underline{\hat{\vect{B}}}_\mathrm{free}(\vect{r},\omega,t)
\bigr\rangle\bigr]
\biggr\}_{\vect{r}=\hat{\vect{r}}_{A}(t)}+\mathrm{C.c.} \end{multline}
and
\begin{equation} \label{4.60b} \vect{F}_\mathrm{source}(t)
=\vect{F}_\mathrm{source}^\mathrm{el}(t) +\vect{F}_\mathrm{source}^\mathrm{mag}(t) \end{equation}
where the components
\begin{multline} \label{4.61} \vect{F}_\mathrm{source}^\mathrm{el}(t) =\biggl\{\frac{\mathrm{i}\mu_0}{\pi}\int_0^\infty\mathrm{d}\omega\,\omega^2
\int_{t_0}^t\mathrm{d}\tau\,\mathrm{e}^{-\mathrm{i}\omega(t-\tau)}\\[.5ex]
\times\bm{\nabla}\bigl\langle\hat{\vect{d}}(t)\!\cdot\!
\mathrm{Im}\,\ten{G}[\vect{r},\hat{\vect{r}}_{A}(\tau),\omega]\!\cdot\!
\hat{\bf d}(\tau)\bigr\rangle
\biggr\}_{\vect{r}=\hat{\vect{r}}_{A}(t)}+\mathrm{C.c.} \end{multline}
and
\begin{multline} \label{4.62} \vect{F}_\mathrm{source}^\mathrm{mag}(t) =\biggl\{\frac{\mu_0}{\pi}\int_{0}^\infty
\mathrm{d}\omega\,\omega\,
\frac{\mathrm{d}}{\mathrm{d} t}\int_{t_0}^t\mathrm{d}\tau\,\mathrm{e}^{-\mathrm{i}\omega(t-\tau)}
\\[.5ex] \times\bigl\langle\hat{\vect{d}}(t)\!\times\!\bigl(\bm{\nabla}\!\times\!
\mathrm{Im}\,\ten{G}[\vect{r},\hat{\vect{r}}_{A}(\tau),\omega]
\bigr)\!\cdot\!\hat{\vect{d}}(\tau)\bigr\rangle
\biggr\}_{\vect{r}=\hat{\vect{r}}_{A}(t)}+\mathrm{C.c.} \end{multline}
are related to the source-field parts of the electric and the induction field, respectively.
While Eq.~(\ref{4.59}) together with Eqs.~(\ref{4.60})--(\ref{4.62})
gives the force on a (non-magnetic) atom subject to an arbitrary electromagnetic field, the pure dispersion force can be obtained by considering the case where the (body-assisted) electromagnetic field is initially prepared in the ground state $|\{0\}\rangle$ so that
\begin{equation} \label{4.62-1}
\langle\{0\}|\cdots\underline{\hat{\vect{E}}}_\mathrm{free}
(\vect{r},\omega,t)|\{0\}\rangle
= \langle\{0\}|\cdots\underline{\hat{\vect{B}}}_\mathrm{free}
(\vect{r},\omega,t)|\{0\}\rangle=0 \end{equation}
[recall Eq.~(\ref{2.36})] which implies that $\vect{F}_\mathrm{free}(t)$ $\!=$ $\!\vect{0}$. Hence, Eq.~(\ref{4.59}) simply reduces to
\begin{equation} \label{4.65} \vect{F}(t)= \vect{F}_\mathrm{source}(t) \end{equation}
in this case. In particular, for chosen atomic position, $\hat{\vect{r}}_A$ may be regarded as a time-independent c-number parameter [$\hat{\vect{r}}_A(t)\mapsto\vect{r}_A$], so that the expectation values to be taken in Eqs.~(\ref{4.61}) and (\ref{4.62}) only refer to the internal state of the atom. It should be pointed out that the concept is not restricted to the calculation of the mean force but can be extended to higher-order force moments (for a discussion of force fluctuations, see also Ref.~\cite{0072}).
\subsubsection{Weak atom--field coupling} \label{sec4.2.2}
The remaining task now consists in the determination of the dipole--dipole correlation function
\begin{equation} \label{4.66} \left\langle\hat{\bf d}(t)\tprod\hat{\bf d}(\tau)\right\rangle =\sum_{m,n}\sum_{m',n'}\vect{d}_{mn}\tprod\vect{d}_{m'n'} \left\langle\hat{A}_{mn}(t)\hat{A}_{m'n'}(\tau)\right\rangle \end{equation}
in Eqs.~(\ref{4.61}) and (\ref{4.62}) [$\hat{A}_{mn}$ $\!=$
$\!|m\rangle\langle n|$, recall Eq.~(\ref{2.84})]. To that end, the problem of the internal atomic dynamics must be solved. Let first consider the case of weak atom--field coupling where the Markov approximation can by used to considerably simplify the problem. Under the assumption that the relevant atomic transition frequencies are well separated from one another so that diagonal and off-diagonal density matrix elements evolve independently, application of the quantum-regression theorem (see,~e.g., Ref.~\cite{0605}) yields the familiar result
\begin{equation} \label{4.67} \left\langle\hat{A}_{mn}(t)\hat{A}_{m'n'}(\tau)\right\rangle
=\delta_{nm'}\left\langle\hat{A}_{mn'}(\tau)\right\rangle
\mathrm{e}^{\{\mathrm{i}\tilde{\omega}_{mn}(\vect{r}_{A})
-[\Gamma_m(\vect{r}_{A})
+\Gamma_n(\vect{r}_{A})]/2\}(t-\tau)} \end{equation}
($t$ $\!\ge$ $\!\tau$, $m$ $\!\neq$ $n$). Here,
\begin{equation} \label{4.68} \tilde{\omega}_{mn}(\vect{r}_{A}) =\omega_{mn}+\delta\omega_m(\vect{r}_{A})
-\delta\omega_n(\vect{r}_{A}) \end{equation}
are the atomic transition frequencies including the position-dependent energy-level shifts\footnote{The Lamb shifts observed in free space are thought of as being already included in the frequencies $\omega_{mn}$.}
\begin{gather} \label{4.69} \delta\omega_m(\vect{r}_{A})
=\sum_k \delta\omega_m^k(\vect{r}_{A}),\\[.5ex] \label{4.70} \delta\omega_m^k(\vect{r}_{A})=\frac{\mu_0}{\pi\hbar}\,
\mathcal{P}\int_0^\infty\mathrm{d}\omega\,\omega^2\,
\frac{\vect{d}_{km}\!\cdot\!\mathrm{Im}\,
\ten{G}^{(1)}(\vect{r}_{A},\vect{r}_{A},\omega)\!\cdot\!
\vect{d}_{mk}}{\tilde{\omega}_{mk}(\vect{r}_{A})-\omega}\, \end{gather}
(cf.~also Refs.~\cite{0186,0439,0187}) which are due to the interaction of the atom with the body-assisted electromagnetic field, and similarly,
\begin{gather} \label{4.71} \Gamma_m(\vect{r}_{A})
= \sum_k \Gamma_m^k(\vect{r}_{A}),\\[.5ex] \label{4.72} \Gamma_m^k(\vect{r}_{A})=\frac{2\mu_0}{\hbar}\,
\Theta[\tilde{\omega}_{mk}(\vect{r}_{A})]
\tilde{\omega}_{mk}^2(\vect{r}_{A})\vect{d}_{km}\!\cdot\!
\mathrm{Im}\,\ten{G}[\vect{r}_{A},\vect{r}_{A},
\tilde{\omega}_{mk}(\vect{r}_{A})]\!\cdot\!\vect{d}_{mk} \end{gather}
are the position-dependent level widths. Note that the position-dependent energy shifts $\hbar\delta\omega_m(\vect{r}_{A})$ as given by Eq.~(\ref{4.69}) together with Eq.~(\ref{4.70}) reduce to those obtained by leading-order perturbation theory, Eqs.~(\ref{4.15x})--(\ref{4.17x}), if the frequency shifts in the denominator on the r.h.s. of Eq.~(\ref{4.70}) are ignored.
Substituting Eqs.~(\ref{4.66}) and (\ref{4.67}) into Eqs.~(\ref{4.60b})--(\ref{4.62}), one can then show that the force on an atom that is initially prepared in an arbitrary state can be represented in the form \cite{0008,0012,0018}
\begin{equation} \label{4.73} \vect{F}(t)
=\sum_{m,n}\sigma_{nm}(t)\vect{F}_{mn}(\vect{r}_{A}) \end{equation}
where the atomic density matrix elements $\sigma_{nm}(t)$ $\!=$ $\!\langle\hat{A}_{mn}(t)\rangle$ solve the intra-atomic master equation together with the respective initial condition, and we have
\begin{equation} \label{4.74} \vect{F}_{mn}(\vect{r}_{A}) =\vect{F}_{mn}^\mathrm{el,or}(\vect{r}_{A})
+\vect{F}_{mn}^\mathrm{el,r}(\vect{r}_{A})
+\vect{F}_{mn}^\mathrm{mag,or}(\vect{r}_{A})
+\vect{F}_{mn}^\mathrm{mag,r}(\vect{r}_{A}) \end{equation}
with the various electric/magnetic, off-resonant/resonant force components being given as follows:
\begin{multline} \label{4.75} \vect{F}_{mn}^\mathrm{el,or}(\vect{r}_{A})
=-\frac{\hbar\mu_0}{2\pi}
\int_0^\infty\mathrm{d}\xi\,\xi^2\\[.5ex] \times\bigl(\bm{\nabla}\,\mathrm{Tr}\bigl\{
[\bm{\alpha}_{mn}(\vect{r}_{A},\mathrm{i}\xi)
+\bm{\alpha}_{mn}(\vect{r}_{A},-\mathrm{i}\xi)]
\!\cdot\!\ten{G}^{(1)}(\vect{r}_{A},\vect{r},\mathrm{i}\xi)
\bigr\}\bigr)_{\vect{r}=\vect{r}_{A}}, \end{multline}
\begin{multline} \label{4.76} \vect{F}_{mn}^\mathrm{el,r}(\vect{r}_{A}) =\mu_0\sum_{k}\Theta({\tilde{\omega}}_{nk})
\Omega^2_{mnk}(\vect{r}_{A})\\[.5ex] \times\bigl\{\bm{\nabla}\vect{d}_{mk}\!\cdot\!
\ten{G}^{(1)}[\vect{r},\vect{r}_{A},\Omega_{mnk}(\vect{r}_{A})]
\!\cdot\!\vect{d}_{kn}\bigr\}_{\vect{r}=\vect{r}_{A}}
+\mathrm{C.c.}, \end{multline}
\begin{multline} \label{4.77} \vect{F}_{mn}^\mathrm{mag,or}(\vect{r}_{A}) =\frac{\hbar\mu_0}{2\pi}
\int_0^\infty\mathrm{d}\xi\,\xi^2\,\mathrm{Tr}\biggl\{\biggl[
\frac{\tilde{\omega}_{mn}(\vect{r}_{A})}{\mathrm{i}\xi}\,
\bm{\alpha}_{mn}^\mathsf{T}(\vect{r}_{A},\mathrm{i}\xi)\\[.5ex] -\,\frac{\tilde{\omega}_{mn}(\vect{r}_{A})}{\mathrm{i}\xi}\,
\bm{\alpha}_{mn}^\mathsf{T}(\vect{r}_{A},-\mathrm{i}\xi)\biggr]
\!\times\!\bigl[\bm{\nabla}\!\times\!
\ten{G}^{(1)}(\vect{r},\vect{r}_{A},\mathrm{i}\xi)\bigr]
\biggr\}_{\vect{r}=\vect{r}_{A}}, \end{multline}
\begin{multline} \label{4.78} \vect{F}_{mn}^\mathrm{mag,r}(\vect{r}_{A}) =\mu_0\sum_{k}\Theta({\tilde{\omega}_{nk}})
\tilde{\omega}_{mn}(\vect{r}_{A})
\Omega_{mnk}(\vect{r}_{A})\\[.5ex]
\times\bigl(\vect{d}_{mk}\!\times\!\bigl\{
\bm{\nabla}\!\times\!\ten{G}^{(1)}[\vect{r},\vect{r}_{A},
\Omega_{mnk}(\vect{r}_{A})]\!\cdot\!\vect{d}_{kn}
\bigr\}\bigr)_{\vect{r}=\vect{r}_{A}}+\mathrm{C.c.} \end{multline}
Here, $\Omega_{mnk}(\vect{r}_{A})$ and $\bm{\alpha}_{mn}(\vect{r}_{A},\omega)$, respectively, are the complex atomic transition frequencies and the generalized polarizability tensor:
\begin{equation} \label{4.79} \Omega_{mnk}(\vect{r}_{A}) =\tilde{\omega}_{nk}(\vect{r}_{A})
+\mathrm{i}[\Gamma_m(\vect{r}_{A})+\Gamma_k(\vect{r}_{A})]/2, \end{equation}
\begin{equation} \label{4.80} \bm{\alpha}_{mn}(\vect{r}_{A},\omega) =\frac{1}{\hbar}\sum_k\left[
\frac{\vect{d}_{mk}\tprod\vect{d}_{kn}}
{-\Omega_{mnk}(\vect{r}_{A})-\omega}
+\frac{\vect{d}_{kn}\tprod\vect{d}_{mk}}
{-\Omega_{nmk}^\ast(\vect{r}_{A})+\omega}
\right]. \end{equation}
Equations~(\ref{4.73})--(\ref{4.78}) show that the force components $\vect{F}_{mn}(\vect{r}_A)$ ($m$ $\!\neq$ $\!n$) associated with (non-vanishing) off-diagonal elements of the atomic density matrix contain contributions arising from the interaction of the atom with both the electric and the magnetic field, where the magnetic force components display a vector structure which is entirely different from that of the electric ones. Since under the assumptions made, diagonal and off-diagonal density matrix elements are not coupled to each other so that
\begin{equation} \label{4.83} \sigma_{nm}(t) =\mathrm{e}^{\{\mathrm{i}\tilde{\omega}_{mn}(\vect{r}_{A})
-[\Gamma_m(\vect{r}_{A})+\Gamma_n(\vect{r}_{A})]/2\}(t-t_0)}
\sigma_{nm}(t_0)
\end{equation}
($m$ $\!\neq$ $\!n$), force components associated with off-diagonal density-matrix elements can only be observed if the atom is initially prepared in an at least partially coherent superposition of energy eigenstates. Accordingly, if the atom is initially prepared in an incoherent superposition of energy eigenstates, then only force components $\vect{F}_{mm}(\vect{r}_A)$ which are associated with diagonal density-matrix elements and which are electrical by their nature,
\begin{equation} \label{4.80-1} \vect{F}_{mm}(\vect{r}_A)
=\vect{F}_{mm}^\mathrm{el,or}(\vect{r}_A)
+\vect{F}_{mm}^\mathrm{el,r}(\vect{r}_A)
\equiv\vect{F}_{mm}^\mathrm{or}(\vect{r}_A)
+\vect{F}_{mm}^\mathrm{r}(\vect{r}_A), \end{equation}
are observed, with the density matrix elements obeying the balance equations
\begin{equation} \label{4.84} \dot{\sigma}_{mm}(t) =-\Gamma_m(\vect{r}_{A})\sigma_{mm}(t)
+\sum_k\Gamma_k^m(\vect{r}_{A})\sigma_{kk}(t). \end{equation}
\begin{figure}\label{fig6}
\end{figure}
In particular, when the level shifts and broadenings are neglected, then the force components $\vect{F}_{mm}^\mathrm{or}(\vect{r}_A)$ and $\vect{F}_{mm}^\mathrm{r}(\vect{r}_A)$ as follow from Eqs.~(\ref{4.75}) and (\ref{4.76}) obviously reduce to those that are obtained from the perturbative potential (\ref{4.15x})--(\ref{4.17x}) by means of Eq.~(\ref{4.3}). Note that the gradient in Eqs.~(\ref{4.75}) and (\ref{4.76}) acts only on the Green tensor and not on the additional position-dependent quantities, so that this result cannot be derived from a potential in the usual way. Since the force components associated with excited-state density matrix elements are transient, they are only observable on time scales of the order of magnitude of the respective decay times $\Gamma_m^{-1}(\vect{r}_A)$ which are known to sensitively depend on the atomic position \cite{0605}. Needless to say that the force $\vect{F}(t)$ that acts on an initially excited atom approaches the ground-state force $\vect{F}_{00}(\vect{r}_A)$ after sufficiently long times, $\lim_{t\to\infty}\langle\vect{F}(t)\rangle$ $\!=$ $\vect{F}_{00}(\vect{r}_{A})$.
In order to illustrate the effect of the body-induced level shifting and broadening on the force, let us consider a two-level atom which is situated at distance $z_{A}$ very close to a dielectric half space. By means of the respective Green tensor (App.~\ref{appA}), it turns out that in the non-retarded limit the shift and width of the transition frequency are determined by
\begin{equation} \label{4.85} \delta\omega(z_{A})=\delta\omega_1(z_{A})-\delta\omega_0(z_{A}) =-\frac{\vect{d}_{01}^2+(\vect{d}_{01}\!\cdot\!\vect{e}_z)^2}
{32\pi\hbar\varepsilon_0 z_{A}^3}\,
\frac{|\varepsilon[\omega_{10}+\delta\omega(z_{A})]|^2-1}
{|\varepsilon[\omega_{10}+\delta\omega(z_{A})]+1|^2} \end{equation}
and \begin{equation} \label{4.86} \Gamma(z_{A})=\Gamma_1(z_{A}) =\frac{\vect{d}_{01}^2+(\vect{d}_{01}\!\cdot\!\vect{e}_z)^2}
{8\pi\hbar\varepsilon_0 z_{A}^3}\, \frac{\mathrm{Im}\,\varepsilon[\omega_{10}+\delta\omega(z_{A})]}
{|\varepsilon[\omega_{10}+\delta\omega(z_{A})]+1|^2}\,, \end{equation}
respectively, where the transition-dipole matrix element has been assumed to be real and the (small) off-resonant contribution to the frequency shift has been omitted. Note that due to the appearance of the frequency shift on the r.h.s. of Eq.~(\ref{4.85}), this equation determines the shift only implicitly. The dominant contribution to the force on the atom in the upper state is the resonant one, i.e., $\vect{F}_{11}(\vect{r}_A)$ $\!\simeq$ $\!\vect{F}_{11}^\mathrm{r}(\vect{r}_A)$. Substituting the half-space Green tensor (App.~\ref{appA}) into Eqs.~(\ref{4.76}) and (\ref{4.80-1}), one can show that [$\vect{F}_{11}^\mathrm{r}(\vect{r}_A)$ $\!=$ $\!F_{11}^\mathrm{r}(z_A)\vect{e}_z$] \cite{0008,0012,0018}
\begin{equation} \label{4.87} F_{11}^\mathrm{r}(z_{A}) =-\frac{3[\vect{d}_{01}^2+(\vect{d}_{01}\!\cdot\!\vect{e}_z)^2]}
{32\pi\hbar\varepsilon_0 z_{A}^4}\,
\frac{|\varepsilon[\Omega_{110}(z_{A})]|^2-1}
{|\varepsilon[\Omega_{110}(z_{A})]+1|^2} \end{equation}
where, according to Eq.~(\ref{4.79}),
\begin{equation} \label{4.88} \Omega_{110}(z_{A})=\tilde{\omega}_{10}(z_{A})
+\mathrm{i}\Gamma(z_{A})/2
=\omega_{10}+\delta\omega(z_{A})
+\mathrm{i}\Gamma(z_{A})/2. \end{equation}
In particular, for a medium whose permittivity is of Drude--Lorentz type, Eq.~(\ref{4.35}) leads to ($\gamma_e,\Gamma$ $\!\ll$ $\!\omega_{\mathrm{T}e}$)
\begin{equation} \label{4.89} \varepsilon[\Omega_{110}(z_{A})] =1+\frac{\omega_{\mathrm{P}e}^2}{\omega_{\mathrm{T}e}^2 -\tilde{\omega}_{10}^2(z_{A}) -\mathrm{i} [\Gamma(z_{A})+\gamma_e]\tilde{\omega}_{10}(z_{A})}\,, \end{equation}
showing that the absorption parameter of the half-space medium, $\gamma_e$, is replaced with the total absorption parameter, i.e., the sum of $\gamma_e$ and the spon\-tan\-eous-decay constant $\Gamma(z_{A})$ of the atom. Figure~\ref{fig6} displays the resonant component of the force on a two-level atom in the upper state placed near a (single-resonance) dielectric half space as a function of the unperturbed transition frequency $\omega_{10}$.
\begin{figure}
\caption{ The off-resonant component of the force on a two-level atom in the upper state placed in front of a dielectric half space, Eq.~(\ref{4.90}), is shown as a function of the unperturbed transition frequency (solid line), the parameters being the same as in Fig.~\ref{fig6}. For comparison, the perturbative result is also shown (dashed lines). The inset displays the difference between the force with and without consideration of level broadening (solid lines). In addition, the same difference is displayed when the level shifts are ignored (dashed lines). }
\label{fig7}
\end{figure}
It is seen that in the vicinity of the (surface-plasmon) frequency \mbox{$\omega_\mathrm{S}$ $\!=$ $\!\sqrt{\omega_{\mathrm{T}e}^2+\omega_{\mathrm{P}e}^2/2}$}, an enhanced force is observed which is attractive (repulsive) for red (blue) detuned atomic transition frequencies $\omega_{10}$ $\!<$ $\!\omega_\mathrm{S}$ ($\omega_{10}$ $\!>$ $\!\omega_\mathrm{S}$)---a result already known from perturbation theory \cite{0042}. However, it is also seen that due to body-induced level shifting and broadening the absolute value of the force can be noticeably reduced. Interestingly, the positions of the extrema of the force remain nearly unchanged, because level shifting and broadening give rise to competing effects that almost cancel.
The calculation of the off-resonant component of the force, Eq.~(\ref{4.80-1}) together with Eq.~(\ref{4.75}), leads to [$\vect{F}_{11}^\mathrm{or}(\vect{r}_A)$ $\!=$ $F_{11}^\mathrm{or}(z_A)\vect{e}_z$]
\begin{multline} \label{4.90} F_{11}^\mathrm{or}(z_{A}) =\frac{3[\vect{d}_{01}^2+(\vect{d}_{01}\!\cdot\!\vect{e}_z)^2]}
{32\pi^2\hbar\varepsilon_0 z_{A}^4}
\int_0^\infty\mathrm{d}\xi\,
\frac{\varepsilon(\mathrm{i}\xi)-1}{\varepsilon(\mathrm{i}\xi)+1}\\[.5ex] \times\frac{\tilde{\omega}_{10}(z_{A})}
{\tilde{\omega}_{10}^2(z_{A})+[\xi+\Gamma(z_{A})/2]^2}\,
\frac{\tilde{\omega}_{10}^2(z_{A})+\xi^2+\Gamma^2(z_{A})/4}
{\tilde{\omega}_{10}^2(z_{A})+[\xi-\Gamma(z_{A})/2]^2}\,. \end{multline}
Equation~(\ref{4.90}) reveals that the off-resonant component of the force is only weakly influenced by the level broadening [the leading-order dependence being $O(\Gamma^2)$] which is in agreement with the physical requirement that the virtual emission and absorption processes governing the off-resonant component should be only weakly affected by decay-induced broadening. Formally, the absence of a linear-order term $O(\Gamma)$ is due to the fact that the atomic polarizability (\ref{4.80}) enters the off-resonant force components (\ref{4.75}) only in the combination $[\bm{\alpha}_{mm}(\vect{r}_{A},\mathrm{i}\xi) +\bm{\alpha}_{mm}(\vect{r}_{A},-\mathrm{i}\xi)]$, which could not have been anticipated from the perturbative result (\ref{4.16x}) [where in fact, $\bm{\alpha}_{m}(\vect{r}_{A},\mathrm{i}\xi)$ and $\bm{\alpha}_{m}(\vect{r}_{A},-\mathrm{i}\xi)$ coincide, recall Eq.~(\ref{4.18x})]. The effects of level shifting and broadening are illustrated in Fig.~\ref{fig7}. Compared to the perturbative result, the frequency shift has the effect of raising or lowering the force for $\omega_{10}$ $\!<$ $\!\omega_\mathrm{S}$ or $\omega_{10}$ $\!>$ $\!\omega_\mathrm{S}$, respectively, whereas the effect of broadening is not visible in the curves. Only by plotting the difference between the results with and without broadening, a slight reduction of the force becomes visible in the vicinity of $\omega_\mathrm{S}$ where $\Gamma$ is largest. Since this behavior is generally typical of off-resonant components, the perturbative result may be regarded as a good approximation for the forces on ground-state atoms where no resonant components are present.
\subsubsection{Strong atom--field coupling} \label{sec4.2.3}
Strong atom--field coupling may occur if an initially excited atom interacts resonantly with a sharply peaked (quasi-)mode of a body-assisted field, as it is observed in cavity-like systems. In this case, the atom--field dynamics can no longer be described within the Markov approximation. Typically, a single atomic transition is in resonance with such a mode, so the (resonant part of the)
dispersion force can be studied, to a good approximation, by employing the two-level model in rotating-wave approximation with respect to the interaction Hamiltonian (\ref{2.90-1}) (see, e.g., Ref.~\cite{0605}). To be more specific, let us consider the interaction of a two-level atom initially prepared in the upper state $|1\rangle$ with the body-assisted electromagnetic field in the ground state
$|\{0\}\rangle$ and calculate the electric part of the resonant component of the force acting on the atom, i.e., in the Schr\"{o}dinger picture,
\begin{equation} \label{4.91} \vect{F}(t)
\simeq\langle\psi(t)|\bigl\{\bm{\nabla}
\bigl[\hat{\vect{d}}\!\cdot\!\hat{\vect{E}}(\vect{r})
\bigr]\bigr\}_{\vect{r}=\vect{r}_{A}}|\psi(t)\rangle, \end{equation}
with the state vector $|\psi(t)\rangle$ [$|\psi(t\!=\!t_0)\rangle$
$\!=$ $\!|\{0\}\rangle|1\rangle$] being represented in the form
\begin{multline} \label{4.92}
|\psi(t)\rangle=\psi_1(t)|\{0\}\rangle|1\rangle\\[.5ex]
+\sum_{\lambda={e},{m}}
\int\mathrm{d}^3r\int_0^\infty\mathrm{d}\omega\,\frac{\psi_0(\omega,t)}
{\hbar g(\vect{r}_{A},\omega)}\,\vect{d}_{01}\!\cdot\!
\ten{G}^\ast_\lambda(\vect{r}_{A},\vect{r},\omega)\!\cdot\!
|\vect{1}(\vect{r}_{A},\omega)\rangle|0\rangle. \end{multline}
It is normalized to unity provided that
\begin{equation} \label{4.93}
|\psi_1(t)|^2+\int_0^\infty\mathrm{d}\omega\,|\psi_0(\omega,t)|^2=1 \end{equation}
and
\begin{equation} \label{4.94} g^2(\vect{r}_{A},\omega)
=\frac{\mu_0}{\pi\hbar}\,\omega^2
\vect{d}_{10}\!\cdot\!
\mathrm{Im}\ten{G}(\vect{r}_{A},\vect{r}_{A},\omega)
\!\cdot\!\vect{d}_{01}. \end{equation}
Substituting Eq.~(\ref{4.92}) into the Schr\"{o}dinger equation
\begin{equation} \label{4.95}
\mathrm{i}\hbar\frac{\partial}{\partial t}|\psi(t)\rangle
=\hat{H}|\psi(t)\rangle, \end{equation}
with $\hat{H}$ being given according to Eq.~(\ref{2.82-1}) together with Eqs.~(\ref{2.83}), (\ref{2.84}) and (\ref{2.90-1}), one obtains the following coupled differential equations for $\psi_1(t)$ and $\psi_0(\omega,t)$:
\begin{align} \label{4.96} &\dot{\psi}_1(t)=-\frac{\mathrm{i}}{\hbar}\,E_1\psi_1(t)
+\mathrm{i}\int_0^\infty\mathrm{d}\omega\,g(\vect{r}_{A},\omega)
\psi_0(\omega,t),\\[.5ex] \label{4.97} &\dot{\psi}_0(\omega,t)=-\frac{\mathrm{i}}{\hbar}\,(E_0+\hbar\omega)
\psi_0(\omega,t)+\mathrm{i} g(\vect{r}_{A},\omega)\psi_1(t). \end{align}
Equation~(\ref{4.97}) together with the initial condition $\psi_0(\omega,t\!=\!t_0)$ $\!=$ $\!0$ can be formally integrated in a straightforward way. Inserting the result into Eqs.~(\ref{4.92}) and (\ref{4.96}) then yields
\begin{multline} \label{4.98}
|\psi(t)\rangle
=\psi_1(t)|\{0\}\rangle|1\rangle
+\frac{\mathrm{i}}{\hbar}\sum_{\lambda={e},{m}}\int\mathrm{d}^3r
\int_0^\infty\mathrm{d}\omega
\int_{t_0}^t\mathrm{d}\tau\,
\mathrm{e}^{-\mathrm{i}(E_0/\hbar+\omega)(t-\tau)}\psi_1(\tau)\\[1ex]
\times\vect{d}_{01}\!\cdot\!
\ten{G}^\ast_\lambda(\vect{r}_{A},\vect{r},\omega)\!\cdot\!
|\vect{1}(\vect{r}_{A},\omega)\rangle|0\rangle \end{multline}
and
\begin{equation} \label{4.99} \dot{\psi}_1(t)=-\mathrm{i}\,\frac{E_1}{\hbar}\,\psi_1(t)
-\int_0^\infty\mathrm{d}\omega\,g^2(\vect{r}_{A},\omega)
\int_{t_0}^t\mathrm{d}\tau\,\mathrm{e}^{-\mathrm{i}(E_0/\hbar+\omega)(t-\tau)}
\psi_1(\tau), \end{equation}
respectively. Combining Eqs.~(\ref{4.91}) and (\ref{4.98}) and making use of the integral relation (\ref{2.30b}), one finds that
\begin{multline} \label{4.100} \vect{F}(t) =\frac{\mathrm{i}\mu_0}{\pi}\int_0^\infty\mathrm{d}\omega\,\omega^2
\left\{\bm{\nabla}\left[\vect{d}_{10}\!\cdot\!
\mathrm{Im}\,\ten{G}^{(1)}(\vect{r},\vect{r}_{A},\omega)
\!\cdot\!\vect{d}_{01}\right]\right\}_{\vect{r}=\vect{r}_{A}}
\\[.5ex] \times\int_{t_0}^t\mathrm{d}\tau\,\psi_1^\ast(t)\psi_1(\tau)
\mathrm{e}^{-\mathrm{i}(E_0/\hbar+\omega)(t-\tau)}
+\mathrm{C.c.} \end{multline}
Since so far nothing has been said about the strength of the atom--field coupling, Eq.~(\ref{4.100}) gives the electric part of the resonant component of the dispersion force on a two-level atom which is initially prepared in the upper state for arbitrary coupling strengths. Let us now approximate that part of the excitation spectrum of the body-assisted electromagnetic field which may give rise to strong atom--field coupling in the resonant transition by a quasi-mode (labeled by $\nu$) of Lorentzian shape,
\begin{equation} \label{4.101} g^2(\vect{r}_{A},\omega)=g^2(\vect{r}_{A},\omega_\nu)\,
\frac{\gamma_\nu^2/4}{(\omega-\omega_\nu)^2+\gamma_\nu^2/4} +
g'{^2}(\vect{r}_{A},\omega) \end{equation}
and assume that the effect of the residual part of the field which is described by the term $g'^{2}(\vect{r}_{A},\omega)$ is weakly coupled to the atom, so that it can be treated in the Markov approximation. {F}rom Eq.~(\ref{4.99}) it then follows that \cite{0719}
\begin{equation} \label{4.103} \psi_1(t)=\mathrm{e}^{[-\mathrm{i} E_1/\hbar-\mathrm{i}\delta\omega'_1(\vect{r}_{A})
-\Gamma_1'(\vect{r}_{A})/2](t-t_0)}\phi_1(t) \end{equation}
where $\phi_1(t)$ is the solution to the differential equation
\begin{equation} \label{4.107} \ddot{\phi}_1(t)
+\left\{\mathrm{i}\Delta\omega(\vect{r}_{A})+\left[\gamma_\nu
-\Gamma_1'(\vect{r}_{A})\right]/2\right\}
\dot{\phi}_1(t)
+{\textstyle\frac{1}{4}}\Omega_\mathrm{R}^2(\vect{r}_{A})
\phi_1(t)=0 \end{equation}
together with the initial conditions $\phi_1(t\!=\!t_0)$ $\!=$ $\!1$, $\dot{\phi}_1(t\!=\!t_0)$ $\!=$ $\!0$. Here, \mbox{$\Delta\omega(\vect{r}_{A})$ $\!=$ $\!\omega_\nu-\tilde{\omega}'_{10}(\vect{r}_{A})$} and $\Omega_\mathrm{R}(\vect{r}_{A})$ $\!=$ $\!\sqrt{2\pi\gamma_\nu g^2(\vect{r}_{A},\omega_\nu)}$, respectively, are the detuning and the vacuum Rabi frequency and
\begin{equation} \label{4.104} \delta\omega_1'(\vect{r}_{A}) =\delta\omega_1(\vect{r}_{A})
+\frac{\Omega_\mathrm{R}^2(\vect{r}_{A})}{4}\,
\frac{\Delta\omega(\vect{r}_{A})}
{[\Delta\omega(\vect{r}_{A})]^2+\gamma_\nu^2/4} \end{equation}
and
\begin{equation} \label{4.105} \Gamma_1'(\vect{r}_{A})=\Gamma_1(\vect{r}_{A})
-\frac{\Omega_\mathrm{R}^2(\vect{r}_{A})}{4}\,
\frac{\gamma_\nu}{[\Delta\omega(\vect{r}_{A})]^2
+\gamma_\nu^2/4}\,, \end{equation}
respectively, are the shift and width of the upper level associated with the residual part of the field where $\delta\omega_1(\vect{r}_{A})$ and $\Gamma_1(\vect{r}_{A})$, are defined according to Eqs.~(\ref{4.69})--(\ref{4.72}) with the shifted transition frequency being given by\footnote{Note that contrary to Eq.~(\ref{4.68}), the ground-state shift is absent here as a consequence of the rotating-wave approximation.}
\begin{equation} \label{4.105-1} \tilde{\omega}'_{10}(\vect{r}_{A})
=\omega_{10}+\delta\omega_1'(\vect{r}_{A}) \end{equation}
in place of Eq.~(\ref{4.68}). Equation (\ref{4.107}) can easily be solved to obtain
\begin{equation} \label{4.108} \phi_1(t)=c_+(\vect{r}_{A})\mathrm{e}^{\Omega_+(\vect{r}_{A})(t-t_0)}
+c_-(\vect{r}_{A})\mathrm{e}^{\Omega_-(\vect{r}_{A})(t-t_0)} \end{equation}
where
\begin{equation} \label{4.109} c_{\pm}(\vect{r}_{A})=\frac{\Omega_{\mp}(\vect{r}_{A})}
{\Omega_{\mp}(\vect{r}_{A})-\Omega_{\pm}(\vect{r}_{A})} \end{equation}
and
\begin{align} \label{4.110} \Omega_\pm(\vect{r}_{A})
=&-{\textstyle\frac{1}{2}}\bigl\{
\mathrm{i}\Delta\omega(\vect{r}_{A})\!+\![\gamma_\nu\!
-\!\Gamma_1'(\vect{r}_{A})]/2\bigr\}
\nonumber\\[.5ex] &\mp{\textstyle\frac{1}{2}}
\sqrt{\bigl\{\mathrm{i}\Delta\omega(\vect{r}_{A})\!+\![\gamma_\nu\!
-\!\Gamma_1'(\vect{r}_{A})]/2\bigr\}^2
-\Omega_\mathrm{R}^2(\vect{r}_{A})}\,. \end{align}
Combination of Eqs.~(\ref{4.100}), (\ref{4.103}) and (\ref{4.108}) then leads to the sought force \cite{0719}:
\begin{equation} \label{4.111} \vect{F}(t) =\frac{\mu_0}{\pi}\int_0^\infty\mathrm{d}\omega\,\omega^2
s(\vect{r}_{A},\omega,t-t_0)\Bigl\{\bm{\nabla}\bigl[\vect{d}_{10}
\!\cdot\!\mathrm{Im}\,\ten{G}^{(1)}(\vect{r},\vect{r}_{A},\omega)
\!\cdot\!\vect{d}_{01}\bigr]\Bigr\}_{\vect{r}=\vect{r}_{A}}\!\!\!
+\mathrm{C.c.} \end{equation}
with
\begin{multline} \label{4.112} s(\vect{r}_{A},\omega,t)\\[.5ex]
=|c_+(\vect{r}_{A})|^2\,
\frac{\mathrm{e}^{[-\Gamma_1'(\vect{r}_{A})+\Omega_+^\ast(\vect{r}_{A})
+\Omega_+(\vect{r}_{A})]t}
-\mathrm{e}^{\{\mathrm{i}[\tilde{\omega}'_{10}(\vect{r}_{A})-\omega]
-\Gamma_1'(\vect{r}_{A})/2+\Omega_+^\ast(\vect{r}_{A})\}t}}
{\omega-\tilde{\omega}'_{10}(\vect{r}_{A})
+\mathrm{i}\Gamma_1'(\vect{r}_{A})/2-\mathrm{i}\Omega_+(\vect{r}_{A})}
\hspace{6ex}\\[.5ex] +c_+^\ast(\vect{r}_{A}) c_-(\vect{r}_{A})\,
\frac{\mathrm{e}^{[-\Gamma_1'(\vect{r}_{A})+\Omega_+^\ast(\vect{r}_{A})
+\Omega_-(\vect{r}_{A})]t}
-\mathrm{e}^{\{\mathrm{i}[\tilde{\omega}'_{10}(\vect{r}_{A})-\omega]
-\Gamma_1'(\vect{r}_{A})/2+\Omega_+^\ast(\vect{r}_{A})\}t}}
{\omega-\tilde{\omega}'_{10}(\vect{r}_{A})
+\mathrm{i}\Gamma_1'(\vect{r}_{A})/2-\mathrm{i}\Omega_-(\vect{r}_{A})}\\[.5ex] +c_-^\ast(\vect{r}_{A}) c_+(\vect{r}_{A})\,
\frac{\mathrm{e}^{[-\Gamma_1'(\vect{r}_{A})+\Omega_-^\ast(\vect{r}_{A})
+\Omega_+(\vect{r}_{A})]t}
-\mathrm{e}^{\{\mathrm{i}[\tilde{\omega}'_{10}(\vect{r}_{A})-\omega]
-\Gamma_1'(\vect{r}_{A})/2+\Omega_-^\ast(\vect{r}_{A})\}t}}
{\omega-\tilde{\omega}'_{10}(\vect{r}_{A})
+\mathrm{i}\Gamma_1'(\vect{r}_{A})/2 -\mathrm{i}\Omega_+(\vect{r}_{A})}\\[.5ex]
+|c_-(\vect{r}_{A})|^2\,
\frac{\mathrm{e}^{[-\Gamma_1'(\vect{r}_{A})+\Omega_-^\ast(\vect{r}_{A})
+\Omega_-(\vect{r}_{A})]t}
-\mathrm{e}^{\{\mathrm{i}[\tilde{\omega}'_{10}(\vect{r}_{A})-\omega]
-\Gamma_1'(\vect{r}_{A})/2+\Omega_-^\ast(\vect{r}_{A})\}t}}
{\omega-\tilde{\omega}'_{10}(\vect{r}_{A})
+\mathrm{i}\Gamma_1'(\vect{r}_{A})/2-\mathrm{i}\Omega_-(\vect{r}_{A})}\,. \end{multline}
Let us first make contact with result obtained in the limit of weak atom--field coupling where the first term under the square root in Eq.~(\ref{4.110}) is much larger than the second one. This is the case when for given transition dipole moment, the spectrum of the field in the resonance region is sufficiently flat,
\begin{equation} \label{4.113} \gamma_\nu \gg 2\Omega_\mathrm{R}(\vect{r}_{A}) \end{equation}
or when the atomic transition is sufficiently far detuned from the field resonance,
\begin{equation} \label{4.113-0}
|\Delta\omega(\mathbf{r}_{A})|\gg
2\Omega^2_\mathrm{R}(\mathbf{r}_{A})/\gamma_\nu. \end{equation}
By means of Taylor expansion it can then be shown that
\begin{align} \label{4.113-1} \Omega_\pm(\vect{r}_{A})\simeq\begin{cases} -\mathrm{i}\Delta\omega(\vect{r}_{A})
-[\gamma_\nu-\Gamma_1'(\vect{r}_{A})]/2,\\ \\[-1.5ex]
{\displaystyle\frac{\mathrm{i}\Omega_\mathrm{R}^2(\vect{r}_{A})}{4}\,
\frac{\Delta\omega(\vect{r}_{A})}
{[\Delta\omega(\vect{r}_{A})]^2+\gamma_\nu^2/4}
-\frac{\Omega_\mathrm{R}^2(\vect{r}_{A})}{8}\,
\frac{\gamma_\nu}{[\Delta\omega(\vect{r}_{A})]^2
+\gamma_\nu^2/4}}\,, \end{cases} \end{align}
so $c_+(\vect{r}_{A})$ $\!\simeq$ $\!1$, $c_-(\vect{r}_{A})$ $\!\simeq$ $\!0$ and Eq.~(\ref{4.100}) [together with Eqs.~(\ref{4.103}) and (\ref{4.108})] approximates to \cite{0719}
\begin{align} \label{4.115} \vect{F}(t)=&\;\mathrm{e}^{-\Gamma_1(\vect{r}_{A})(t-t_0)}\,
\frac{\mu_0}{\pi}\int_0^\infty\mathrm{d}\omega\,\omega^2
\frac{\left[\bm{\nabla}\vect{d}_{10}\!\cdot\!
\mathrm{Im}\,\ten{G}^{(1)}(\vect{r},\vect{r}_{A},\omega)
\!\cdot\!\vect{d}_{01}\right]_{\vect{r}=\vect{r}_{A}}}
{\omega-\tilde{\omega}_{10}(\vect{r}_{A})
-\mathrm{i}\Gamma_1(\vect{r}_{A})/2}
+\mathrm{C.c.}\nonumber\\[.5ex] \simeq&\;\mathrm{e}^{-\Gamma_1(\vect{r}_{A})(t-t_0)}\vect{F}_1(\vect{r}_A) \end{align}
with
\begin{equation} \label{4.115-1} \vect{F}_1(\vect{r}_A) =\mu_0\Omega^2_{10}(\vect{r}_{A})
\bigl\{\bm{\nabla}\vect{d}_{10}\!\cdot\!
\ten{G}^{(1)}[\vect{r},\vect{r}_{A},\Omega_{10}(\vect{r}_{A})]
\!\cdot\!\vect{d}_{01}\bigr\}_{\vect{r}=\vect{r}_{A}}
+\mathrm{C.c.} \end{equation}
and
\begin{equation} \label{4.115-2} \Omega_{10}(\vect{r}_{A}) =\tilde{\omega}_{10}(\vect{r}_{A})+\mathrm{i}\Gamma_1(\vect{r}_{A})/2 \end{equation}
which corresponds to the term $\sigma_{11}(t)\vect{F}_{11}^\mathrm{el,r}(\vect{r}_A)$ in Eqs.~(\ref{4.73}) and (\ref{4.74}) with $\vect{F}_{11}^\mathrm{el,r}(\vect{r}_A)$ being given according to Eq.~(\ref{4.76}).
The strong-coupling limit is realized if the spectrum of the field features a sufficiently sharp peak and the atomic transition is near resonant with this peak, such that
\begin{equation} \label{4.116} \gamma_\nu\le 2\Omega_\mathrm{R}(\vect{r}_{A})
\quad\mathrm{and}\quad
|\Delta\omega(\mathbf{r}_{A})|\ll
2\Omega^2_\mathrm{R}(\mathbf{r}_{A})/\gamma_\nu. \end{equation}
In this case, the square root in Eq.~(\ref{4.110}) becomes approximately real,
\begin{equation} \label{4.117} \Omega_{\pm}(\vect{r}_{A})
\simeq -{\textstyle\frac{1}{2}}\left\{\mathrm{i}
\Delta\omega(\vect{r}_{A})
+{\textstyle\frac{1}{2}}[\gamma_\nu-\Gamma_1'(\vect{r}_{A})]\right\}
\mp{\textstyle\frac{1}{2}}\mathrm{i}\Omega(\vect{r}_{A}) \end{equation}
where
\begin{equation} \label{4.118} \Omega(\vect{r}_{A})=\sqrt{\Omega_\mathrm{R}^2(\vect{r}_{A})
+[\Delta\omega(\vect{r}_{A})]^2
-[\gamma_\nu-\Gamma_1'(\vect{r}_{A})]^2/4} \end{equation}
so that the coefficients $c_\pm(\vect{r}_{A})$ [Eq.~(\ref{4.109})] can be given in the form
\begin{equation} \label{4.119} c_\pm(\vect{r}_{A})
=\frac{\Omega(\vect{r}_{A})\mp
\Delta\omega(\vect{r}_{A})\pm\mathrm{i}
[\gamma_\nu\!-\!\Gamma'_1(\vect{r}_{A})]/2}
{2\Omega(\vect{r}_{A})}\,. \end{equation}
Substituting Eqs.~(\ref{4.117}) and (\ref{4.119}) into Eq.~(\ref{4.111}) [together with Eq.~(\ref{4.112})], one finds that for real dipole matrix elements $\vect{F}(t)$ approximates to \cite{0719}
\begin{multline} \label{4.121} \vect{F}(t) =2\mathrm{e}^{-[\gamma_\nu + \Gamma_1'(\vect{r}_{A})](t-t_0)/2}
\sin^2[\Omega(\vect{r}_{A})(t-t_0)/2]\\
\times\frac{[\Delta\omega(\vect{r}_{A})]^2
-[\gamma_\nu\!-\!\Gamma'_1(\vect{r}_{A})]^2/4}
{\Omega_\mathrm{R}^2(\vect{r}_{A})+
[\Delta\omega(\vect{r}_{A})]^2
-[\gamma_\nu\!-\!\Gamma'_1(\vect{r}_{A})]^2/4}\,
\mathbf{F}_1(\vect{r}_{A}) \end{multline}
where $\mathbf{F}_1(\vect{r}_{A})$ is given according to Eq.~(\ref{4.115-1}) with
\begin{equation} \label{4.122} \Omega'_{10}(\vect{r}_{A})
=\tilde{\omega}'_{10}(\vect{r}_{A})+\mathrm{i}\Gamma'_1(\vect{r}_{A})/2 \end{equation}
in place of Eq.~(\ref{4.115-2}). Note that $[\gamma_\nu+\Gamma_1'(\vect{r}_A)]/2$ $\!=$ $\!\gamma_\nu/2$ if
$|\Delta\omega(\vect{r}_A)|$ $\!\ll$ $\gamma_\nu/2$ and $[\gamma_\nu+\Gamma_1'(\vect{r}_A)]/2$ $\!=$ $\!\Gamma_1(\vect{r}_A)/2$ if $\gamma_\nu/2$ $\!\ll$
$\!|\Delta\omega(\vect{r}_A)|$ $\!\ll$ $\!2\Omega^2_\mathrm{R}(\mathbf{r}_{A})/\gamma_\nu$.
Comparing Eq.~(\ref{4.121}) with Eq.~(\ref{4.115}), we see that while the resonant component of the force in the limit of weak atom--field coupling simply exponentially decreases as a function of time, Rabi oscillations of the force are typically observed in the strong-coupling limit---in agreement with the well-known features of spontaneous emission in the two coupling regimes. As a consequence of the appearance of Rabi oscillations, the magnitude of the force changes periodically; for appropriate spatial structure of the resonantly interacting quasi-mode, the atom may be trapped with the trap being set by the atom itself, cf.~also Refs.~\cite{0409,0410}. Rabi oscillations do not occur if the system is initially prepared in a dressed state; in this case the (exponentially decaying) force is simply given by the gradient of the position-dependent part $\pm\hbar\Omega(\vect{r}_{A})/2$ [recall Eq.~(\ref{4.118})] of the respective dressed-state energy \cite{0179,0407}.
\section{Concluding remarks}\vspace*{-1ex} \label{sec5}
Dispersion forces are a particular signature of the quantum nature of the interaction of matter with the electromagnetic field. As soon as the interacting matter consists of a large number of elementary atomic particles, exact microscopic calculations become very involved. Therefore most theoretical approaches to dispersion forces make use of assumptions typical of macroscopic electrodynamics by introducing---sooner oder later---familiar macroscopic concepts such as boundary conditions at surfaces of discontinuity and/or constitutive relations averaged over a sufficiently large number of the elementary constituents of the respective material objects. Macroscopic electrodynamics whose applicability surprisingly ranges even to nano-structures, has the benefit of being universally valid, because it uses only very general physical properties, without the need of involved ab initio calculations. Moreover, all the relevant quantities used for characterizing the material objects can easily be inferred from measurements. This concept does not only apply to classical electrodynamics but also to QED. Macroscopic QED has been well elaborated for the case of locally responding media described in terms of complex-valued, position- and frequency-dependent permittivities and permeabilities. It can be extended to arbitrary linear media, including spatially dispersing media, since the description of the quantized field in terms of current densities and the Green tensor associated with the macroscopic Maxwell equations is independent of the particular medium description. When supplemented with standard atom--field coupling terms, a powerful tool for studying medium-assisted quantum effects in QED is obtained. Clearly, the applicability of the theory is restricted to body--body and body--atom separations that are sufficiently large compared with the length scale on which the atomistic structure of the bodies begins to play a role.
In particular, the so established macroscopic QED provides a unified approach to the various types of dispersion forces---an approach which incorporates the benefits of normal-mode and linear-response approaches, while exactly taking into account real material properties. In particular, dispersion forces between electrically neutral, unpolarized and unmagnetized ground-state bodies simply reflect the forces which are due to the action of the fluctuating body-assisted electromagnetic vacuum on the fluctuating charge and current densities of the bodies. Since all the relevant characteristics of the bodies enter the so obtained force formulas via the Green tensor of the associated macroscopic Maxwell equations, they are valid for arbitrary (linear) bodies. Both the Casimir stress and the Casimir force density can thus be introduced in a natural way. Moreover, by appropriate Born-series expansions of the Green tensor, relations between dispersion forces on bodies and dispersion forces on atoms can be established which clearly demonstrate the common origin of all these forces. In particular, the force on an atom in the presence of arbitrary bodies as well as the force between two atoms can be obtained as limiting cases of the body--body force.
In this article we have restricted our attention to ground-state bodies, i.e, to bodies that at zero temperature interact with the electromagnetic field where the effect of dispersion forces is purely quantum by nature. As outlined, an extension of the central results to include equilibrium systems at finite temperatures can be obtained in a straightforward way by simply replacing the vacuum averages by thermal averages. In this context it should be pointed out that the central assumption of linear-response theories according to which the thermal average of the field fluctuations is related to the imaginary part of the field response function, is explicitly fulfilled within the framework of macroscopic QED.\vspace*{-1ex}\pagebreak
As we have seen, dispersion forces on ground-state atoms turn out to be limiting cases of dispersion forces on macroscopic bodies, so the corresponding formulas can be obtained without explicitly addressing the underlying atom--field interaction. Of course, they can also be derived by explicitly solving the quantum-mechanical problem of individual atoms interacting with the electromagnetic field, with the presence of macroscopic bodies being again described within the framework of macroscopic QED. For ground-state atoms where only virtual transitions occur, this leads to results that agree with the ones obtained from the purely macroscopic approach, as expected. In fact, explicitly addressing the atom--field interaction is more flexible, because it can also be applied to non-equilibrium systems, such as initially excited atoms where also real transitions are involved in the atom--field interaction. In this case, a dynamical description is in general preferred to be employed, leading to time-dependent expressions for the forces, according to the temporal evolution of the atomic quantum state. In particular for weak atom--field coupling, the force on an initially excited atom is a sum of components whose temporal evolution follows that of the associated atomic density matrix elements which is in turn governed by the familiar master equation of an atomic system undergoing radiative damping. For strong atom--field coupling, damped Rabi oscillations may occur which periodically change the magnitude of the force. The dynamical approach could serve as a starting point for studying dispersion forces on bodies that are not in thermal equilibrium, by appropriately modeling such bodies as collections of excited atoms \cite{0333,0522}.\vspace*{-1ex}
Including linearly responding bodies in macroscopic QED has the advantage that from the very beginning of all calculations the effect of the bodies is taken into account in a consistent manner, without the need to specify the properties of the bodies at an early stage. In this way very general results of broad applicability can be obtained. This naturally applies not only to dispersion forces, but also to other quantum phenomena of radiation--matter interaction which are strongly influenced by the presence of macroscopic bodies---phenomena that may be subsumed under the term Casimir effect in the broadest sense of the word. Typical examples are the enhancement and inhibition of spontaneous emission, resonant energy transfer between atoms or molecules and the wide field of cavity-QED effects.
\ack We would like to acknowledge fruitful collaboration with Ho Trung Dung, T. Kampf, C. Raabe and H. Safari. Furthermore, we are grateful to L. Arntzen, G. Barton, I. Bondarev, M. DeKieviet, A. Guzm\'{a}n, A. Lambrecht, S. Linden, E. Shamonina, Y. Sherkunov, L. Rizzuto, M. S. Toma\v{s} and C. V\'{i}llarreal for stimulating discussions.
\appendix
\section{Overview over scenarios} \label{app1}
In order to provide for an overview over the various theoretical works on dispersion forces, references containing work on body--body, atom--body and atom--atom forces are given in separate tables. In the tables, the references are structured according to the scenarios considered, regardless of the methods used to address these scenarios.
\begin{table}[!h!] \begin{center}
\begin{tabular}{|c||c|c|c|} \hline Material $\rightarrow$ & & &Magneto- \\ \cline{1-1} Geometry $\downarrow$ & \raisebox{2.5ex}[0pt]{Perfect cond.} &\raisebox{2.5ex}[0pt]{Electric} &electric \\ \hline\hline &\cite{0373,0068,0131,0746} &\cite{0120,0650,0649,0341,0630,0631, 0651,0333,0522} &\cite{0122,0123} \\ &\cite{0602,0677,0674} &\cite{0611,0652,0642,0641,0656,0640,0668,0667,0626} &\cite{0659,0134} \\ Half space &\cite{0637,0744} &\cite{0644,0132,0676,0690,0607,0657,0680,0628,0621,0604} &\cite{0124,0133} \\ + half space &\cite{0616}$^\mathrm{T}$ &\cite{0675,0672,0687,0048,0638,0606,0603}, \cite{0666}$^\mathrm{T}$ &\cite{0125}$^\mathrm{T}$ \\ &\cite{0747,0748,0749}$^\mathrm{L}$ &\cite{0688,0625,0645,0682,0681, 0057,0264,0646,0691,0689,0683}$^\mathrm{T}$ &\cite{0126}$^\mathrm{T}$ \\ & &\cite{0673}$^\mathrm{L}$, \cite{0684,0743}$^\mathrm{Tq}$ &\cite{0127}$^\mathrm{T}$ \\ \hline Half space & & &\cite{0670,0669} \\ + plate & & &\cite{0661,0198} \\ \hline Half space &\cite{0695,0637,0744} &\cite{0641,0638}, \cite{0625}$^\mathrm{T}$ & \\ + sphere & & & \\ \hline Half space &\cite{0750,0695} & & \\ + cylinder & & & \\ \hline Plate & &\cite{0197,0612,0678,0665,0203,0647,0660} & \\ + plate & &\cite{0639}$^\mathrm{Tq}$ & \\ \hline Plate & &\cite{0071} & \\ + sphere & & & \\ \hline Sphere &\cite{0124,0637} &\cite{0343,0654,0377} & \\ + sphere & & & \\ \hline \end{tabular} \end{center} \vspace*{2ex} \caption{ Schedular summary of references associated with theoretical work on body--body forces. The superscripts indicate that finite temperature (T), lateral forces (L) and/or torques (Tq) are included. } \label{tab4} \end{table}
\begin{table}[!h!] \begin{center}
\begin{tabular}{|c||c|c|c|} \hline Material $\rightarrow$ & & &Magneto- \\ \cline{1-1} Geometry $\downarrow$ & \raisebox{2.5ex}[0pt]{Perfect conductor} &\raisebox{2.5ex}[0pt]{Electric} &electric \\ \hline\hline &\cite{0030,0022,0288,0282,0289} &\cite{0119,0120,0288,0282,0277,0025,0023,0026,0027} &\cite{0043,0392} \\ &\cite{0325,0412,0047,0055,0054,0056,0320} &\cite{0269,0281,0273,0286,0276,0341,0348,0272,0283,0290,0036} &\cite{0012,0018} \\ &\cite{0321,0072,0035,0039} &\cite{0285,0287,0295,0051,0053,0031,0039} &\cite{0019}, \cite{0330}$^\mathrm{M}$ \\ &\cite{0041}, \cite{0367,0061}$^\mathrm{E}$ &\cite{0041,0077,0653,0400,0280,0048,0388} & \\ \raisebox{2.5ex}[0pt]{Half space} &\cite{0062,0296,0235}$^\mathrm{E}$ &\cite{0020},\cite{0347,0028,0333,0522,0296}$^\mathrm{E}$ & \\ &\cite{0042}$^\mathrm{E}$, \cite{0095}$^\mathrm{M}$ &\cite{0308,0331,0042,0044,0045,0443}$^\mathrm{E}$ & \\ &\cite{0376,0037}$^\mathrm{T}$ &\cite{0008,0012,0018}$^\mathrm{E}$, \cite{0275}$^\mathrm{M}$, \cite{0057}$^\mathrm{T}$ & \\ & &\cite{0264,0046,0394,0399}$^\mathrm{T}$ & \\ \hline Plate & &\cite{0391}$^\mathrm{T}$ &\cite{0012,0019} \\ \hline &\cite{0110,0346} &\cite{0110,0270,0060,0069,0077,0349,0372,0305} &\cite{0113} \\ \raisebox{2.5ex}[0pt]{Sphere} & &\cite{0017} & \\ \hline & &\cite{0040,0397,0284,0077,0395,0316,0305} & \\ \raisebox{2.5ex}[0pt]{Cylinder} & &\cite{0186,0439,0187}, \cite{0442}$^\mathrm{E}$, \cite{0391}$^\mathrm{T}$ & \\ \hline &\cite{0070,0321,0032,0314} &\cite{0032,0314}, \cite{0033}$^\mathrm{T}$ &\cite{0012,0019} \\ \raisebox{2.5ex}[0pt]{Planar} &\cite{0327,0292,0278,0301,0063,0315}$^\mathrm{E}$ & & \\ \raisebox{2.5ex}[0pt]{Cavity} &\cite{0293}$^\mathrm{E,M}$, \cite{0034}$^\mathrm{T}$ & & \\ \hline Spher. cav. & &\cite{0393,0396,0313,0305}, \cite{0441}$^\mathrm{E}$ & \\ \hline Cyl. cav. & &\cite{0316,0305} & \\ \hline Parab. cav. &\cite{0390,0389} & & \\ \hline \end{tabular} \end{center} \vspace*{2ex} \caption{ Schedular summary of references associated with theoretical work on atom--body forces. Unless otherwise stated, nonmagnetic ground-state atoms are considered. The superscripts indicate that excited atoms (E), magnetic atoms (M) and/or finite temperature (T) are included.} \label{tab3} \end{table}
\begin{table}[!h!] \begin{center}
\begin{tabular}{|c||c|c|c|} \hline Material $\rightarrow$ & & &Magneto- \\ \cline{1-1} Geometry $\downarrow$ & \raisebox{2.5ex}[0pt]{Perfect conductor} &\raisebox{2.5ex}[0pt]{Electric} &electric \\ \hline\hline
&\multicolumn{3}{c|}{ \cite{0030,0374,0515,0510,0507,0119,0120, 0514,0513,0237,0011,0521,0325,0498, 0047,0055,0054,0056,0320,0051,0323, 0007,0067,0201,0200,0490,0035,0048,0491,0020}}\\ Free space
&\multicolumn{3}{c|}{ \cite{0527,0526,0493,0099,0098,0333,0522}$^\mathrm{E}$, \cite{0496,0497}$^\mathrm{E,M}$, \cite{0499,0121,0189,0492,0089,0095, 0094,0097,0537,0096,0491}$^\mathrm{M}$}\\
&\multicolumn{3}{c|}{ \cite{0104}$^\mathrm{M,T}$, \cite{0103,0376,0105,0101,0057,0264,0037,0102}$^\mathrm{T}$ } \\ \hline & &\cite{0351} &\cite{0009,0113,0739}\\ \raisebox{2.5ex}[0pt]{Bulk medium} & & &\cite{0669}$^\mathrm{M}$ \\ \hline Half Space &\cite{0361,0367,0309,0679}, \cite{0093}$^\mathrm{T}$ &\cite{0518,0036,0107,0653,0523} &\cite{0009,0491} \\ \hline Planar cavity &\cite{0092}, \cite{0093}$^\mathrm{T}$ &\cite{0107}, \cite{0108}$^\mathrm{T}$ & \\ \hline \end{tabular} \end{center} \vspace*{2ex} \caption{ Schedular summary of references associated with theoretical work on atom--atom forces, possibly in the presence of bodies. Unless otherwise stated, nonmagnetic ground-state atoms are considered. The superscripts indicate that excited atoms (E), magnetic atoms (M) and/or finite temperature (T) are included.} \label{tab2} \end{table}
\section{Green tensors} \label{appA}
The Green tensor in free space is given by \cite{0003}
\begin{equation} \label{A.1} \ten{G}_\mathrm{free}(\vect{r},\vect{r}',\mathrm{i}\xi)
=\frac{1}{3}\Bigl(\frac{c}{\xi}\Bigr)^2\delta(\bm{\rho})\ten{I}
+\frac{c^2\mathrm{e}^{-\xi\rho/c}}{4\pi\xi^2\rho^3}
\bigl[a(\xi\rho/c)\ten{I}
-b(\xi\rho/c)
\vect{e}_\rho\tprod\vect{e}_\rho\bigr] \end{equation}
($\bm{\rho}$ $\!=$ $\!\vect{r}-\vect{r}'$; $\rho$ $\!=$
$\!|\bm{\rho}|$; $\vect{e}_\rho$ $\!=$ $\!\bm{\rho}/\rho$) where
\begin{equation} \label{A.2} a(x)=1+x+x^2,\qquad b(x)=3+3x+x^2. \end{equation}
The scattering Green tensor for the planar magneto-electric structure characterized by Eqs.~(\ref{3.27}) and (\ref{3.28}) is given by \cite{0217,0215}
\begin{equation} \label{A.3} \ten{G}^{(1)}(\vect{r},\vect{r}',\mathrm{i}\xi) =\int\mathrm{d}^2q\,\mathrm{e}^{\mathrm{i}\vect{q}\,\!\cdot\!\,(\vect{r}-\vect{r}')} \ten{G}^{(1)}(\vect{q},z,z',\mathrm{i}\xi)\quad\mbox{for }0<z,z'<d \end{equation}
($\vect{q}\perp\vect{e}_z$) with
\begin{multline} \label{A.4} \ten{G}^{(1)}(\vect{q},z,z',\mathrm{i}\xi)
=\frac{\mu(\mathrm{i}\xi)}{8\pi^2b}
\sum_{\sigma=s,p}\biggl\{\frac{r_{\sigma -}r_{\sigma +}
\mathrm{e}^{-2bd}}{D_\sigma}
\Bigl[\vect{e}_\sigma^+\tprod\vect{e}_\sigma^+\mathrm{e}^{-b(z-z')}
+\vect{e}_\sigma^-\tprod\vect{e}_\sigma^-\mathrm{e}^{b(z-z')}\Bigr]\\[.5ex] +\,\frac{1}{D_\sigma}
\Bigl[\vect{e}_\sigma^+\tprod\vect{e}_\sigma^-r_{\sigma -}
\mathrm{e}^{-b(z+z')}
+\vect{e}_\sigma^-\tprod\vect{e}_\sigma^+r_{\sigma +}
\mathrm{e}^{-2bd}e^{b(z+z')}\Bigr]\biggr\}. \end{multline}
Here, $b$ and $D_\sigma$ are defined by Eqs.~(\ref{3.32}) and (\ref{3.33}), respectively,
\begin{equation} \label{A.5} \vect{e}_s^\pm=\vect{e}_q\!\times\!\vect{e}_z,
\quad\vect{e}_p^\pm=-\frac{1}{k}(\mathrm{i} q\vect{e}_z
\pm b\vect{e}_q) \end{equation}
($\vect{e}_q$ $\!=$ $\!\vect{q}/q$, $q$ $\!=$ $\!|\vect{q}|$) with
\begin{equation} \label{A.6} k=\frac{\xi}{c}\sqrt{\varepsilon(\mathrm{i}\xi)\mu(\mathrm{i}\xi)} \end{equation}
are the polarization vectors for $s$- and $p$-polarized waves propagating in the positive ($+$) and negative ($-$) $z$-directions, and $r_\sigma^\pm$ $\!=$ $\!r_\sigma^\pm(\xi,q)$ with $\sigma$ $\!=$ $\!s,p$ describe the reflection of these waves at the right ($+$) and left ($-$) walls, respectively.
In particular, assume that both walls are multi-slab magneto-electrics consisting of $N_\pm$ homogeneous layers of thicknesses $d^j_\pm$ ($j$ $\!=$ $1,\ldots,N_\pm$) with $d^{N_\pm}_\pm$ $\!=$ $\!\infty$, permittivity $\varepsilon^j_\pm(\omega)$ and permeability $\mu^j_\pm(\omega)$. In this case the reflection coefficients can be obtained from the recurrence relations [$r_{\sigma\pm}$ $\!\equiv$ $\!r_{\sigma\pm}^0$; $d$ $\!\equiv$ $\!d^0_\pm$; $\varepsilon(\omega)$ $\!\equiv$ $\varepsilon^0_\pm(\omega)$; $\mu(\omega)$ $\!\equiv$ $\mu^0_\pm(\omega)$]
\begin{align} \label{A.7} &r_{s\pm}^j= \frac{(\mu^{j+1}_\pm b^j_\pm-\mu^j_\pm b^{j+1}_\pm ) +(\mu^{j+1}_\pm b^j_\pm+\mu^j_\pm b^{j+1}_\pm ) \,\mathrm{e}^{-2b^{j+1}_\pm d^{j+1}_\pm }r_{s\pm}^{j+1}} {(\mu^{j+1}_\pm b^j_\pm+\mu^j_\pm b^{j+1}_\pm ) +(\mu^{j+1}_\pm b^j_\pm-\mu^j_\pm b^{j+1}_\pm ) \,\mathrm{e}^{-2b^{j+1}_\pm d^{j+1}_\pm }r_{s\pm}^{j+1}}\,,\\[.5ex] \label{A.8} &r_{p\pm}^j= \frac{(\varepsilon^{j+1}_\pm b^j_\pm-\varepsilon^j_\pm b^{j+1}_\pm ) +(\varepsilon^{j+1}_\pm b^j_\pm+\varepsilon^j_\pm b^{j+1}_\pm ) \,\mathrm{e}^{-2b^{j+1}_\pm d^{j+1}_\pm }r_{p\pm}^{j+1}} {(\varepsilon^{j+1}_\pm b^j_\pm+\varepsilon^j_\pm b^{j+1}_\pm ) +(\varepsilon^{j+1}_\pm b^j_\pm-\varepsilon^j_\pm b^{j+1}_\pm ) \,\mathrm{e}^{-2b^{j+1}_\pm d^{j+1}_\pm }r_{p\pm}^{j+1}} \end{align}
($j$ $\!=$ $\!0,\ldots,N_\pm$ $\!-$ $\!1$) with $r_{\sigma\pm}^{N_\pm}$ $\!=$ $\!0$ where
\begin{equation} \label{A.9} b^j_\pm = \sqrt{\frac{\xi^2}{c^2}\ \varepsilon^j_\pm(\mathrm{i}\xi)\mu^j_\pm(\mathrm{i}\xi)+q^2}\,. \end{equation}
For single, semi-infinite slabs, Eqs.~(\ref{A.7}) and (\ref{A.8}) reduce to the Fresnel coefficients
\begin{equation} \label{A.10} r_{s\pm}= \frac{\mu^1_\pm b - \mu b^1_\pm} {\mu^1_\pm b + \mu b^1_\pm}\,,\qquad r_{p\pm}= \frac{\varepsilon^1_\pm b - \varepsilon b^1_\pm} {\varepsilon^1_\pm b +\varepsilon b^1_\pm}\,. \end{equation}
\end{document} | arXiv |
Peter Lorimer (mathematician)
Peter James Lorimer (16 April 1939 – 7 February 2010) was a New Zealand mathematician. His research concerned group theory, combinatorics, and Ramsey theory.
Peter Lorimer
Born
Peter James Lorimer
(1939-04-16)16 April 1939
Christchurch, New Zealand
Died7 February 2010(2010-02-07) (aged 70)
Auckland, New Zealand
Alma materMcGill University
Scientific career
FieldsMathematics
InstitutionsUniversity of Canterbury, University of Auckland
ThesisA study of T2-groups. (1963)
Doctoral advisorHans Schwerdtfeger
Academic career
Born in Christchurch, Lorimer did a BSc / MSc in mathematics at the University of Auckland and won a Commonwealth Scholarship to do a PhD at McGill University in Montreal, which he completed in 1963 under the supervision of Hans Schwerdtfeger.[1] He returned to New Zealand to lecture, first at University of Canterbury and then at University of Auckland.[2][3]
References
1. Peter Lorimer at the Mathematics Genealogy Project
2. "In memoriam: Peter James Lorimer (16.04.1939 – 7.02.2010) – MathsDept". Math.auckland.ac.nz. Retrieved 30 July 2014.
3. "Peter James Lorimer « Obituaries « Fellowship « The Academy « Our Organisation « Royal Society of New Zealand". Royalsociety.org.nz. Retrieved 30 July 2014.
External links
• institutional homepage
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| Wikipedia |
Debris Disk Around a Dead Star
by Paul Gilster on February 13, 2007
Our Solar System in the distant future may look something like the Helix nebula today. That's because in about five billion years, the Sun will have become a white dwarf, its inner planets swallowed up by its earlier expansion, its outer planets, asteroids and comets surviving in distant orbits and colliding with each other to form a ring of dusty debris. The Sun will undergo, in other words, a kind of rejuvenation, experiencing what scientists call 'late bombardment' in a system that has become dynamically young again.
Such a disk has now been found in the Helix nebula, some 700 light years away in Aquarius. It took the infrared tools of the Spitzer Space Telescope to sort out the glow of the dusty disk that circles the remnant white dwarf between 35 and 150 AU out. The assumption is that the disk is the result of smashups in the outer system, presumably involving objects like those in our Kuiper Belt or comets from an Oort-like cloud.
Image: Spitzer's infrared view of the Helix nebula. Infrared light from the outer gaseous layers is represented in blues and greens. The white dwarf is visible as a tiny white dot in the center of the picture. The red color in the middle of the eye denotes the final layers of gas blown out when the star died. The brighter red circle in the very center is the glow of a dusty disk circling the white dwarf (the disk itself is too small to be resolved). Credit: NASA/JPL-Caltech/K. Su (Univ. of Arizona).
A dusty disk found last year around the white dwarf G29-38 had shown that objects like these could survive around dead stars, though the disk around G29-38 was much closer to its star. We obviously have much to learn about debris disks in such settings. And what exactly does happen when a Sol-like star becomes a red giant? Here's a snippet from the paper on this work (references edited out; see the preprint):
It has been established that any planet closer than ~1 AU will be engulfed by an expanding red giant…, while planets outside ~5 AU from the Sun will survive post-main-sequence evolution…, and the orbits of surviving planets and most of the Kuiper Belt objects (KBOs) and Oort Cloud comets will expand adiabatically and remain bound to the solar system… The re-stabilized KBOs and Oort Cloud can later become the source of objects that go into the inner part of the system, either plunging into the white dwarf or breaking up due to tidal destruction, and they can populate the inner system with dust.
The newfound debris disk may be solving a different mystery as well. The Helix nebula's white dwarf is known to be emitting highly energetic x-rays, leading some astronomers to believe it was accreting matter from a hidden companion star. But disk material falling onto the star and triggering the outbursts seems to be a more satisfactory answer. "The high-energy X-rays were an unsolved mystery, said You-Hua Chu (UIUC). "Now, we might have found an answer in the infrared."
The paper on this work is Su et al., "A Debris Disk around the Central Star of the Helix Nebula?," which will appear in Astrophysical Journal Letters. The preprint is already available online.
djlactin February 14, 2007, 1:19
beautiful image! what's the scale?
Administrator February 14, 2007, 8:04
The image scale: 31.5 x 23.7 arcmin, as per Spitzer info.
ljk February 14, 2007, 11:18
Astrophysics, abstract
astro-ph/0701474
From: Carl H. Gibson [view email]
Date (v1): Tue, 16 Jan 2007 17:50:50 GMT (892kb)
Date (revised v2): Tue, 13 Feb 2007 13:25:36 GMT (825kb)
Interpretation of the Helix Planetary Nebula using Hydro-Gravitational-Dynamics: Planets and Dark Energy
Authors: Carl H. Gibson (UCSD), Rudolph E. Schild (Harvard)
Comments: 46 pages 11 figures, see this http URL for further information and higher resolution figures
Hubble Space Telescope (HST/ACS) images of the Helix Planetary Nebula (NGC 7293) are interpreted using the hydro-gravitational-dynamics theory (HGD) of Gibson 1996-2006. HGD predicts that baryonic-dark-matter (BDM) dominates the mass of galaxies (Schild 1996) as Jovian (promordial-fog-particle, PFP) Planets (JPPs) in proto-globular-star-cluster (PGC) clumps within galaxy halo diameters surrounding its stars. From HGD, supernova Ia (SNe Ia) events normally occur in planetary nebulae (PNe) within PGCs where binary clustering cascades of merging planets produce central binary star systems. As central stars of PNe, binaries exchange mass and accrete JPPs to grow white-dwarfs to $1.44 M_{\sun}$ instability within ionized (Oort cloud) cavities bounded by evaporating JPPs. SNe Ia events are thus intermittently obscured by radiation-inflated JPP atmospheres producing systematic SNe Ia distance errors, so the otherwise mysterious "dark energy" concept is unnecessary. HST/ACS and WFPC2 Helix images show $>7000$ cometary globules, here interpreted as gas-dust cocoons of JPPs evaporated by beamed radiation from its white-dwarf plus companion central binary star system. Mass for growing the stars, the PNe, and possibly a SNe Ia event, is accreted gravitationally from ambient BDM JPPs. Measured JPP masses $\approx 3 \times 10^{25}$ kg with spacing $\approx 10^{14}$ m support the HGD prediction that the density $\rho$ of galaxy star forming regions fossilize the density $\rho_{0} \approx (3-1) \times 10^{-17}$ kg m$^{-3}$ existing at 30,000 years in the plasma-epoch, when proto-superclusters fragmented in the expanding universe giving the first gravitational structures.
http://arxiv.org/abs/astro-ph/0701474
ljk March 16, 2007, 9:39
From: Amaya Moro-Martin [view email]
Date: Thu, 15 Mar 2007 04:07:54 GMT (694kb)
Extra-Solar Kuiper Belt Dust Disks
Authors: Amaya Moro-Martin, Mark C. Wyatt, Renu Malhotra, David E. Trilling
Comments: 18 pages, 5 figures. Chapter from the book "Kuiper Belt", edited by A. Barucci, H. Boehnhardt, D. Cruikshank and A. Morbidelli. Forthcoming
The dust disks observed around mature stars are evidence that plantesimals are present in these systems on spatial scales that are similar to that of the asteroids and the KBOs in the Solar System. These dust disks (a.k.a. "debris disks") present a wide range of sizes, morphologies and properties. It is inferred that their dust mass declines with time as the dust-producing planetesimals get depleted, and that this decline can be punctuated by large spikes that are produced as a result of individual collisional events. The lack of solid state features indicate that, generally, the dust in these disks have sizes larger than approximately 10 microns, but exceptionally, strong silicate features in some disks suggest the presence of large quantities of small grains, thought to be the result of recent collisions. Spatially resolved observations of debris disks show a diversity of structural features, such as inner cavities, warps, offsets, brightness asymmetries, spirals, rings and clumps. There is growing evidence that, in some cases, these structures are the result of the dynamical perturbations of a massive planet. Our Solar System also harbors a debris disk and some of its properties resemble those of extra-solar debris disks. From the cratering record, we can infer that its dust mass has decayed with time, and that there was at least one major "spike" in the past during the Late Heavy Bombardment. This offers a unique opportunity to use extra-solar debris disks to shed some light in how the Solar System might have looked in the past. Similarly, our knowledge of the Solar System is influencing our understanding of the types of processes which might be at play in the extra-solar debris disks.
ljk May 9, 2007, 21:49
Planetary embryos and planetesimals residing in thin debris disks
Authors: Alice C. Quillen (Rochester), Alessandro Morbidelli (Nice), Alex Moore (Rochester)
(Submitted on 9 May 2007)
Abstract: We consider constraints on the planetesimal population residing in the disks of AU Microscopii, Beta Pictoris and Fomalhaut taking into account their observed thicknesses and normal disk opacities. We estimate that bodies of radius 5, 180 and 70 km are responsible for initiating the collisional cascade accounting for the dust production for AU-Mic, Beta-Pic and Fomalhaut's disks, respectively, at break radii from the star where their surface brightness profiles change slope. Larger bodies, of radius 1000km and with surface density of order 0.01 g/cm^2, are required to explain the thickness of these disks assuming that they are heated by gravitational stirring. A comparison between the densities of the two sizes suggests the size distribution in the largest bodies is flatter than that observed in the Kuiper belt. AU Mic's disk requires the shallowest size distribution for bodies with radius greater than 10km suggesting that the disk contains planetary embryos experiencing a stage of runaway growth.
submitted to MNRAS
Astrophysics (astro-ph)
arXiv:0705.1325v1 [astro-ph]
From: Alice C. Quillen [view email]
[v1] Wed, 9 May 2007 16:50:17 GMT (21kb)
ljk July 10, 2007, 12:18
Dynamics of Exozodiacal Clouds
Authors: M. Kuchner, C. Stark, O. Absil, J.-C. Augereau, P. Thebault
(Submitted on 9 Jul 2007)
Abstract: The inner Solar System contains a cloud of small (1-100 micron) dust grains created when small bodies-asteroids, comets, and Kuiper belt objects-collide and outgas. This dust cloud, the zodiacal cloud probably has extrasolar analogs, exozodiacal clouds. Exozodiacal clouds are related to debris disks, clouds of rocks and dust orbiting main sequence stars thought to represent the debris left over from planet formation. Some debris disks appear to contain distinct inner clouds that could be considered massive exozodiacal clouds (e.g. Koerner et al. 1998, Absil et al. 2006).
This white paper addresses the need for future theoretical work on the dynamics of exozodiacal clouds. This theoretical work should help us discover new planets and understand exozodiacal clouds as astrophysical noise. So far, observations of nearby stars have not provided good constraints on the structures of exozodiacal clouds. But future observations probably will demand a better theoretical understanding of these systems.
Comments: ExoPlanet Task Force White Paper
From: Philippe Thebault [view email]
[v1] Mon, 9 Jul 2007 15:13:48 GMT (1837kb)
ljk August 2, 2007, 15:48
The Chemical Composition of an Extrasolar Minor Planet
Authors: B. Zuckerman (1), D. Koester (2), C. Melis (1), B. Hansen (1), M. Jura (1) ((1) UCLA, (2) University of Kiel)
(Submitted on 1 Aug 2007)
Abstract: We report the relative abundances of 17 elements in the atmosphere of the white dwarf star GD 362, material that, very probably, was contained previously in a large asteroid or asteroids with composition similar to the Earth/Moon system. The asteroid may have once been part of a larger parent body not unlike one of the terrestrial planets of our solar system.
Comments: ApJ, in press
From: Michael Jura [view email]
[v1] Wed, 1 Aug 2007 17:02:53 GMT (44kb)
EF Cha: Warm Dust Orbiting a Nearby 10 Myr Old Star
Authors: Joseph H. Rhee, Inseok Song, B. Zuckerman
(Submitted on 8 Jun 2007 (v1), last revised 31 Jul 2007 (this version, v2))
Abstract: Most Vega-like stars have far-infrared excess (60 micron or longward in IRAS, ISO, or Spitzer MIPS bands) and contain cold dust (less than ~150K) analogous to the Sun's Kuiper-Belt region. However, dust in a region more akin to our asteroid belt and thus relevant to the terrestrial planet building process is warm and produces excess emission in mid-infrared wavelengths. By cross-correlating Hipparcos dwarfs with the MSX catalog, we found that EF Cha, a member of the recently identified, ~10 Myr old, "Cha-Near" Moving Group, possesses prominent mid-infrared excess. N-band spectroscopy reveals a strong emission feature characterized by a mixture of small, warm, amorphous and possibly crystalline silicate grains. Survival time of warm dust grains around this A9 star is less than ~ 1E5 yrs, much less than the age of the star. Thus, grains in this extra-solar terrestrial planetary zone must be of "second generation" and not a remnant of primodial dust and are suggestive of substantial planet formation activity. Such second generation warm excess occurs around ~ 13% of the early-type stars in nearby young stellar associations.
Comments: New Spitzer MIPS data added; 14 pages, 1 figure, ApJ in press
From: Joseph Rhee [view email]
[v1] Fri, 8 Jun 2007 22:05:08 GMT (24kb)
[v2] Tue, 31 Jul 2007 23:05:31 GMT (25kb)
ljk February 28, 2008, 7:19
Pollution of single white dwarfs by accretion of many small asteroids
Authors: M. Jura (UCLA)
(Submitted on 27 Feb 2008)
Abstract: Extrapolating from the solar system's asteroid belt, we propose that externally-contaminated white dwarfs without an infrared excess may be experiencing continuous accretion of gas-phase material that ultimately is derived from the tidal destruction of multiple small asteroids. If this scenario is correct, then observations of metal-polluted white dwarfs may lead to determining the bulk elemental compositions of ensembles of extrasolar minor planets.
Comments: AJ, in press, 19 pages, 4 figures
[v1] Wed, 27 Feb 2008 19:46:55 GMT (28kb)
ljk March 10, 2008, 22:32
Astronomers at the University of Rochester have announced that low-mass stars, and maybe even super-Jupiter-sized planets might actually be responsible for the beautiful puffy nebulae. Their research appears in the latest editions of the Astrophysical Journal Letters and Monthly Notices of the Royal Astronomical Society.
http://www.universetoday.com/2008/03/10/planets-might-actually-shape-planetary-nebulae-plus-a-gallery/
Next post: Celestial Postage
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\begin{document}
\title{Bounded variation spaces with generalized Orlicz growth related to image denoising}
\date{\today}
\author{Michela Eleuteri} \address{Michela Eleuteri, Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università degli Studi di Modena e Reggio Emilia, Italy} \email{\texttt{[email protected]}}
\author{Petteri Harjulehto}
\address{Petteri Harjulehto,
Department of Mathematics and Statistics, FI-00014 University of Helsinki, Finland}
\email{\texttt{[email protected]}}
\author{Peter Hästö}
\address{Peter Hästö, Department of Mathematics and Statistics, FI-20014 University of Turku, Finland}
\email{\texttt{[email protected]}}
\begin{abstract} Motivated by the image denoising problem and the undesirable stair-casing effect of the total variation method, we introduce bounded variation spaces with generalized Orlicz growth. Our setup covers earlier variable exponent and double phase models. We study the norm and modular of the new space and derive a formula for the modular in terms of the Lebesgue decomposition of the derivative measure and a location dependent recession function. We also show that the modular can be obtained as the $\Gamma$-limit of uniformly convex approximating energies. \end{abstract}
\keywords{Generalized bounded variation, generalized Orlicz space, Musielak--Orlicz space, non-standard growth, Gamma-convergence, minimizer, image denoising, variable exponent, double phase.} \subjclass[2020]{35J60; 26B30, 35B40, 35J25, 46E35, 49J27, 49J45.}
\maketitle
\section{Introduction}
In PDE-based image processing, a function $u:\Omega\to \mathbb{R}$ represents the gray-scale intensity at each location of an image. Edges of objects correspond to discontinuities of $u$ and make this field challenging for function spaces and the calculus of variations. The space ${\rm BV}$ of functions of bounded variation has proven to be useful in the field. We refer to the book \cite{AubK06} by Aubert and Kornprobst for an overview. The classical ROF image restoration/denoising model \cite{RudOF92} calls for minimizing the energy \[
\inf_{u\in{\rm BV}(\Omega)} \int_\Omega |Du| + |u-f|^2\, dx, \] where $f\in L^2(\Omega)$ is the given, corrupted input image that is to be restored.
The \textit{fidelity term $|u-f|^2$} forces $u$ to be close to $f$ on average, whereas the
\textit{regularizing term $|Du|$} limits the variation of $u$. This model suffers from a stair-casing effect that leads to piecewise constant minimizers \cite{ChaL97, Jal16}. For a recent overview of autonomous variants of the model we refer to \cite{PagPRV_pp}.
Image restoration has also been approached with non-autonomous energies that treat different locations differently. The first such model, by Chen, Levine and Rao \cite{CheLR06}, involves the minimization of \begin{equation}\label{eq:CLR}
\min_{u\in {\rm BV}(\Omega)} \int_\Omega \varphi_{clr}(x, |Du|)+ |u-f|^2\, dx, \end{equation} where the regularizing term has variable exponent growth for small energies and is given by \[ \varphi_{clr}(x,t):= \begin{cases} \tfrac1{p(x)}t^{p(x)}, &\text{when } t\in [0,1], \\ t - 1+\tfrac1{p(x)}, &\text{when } t>1. \end{cases} \] The variable exponent $p:\Omega\to (1,2]$ is a function bounded away from $1$ (i.e.\ $p^-:=\inf p >1$) which should be chosen close to $2$ in smooth areas of the image and close to $1$ near likely edges to avoid stair-casing as well as blurring. Since $\varphi(x,t)\sim t$ as $t\to\infty$, this model can be analyzed in the classical ${\rm BV}$-space. Furthermore, using the Lebesgue decomposition of the derivative measure $Du$, Chen, Levine and Rao define \[
\int_\Omega \varphi_{clr}(x, |Du|)\, dx :=
\int_\Omega \varphi_{clr}(x, |\nabla^a u|)\, dx + |D^su|(\Omega), \] where $\nabla^a u$ is the density of the absolutely continuous part of the derivative. They prove for instance that \begin{equation}\label{eq:CLRduality}
\int_\Omega \varphi_{clr}(x, |Du|)\, dx =
\sup_{w\in C^1_0(\Omega; {\mathbb{R}^n}), |w|\leqslant 1} \int_\Omega u \divop w - \tfrac1{p'(x)} |w|^{p'(x)}\, dx \end{equation} and use this duality formulation to prove existence and properties of minimizers of \eqref{eq:CLR}.
The reason why we call this a duality formulation and the rationale behind the term $\tfrac1{p'(x)} |w|^{p'(x)}$ will become clear once we introduce a more general framework.
Subsequently, Li, Li and Pi \cite{LiLP10} proposed an image restoration model in the variable exponent space $W^{1,{p(\cdot)}}(\Omega)$ with energy $\varphi_{p(\cdot)}(x,t):=t^{p(x)}$ and $p^->1$. The last restriction implies that the problem involves only reflexive Sobolev spaces and that the minimizers are $C^{1,\alpha}$, so theoretically it is ill-suited to the image processing context. Harjulehto, H\"ast\"o, Latvala and Toivanen \cite{HarHL08, HarHLT13} considered the same energy without the restriction $p^->1$. In this case, a relaxation procedure shows that the ``correct'' energy for ${\rm BV}$-functions is \[
\int_\Omega \varphi_{p(\cdot)}(x, |Du|)\, dx :=
\int_\Omega \varphi_{p(\cdot)}(x, |\nabla^a u|)\, dx + |D^su|(\{p=1\}) \]
provided $|D^su|(\{p>1\})=0$, analogously to the Chen--Levine--Rao formula \eqref{eq:CLRduality}.
More recently, double phase energies have attracted the attention of many in the field of non-standard growth \cite{BaaBL22, BarCM18, ColM15a, DeF22, FarFW22, LiuP22, MizS21}. Most important for image processing is the version $\varphi_{dp}(x,t):= t + a(x)t^2$ with $a\geqslant 0$ and powers $1$ and $2$. Harjulehto and H\"ast\"o \cite{HarH21} considered this energy with the interpretation \[
\int_\Omega \varphi_{dp}(x, |Du|)\, dx :=
\int_\Omega \varphi_{dp}(x, |\nabla^a u|)\, dx + |D^su|(\{a=0\}) \]
provided $|D^su|(\{a>0\})=0$. For instance they showed that it is the $\Gamma$-limit as $\varepsilon\to 0^+$ of the uniformly convex approximating energies given by $\varphi_\varepsilon(x,t):=t^{1+\varepsilon} + a(x)t^2$.
The purpose of the present article is to introduce a general model which covers all these cases as well as countless variants like the perturbed variable exponent model and the Orlicz double phase model (see \cite{HasO22a, HasO22b} for a list on variants with references). Generalized Orlicz spaces, also known as Musielak--Orlicz spaces, have been widely studied recently (see, e.g., \cite{ChlGSW21, HadSSV_pp, HurOS_pp, WanZ22, WeiXY22}). We consider a generalized $\Phi$-function $\varphi:\Omega\times [0,\infty)\to[0,\infty]$ which may have linear growth at infinity at some points and superlinear growth at others.
The dual space in the linear case is $L^\infty$ which can be seen in the restriction $|w|\leqslant 1$ in \eqref{eq:CLRduality}. This space lacks several nice properties but it is nevertheless very concrete. However, to deal with the general case we consider the space $L^{\varphi^*}(\Omega)$ given by the conjugate function $\varphi^*$. Now the linearity of $\varphi$ means that $\varphi^*$ is not doubling; in fact, it is not even finite. Consequently, we can neither use the theory of doubling $\Phi$-functions, nor the concreteness of the space $L^\infty(\Omega)$. Fortunately, the theory of non-doubling variable exponent and generalized Orlicz spaces has been developed in \cite{DieHHR11, HarH19} and we know for instance that the maximal operator is bounded irrespective of doubling. Nevertheless, we need new types of approximation estimates that handle the transition between the $L^1$-, $L^p$- and $L^\infty$-regimes without extra constants which can ruin an argument in the non-doubling case. These techniques require subtly stronger assumptions on $\varphi$, as the usual \hyperref[def:a1]{{\normalfont(A1)}}{} does not suffice (see Example~\ref{eg:weightNeeded}).
Duality is a commonly used strategy in ${\rm BV}$-spaces and image restoration. We use it to define appropriate norms $V_\varphi$ and modulars $\varrho_{V,\varphi}$ and study their properties in Section~\ref{sect:basic}. To our knowledge, this is the first time that the duality approach has been used to define a modular in a Sobolev-type space. In Section~\ref{sect:approximation}, we consider approximation with respect to $V_\varphi$ and the new space ${\rm BV}^\varphi(\Omega)$ which generalizes ${\rm BV}(\Omega)$. The main result (Theorem~\ref{thm:exactFormula}) provides the formula \[
\varrho_{V,\varphi}(u) = \varrho_\varphi(|\nabla^a u|) + \int_\Omega \varphi'_\infty \, d|D^su| \] for the modular in terms of the \textit{recession function $\varphi'_\infty:\Omega\to[0,\infty]$} defined by \[ \varphi'_\infty(x):= \limsup_{t\to \infty} \frac{\varphi(x, t)} t. \] This function is often used in relaxation including in image processing (see, e.g., \cite{AmeGZ14,PagPRV_pp}). However, since we consider the non-autonomous case, our recession function depends on $x$ and so acts as a weight on the singular part of the function. For instance in the case $\varphi(x,t):=t^{p(x)}$ we have $\varphi'_\infty=1$ in the set $\{p=1\}$ and $\varphi'_\infty=\infty$ elsewhere. This example shows that the continuity of $\varphi$ does not ensure the continuity of $\varphi'_\infty$. Furthermore, this makes the non-autonomous case much more difficult than the autonomous case, where the space ${\rm BV}^\varphi$ reduces to classical ${\rm BV}$- or Sobolev spaces (see Corollary~\ref{cor:Orlicz}).
Using this formula we conclude the paper in Section~\ref{sect:Gamma} by showing the $\Gamma$-convergence of regularized functionals from \cite{EleHH_pp} to $\varrho_{V,\varphi}$. We start with background (Section~\ref{sect:background}) and auxiliary results (Section~\ref{sect:auxiliary}). A critical tool of independent interest is the Young convolution inequality with asymptotically sharp constants (Corollary~\ref{cor:convolution}).
\section{Background} \label{sect:background}
\subsection*{Notation and terminology}
Throughout the paper we always consider a bounded domain $\Omega \subset {\mathbb{R}^n}$, i.e.\ an open and connected set. By $p':=\frac p {p-1}$ we denote the H\"older conjugate exponent of $p\in [1,\infty]$. The notation $f\lesssim g$ means that there exists a constant $c>0$ such that $f\leqslant c g$. The notation $f\approx g$ means that $f\lesssim g\lesssim f$ whereas $f\simeq g$ means that $f(t/c)\leqslant g(t)\leqslant f(ct)$ for some constant $c \geqslant 1$. By $c$ we denote a generic constant whose value may change between appearances. A function $f$ is \textit{almost increasing} (more precisely, $L$-almost increasing) if there exists $L \geqslant 1$ such that $f(s) \leqslant L f(t)$ for all $s \leqslant t$. \textit{Almost decreasing} is defined analogously. By \textit{increasing} we mean that the inequality holds for $L=1$ (some call this non-decreasing), similarly for \textit{decreasing}.
Consider a function $\|\cdot\|: X \to [0, \infty]$ on a real vector space $X$ and the following conditions: \begin{itemize}
\item[(N1)] $\|f\|=0$ implies that $f=0$.
\item[(N2)] $\|af\| = |a| \|f\|$ for all $f \in X$ and $a \in \mathbb{R}$;
\item[(N3)] $\|f+g\| \leqslant \|f\|+ \|g\|$ for all $f, g \in X$.
\item[(N3$'$)] $\|f+g\| \lesssim \|f\|+ \|g\|$ for all $f, g \in X$. \end{itemize}
We use the following terminology for $\|\cdot\|$:\\[4pt] \centerline{ \begin{tabular}{rcccc} &(N1)& (N2)& (N3) & (N3$'$) \\ \hline \textit{quasi-seminorm} & &\checkmark&&\checkmark\\ \textit{seminorm} & &\checkmark&\checkmark&\\ \textit{quasinorm} & \checkmark&\checkmark&&\checkmark\\ \textit{norm} &\checkmark &\checkmark & \checkmark &\\ \end{tabular}}
\subsection*{Generalized Orlicz spaces}
We first define types of modulars that generate our spaces. Note that our terminology differs from Musielak \cite{Mus83}. Our justification is the following: a quasi-semimodular generates a quasi-seminorm, a semimodular generates a seminorm, etc.
\begin{defn}\label{def:quasiConvexsemimodular} Let $X$ be a real vector space. A function $\varrho:X \to [0,\infty]$ is called a \textit{quasi-semimodular} on $X$ if: \begin{enumerate} \item $\varrho(0_{X})=0$; \item the function $\lambda\mapsto\varrho(\lambda x)$ is increasing on $[0,\infty)$ for every $x\in X$; \item $\varrho (-x)=\varrho(x)$ for every $x\in X$; \item there exists $\beta\in (0,1]$ such that $\varrho(\beta(\alpha x +(1-\alpha)y) ) \leqslant \alpha\varrho(x) + (1-\alpha)\varrho(y)$ for every $x,y\in X$ and every $\alpha \in [0,1] $. \end{enumerate} If (4) holds with $\beta=1$, then $\varrho$ is a \textit{semimodular}. A (quasi-)semimodular is called a \textit{(quasi)modular} provided $\varrho(x)=0$ if and only if $x=0_X$.
\end{defn}
\begin{defn}\label{def:modularSpace} If $\varrho$ is a quasi-semimodular in $X$, then the \textit{modular space
$X_\varrho:=\{x \in X \mid \|x\|_\varrho<\infty\}$} is defined by the quasi-seminorm \[
\|x\|_{\varrho}:= \inf \bigg\{\lambda >0 \,\Big|\,\varrho\Big(\frac{x}{\lambda}\Big) \leqslant 1 \bigg\}. \] \end{defn}
The next definitions are from \cite{HarH19}. Our previous works were based on conditions defined for almost every point $x\in \Omega$. In this article we also use singular measures, so the assumptions are adjusted to hold for every point, following \cite{HasJR_pp}. We denote by $L^0(\Omega)$ the set of measurable functions in $\Omega$.
\begin{defn} \label{def2-1} We say that $\varphi: \Omega\times [0, \infty) \to [0, \infty]$ is a \textit{weak $\Phi$-function}, and write $\varphi \in \Phi_{\text{\rm w}}(\Omega)$, if the following conditions hold for every $x \in \Omega$: \begin{itemize} \item
$\varphi(\cdot, |f|)$ is measurable for every $f\in L^0(\Omega)$. \item $t \mapsto \varphi(x, t)$ is increasing. \item $\displaystyle \varphi(x, 0) = \lim_{t \to 0^+} \varphi(x,t) =0$ and $\displaystyle \lim_{t \to \infty}\varphi(x,t)=\infty$. \item $t \mapsto \frac{\varphi(x, t)}t$ is $L$-almost increasing on $(0,\infty)$ with constant $L$ independent of $x$. \end{itemize} If $\varphi\in\Phi_{\text{\rm w}}(\Omega)$ is additionally convex and left-continuous with respect to $t$ for every $x\in\Omega$, then $\varphi$ is a \textit{convex $\Phi$-function} and we write $\varphi \in \Phi_{\text{\rm c}}(\Omega)$. If $\varphi$ does not depend on $x$, then we omit the set and write $\varphi \in \Phi_{\text{\rm w}}$ or $\varphi \in \Phi_{\text{\rm c}}$. \end{defn}
Since the range of $\varphi$ is $[0,\infty]$, convexity can be defined as usual by the inequality \[ \varphi(x, \theta t+(1-\theta)s)\leqslant \theta\varphi(x, t)+(1-\theta)\varphi(x,s) \] including the case $\infty\leqslant\infty$. As we deal with conjugates of linear growth at infinity, it is crucial that we allow extended real-valued $\Phi$-functions. Chlebicka, Gwiazda and colleagues (e.g.\ \cite{BorC22, ChlGSW21}) have considered the case of non-doubling N-functions; however, this is not sufficient here since N-functions exclude $L^1$- and $L^\infty$-spaces which are needed.
\begin{defn} \label{rhophi} Let $\varphi \in \Phi_{\text{\rm w}}(\Omega)$ and $\displaystyle
\varrho_\varphi(f) := \int_{\Omega} \varphi (x, |f|) \, dx$ for all $f \in L^0(\Omega)$. The set \[ L^{\varphi}(\Omega) := (L^0(\Omega))_{\varrho_\varphi} = \big\{f \in L^0(\Omega) \mid \varrho_\varphi(\lambda f) < \infty \quad \textnormal{for some }\lambda > 0\big\} \] with quasinorm given by $
\|f\|_\varphi :=
\|f\|_{\varrho_\varphi}
$
is called a \textit{generalized Orlicz space}. We use the abbreviation $\|v\|_\varphi := \big\| |v|\big\|_\varphi$ for vector-valued functions. \end{defn}
We observe that $\|\cdot\|_\varphi$ is a quasinorm in $L^\varphi(\Omega)$ if $\varphi \in \Phi_{\text{\rm w}}(\Omega)$, and a norm if $\varphi \in \Phi_{\text{\rm c}}(\Omega)$ \cite[Lemma~3.2.2]{HarH19}. We define two Sobolev spaces; the space $L^{1,\varphi}$ is sometimes denoted by $V^1L^\varphi$, indicating the that first variation $\nabla u$ belongs to $L^\varphi$. Note that $W^{1, \varphi}(\Omega) = L^{1, \varphi} (\Omega) \cap L^\varphi(\Omega)$.
\begin{defn} Let $\varphi \in \Phi_{\text{\rm w}}(\Omega)$. A function $u \in W^{1,1}(\Omega)$ belongs to
the \textit{Sobolev space} $W^{1, \varphi}(\Omega)$ if $|u|, |\nabla u| \in L^\varphi(\Omega)$ and to
the \textit{Sobolev space} $L^{1, \varphi}(\Omega)$ if $|\nabla u| \in L^\varphi(\Omega)$. The spaces are equipped with the (quasi)norms \[
\|u\|_{W^{1, \varphi}(\Omega)} := \|u\|_\varphi + \|\nabla u\|_\varphi \quad\text{and}\quad
\|u\|_{L^{1, \varphi}(\Omega)} := \|u\|_{L^1(\Omega)} + \|\nabla u\|_{L^{\varphi}(\Omega)}. \] \end{defn}
When $\varphi$ in a sub- or superscript is replaced by a real number (e.g., $L^{1,p}$ or $\varrho_2$), this is an abbreviation for the $\Phi$-function $\varphi(x,t)\equiv t^p$.
\section{Auxiliary results} \label{sect:auxiliary}
\subsection*{Regularity conditions for harmonic analysis and PDE}
We say that $\omega: [0, \infty) \to [0, \infty]$ is a \textit{modulus on continuity} if it is increasing and $\omega (0) = \lim_{t \to 0^+} \omega(t)=0$. Note that we do not require concavity and allow extended real values.
For $\varphi:\Omega\times [0,\infty)\to [0,\infty)$ and $p,q>0$ we define some conditions: \begin{itemize}[leftmargin=4em] \item[(A0)]\label{def:a0} There exists $\beta \in(0, 1]$ such that $\varphi(x, \beta) \leqslant 1 \leqslant \varphi(x, \frac1\beta)$ for every \ $x \in \Omega$.
\item[(A1)]\label{def:a1}
For every $K>0$ there exists $\beta \in (0,1]$ such that, for every $x,y\in \Omega$, \[
\varphi(x,\beta t) \leqslant \varphi(y,t)+1 \quad\text{when}\quad \varphi(y, t) \in \bigg[0, \frac{K}{|x-y|^n}\bigg]. \] \item[(VA1)]\label{def:va1}
For every $K>0$ there exists a modulus of continuity $\omega$ such that, for every $x,y\in \Omega$, \[
\varphi(x,\tfrac t{1+\omega(|x-y|)}) \leqslant \varphi(y,t)+\omega(|x-y|) \quad\text{when}\quad \varphi(y, t) \in \bigg[0,
\frac{K}{|x-y|^n}\bigg]. \]
\item[(aInc)$_p$] \label{def:aInc} There exists $L_p\geqslant 1$ such that $t \mapsto \frac{\varphi(x,t)}{t^{p}}$ is $L_p$-almost increasing in $(0,\infty)$ for every $x\in\Omega$.
\item[(aDec)$_q$] \label{def:aDec} There exists $L_q\geqslant 1$ such that $t \mapsto \frac{\varphi(x,t)}{t^{q}}$ is $L_q$-almost decreasing in $(0,\infty)$ for every $x\in\Omega$. \end{itemize}
We say that \ainc{} holds if \ainc{p} holds for some $p>1$, and similarly for \adec{}.
If $\varphi\in \Phi_{\text{\rm w}}(\Omega)$, then $\varphi(\cdot ,1)\approx 1$ implies \hyperref[def:a0]{{\normalfont(A0)}}{}, and if $\varphi$ satisfies \adec{}, then \hyperref[def:a0]{{\normalfont(A0)}}{} and $\varphi(\cdot ,1)\approx 1$ are equivalent. For instance, $\varphi(x, t)=t^p$ always satisfies \hyperref[def:a0]{{\normalfont(A0)}}{}, since $\varphi(x, 1) \equiv 1$. Assumption \hyperref[def:a1]{{\normalfont(A1)}}{} is an almost continuity condition; in the variable exponent case $\varphi(x, t):=t^{p(x)}$ it corresponds to $\log$-Hölder continuity of $\frac1p$ \cite[Proposition 7.1.2]{HarH19}. Finally, \ainc{} and \adec{} are quantitative versions of the $\nabla_2$ and $\Delta_2$ conditions and measure lower and upper growth rates.
Note that the definition of \hyperref[def:a1]{{\normalfont(A1)}}{} differs slightly from \cite{HarH19, Has15}, where it is assumed that \[
\varphi(x,\beta t) \leqslant \varphi(y,t) \quad\text{when}\quad \varphi(y, t) \in \bigg[1, \frac{1}{|B|}\bigg] \] and $x$ and $y$ belong to the ball $B$. If $\varphi$ satisfies \hyperref[def:a0]{{\normalfont(A0)}}{}, then this is equivalent to \[
\varphi(x,\beta t) \leqslant \varphi(y,t)+1 \quad\text{when}\quad \varphi(y, t) \in \bigg[0, \frac{1}{|B|}\bigg] \] and if $\varphi$ satisfies \adec{}, then we can equivalently add in the $K$, as well, see \cite{Has_pp, HasO22b}.
The ``vanishing \hyperref[def:a1]{{\normalfont(A1)}}{}'' condition \hyperref[def:va1]{{\normalfont(VA1)}}{} is a continuity condition for $\varphi$ which was introduced to prove maximal regularity of minimizers \cite{HasO22a}. In the variable exponent case it corresponds to vanishing $\log$-H\"older continuity. We need the following weaker version of \hyperref[def:va1]{{\normalfont(VA1)}}{} where at least one of the points has to belong to the set $\{\varphi'_\infty<\infty\}$ defined using the recession function:
\begin{defn} We say that $\varphi \in \Phi_{\text{\rm w}}(\Omega)$ satisfies \textit{restricted \hyperref[def:va1]{{\normalfont(VA1)}}{}} if it satisfies \hyperref[def:a1]{{\normalfont(A1)}}{} and for every $K>0$ there exists a modulus of continuity $\omega$ such that \begin{equation*}
\varphi(x,\tfrac t{1+\omega(|x-y|)}) \leqslant \varphi(y,t)+\omega(|x-y|) \quad\text{when}\quad \varphi(y, t) \in \bigg[0,
\frac{K}{|x-y|^n}\bigg] \end{equation*} for every $x,y\in \Omega$ with $\varphi'_\infty(x) <\infty$ or $\varphi'_\infty(y) <\infty$. \end{defn}
In \cite[Section~3]{HarHL08}, it was shown that $\log$-H\"older continuity in the variable exponent case was not sufficient for ${\rm BV}$-type spaces and a strong $\log$-H\"older continuity condition was introduced. As mentioned above, $\varphi(x,t) = t^{p(x)}$ satisfies \hyperref[def:a1]{{\normalfont(A1)}}{} if and only if $\frac1p$ is $\log$-H\"older continuous. We now prove a corresponding connection between restricted \hyperref[def:va1]{{\normalfont(VA1)}}{} and strong $\log$-H\"older continuity. For simplicity, only the case of finite exponents is considered.
\begin{prop}\label{prop:strong} Let $\varphi(x,t):=t^{p(x)}$ be a variable exponent energy with $p:\Omega\to[1,\infty)$. Then restricted \hyperref[def:va1]{{\normalfont(VA1)}}{} is equivalent to the \emph{strong $\log$-H\"older continuity} of $\frac1p$, i.e.\ $\log$-H\"older continuity with \[
\lim_{x\to y} \big|1-\tfrac1{p(x)}\big| \log \tfrac{1}{|x-y|} = 0 \] uniformly in $y\in \{p=1\}$. \end{prop}
\begin{proof} The connection between $\log$-H\"older continuity and \hyperref[def:a1]{{\normalfont(A1)}}{} was established in \cite[Proposition 7.1.2]{HarH19}, so it only remains to consider the vanishing $\log$-H\"older continuity around the set $\{p=1\}$. Suppose that $p(y)=1$ or, equivalently, $\varphi'_\infty(y)<\infty$. Then $\varphi(y,t)=t^1=t$.
First we assume restricted \hyperref[def:va1]{{\normalfont(VA1)}}{} with $K=1$ and modulus of continuity $\omega$. Choosing $t := |x - y|^{-n}\geqslant 1$ and denoting $r:=|x-y|$, we have \[ \big(\tfrac{t}{1 + \omega(r)}\big)^{p(x)} = \varphi(x,\tfrac t{1+\omega(r)}) \leqslant \varphi(y,t)+\omega(r) \leqslant (1+\omega(r)) t. \] Taking the logarithm of the equivalent inequality $t^{p(x)-1}\leqslant (1+\omega(r))^{p(x)+1}$, we find that \[
\big|1-\tfrac1{p(x)}\big| \log \tfrac{1}{|x-y|} \leqslant \tfrac{1+ p(x)}{np(x)} \log(1 + \omega(r)) \leqslant \tfrac{2}{n} \log(1 + \omega(r)) \to 0 \] as $r\to 0^+$. Thus $p$ is strongly $\log$-H\"older continuous.
Assume conversely that $p$ is strongly $\log$-H\"older continuous so that \[ \omega_p(r):=
\sup_{y\in\{p=1\},\, x\in B_r(y)} \big|1-\tfrac1{p(x)}\big| \log \tfrac{1}{|x-y|} \to 0 \] as $r\to 0^+$. To establish the restricted \hyperref[def:va1]{{\normalfont(VA1)}}{}-condition when $p(y) = 1$, it is enough that \[ t^{p(x) - 1} \leqslant (1 + \omega(r))^{p(x)}. \]
The inequality is trivial if $t \in [0, 1]$. So let $t>1$. The left-hand side is increasing in $t$ so the worst case is when $t = \frac{K}{|x-y|^n}$ and we can choose $\omega$ based on the estimate \[ \begin{split} t^{1-\frac1{p(x)}}-1 &
\leqslant \Big(\frac{K}{r^n}\Big)^{\frac{\omega_p(r)}{\log \frac1r}}-1 = e^{\frac{\log K + n\log \frac1r}{\log \frac1r} \omega_p(r)}-1 \leqslant e^{(\log K + n) \omega_p(r)} - 1 =: \omega(r) \end{split} \] when $r\leqslant \frac1e$. The strong $\log$-H\"older continuity ensures that this tends to zero when $x\to y$. On the other hand, if $p(x) = 1$ in the \hyperref[def:va1]{{\normalfont(VA1)}}{}-condition, then we need \[ \frac{t}{1 + \omega(r)} \leqslant t \leqslant t^{p(y)} + \omega(r), \] which holds since $\sup_{t\geqslant 0} (t - t^{p(y)}) = p(y)^{-p'(y)}(p(y)-1) \leqslant 1-\frac1{p(y)} \leqslant \omega_p(r) \leqslant \omega(r)$ for all small $r>0$. \end{proof}
\subsection*{Inequalities with sharp constants}
Analogues of Jensen's inequality \cite[Theorem~4.3.2]{HarH19} and Young's convolution inequality \cite[Lemma~4.4.6]{HarH19} are known in the generalized Orlicz space under the \hyperref[def:a1]{{\normalfont(A1)}}{} assumption, but only with constants $\beta\ll 1$. Here we show that the \hyperref[def:va1]{{\normalfont(VA1)}}{} assumption lets us choose the constant $\beta\to 1^-$ at the price of restricting to a small ball. The next result is an improvement of \cite[Theorem~2.3]{HasJR_pp}. Note that we do not assume \adec{}. This makes the proof more difficult but is critical to the application in this article.
\begin{thm}[Jensen's inequality]\label{thm:jensen} If $\varphi\in \Phi_{\text{\rm c}}(\Omega)$ satisfies \hyperref[def:va1]{{\normalfont(VA1)}}{} and $\mu$ is a probability measure in the ball $B=B_r$
with $|B|\, \|\frac{d\mu}{dx}\|_\infty=:m<\infty$, then \[
\varphi_B^-\bigg(\frac1{1+\omega(r)} \int_{B\cap \Omega} |f|\, d\mu\bigg) \leqslant \int_{B\cap \Omega} \varphi(x, f)\, d\mu + \omega(r), \] where $\omega$ be the modulus of continuity from \hyperref[def:va1]{{\normalfont(VA1)}}{} with $K:=m\varrho_\varphi(f)+2$
and $r>0$ is so small that $\omega(r)\leqslant \frac1{|B|}$. \end{thm} \begin{proof}
We define $t_0:= \int_{B\cap \Omega}|f|\, d\mu$. By \cite[Lemma~4.3.1]{HarH19}, there exists $\beta>0$ such that \[
\varphi_B^-\bigg(\beta\int_{B\cap \Omega} |f|\, d\mu\bigg) \leqslant \int_{B\cap \Omega} \varphi(x, f)\, d\mu. \] If $t_0=\infty$, this implies that the right-hand side of the claim is infinite so there is nothing to prove. Thus we may assume that $t_0<\infty$.
Denote by $\varphi'$ the left-continuous function, increasing in $s$, with \[ \varphi(x,t) = \int_0^t \varphi'(x,s)\, ds. \] Such a function exists since $\varphi$ is convex in the second variable. Fix $x_0\in B$ with \[ \tfrac1{1+\omega(r)} \varphi'\big(x_0, \tfrac1{1+\omega(r)} t_0\big)\leqslant (\varphi')_B^-\big( \tfrac1{1+\omega(r)} t_0\big) \]
and assume $\beta\leqslant \frac1{1+\omega(r)}$ is so small that $\varphi(x_0,\beta t_0) \leqslant \frac K{|B|}$.
We define $\psi\in\Phi_{\text{\rm c}}$ by \[ \psi(t):= \int_0^t \varphi'(x_0, \min\{s, \beta t_0\})\, ds\,; \] $\psi$ is convex since $\psi'$ is increasing. Furthermore, $\psi(\beta t)=\varphi(x_0, \beta t)$ if $t\leqslant t_0$. When $t\leqslant t_0$ we consider two cases to show that \[ \psi(\beta t)
\leqslant \varphi(x, t)+\omega(r) \]
for $x\in B$: if $\varphi(x, t)\leqslant \frac K{|B|}$ this follows from \hyperref[def:va1]{{\normalfont(VA1)}}{} and otherwise it follows
from $\varphi(x_0, \beta t) \leqslant \frac K{|B|}\leqslant \varphi(x, t)$. When $t> t_0$ we estimate \begin{align*} \psi(\beta t) &= \psi(\beta t_0) + \beta(t-t_0)\varphi'(x_0, \beta t_0) \leqslant \varphi(x, t_0)+\omega(r) + (t-t_0)(\varphi')_B^-(t_0) \\ &\leqslant \varphi(x, t_0)+\omega(r) + (t-t_0) \varphi'(x, t_0) \leqslant \varphi(x, t) + \omega(r), \end{align*} where we also used the convexity of $\varphi$ in the last step.
It follows from Jensen's inequality for $\psi$ that \[ \varphi\big(x_0, \beta t_0\big) =
\psi\bigg(\beta\int_{B\cap \Omega} |f|\, d\mu\bigg) \leqslant
\int_{B\cap \Omega} \psi(\beta|f|)\, d\mu \leqslant
\int_{B\cap \Omega} \varphi(x, |f|)\, d\mu + \omega(r). \] This is the claim once we show that we can choose $\beta=\frac1{1+\omega(r)}$.
Since the integral on the right-hand side can be estimated by $\frac m{|B|}\varrho_\varphi(f)=\frac{K-2}{|B|}$ and
$\omega(r)\leqslant \frac1{|B|}$, the inequality gives $\varphi(x_0, \beta t_0) \leqslant \frac{K-1}{|B|}$.
To summarize, we have shown that $\varphi(x_0, \beta t_0) \leqslant \frac{K}{|B|}$ implies
$\varphi(x_0, \beta t_0) \leqslant \frac{K-1}{|B|}$.
We next investigate how large we can make $\beta$. Consider the set \[
\Theta := \Big\{\theta \in (0,1] \,\Big|\, \varphi\big(x_0, \tfrac\theta{1+\omega(r)} t_0\big) \leqslant \tfrac K{|B|}\Big\}. \] Since $\varphi(x_0, t)\to 0$ when $t\to 0^+$, the set is non-empty. If $\theta_k\in\Theta$ with $\theta_k\nearrow \theta_0$, then the left-continuity of $\varphi(x_0,\cdot)$ implies that $\theta_0\in \Theta$. If $\sup \Theta = 1$, then this means that the previous Jensen inequality holds for $\beta=\frac1{1+\omega(r)}$ and the claim is proved. Suppose then that $\theta_0:=\sup \Theta \in (0,1)$. For $\theta_0<\theta$, this implies that \[ \varphi\big(x_0, \tfrac{\theta_0 }{1+\omega(r)}t_0\big) \leqslant
\tfrac{K-1}{|B|} <
\tfrac K{|B|} < \varphi\big(x_0, \tfrac\theta{1+\omega(r)}t_0\big). \] Since $\varphi(x_0,\cdot)$ is convex, such discontinuity is only possible if the right-hand side equals infinity for every $\theta>\theta_0$. If $\varphi(x_0, \frac1{1+\omega(r)} t_0)=\infty$, then \[ \infty = \varphi'(x_0, \tfrac1{1+\omega(r)} t_0) \leqslant (1+\omega(r)) (\varphi')_B^-(\tfrac1{1+\omega(r)} t_0) \] by the choice of $x_0$. It follows that $\varphi(x, t_0)=\infty$ for every $x\in B$.
The set $A:=\{x\in B\mid |f(x)|\geqslant t_0\}$
has positive $\mu$-measure since $t_0$ is the $\mu$-average of $|f|$. Thus also
$\int_{B\cap \Omega} \varphi(x, |f|)\, d\mu =\infty$, so the claim holds in the form $\infty\leqslant\infty$ in this case. \end{proof}
The convolution in the next result should be understood as \[ f*\eta(x) := \int_\Omega f(y)\eta(x-y)\, dy \] to account for the fact that $f$ and $\varphi$ are only defined in $\Omega$. Extending $\varphi$ outside $\Omega$ while preserving \hyperref[def:va1]{{\normalfont(VA1)}}{} is non-trivial, but luckily that is not needed here.
\begin{cor}[Young's convolution inequality]\label{cor:convolution} Let $\varphi\in \Phi_{\text{\rm c}}(\Omega)$ satisfy \hyperref[def:va1]{{\normalfont(VA1)}}{} and $\eta$ be the standard mollifier. Then there exists a modulus of continuity $\omega$ such that \[ \varrho_\varphi\big(\tfrac1{1+\omega(\delta)} f*\eta_\delta\big) \leqslant \varrho_\varphi(f) + \omega(\delta) \] for every $\delta>0$. \end{cor} \begin{proof} We may assume that $\varrho_\varphi(f)<\infty$ since otherwise there is nothing to prove. Let $\omega$ be the modulus of continuity from \hyperref[def:va1]{{\normalfont(VA1)}}{} with $K:=m\varrho_\varphi(f)+2$
and let $r>0$ be so small that $\omega(r)\leqslant \frac1{|B_r|}$. Thus Theorem~\ref{thm:jensen} yields \[
\varphi_{B_r}^-\bigg(\frac{|f*\eta_r(x)|}{1+\omega(r)} \bigg) \leqslant (\varphi(\cdot, f)*\eta_r)(x) + \omega(r). \] This yields \[
\varphi_{B_r}^-\bigg(\frac{|f*\eta_r(x)|}{1+\omega(r)} \bigg)
\leqslant \frac m{|B_r|} \int_{B_r\cap \Omega} \varphi(x, f) \, dx + \frac1{|B_r|} < \frac{K}{|B_r|}. \] Thus we obtain by \hyperref[def:va1]{{\normalfont(VA1)}}{} that \[
\varphi\bigg(x, \frac{|f*\eta_r(x)|}{(1+\omega(r))^2} \bigg) \leqslant
\varphi_{B_r}^-\bigg(\frac{|f*\eta_r(x)|}{1+\omega(r)} \bigg) + \omega(r) \leqslant (\varphi(\cdot, f)*\eta_r)(x) + 2\omega(r). \]
We integrate this over $\Omega$ and use Fubini's Theorem to conclude that \[ \varrho_\varphi\big(\tfrac1{(1+\omega(r))^2} f*\eta_r\big) \leqslant
\int_\Omega \varphi(x, f)*\eta_r \, dx + 2|\Omega| \omega(r) \leqslant
\int_\Omega \varphi(x, f) \, dx + 2|\Omega|\, \omega(r). \] This gives the claim with the modulus of continuity
$\hat\omega(r):= \max\{2\omega(r)+\omega(r)^2, 2|\Omega|\, \omega(r)\}$; when
$\omega(r)>\frac1{|B_r|}$ we set $\hat\omega(r):=\infty$. \end{proof}
\subsection*{Associate spaces and conjugate modulars}
The associate space is a variant of the dual function space which works better at the end-points $p=1$ and $p=\infty$. We define the \textit{associate space $(L^\varphi(\Omega))'\subset L^0(\Omega)$} by the norm \[
\|u\|_{(L^\varphi(\Omega))'} :=
\sup_{\|v\|_\varphi \leqslant 1} \int_\Omega uv\, dx. \] According to \cite[Theorem~3.4.6]{HarH19}, $(L^\varphi(\Omega))'= L^{\varphi^*}(\Omega)$ for $\varphi\in \Phi_{\text{\rm w}}(\Omega)$, where \[ \varphi^*(x, t) := \sup_{s\geqslant 0} (st - \varphi(x, s)). \] The conjugate function $\varphi^*$ has the following properties: \begin{itemize} \item If $\varphi \in \Phi_{\text{\rm w}}(\Omega)$, then $\varphi^*\in \Phi_{\text{\rm c}}(\Omega)$, so $\varphi^*$ is always convex and left-continuous \cite[Lemma~2.4.1]{HarH19}. \item For $p,q\in (1,\infty)$, $\varphi$ satisfies \ainc{p} or \adec{q} if and only if $\varphi^*$ satisfies \adec{p'} or \ainc{q'}, respectively \cite[Proposition~2.4.9]{HarH19}. \item If $\varphi \in \Phi_{\text{\rm c}}(\Omega)$, then $\varphi^*(x, \frac{\varphi(x, t)}{t}) \leqslant \varphi(x, t)$ \cite[p.~35]{HarH19} and $\varphi^{**}=\varphi$ \cite[Corollary~2.6.3]{DieHHR11}. \item If $\varphi$ satisfies \hyperref[def:a0]{{\normalfont(A0)}}{} or \hyperref[def:a1]{{\normalfont(A1)}}{}, then so does $\varphi^*$ \cite[Lemmas~3.7.6 and 4.1.7]{HarH19}. \end{itemize} It is well-known that ``Young's equality'' \[ t\varphi'(t) = \varphi(t) + \varphi^*(\varphi'(t)) \] holds when $\varphi\in\Phi_{\text{\rm c}}$ is continuously differentiable. In fact, we can prove it for any sub-gradient even without assuming convexity:
\begin{lem}\label{lem:dual-equality} Let $\varphi\in\Phi_{\text{\rm w}}$. If $\varphi(s)\geqslant \varphi(s_0)+k(s-s_0)$ for all $s\geqslant 0$, then $\varphi^*(k)=ks_0-\varphi(s_0)$. \end{lem} \begin{proof} We observe that \[ ks_0-\varphi(s_0) \leqslant \sup_{s\geqslant 0}(sk-\varphi(s)) \leqslant \sup_{s\geqslant 0}(sk-(\varphi(s_0)+k(s-s_0))) = ks_0-\varphi(s_0). \] Therefore, $\varphi^*(k)=\sup_{s\geqslant 0}(sk-\varphi(s))=ks_0-\varphi(s_0)$. \end{proof}
Every $\varphi\in\Phi_{\text{\rm c}}$ can be represented as \[ \varphi(t) = \int_0^t \varphi'(\tau)\, d\tau. \] The function $\varphi':\Omega\times[0,\infty)\to [0,\infty)$ can be the left-continuous left-derivative, the right-continuous right-derivative, or something in between. In any case, $\varphi(s)\geqslant \varphi(s_0)+\varphi'(s_0)(s-s_0)$ and so the previous lemma implies that \[ \varphi^*(\varphi'(t))= t \varphi'(t)-\varphi(t). \] Additionally, it is known that $\varphi(t)\approx t \varphi'(t)$ if $\varphi$ satisfies \adec{} \cite[Lemma 3.3]{HarHJ23}.
\subsection*{Functions of bounded variation}
A function $u \in L^1(\Omega)$ has \textit{bounded variation}, denoted $u \in {\rm BV}(\Omega)$, if \[
V(u, \Omega) := \sup \bigg\{ \int_\Omega u \divop w \, dx \, \Big|\, w\in C^1_0(\Omega;{\mathbb{R}^n}),
|w|\leqslant 1 \bigg\} < \infty. \] Such functions have weak first derivatives which are Radon measures which we denote $Du$.
By \cite[Proposition~3.6]{AmbFP00}, $V(u,\Omega)$ equals the total variation $|Du|(\Omega)$ of the measure $Du$, defined as \[
|Du|(A):=\sup_{\cup A_i = A} \sum_i |Du(A_i)| \] where the supremum is taken over finite partitions of $A$ by measurable sets $A_i$. Furthermore, we use the Lebesgue decomposition \begin{equation*} Du = D^a u + D^s u, \end{equation*} where $D^a u$ is the absolutely continuous part of the derivative and $D^s u$ is the singular part. The density of $D^a u$ is the vector valued function $\nabla^a u$ such that \[ \int_\Omega w \cdot d D^a u = \int_\Omega w \cdot \nabla^a u \, dx \] for all $w \in C^\infty_0(\Omega; {\mathbb{R}^n})$. The space ${\rm BV}$ has the following compactness-type property \cite[Proposition~3.13]{AmbFP00}:
if $\sup_i \big(\|u_i\|_{L^1(\Omega)} + |Du_i|(\Omega)\big)<\infty$, then there exists a subsequence and $u\in {\rm BV}(\Omega)$ such that \begin{equation*} u_{i_j}\to u\text{ in } L^1(\Omega) \quad\text{and}\quad
|Du|(\Omega)\leqslant \liminf_{j \to \infty} |Du_{i_j}|(\Omega). \end{equation*} We refer to \cite{AmbFP00} for more information about ${\rm BV}$ spaces.
The next lemma shows that the equality $V(u,\Omega)=|Du|(\Omega)$ holds separately for the singular part.
\begin{lem}\label{lem:singularPart} If $u\in {\rm BV}(\Omega)$, then
$\displaystyle|D^su|(\Omega)=\sup\bigg\{ \int_\Omega w\cdot dD^su \,\Big|\, w \in C^1_0(\Omega; {\mathbb{R}^n}), |w|\leqslant 1\bigg\}$. \end{lem} \begin{proof}
By $|Du|(\Omega)=V(u,\Omega)$, the definition of the weak derivative and $Du=D^au+D^su$, we see that \begin{align*}
|Du|(\Omega) &=
\sup_{w \in C^1_0(\Omega; {\mathbb{R}^n}), |w|\leqslant 1}\bigg( \int_\Omega w\cdot dD^au + \int_\Omega w\cdot dD^su \bigg) \\
&\leqslant
\sup_{w \in L^\infty(\Omega; {\mathbb{R}^n}), |w|\leqslant 1}\int_\Omega w\cdot dD^au + \sup_{w \in L^\infty(\Omega; {\mathbb{R}^n}), |w|\leqslant 1}\int_\Omega w\cdot dD^su \\ &=
|D^au|(\Omega) + |D^su|(\Omega). \end{align*}
Since $|D^au|(\Omega) + |D^su|(\Omega) = |Du|(\Omega)$ as $D^a$ and $D^s$ are mutually singular, each inequality has to be an equality, and so the claim follows. \end{proof}
\section{Basic properties of dual norms and modulars} \label{sect:basic}
In this section we use a duality approach to define a norm and a modular. The ``dual norm'' $V_\varphi$ is related to the associate space and H\"older's inequality, whereas the ``dual modular'' $\varrho_{V,\varphi}$ is related to Young's inequality. Note that $V_\varphi$ is not the norm generated by $\varrho_{V,\varphi}$; their relationship is explored in Lemma~\ref{lem:equivalence}.
\begin{defn}\label{defn:V-rho} Let $\varphi \in \Phi_{\text{\rm w}}(\Omega)$. For $u\in L^1(\Omega)$, we define the ``dual norm'' \[ V_\varphi(u, \Omega):=
V_\varphi(u):=\sup\bigg\{ \int_\Omega u \divop w \, dx \,\Big|\, w \in C^1_0(\Omega; {\mathbb{R}^n}), \| w \|_{\varphi^*}\leqslant 1 \bigg\} \] and the ``dual modular'' \[
\varrho_{V,\varphi}(u):=\sup\bigg\{ \int_\Omega u \divop w - \varphi^*(x, |w|)\, dx \,\Big|\, w \in C^1_0(\Omega; {\mathbb{R}^n})\bigg\}. \] We say that $u\in L^\varphi(\Omega)$ belongs to ${\rm BV}^\varphi(\Omega)$ if \[
\|u\|_{{\rm BV}^\varphi}:= \|u\|_\varphi + V_\varphi(u) < \infty. \] \end{defn}
The next example shows that this definition is an extension of the ordinary ${\rm BV}$-space, in which the norm and modular coincide. Example~\ref{eg:weightNeeded} shows that interesting things can happen in the non-autonomous case, which do not appear at all when $\varphi$ is independent of $x$.
\begin{eg} Let $\varphi(x,t):= t$ and consider the corresponding functions $V_1$ and $\varrho_{V,1}$. Then $\varphi^*(x,t)=\infty \chi_{(1,\infty)}(t)$ so that $\varrho_{\varphi^*}(w)<\infty$ if and only if $w\leqslant 1$ almost everywhere, in which case $\varrho_{\varphi^*}(w)=0$. Hence
$V_1(u) = \varrho_{V,1}(u) = |Du|(\Omega) =V(u,\Omega)$. \end{eg}
\begin{eg}\label{eg:weightNeeded} Let $\varphi(x,t):= \frac1{p(x)}t^{p(x)}$ for $p:\mathbb{R}\to[1,\infty)$. Then $\varphi^*(x,t)=\frac1{p'(x)}t^{p'(x)}$ when $p(x)>1$ and $\varphi^*(x,t)=\infty \chi_{(1,\infty)}(t)$ when $p(x)=1$. Consider the Heaviside function $h=\chi_{(0,\infty)}$ so that $Dh=\delta_{\{0\}}$, the Dirac measure. Now \[ \varrho_{V,\varphi}(h) =
\sup\big\{ w(0) - \varrho_{\varphi^*}(|w|) \,|\, w \in C^1_0(\Omega)\big\}. \] Since $w$ is continuous, there exists for every $\varepsilon\in (0,w(0))$ a number $\delta>0$ such that \[ \varrho_{\varphi^*}(w) \geqslant \int_{-\delta}^\delta \frac1{p'(x)}(w(0)-\varepsilon)^{p'(x)}. \]
Suppose first that $p(x):=1+\frac {c_{\log}}{\log(1/|x|)}$ for small $|x|$. Then
$p'(x)=\frac {\log(1/|x|)}{c_{\log}} + 1$ and \[
(w(0)-\varepsilon)^{p'(x)} = |x|^{-\frac{\log(w(0)-\varepsilon)}{c_{\log}}} (w(0)-\varepsilon). \] Hence the previous integral converges if $\log w(0)<c_{\log}$ and diverges if $\log w(0)>c_{\log}$ (when $\varepsilon\to 0^+$). On the other hand, we can choose $w$ such that $0\leqslant w\leqslant w(0)\chi_{[-\delta, \delta]}$. From this we see that $\inf_w \varrho_{\varphi^*}(w) = 0$ when $\log w(0)<c_{\log}$. It follows that \[ \varrho_{V,\varphi}(h) = e^{c_{\log}}. \]
In the same way we can show that $\varrho_{V,\varphi}(h)=1$ if $p(x):=1+|x|^\alpha$ for some $\alpha>0$. This example shows that $\varrho_{V,\varphi}(h)$ depends on the behavior of the exponent in a neighborhood of $0$, even though the support of the derivative is only $\{0\}$. \end{eg}
\begin{rem}\label{rem:restrictedTestFunction}
If $\varrho_{\varphi^*}(|w|)=\infty$, then
$\int_\Omega u \divop w - \varphi^*(x, |w|)\, dx = - \infty$ since $\int_\Omega u \divop w \,dx $ is finite as $u\in L^1(\Omega)$ and $w \in C^1_0(\Omega; {\mathbb{R}^n})$. Testing with $w\equiv 0$, we see that the supremum in $\varrho_{V,\varphi}$ is always non-negative.
Therefore test-functions $w$ with $\varrho_{\varphi^*}(|w|)=\infty$ can be omitted and we obtain the alternative, equivalent formulation \[ \varrho_{V,\varphi}(u) =
\sup\bigg\{ \int_\Omega u \divop w - \varphi^*(x, |w|)\, dx \,\Big|\, w \in C^1_0(\Omega; {\mathbb{R}^n}), \varrho_{\varphi^*}(|w|)<\infty \bigg\}. \]
Note that $\varrho_{\varphi^*}(|w|)<\infty$ does not follow from $w \in C^1_0(\Omega; {\mathbb{R}^n})$ as $\varphi^*$ does not satisfy \adec{}. \end{rem}
\begin{rem} In our definition we use test-functions from $C^1_0(\Omega;{\mathbb{R}^n})$. This corresponds to the definition of the usual ${\rm BV}$-space. An alternative in duality formulations (e.g.\ \cite{AmeGZ14, CheLR06}) is $C^1(\Omega;{\mathbb{R}^n})$, which means that the boundary values of the function $u$ also influence the norm and
modular. The restriction $|w|\leqslant 1$ carries over nicely to the boundary and leads to a boundary term in
$L^1(\partial \Omega)$. This is not the case with $\varrho_{\varphi^*}(|w|)<\infty$. It seems that an additional boundary term for $w$ of fractional Sobolev space-type is needed in $\varrho_{V,\varphi}$ if we want to obtain appropriate boundary values in the generalized Orlicz case. This remains a problem for future research. \end{rem}
Let $w \in C^1_0(\Omega; {\mathbb{R}^n})$ with $|w|\leqslant 1$. Since $\Omega$ is bounded and \hyperref[def:a0]{{\normalfont(A0)}}{} for $\varphi$ implies \hyperref[def:a0]{{\normalfont(A0)}}{} for $\varphi^*$, we find that $w \in L^{\varphi^*}(\Omega)$ and
$\|w\|_{\varphi^*} \leqslant c\, \|w\|_\infty \leqslant c$, see Corollary 3.7.10 in \cite{HarH19}. By the definition of $V_\varphi$, \[
\int_\Omega u \divop w \, dx = \|w\|_{\varphi^*} \int_\Omega u \divop \tfrac w{ \|w\|_{\varphi^*}} \, dx
\leqslant \|w\|_{\varphi^*} V_\varphi(u) \] and taking supremum over all such $w$, we obtain that $V(u,\Omega)\leqslant c V_\varphi(u)$. Since $\Omega$ is bounded and $\varphi$ satisfies \hyperref[def:a0]{{\normalfont(A0)}}{}, we have $L^\varphi(\Omega) \hookrightarrow L^1(\Omega)$ by \cite[Corollary 3.7.9]{HarH19}. Thus ${\rm BV}^\varphi(\Omega) \hookrightarrow {\rm BV}(\Omega)$ provided $\varphi$ satisfies \hyperref[def:a0]{{\normalfont(A0)}}{}.
\begin{lem} If $\varphi \in \Phi_{\text{\rm w}}(\Omega)$, then $V_\varphi$ is a seminorm
and $\|\cdot\|_{{\rm BV}^\varphi}$ is a quasinorm in ${\rm BV}^\varphi(\Omega)$.
Moreover, if $\varphi \in \Phi_{\text{\rm c}}(\Omega)$, then $\|\cdot\|_{{\rm BV}^\varphi}$ is a norm. \end{lem}
\begin{proof}
The homogeneity property $V_\varphi(au)= |a|V_\varphi(u)$ is clear. Let us show that $V_\varphi$ satisfies the triangle inequality. If $u, v \in L^1(\Omega)$, then \[ \int_\Omega (u + v) \divop w \, dx = \int_\Omega u \divop w \, dx + \int_\Omega v \divop w \, dx \leqslant V_\varphi(u) + V_\varphi(v) \]
for $w \in C^1_0(\Omega; {\mathbb{R}^n})$ with $\|w\|_{\varphi^*}\leqslant 1$. By taking the supremum over such $w$ we have \[ V_\varphi(u+v) \leqslant V_\varphi(u) + V_\varphi(v). \]
Note that $\|\cdot \|_\varphi$ is a quasinorm if $\varphi \in \Phi_{\text{\rm w}}(\Omega)$, and a norm if $\varphi \in \Phi_{\text{\rm c}}(\Omega)$. Combining these two results, we obtain
the (quasi)triangle inequality for the sum that is ${\|\cdot\|}_{{\rm BV}^\varphi(\Omega)}$. These properties also imply that ${\rm BV}^\varphi(\Omega)$ is a vector space. \end{proof}
From the next lemma it follows that the sum $\varrho_\varphi + \varrho_{V,\varphi}$ is a quasi-semimodular, and a quasimodular if $\varphi$ satisfies \adec{}. Note that the convexity of $\varphi$ is not required.
\begin{lem}\label{lem:pseudo-modular} If $\varphi\in\Phi_{\text{\rm w}}(\Omega)$, then $\varrho_{V,\varphi}$ is a left-continuous semimodular in $L^1(\Omega)$. \end{lem} \begin{proof} Since $w=0$ is a possible test function, we see that $\varrho_{V,\varphi}\geqslant 0$.
If $u=0$ a.e., then the integrand in $\varrho_{V,\varphi}$ is the non-positive function $-\varphi^*(x, |w|)$, so that $\varrho_{V,\varphi}(0)\leqslant 0$. Thus $\varrho_{V,\varphi}(0)=0$ and property (1) from the definition of semimodular holds. If $w$ is a test function, then so is $-w$ and hence $\varrho_{V,\varphi}(-u)=\varrho_{V,\varphi}(u)$. Thus property (3) holds, as well.
To show that $\lambda \mapsto \varrho(\lambda x)$ is increasing we let $\lambda \in (0, 1)$ and $w \in C^1_0(\Omega; {\mathbb{R}^n})$. Since $\varphi^*$ is increasing, \[
\int_\Omega \lambda u \divop w - \varphi^*(x, |w|)\, dx
\leqslant \int_\Omega u \divop(\lambda w) - \varphi^*(x, |\lambda w|)\, dx \leqslant \varrho_{V,\varphi}(u), \] as $\lambda w \in C^1_0(\Omega; {\mathbb{R}^n})$. Taking the supremum over $w$, we get $\varrho_{V,\varphi}(\lambda u) \leqslant \varrho_{V,\varphi}(u)$.
Let us prove that $\varrho_{V,\varphi}$ is convex. Let $u, v \in L^1(\Omega)$, $\theta \in (0, 1)$ and $w\in C^1_0(\Omega; {\mathbb{R}^n})$. Then \[ \begin{split}
&\int_\Omega (\theta u + (1-\theta) v) \divop w - \varphi^*(x, |w|)\, dx\\
& \qquad = \theta\int_\Omega u \divop w - \varphi^*(x, |w|)\, dx
+(1-\theta)\int_\Omega v \divop w - \varphi^*(x, |w|)\, dx\\ &\qquad \leqslant \theta \varrho_{V,\varphi}(u) + (1-\theta) \varrho_{V,\varphi}( v). \end{split} \] The claim follows when we take the supremum over $w\in C^1_0(\Omega; {\mathbb{R}^n})$.
Finally, we show that $\varrho$ is left-continuous. Since $\lambda\mapsto \varrho_{V,\varphi}(\lambda u)$ is increasing, $\varrho_{V,\varphi}(\lambda u) \leqslant \varrho_{V,\varphi}(u)$ for $\lambda \in(0, 1)$. We next consider the opposite inequality at the limit. Let first $\varrho_{V,\varphi}(u)<\infty$ and fix $\varepsilon>0$. By the definition of $\varrho_{V,\varphi}$ and Remark~\ref{rem:restrictedTestFunction} there exists a test function
$w\in C^1_0(\Omega; {\mathbb{R}^n})$ with $\varrho_{\varphi^*}(|w|)<\infty$ such that \[ \int_\Omega u \divop w \, dx \geqslant
\varrho_{V,\varphi}(u) - \varepsilon + \varrho_{\varphi^*}(|w|). \]
Multiplying the inequality by $\lambda\in (0,1)$ and subtracting $\varrho_{\varphi^*}(|w|)$, we obtain that \[ \varrho_{V,\varphi}(\lambda u) \geqslant
\lambda (\varrho_{V,\varphi}(u) - \varepsilon) + (\lambda-1)\varrho_{\varphi^*}(|w|). \] Hence \[ \lim_{\lambda\to 1^-}\varrho_{V,\varphi}(\lambda u) \geqslant \varrho_{V,\varphi}(u) - \varepsilon. \] The claim follows from this as $\varepsilon\to 0^+$. The case $\varrho_{V,\varphi}(u)=\infty$ is proved similarly, we only need to replace $\varrho_{V,\varphi}(u) - \varepsilon$ by $\frac1\varepsilon$. \end{proof}
From the previous lemma it follows that $\varrho_{V,\varphi}$ defines a seminorm by the Luxenburg method (Definition~\ref{def:modularSpace}); for a proof see \cite{HarHJ_pp}. We next show that this seminorm is comparable to $V_\varphi$.
\begin{lem}\label{lem:equivalence} If $\varphi\in\Phi_{\text{\rm w}}(\Omega)$ and $u\in {\rm BV}^\varphi(\Omega)$, then \[
\| u\|_{\varrho_{V,\varphi}} \leqslant V_\varphi(u) \leqslant 2 \| u\|_{\varrho_{V,\varphi}}. \] \end{lem} \begin{proof}
If $V_\varphi(u)=0$, then $\varrho_{V,\varphi}(\frac u\lambda)=0$ for every $\lambda>0$ so that $\| u\|_{\varrho_{V,\varphi}} = 0$. The case $V_\varphi(u)=\infty$ is excluded by the assumption $u\in {\rm BV}^\varphi(\Omega)$.
Since the claim is homogeneous, the case $V_\varphi(u)\in (0,\infty)$ reduces to $V_\varphi(u)=1$. By the definition of $V_\varphi$, it then follows that \[ \int_\Omega u \divop w \, dx \leqslant
V_\varphi(u) \|w\|_{\varphi^*} =
\|w\|_{\varphi^*} \leqslant
1+\varrho_{\varphi^*}(|w|); \] the last step is a general property of the Luxemburg norm, see \cite[Corollary~2.1.15]{DieHHR11}. Thus \[ \varrho_{V,\varphi}(u)
\leqslant
\sup_{w \in C^1_0(\Omega; {\mathbb{R}^n}), \varrho_{\varphi^*}(|w|)<\infty}\big(1+\varrho_{\varphi^*}(|w|) - \varrho_{\varphi^*}(|w|) \big) = 1, \]
and so $\|u\|_{\varrho_{V,\varphi}}\leqslant 1$. This concludes the proof of the first inequality.
We next establish the opposite inequality $2\|u\|_{\varrho_{V,\varphi}} \geqslant 1$, which is equivalent to $\varrho_{V,\varphi}(\frac{2u}\lambda)\geqslant 1$ for every $\lambda<1$
. Since $\varrho_{\varphi^*}(|w|)\leqslant 1$ when $\|w\|_{\varphi^*}\leqslant 1$, we conclude that \begin{align*} \varrho_{V,\varphi}(\tfrac{2u}\lambda) &\geqslant
\sup\bigg\{ \int_\Omega \tfrac{2u}\lambda \divop w - \varphi^*(x, |w|)\, dx \,\Big|\, w \in C^1_0(\Omega; {\mathbb{R}^n}), \|w\|_{\varphi^*}\leqslant 1\bigg\}\\ &\geqslant \tfrac 2\lambda V_\varphi(u) - 1 > 1. \qedhere \end{align*} \end{proof}
The following result is the counterpart of Theorem 5.2 in \cite{EvaG92}, see also Theorem 1.9 in \cite{Giusti}.
\begin{lem}[Weak lower semicontinuity]\label{lem:sequence-in-BV}
Let $\varphi \in \Phi_{\text{\rm w}}(\Omega)$, $u, u_k \in L^1(\Omega)$ with $u_k \rightharpoonup u$ in $L^1(\Omega)$. Then \[ V_\varphi(u) \leqslant \liminf_{k\to \infty} V_\varphi(u_k) \qquad\text{and}\qquad \varrho_{V,\varphi}(u) \leqslant \liminf_{k\to \infty} \varrho_{V,\varphi}(u_k). \] \end{lem} \begin{proof} If $w \in C^1_0(\Omega; {\mathbb{R}^n})$, then $\divop w \in L^\infty(\Omega)$
and weak convergence in $L^1(\Omega)$ with $\|w\|_{\varphi^*}\leqslant 1$ give \[ \int_\Omega u \divop w \, dx = \lim_{k \to \infty} \int_\Omega u_k \divop w \, dx\leqslant \liminf_{k \to \infty} V_\varphi(u_k). \] The first inequality of the claim follows by taking the supremum over all such $w$. Subtracting
$\varrho_{\varphi^*}(|w|)<\infty$ from both sides of the equality similarly gives the second inequality. \end{proof}
\section{Approximation properties of the dual norm} \label{sect:approximation}
In this section we prove a compactness-type property of ${\rm BV}^\varphi$ and estimate
the $V_\varphi$-norm of $W^{1,1}_{\rm loc}$-functions by $\|\nabla u\|_\varphi$. We first connect the norm $V_\varphi$ with the associate space $(L^{\varphi^*})'$-norm of the gradient. One crucial difference between these norms is that in the associate space norm we test with functions in $L^{\varphi^*}$ whereas in $V_\varphi$ the test functions are smooth. Thus some approximation is needed, but we cannot use density in $L^{\varphi^*}(\Omega)$ since $\varphi^*$ is not, in general, doubling. We start with a property of lower semicontinuous functions. Although the result is known, we did not find a reference for these exact properties of the approximating functions, so we provide a proof for completeness.
\begin{lem}\label{lem:lsc-approx} Let $f:\Omega\to [0,\infty]$ be lower semicontinuous. Then there exist functions $w_i\in C^1_0(\Omega)$ with $0\leqslant w_i\leqslant f$ and $w_i\to f$. \end{lem} \begin{proof} We first define \[ f_i := \sum_{k=1}^\infty 2^{-i} \chi_{\{f>2^{-i}k\}}. \] If $f(x)\in (2^{-i}k, 2^{-i}(k+1)]$, then $f_i(x)= 2^{-i}k$. Hence $0\leqslant f_i\leqslant f$ and $f_i\nearrow f$. Thus it suffices to approximate $f_i$ and use a diagonal argument. Since $\{f>2^{-i}k\}$ is open, we can find non-negative functions $w_j^{k,i}\in C^1_0(\{f>2^{-i}k\})$ with $w_j^{k,i} \nearrow \chi_{\{f>2^{-i}k\}}$ as $j\to\infty$. Set \[ w_j^i := \sum_{k=1}^j 2^{-i} w_j^{k,i}. \] Since each sum is finite, $w_j^i\in C^1_0(\Omega)$. Furthermore, $w_j^i \nearrow f_i$ as $j\to\infty$. \end{proof}
In Theorem~\ref{thm:BV-gradient}(1) we assume that $C^1_0(\Omega; {\mathbb{R}^n})$ is dense in $L^{\varphi^*}(\Omega; {\mathbb{R}^n})$. If $\varphi^*$ satisfies \hyperref[def:a0]{{\normalfont(A0)}}{} and \adec{}, then this holds by \cite[Theorem~3.7.15]{HarH19}. Furthermore, $\varphi^*$ satisfies these conditions if and only if $\varphi$ satisfies \hyperref[def:a0]{{\normalfont(A0)}}{} and \ainc{}. In other words, this is exactly the opposite of the linear growth case that we are interested in. However, in this case we can give an exact formula for the variation $V_\varphi$ in terms of the norm of the associate space.
The case when $\varphi^*$ does not satisfy \adec{} is more interesting and involves the technical difficulties that we expect with ${\rm BV}$-type spaces. Now we need to approximate not the test function but the function itself so the regularity of $\varphi$ matters.
\begin{thm}\label{thm:BV-gradient} Let $\varphi \in \Phi_{\text{\rm w}}(\Omega)$ and $u \in W^{1, 1}_{\rm loc}(\Omega)$.
Then $V_\varphi(u) \leqslant \| \nabla u \|_{(L^{\varphi^*}(\Omega))'} $. \begin{enumerate} \item[(1)] If $C^1_0(\Omega; {\mathbb{R}^n})$ is dense in $L^{\varphi^*}(\Omega; {\mathbb{R}^n})$,
then $V_\varphi(u) = \| \nabla u \|_{(L^{\varphi^*}(\Omega))'}$. \item [(2)] If $\varphi$ satisfies \hyperref[def:a0]{{\normalfont(A0)}}{}, \hyperref[def:a1]{{\normalfont(A1)}}{} and \adec{},
then $V_\varphi(u) \approx \| \nabla u \|_\varphi$. \end{enumerate} \end{thm} \begin{proof} Since $u \in W^{1, 1}_{\rm loc}(\Omega)$, it follows from the definition of $V_\varphi$ and integration by parts that \begin{equation}\label{eq:integrationByParts}
V_\varphi(u)=\sup\bigg\{ \int_\Omega \nabla u \cdot w \, dx \,\Big|\, w \in C^1_0(\Omega; {\mathbb{R}^n}), \| w \|_{\varphi^*}\leqslant 1 \bigg\}. \end{equation} The definition of the associate space norm implies that \[ \int_\Omega \nabla u \cdot w \, dx \leqslant
\int_\Omega |\nabla u|\, |w| \, dx \leqslant
\| \nabla u \|_{(L^{\varphi^*}(\Omega))'} \|w\|_{L^{\varphi^*}(\Omega)}. \]
Taking the supremum over $w\in C^1_0(\Omega; {\mathbb{R}^n})$ with $\|w\|_{L^{\varphi^*}(\Omega)}\leqslant 1$, we conclude that
$V_\varphi(u) \leqslant \| \nabla u \|_{(L^{\varphi^*}(\Omega))'}$.
Under assumption (1), we next show the opposite inequality,
$ \| \nabla u \|_{(L^{\varphi^*}(\Omega))'} \leqslant V_\varphi(u)$.
Let $ w \in L^{\varphi^*}(\Omega; {\mathbb{R}^n})$ with $\|w\|_{\varphi^*}=1$ and let $(w_j)$ be a sequence from $C^1_0(\Omega; {\mathbb{R}^n})$ with $w_j \to w$ in $L^{\varphi^*}(\Omega; {\mathbb{R}^n})$ and pointwise a.e.
Since also $w_j/\|w_j\|_{\varphi^*} \to w$ in $L^{\varphi^*}(\Omega; {\mathbb{R}^n})$, we may assume that
$\|w_j\|_{\varphi^*}=1$. By Fatou's Lemma, \[ \liminf_{j \to \infty} \int_\Omega \nabla u \cdot w_j \, dx \geqslant \int_\Omega \nabla u \cdot w \, dx, \] so it follows from \eqref{eq:integrationByParts} that \[
V_\varphi(u)\geqslant \sup\bigg\{ \int_\Omega \nabla u \cdot w \, dx \,\Big|\, w \in L^{\varphi^*}(\Omega; {\mathbb{R}^n}), \|
w\|_{\varphi^*}\leqslant 1 \bigg\}. \]
Let $h\in L^{\varphi^*}(\Omega)$. We set $w:=\frac{\nabla u}{|\nabla u|} h$ if $|\nabla u|\neq 0$ and $0$ otherwise. This gives \[
V_\varphi(u)\geqslant \sup\bigg\{ \int_\Omega |\nabla u| \, h \, dx \,\Big|\, h \in L^{\varphi^*}(\Omega), \| h\|_{\varphi^*}\leqslant 1
\bigg\} = \| \nabla u \|_{(L^{\varphi^*}(\Omega))'}. \]
Hence $V_\varphi(u) = \| \nabla u \|_{(L^{\varphi^*}(\Omega))'}$ and the proof of (1) is complete.
Then we prove (2).
Fix $h\in C^1_0(\Omega)$. Since $w_{\varepsilon,\delta}:=(\frac{\nabla u}{|\nabla u|+\varepsilon})* \eta_\delta$ is
bounded and converges to $\frac{\nabla u}{|\nabla u|+\varepsilon}$ in $L^1$ and a.e.\ as $\delta\to 0^+$,
it follows by $L^1$-convergence and dominated convergence with majorant $|\nabla u|\, |h|$ that \[ \lim_{\varepsilon\to 0^+}\lim_{\delta\to 0^+}\int_\Omega \nabla u \cdot (w_{\varepsilon,\delta} h) \, dx =
\lim_{\varepsilon\to 0^+}\int_\Omega \frac{|\nabla u|^2}{|\nabla u|+\varepsilon}\, h \, dx =
\int_\Omega |\nabla u|\, h \, dx. \] Since $w_{\varepsilon,\delta}h\in C^1_0(\Omega;{\mathbb{R}^n})$, this and \eqref{eq:integrationByParts} imply that \[
V_\varphi(u) \geqslant
\sup\bigg\{ \int_\Omega |\nabla u|\, h \, dx \,\Big|\, h \in C^1_0(\Omega), \| h \|_{\varphi^*}\leqslant 1 \bigg\}. \]
Let $g\in L^{\varphi^*}(\Omega)$ with $\| g \|_{\varphi^*}\leqslant 1$. Since $\varphi^*$ satisfies \hyperref[def:a0]{{\normalfont(A0)}}{}, \hyperref[def:a1]{{\normalfont(A1)}}{} and \ainc{}, the Hardy--Littlewood maximal operator $M$ is bounded in $L^{\varphi^*}(\Omega)$, with some constant $c_M>0$ \cite{Has15}. The function $\tilde g:=\frac{Mg}{c_M}\in L^{\varphi^*}(\Omega)$ is lower semi-continuous,
$\| \tilde g \|_{\varphi^*}\leqslant 1$ and $\tilde g\geqslant \frac g{c_M}$. By Lemma~\ref{lem:lsc-approx}, we can find $h_i\in C^1_0(\Omega)$ with $h_i\to \tilde g$
and $0\leqslant h_i\leqslant \tilde g$. By dominated convergence, with $|\nabla u| \,\tilde g$ as a majorant, we find that \[ V_\varphi(u)
\geqslant
\lim_{i\to \infty}\int_\Omega |\nabla u|\, h_i \, dx =
\int_\Omega |\nabla u|\, \tilde g \, dx \geqslant
\frac1{c_M}\int_\Omega |\nabla u|\, g \, dx. \] Since $g$ is arbitrary, this implies that \[ V_\varphi(u) \geqslant \frac1{c_M}
\sup\bigg\{ \int_\Omega |\nabla u|\, g \, dx \,\Big|\, g \in L^{\varphi^*}(\Omega), \|g \|_{\varphi^*}\leqslant 1 \bigg\} =
\frac{ \| \nabla u\|_{(L^{\varphi^*}(\Omega))'} }{c_M}.
\] By Theorem~3.4.6 and Proposition~2.4.5 of \cite{HarH19},
$\| \nabla u \|_{(L^{\varphi^*}(\Omega))'} \approx \| \nabla u\|_\varphi$, so the proof of (2) is complete. \end{proof}
The next lemma is the counterpart of \cite[Theorem 5.3]{EvaG92} and \cite[Theorem 1.17]{Giusti}, albeit with an extra constant $c_\varphi$. The extra constant is expected, since we assume only \hyperref[def:a1]{{\normalfont(A1)}}{}, cf.\ Example~\ref{eg:weightNeeded} and Proposition~\ref{prop:modularDensity}.
\begin{lem}[Approximation by smooth functions]\label{lem:density} Assume that $\varphi \in \Phi_{\text{\rm w}}(\Omega)$ satisfies \hyperref[def:a0]{{\normalfont(A0)}}{}, \hyperref[def:a1]{{\normalfont(A1)}}{} and \adec{}. Then there exists $c_\varphi\geqslant 1$ such that for every $u \in L^\varphi(\Omega)$ we can find $u_k \in C^\infty(\Omega)$ with \[ u_k \to u\text{ in }L^\varphi(\Omega) \quad\text{and}\quad V_\varphi(u) \leqslant \lim_{k \to \infty} V_\varphi(u_k)\leqslant c_\varphi V_\varphi(u). \] If additionally $u\in W^{1,p}(\Omega)$ or $u \in L^p(\Omega)$, $p\in [1,\infty)$, then the sequence can be chosen with $u_k \to u$ in $ W^{1,p}(\Omega)$ or $L^p(\Omega)$ as well. \end{lem} \begin{proof} For $k\in\mathbb{Z}_+$, we define \[
U_k := \bigg\{x \in \Omega \,\Big|\, \qopname\relax o{dist}(x, \partial \Omega) > \frac1{m+k} \bigg\}, \] where $m>0$ is chosen such that $U_1$ is non-empty. Set $V_1:=U_2$ and $V_k := U_{k+1} \setminus \overline{U_{k-1}}$ for $k\geqslant 2$. Let $(\xi_k)$ be a partition of unity subordinate to $(V_k)$, i.e.\ $\xi_k \in C^\infty_0 (V_k)$, $0 \leqslant \xi_k \leqslant 1$ and $\sum_{k=1}^\infty \xi_k=1$ for all $x \in \Omega$.
Let $\varepsilon>0$ and let $\eta$ be the standard mollifier. Choose $\varepsilon_k\in (0,\varepsilon)$ so small that $\operatornamewithlimits{supp}(\eta_{\varepsilon_k} * (u \xi_k)) \subset V_k$, \begin{align}\label{eq:density}
\| \eta_{\varepsilon_k}* (u\xi_k) -u\xi_k \|_\varphi \leqslant \frac{\varepsilon}{2^k} \qquad\text{and}\qquad
\| \eta_{\varepsilon_k}* (u \nabla \xi_k) -u \nabla \xi_k \|_\varphi \leqslant \frac{\varepsilon}{2^k}; \end{align} the last conditions are possible by \hyperref[def:a0]{{\normalfont(A0)}}{}, \hyperref[def:a1]{{\normalfont(A1)}}{} and \adec{}
since $u\in L^\varphi(\Omega)$ and $\xi_k, |\nabla \xi_k|\in L^\infty(\Omega)$ \cite[Theorem~4.4.7]{HarH19} . Let us define \[ u_\varepsilon := \sum_{k=1}^\infty \eta_{\varepsilon_k}* (u\xi_k). \] In a neighborhood of each point there are at most three non-zero terms in the sum, hence $u_\varepsilon \in C^\infty (\Omega)$.
Since $\|\cdot\|_\varphi$ is equivalent to a norm, it satisfies a countable quasitriangle inequality \cite[Corollary~3.2.5]{HarH19}. Using $u = \sum_{k=1}^\infty \xi_ku$ and \eqref{eq:density} with this inequality, we find that \[ \begin{split}
\|u_\varepsilon -u\|_\varphi &\leqslant \Big\|\sum_{k=1}^\infty (\eta_{\varepsilon_k}* (u\xi_k) - \xi_ku) \Big\|_\varphi
\lesssim \sum_{k=1}^\infty \| \eta_{\varepsilon_k}* (u\xi_k) -u\xi_k \|_\varphi \leqslant \sum_{k=1}^\infty\frac{\varepsilon}{2^k} = \varepsilon. \end{split} \] Thus $u_\varepsilon \to u$ in $L^\varphi(\Omega)$ and so Lemma~\ref{lem:sequence-in-BV} yields \[ V_\varphi(u) \leqslant \liminf_{\varepsilon\to 0^+} V_\varphi(u_\varepsilon). \]
If we assume $u \in L^p(\Omega)$ or $|\nabla u|\in L^p(\Omega)$, then we can add to \eqref{eq:density} also the requirement \[
\| \eta_{\varepsilon_k}* (u\xi_k) -u\xi_k \|_p \leqslant \frac{\varepsilon}{2^k} \quad\text{or}\quad
\| \eta_{\varepsilon_k}* (\nabla u \,\xi_k) -\nabla u \,\xi_k \|_p \leqslant \frac{\varepsilon}{2^k} \]
and estimate in the same way $\|u_\varepsilon - u\|_p\leqslant \varepsilon$
or $\|\nabla u_\varepsilon - \nabla u\|_p\leqslant \varepsilon$. Thus also $u_\varepsilon \to u$ in $L^p(\Omega)$ or $W^{1,p}(\Omega)$, as claimed.
Fix $w\in C^1_0(\Omega;{\mathbb{R}^n})$ with $\|w\|_{\varphi^*}\leqslant 1$. Since $w$ has a compact support in $\Omega$, only finitely many of the terms $(\eta_{\varepsilon_k}*(u\xi_k)) \divop w$ are non-zero. Thus the sums below are really finite and can be interchanged with integrals and derivatives. Using the definition of $u_\varepsilon$, Fubini's Theorem in the convolution, the product rule and $\sum_{k=1}^\infty \nabla \xi_k = \nabla \sum_{k=1}^\infty \xi_k = \nabla 1=0$, we conclude that \[ \begin{split} \int_\Omega u_\varepsilon \divop w \, dx &= \sum_{k=1}^\infty \int_\Omega (\eta_{\varepsilon_k}*(u\xi_k)) \divop w \, dx = \sum_{k=1}^\infty \int_\Omega (u\xi_k) \divop (\eta_{\varepsilon_k}* w) \, dx \\ &= \sum_{k=1}^\infty \int_\Omega u \divop (\xi_k(\eta_{\varepsilon_k}* w)) \, dx - \sum_{k=1}^\infty \int_\Omega u \nabla \xi_k \cdot (\eta_{\varepsilon_k}* w) \, dx \\ &= \underbrace{ \sum_{k=1}^\infty \int_\Omega u \divop (\xi_k(\eta_{\varepsilon_k}* w)) \, dx}_{=: I} - \underbrace{ \sum_{k=1}^\infty \int_\Omega w \cdot (\eta_{\varepsilon_k}*(u \nabla \xi_k) - u\nabla \xi_k)\,dx}_{=: II}\,. \end{split} \] For $II$ we obtain by Hölder's inequality and \eqref{eq:density} that \[
|II| \lesssim
\sum_{k=1}^\infty \|w\|_{\varphi^*} \big\| \eta_{\varepsilon_k}*(u \nabla \xi_k) - u\nabla \xi_k \big\|_{\varphi} \leqslant \sum_{k=1}^\infty \frac{\varepsilon}{2^k} = \varepsilon . \] As $\sum_{k=1}^\infty \xi_k(\eta_{\varepsilon_k}* w) \in C^1_0(\Omega; {\mathbb{R}^n})$ is a viable test function (up to a constant), we obtain that \[ \begin{split}
|I| =
\bigg|\int_\Omega u \divop \bigg(\sum_{k=1}^\infty \xi_k\,\eta_{\varepsilon_k}* w\bigg) \, dx \bigg| &\leqslant
V_\varphi(u)
\bigg\|\sum_{k=1}^\infty \xi_k\,\eta_{\varepsilon_k}* w\bigg\|_{\varphi^*}\\ &\lesssim
V_\varphi(u) \| Mw\|_{\varphi^*}. \end{split} \] Since $\varphi^*$ satisfies \hyperref[def:a0]{{\normalfont(A0)}}{}, \hyperref[def:a1]{{\normalfont(A1)}}{} and \ainc{}, maximal operator $M$ is bounded in $L^{\varphi^*}(\Omega)$ \cite{Has15}. So the estimates for $I$ and $II$ give \[
\Big| \int_\Omega u_\varepsilon \divop w \, dx \Big| \lesssim V_\varphi(u) + \varepsilon. \] Hence $V_\varphi(u_\varepsilon) \lesssim V_\varphi(u) + \varepsilon \to V_\varphi(u)$ as $\varepsilon\to 0^+$. By choosing a subsequence we ensure that $\lim_k V_\varphi(u_k)$ exists. \end{proof}
A bounded domain $\Omega\subset{\mathbb{R}^n}$ is a \emph{John domain} if there exist constants $0< \alpha \leqslant \beta<\infty$ and a point $x_0 \in \Omega$ such that each point $x\in \Omega$ can be joined to $x_0$ by a rectifiable curve $\gamma:[0,\ell_\gamma] \to \Omega$ parametrized by arc length with $\gamma(0) = x$, $\gamma(\ell_\gamma) = x_0$, $\ell_\gamma\leqslant \beta\,,$ and \[ t \leqslant \tfrac{\beta}{\alpha} \qopname\relax o{dist}\big(\gamma(t), \partial \Omega \big) \quad \text{for all} \quad t\in[0, \ell_\gamma]. \]
Examples of John domains include convex domains and domains with Lipschitz boundary, but also some domains with fractal boundaries such as the von Koch snowflake. The next compactness-type result is the counterpart of Theorem 5.5 in \cite{EvaG92}, see also Theorem 1.19 in \cite{Giusti}.
\begin{thm}\label{thm:BV-compact-Lphi} Let $\Omega \subset {\mathbb{R}^n}$ be a bounded John domain and $\varphi \in \Phi_{\text{\rm w}}(\Omega)$ satisfy \hyperref[def:a0]{{\normalfont(A0)}}{}, \hyperref[def:a1]{{\normalfont(A1)}}{} and \adec{}.
If $(u_k)$ is a sequence in ${\rm BV}^\varphi(\Omega)$ with $ \sup_k \|u_k\|_{{\rm BV}^\varphi} < \infty$, then there exists a subsequence $(u_{k_j})$ and $u \in {\rm BV}^\varphi(\Omega)$ such that $u_{k_j} \to u$ in $L^\varphi(\Omega)$
and $\|u\|_{{\rm BV}^\varphi}\leqslant \liminf \|u_{k_j}\|_{{\rm BV}^\varphi}$. \end{thm}
\begin{proof} By Lemma~\ref{lem:density}, we may choose functions $ v_k \in C^\infty(\Omega) \cap {\rm BV}^\varphi(\Omega)$ such that \[
\|u_k - v_k\|_\varphi < \tfrac1k \qquad\text{and}\qquad \sup_k V_{\varphi}(v_k) < \infty. \] Theorem~\ref{thm:BV-gradient}(2) for $v_k\in C^\infty(\Omega)$
yields that $\sup_k \|\nabla v_k\|_\varphi < \infty$, so the sequence is bounded in $W^{1,\varphi}(\Omega)$. Since $\Omega$ is a John domain, the compact embedding $W^{1,\varphi}(\Omega) \hookrightarrow \hookrightarrow L^\varphi(\Omega)$ holds \cite{HarH19, HarHJ_pp}, and thus $(v_k)$ has a subsequence $(v_{k_j})$ converging to some $u$ in $L^\varphi(\Omega)$.
Therefore $\|u_{k_j} - v_{k_j}\|_\varphi < \frac1{k_j}$ implies that also $u_{k_j} \to u$ in $L^\varphi(\Omega)$ and, by Lemma~\ref{lem:sequence-in-BV}, $u \in {\rm BV}^\varphi(\Omega)$. \end{proof}
\section{Explicit expression for the dual modular} \label{sect:explicit}
In this section we derive a formula for the ``dual modular'' $\varrho_{V,\varphi}$ from Definition~\ref{defn:V-rho} in terms of $\varrho_\varphi$ of the derivative's absolutely continuous part and the singular part with weight given by the recession function \[ \varphi'_\infty(x)= \limsup_{t\to \infty} \frac{\varphi(x, t)} t. \] Throughout this section, we assume that $\varphi\in \Phi_{\text{\rm c}}(\Omega)$. Then $t\mapsto\frac{\varphi(\cdot, t)} t$ is increasing and the limit superior is a limit. Moreover, if the derivative of $\varphi$ with respect to $t$ exists, then it is increasing by convexity, so $\lim_{t\to \infty} \varphi'(\cdot, t)$ exists and equals $\varphi'_\infty$ by l'H\^{o}pital's rule. The following lemma illustrates the significance of $\varphi'_\infty$.
In \cite[Section~3 and Example A.1]{HarHL08} it was shown that $\log$-H\"older continuity is not sufficient when working in ${\rm BV}^{p(\cdot)}$. Similarly, the \hyperref[def:a1]{{\normalfont(A1)}}{} condition (corresponding to $\log$-Hölder continuity) is not sufficient in the next results in view of Example~\ref{eg:weightNeeded}. Instead, we use the restricted \hyperref[def:va1]{{\normalfont(VA1)}}{} which corresponds to strong $\log$-H\"older continuity (Proposition~\ref{prop:strong}). Note that here we need the inequality at every point, since we will use the estimate with the singular measure $D^su$.
\begin{lem}\label{lem:bound}
Let $\varphi\in \Phi_{\text{\rm c}}(\Omega)$ satisfy restricted \hyperref[def:va1]{{\normalfont(VA1)}}{}. If $w\in C(\Omega)$ with $\varrho_{\varphi^*}(w)<\infty$, then $|w|\leqslant \varphi'_\infty$. \end{lem} \begin{proof} We assume that $w\geqslant 0$ to simplify notation. Suppose to the contrary that $w(x_0)>\varphi'_\infty(x_0)$ for some point $x_0\in\Omega$. Since $\varphi$ is convex, $t\mapsto \frac{\varphi(x_0, t)} t$ is increasing and $\frac{\varphi(x_0, t)} t\leqslant \varphi'_\infty(x_0)$ for every $t>0$. Now $\varphi'_\infty(x_0)<\infty$ and $\varphi(x_0, t) \leqslant \varphi'_\infty(x_0) t$ give $\varphi^*(x_0, s) \geqslant \infty \chi_{(\varphi'_\infty(x_0), \infty)}(s)$ and $\varphi^*(x_0,w(x_0)) = \infty$. From this and $\varrho_{\varphi^*}(w)<\infty$ it follows that $w\leqslant \varphi'_\infty$ almost everywhere. However, $\varphi'_\infty$ need not be continuous, so we cannot directly conclude that the inequality holds everywhere.
Let $\omega$ be from \hyperref[def:va1]{{\normalfont(VA1)}}{} for $K:=1$. Choose $r_0>0$ and $\beta:= \frac1{1+ \omega(r_0)}$ such that $\varphi'_\infty(x_0) < \beta^3 w(x_0) < \beta^2 w(x)$ for every $x\in B(x_0, r_0)$. Note that $\varphi^*(x_0, \beta^3 w(x_0)) = \infty$ and $\varphi(x_0,t)\leqslant \varphi'_\infty(x_0)t\leqslant \beta^2 w(x) t$ for all $t\geqslant 0$. Since $\varphi(x_0,\cdot)$ is finite and convex, it is continuous and we can find $t_x$ with
$\varphi(x_0,t_x)=|x-x_0|^{-n}$. By restricted \hyperref[def:va1]{{\normalfont(VA1)}}{}, \[ \varphi(x, \beta t_x)\leqslant \varphi(x_0, t_x) + \omega(r_0) = \varphi(x_0, t_x)+\tfrac1\beta -1 \leqslant \beta^2 w(x) t_x +\tfrac1\beta. \] By the definition of $\varphi^*$ and the previous inequalities, we obtain that \[ \varphi^*(x, w(x)) \geqslant \beta t_x w(x) - \varphi(x, \beta t_x) \geqslant \beta(1-\beta) w(x)t_x - \tfrac1\beta \geqslant \tfrac{1-\beta}\beta \varphi(x_0, t_x) - \tfrac1\beta. \]
Since $\varphi(x_0, t_x) = |x-x_0|^{-n}$, we conclude that \[ \int_\Omega \varphi^*(x, w)\, dy \gtrsim
\int_{B(x_0,r_0)} |x-x_0|^{-n}\, dy - c = \infty. \] This contradicts the assumption $\varrho_{\varphi^*}(w)<\infty$ and thus the counter-assumption $w(x_0)>\varphi'_\infty(x_0)$ was incorrect and the claim is proved. \end{proof}
We define \[
T^\varphi := \big\{ w \in C^1_0(\Omega; {\mathbb{R}^n})\,\big|\, \varrho_{\varphi^*}(|w|)<\infty \big\}. \]
Then the usual test function space of ${\rm BV}$ is $T^1$ since $\varrho_{\infty}(|w|)<\infty$ if and
only if $|w|\leqslant 1$ a.e. In the next propositions we first consider the singular and absolutely continuous parts of the derivative separately. Then we combine them to handle the whole function in Theorem~\ref{thm:exactFormula}.
\begin{prop}\label{prop:singularPart} Let $\varphi\in \Phi_{\text{\rm c}}(\Omega)$ satisfy \hyperref[def:a0]{{\normalfont(A0)}}{}, \adec{} and restricted \hyperref[def:va1]{{\normalfont(VA1)}}{}. If $u\in {\rm BV}(\Omega)$, then \[
\sup_{w\in T^\varphi} \int_\Omega w\cdot dD^su = \int_\Omega \varphi'_\infty \, d|D^su|. \] \end{prop} \begin{proof} By the definition of the total variation of a measure and Lemma~\ref{lem:bound}, \[ \sup_{w\in T^\varphi} \int_\Omega w\cdot dD^su \leqslant
\sup_{w\in T^\varphi} \int_\Omega |w|\,d|D^su| \leqslant
\int_\Omega \varphi'_\infty \, d|D^su|. \] For the opposite inequality, we define $h_k:\Omega\to[0,\infty]$ by \[ h_k(x):=\lim_{r\to 0^+}\inf_{y\in B(x,r)} \frac{\varphi(y, k)}k. \] Then $h_k$ is lower semicontinuous with $h_k\leqslant \frac{\varphi(\cdot, k)}k\leqslant \varphi'_\infty$. From the first inequality it follows that $\varphi^*(\cdot, h_k) \leqslant \varphi(\cdot, k)$ so $\varrho_{\varphi^*}(h_k)\leqslant \varrho_\varphi(k)<\infty$ since $\varphi$ satisfies \hyperref[def:a0]{{\normalfont(A0)}}{} and \adec{} and $\Omega$ is bounded. Let us show that $h_k\to \varphi'_\infty$. If $\varphi'_\infty(x)=\infty$, then since $\varphi^+(k)<\infty$ we can use \hyperref[def:a1]{{\normalfont(A1)}}{} in all sufficiently small balls to conclude that \[ h_k(x)=\lim_{r\to 0^+}\inf_{y\in B(x,r)} \frac{\varphi(y, k)}k \geqslant \frac{\varphi(x, \beta k) -1 }k \to \beta \varphi'_\infty(x)=\infty \] as $k\to \infty$. If $\varphi'_\infty(x)<\infty$, then we use the same inequality but now with $\beta:= \frac{1}{1+ \omega(r)}$ from the restricted \hyperref[def:va1]{{\normalfont(VA1)}}{} condition; we obtain the desired convergence as $\beta\to 1^-$.
Note that $h_k$ is increasing in $k$ since $\varphi$ is convex. It follows by monotone convergence that \[
\int_\Omega \varphi'_\infty \, d|D^su| =
\lim_{k\to\infty} \int_\Omega h_k\,d|D^su|. \]
Let $\varepsilon>0$ and assume $\int_\Omega \varphi'_\infty \, d|D^su|<\infty$. We can find $h=h_k$ and $K>0$ such that \[
\int_\Omega \varphi'_\infty \, d|D^su| \leqslant
\int_\Omega h\,d|D^su| + \varepsilon \leqslant
\sum_{j=1}^{K^2} \int_\Omega \tfrac1K \chi_{\{h>\frac jK\}}\,d|D^su| + 2\varepsilon =
\sum_{j=1}^{K^2} \tfrac1K |D^su|\big(\{h>\tfrac jK\}\big) + 2\varepsilon. \] Since $h$ is lower semicontinuous, $\{h>\tfrac jK\}$ is open, and hence
by Lemma~\ref{lem:singularPart} we can choose
$w_j\in C^1_0(\{h>\tfrac jK\}; {\mathbb{R}^n})$ with $|w_j|\leqslant 1$ such that \[
\int_\Omega \varphi'_\infty \, d|D^su| \leqslant \sum_{j=1}^{K^2} \frac1K \int_{\{h>\tfrac jK\}} w_j\cdot dD^su + 3\varepsilon = \int_\Omega \bigg(\underbrace{\sum_{j=1}^{K^2} \tfrac1K w_j}_{=:w_\varepsilon}\bigg) \cdot dD^su + 3\varepsilon. \] Note that $w_\varepsilon \in C^1_0(\Omega; {\mathbb{R}^n})$ and \[
|w_\varepsilon| \leqslant \sum_{j=1}^{K^2} \tfrac1K |w_j| \leqslant \sum_{j=1}^{K^2} \tfrac1K \chi_{\{h>\frac jK\}} \leqslant h \]
so that $\varrho_{\varphi^*}(|w_\varepsilon|)<\infty$. Therefore $w_\varepsilon\in T^\varphi$ and \[
\int_\Omega \varphi'_\infty \, d|D^su| \leqslant \int_\Omega w_\varepsilon\cdot dD^su + 3\varepsilon \leqslant \sup_{w\in T^\varphi} \int_\Omega w\cdot dD^su +3\varepsilon. \] The upper bound follows from this as $\varepsilon\to 0^+$.
If $\int_\Omega \varphi'_\infty \, d|D^su|=\infty$, then a similar argument gives $\frac1{3\varepsilon}\leqslant \sup_{w\in T^\varphi} \int_\Omega w\cdot dD^su$ and the claim again follows. \end{proof}
In the next result we assume that $\varphi$ is continuous in both variables. Removing this somewhat unusual requirement is an open problem. Similar to the case of $V_\varphi$ in Theorem~\ref{thm:BV-gradient}(2), the approximation is made much more difficult by the fact that $\varphi^*$ is not doubling.
\begin{prop}\label{prop:exactFormulaAC} Let $\varphi\in \Phi_{\text{\rm c}}(\Omega) \cap C(\Omega\times [0,\infty))$ satisfy \hyperref[def:a0]{{\normalfont(A0)}}{} and \adec{}. If $u\in {\rm BV}(\Omega)$, then \[
\sup_{w\in T^\varphi} \int_\Omega \nabla^a u\cdot w - \varphi^*(x, |w|)\, dx =
\varrho_\varphi(|\nabla^a u|). \] \end{prop}
\begin{proof} The upper bound follows directly from Young's inequality: \[
\sup_{w\in T^\varphi} \int_\Omega \nabla^a u\cdot w - \varphi^*(x, |w|)\, dx \leqslant
\int_\Omega \varphi(x, |\nabla^a u|)\, dx. \] For the lower bound we make several reductions. Choose $g_i\in C(\Omega; {\mathbb{R}^n})\cap L^\varphi(\Omega; {\mathbb{R}^n})$ with $g_i\to \nabla^a u$ pointwise a.e and in $L^1(\Omega; {\mathbb{R}^n})$. Then Fatou's Lemma and $L^1$-convergence yield \[
\int_\Omega \varphi(x, |\nabla^a u|)\, dx\leqslant \liminf_{i \to \infty} \int_\Omega \varphi(x, |g_i|)\, dx \quad\text{and}\quad \lim_{i \to \infty} \int_\Omega g_i\cdot w \, dx = \int_\Omega \nabla^a u\cdot w \, dx \] when $w\in T^\varphi$. Thus it suffices to show that \[
\int_\Omega \varphi(x, |g|)\, dx \leqslant
\sup_{w\in T^\varphi} \int_\Omega g\cdot w - \varphi^*(x, |w|)\, dx \] for $g\in C(\Omega; {\mathbb{R}^n})\cap L^\varphi(\Omega; {\mathbb{R}^n})$.
Furthermore, replacing $w$ by $\frac g{\varepsilon+|g|} |w|$ and letting $\varepsilon\to 0^+$, we see
that $g\cdot \frac g{\varepsilon+|g|} |w|\to |g| \, |w|$ pointwise. Thus by monotone convergence the vector-values of $g$ and $w$ are irrelevant and we need only show that \[
\int_\Omega \varphi(x, |g|)\, dx \leqslant
\sup_{w\in C^1_0(\Omega)} \int_\Omega |g w| - \varphi^*(x, |w|)\, dx \] for $g\in C(\Omega)\cap L^\varphi(\Omega)$. We also exclude test-functions with $\varrho_{\varphi^*}(w)=\infty$ by Remark~\ref{rem:restrictedTestFunction}.
Let $\varphi'$ be the left-derivative of $\varphi$ with respect to the second variable. Then $\varphi'$ is left-continuous and $\varphi(x,s)\geqslant \varphi(x,s_0)+\varphi'(x,s_0)(s-s_0)$ by convexity.
We would like to choose $w:=\varphi'(x, |g|)$ in the previous supremum. However, this function is not in general smooth and we cannot use regular approximation by smooth functions since $\varphi^*$ is not doubling. Instead we define \[ \psi_\varepsilon(x,t) := \int_{-\infty}^\infty \varphi(x, \max\{\tau,0\}) \zeta_\varepsilon(t-\tau)\, d\tau = \varphi*_t\zeta_\varepsilon(x, t), \] where $\zeta_\varepsilon$ is a mollifier in $\mathbb{R}$ with support in $[0, \varepsilon]$. Since $\varphi$ and $\varphi'$ are increasing in the second variable and left-continuous, we see that $\psi_\varepsilon\nearrow \varphi$ and $\psi_\varepsilon'\nearrow \varphi'$ as $\varepsilon\to 0^+$. Furthermore, $\psi_\varepsilon' = \varphi*_t(\zeta_\varepsilon')$ is continuous in $x$ since $\varphi$ is and it continuous in $t$ as a convolution with a smooth function. Let $v_i \in C_0(\Omega)$ with $0\leqslant v_i \leqslant 1$ and $v_i\nearrow 1$. By uniform continuity in $\operatornamewithlimits{supp} v_i$, we can choose $\delta=\delta_{\varepsilon, i}>0$ such that
$\psi_\varepsilon'(x, |g(x)|\,v_i(x))-\varepsilon\leqslant \psi_\varepsilon'(y, |g(y)|\,v_i(y))$ for all $x\in B(y,\delta)$ and $y\in\Omega$. Then \[
w_{\varepsilon, i}:= \max\{\psi_\varepsilon'(\cdot, |g|\, v_i)-\varepsilon, 0\} *_x \eta_\delta
\leqslant \psi_\varepsilon'(\cdot, |g|)\leqslant \varphi'(\cdot, |g|). \]
Now $w_{\varepsilon,i}\to \varphi'(\cdot, |g|)$, so we conclude by Fatou's Lemma that \[
\int_\Omega |g| \varphi'(x, |g|) \, dx \leqslant \liminf_{i\to\infty, \varepsilon\to 0}\int_\Omega |g w_{\varepsilon, i}| \, dx. \] Since $\varphi$ satisfies \hyperref[def:a0]{{\normalfont(A0)}}{} and \adec{}, we see that \[
\varphi^*(x, |w_{\varepsilon,i}|) \leqslant
\varphi^*(x, \varphi'(x, |g|)) \leqslant
|g| \varphi'(x, |g|) \lesssim
\varphi(x, |g|). \] As $g\in L^\varphi(\Omega)$ and $\varphi$ satisfies \adec{}, $\varrho_\varphi(g)<\infty$. Thus dominated convergence with majorant $c \varphi(\cdot, g)$ yields \[
\int_\Omega \varphi^*(x, \varphi'(x, |g|))\, dx =
\lim_{i\to\infty, \varepsilon\to 0}\int_\Omega \varphi^*(x, |w_{\varepsilon,i}|)\, dx. \]
Since $w_\varepsilon$ is a valid test-function and $\varphi'(\cdot, |g|)<\infty$ a.e., this together with ``Young's equality'' (Lemma~\ref{lem:dual-equality}) implies that \begin{align*}
\sup_{w\in C^1_0(\Omega)} \int_\Omega |g w| - \varphi^*(x, |w|)\, dx &\geqslant
\liminf_{i\to\infty, \varepsilon\to 0}\int_\Omega |g| \, |w_{\varepsilon,i}| - \varphi^*(x, |w_{\varepsilon,i}|)\, dx \\ &\geqslant
\int_\Omega |g| \varphi'(x, |g|) - \varphi^*(x, \varphi'(x, |g|))\, dx =
\int_\Omega \varphi(x, |g|) \, dx. \end{align*} This completes the proof of the lower bound. \end{proof}
We next derive a simple, closed form expression for $\varrho_{V,\varphi}$. This is the main result of the paper. Note that the right-hand side expression was also obtained recently in the one-dimensional case for a modular based on the Riesz variation assuming the \hyperref[def:va1]{{\normalfont(VA1)}}{} condition \cite{HasJR_pp}.
\begin{thm}\label{thm:exactFormula} Let $\varphi\in \Phi_{\text{\rm c}}(\Omega) \cap C(\Omega\times [0,\infty))$ satisfy \hyperref[def:a0]{{\normalfont(A0)}}{}, \adec{} and restricted \hyperref[def:va1]{{\normalfont(VA1)}}{}. If $u\in {\rm BV}(\Omega)$, then \[
\varrho_{V,\varphi}(u) = \varrho_\varphi(|\nabla^a u|) + \int_\Omega \varphi'_\infty \, d|D^su|. \] \end{thm} \begin{proof} Since $Du= D^a u + D^su$, integration by parts implies that \[ - \int_\Omega u \divop w\, dx = \int_\Omega w\cdot dDu = \int_\Omega \nabla^a u\cdot w\, dx + \int_\Omega w \cdot dD^su \] for $w\in T^\varphi$. Hence the claim follows from Propositions~\ref{prop:singularPart} and \ref{prop:exactFormulaAC} once we prove that \begin{align*}
&\sup_{w\in T^\varphi} \bigg[\int_\Omega \nabla^a u\cdot w - \varphi^*(x, |w|)\, dx + \int_\Omega w \cdot dD^su\bigg]\\ &\qquad=
\sup_{w\in T^\varphi} \int_\Omega \nabla^a u\cdot w - \varphi^*(x, |w|)\, dx + \sup_{w\in T^\varphi} \int_\Omega w \cdot dD^su. \end{align*} The inequality ``$\leqslant$'' is clear, so we focus on the opposite one.
We assume first that $\varrho_\varphi(|\nabla^a u|) + \int_\Omega \varphi'_\infty \, d|D^su|<\infty$ and fix $\varepsilon>0$. By the definition of supremum we can choose $w_1, w_2\in T^\varphi$ such that \[
\sup_{w\in T^\varphi} \int_\Omega \nabla^a u\cdot w - \varphi^*(x, |w|)\, dx \leqslant
\int_\Omega \nabla^a u\cdot w_1 - \varphi^*(x, |w_1|)\, dx + \varepsilon < \infty \] and \[ \sup_{w\in T^\varphi} \int_\Omega w \cdot dD^su \leqslant \int_\Omega w_2 \cdot dD^su + \varepsilon< \infty. \] Since $u \in {\rm BV}(\Omega)$ and $w_i\in T^\varphi$,
we have $|\nabla^a u |\,| w_i| \in L^1(\Omega)$ and $\varrho_{\varphi^*}(|w_i|)<\infty$. Thus, by the absolute continuity of the integral, we find $\delta>0$ such that \[
\bigg|\int_{\Omega\setminus \Omega_1} \nabla^a u \cdot w_i-\varphi^*(x, |w_i|)\, dx \bigg| \leqslant
\int_{\Omega\setminus \Omega_1} |\nabla^a u|\,|w_i|+\varphi^*(x, |w_i|)\, dx \leqslant \varepsilon
\]
for $i\in \{1,2\}$ and any $\Omega_1\subset \Omega$ with $|\Omega\setminus \Omega_1|<\delta$ and \[
\bigg|\int_{\Omega\setminus \Omega_2} w_2\cdot dD^su \bigg| \leqslant
\int_{\Omega\setminus \Omega_2} \varphi'_\infty\, d|D^su| \leqslant \varepsilon \]
for any $\Omega_2\subset \Omega$ with $|D^su|(\Omega\setminus \Omega_2)<\delta$.
Since $\operatornamewithlimits{supp} D^su$ has Lebesgue measure zero, we can find a finite collection of open rectangles $Q_i\subset\Omega$ with
$|D^su|(\bigcup Q_i) > |D^su|(\Omega)-\delta$ and $|\bigcup 2Q_i|<\delta$. Then we choose $\theta\in C^1_0(\Omega)$ with $0\leqslant \theta\leqslant 1$, $\theta=1$ in $\Omega_2:=\bigcup Q_i$ and $\theta=0$ in $\Omega_1:=\Omega\setminus\bigcup 2Q_i$. Let $w_\varepsilon:=\theta w_2 + (1-\theta)w_1\in C^1_0(\Omega; {\mathbb{R}^n})$. Since $w_\varepsilon$ is a pointwise convex combination,
\[
\varphi^*(\cdot, |w_\varepsilon|) \leqslant \varphi^*(\cdot, \max\{|w_2|, |w_1|\})
\leqslant \varphi^*(\cdot, |w_2|) + \varphi^*(\cdot, |w_1|). \]
This yields that $\varrho_{\varphi^*}(|w_\varepsilon|) \leqslant \varrho_{\varphi^*}(|w_2|) + \varrho_{\varphi^*}(|w_1|)< \infty$,
and so $w_\varepsilon \in T^\varphi$. By Lemma~\ref{lem:bound}, $|w_\varepsilon|\leqslant \varphi'_\infty$. We obtain that \begin{align*} \varrho_{V,\varphi}(u) &\geqslant
\int_\Omega \nabla^a u\cdot w_\varepsilon - \varphi^*(x, |w_\varepsilon|)\, dx + \int_\Omega w_\varepsilon \cdot dD^su \\ & \geqslant
\int_{\Omega_1} \nabla^a u\cdot w_1 - \varphi^*(x, |w_1|)\, dx + \int_{\Omega_2} w_2 \cdot dD^su - c_\theta \\ & \geqslant
\int_{\Omega} \nabla^a u\cdot w_1 - \varphi^*(x, |w_1|)\, dx + \int_{\Omega} w_2 \cdot dD^su - 5\varepsilon, \end{align*} where \begin{align*} c_\theta &:=
\int_{\Omega\setminus\Omega_1} |\nabla^a u|\,|w_\varepsilon| + \varphi^*(x, |w_\varepsilon|)\, dx
+ \int_{\Omega\setminus\Omega_2} |w_\varepsilon|\, d|D^su|
\leqslant 3\varepsilon \end{align*} by the absolute integrability assumptions. By the choice of $w_1$ and $w_2$, \[ \varrho_{V,\varphi}(u) \geqslant
\sup_{w\in T^\varphi} \int_\Omega \nabla^a u\cdot w - \varphi^*(x, |w|)\, dx + \sup_{w\in T^\varphi} \int_\Omega w \cdot dD^su - 7\varepsilon. \] The lower bound follows as $\varepsilon\to 0^+$. This concludes the
proof in the case $\varrho_\varphi(|\nabla^a u|) + \int_\Omega \varphi'_\infty \, d|D^su|<\infty$.
When $\varrho_\varphi(|\nabla^a u|)=\infty$ and $\int_\Omega \varphi'_\infty \, d|D^su|<\infty$, we estimate \begin{align*}
&\sup_{w\in T^\varphi} \bigg[\int_\Omega \nabla^a u\cdot w - \varphi^*(x, |w|)\, dx + \int_\Omega w \cdot dD^su\bigg]\\ &\qquad\geqslant
\sup_{w\in T^\varphi} \int_\Omega \nabla^a u\cdot w - \varphi^*(x, |w|)\, dx - \sup_{w\in T^\varphi} \int_\Omega w \cdot dD^su \\ &\qquad=
\varrho_\varphi(|\nabla^a u|)-\int_\Omega \varphi'_\infty \, d|D^su| = \infty. \end{align*}
Only the case $\int_\Omega \varphi'_\infty \, d|D^su|=\infty$ remains. By the proof of Proposition~\ref{prop:singularPart}, there exists $w_\varepsilon\in C^1_0(\Omega; {\mathbb{R}^n})$ with \[ \int_\Omega w_\varepsilon \cdot dD^su > \frac1\varepsilon \]
and $|w_\varepsilon|\leqslant \frac{\varphi(\cdot, k)}k$ for some $k=k_\varepsilon$. For any $\theta:\Omega\to [0,1]$, we find that \[ \begin{split}
\nabla^a u \cdot (\theta w_\varepsilon) - \varphi^*(\cdot, |\theta w_\varepsilon|) &\geqslant \nabla^a u \cdot (\theta w_\varepsilon) - \varphi^* \Big(\cdot, \frac{\varphi(\cdot, k)}k\Big)
\geqslant -\Big(\frac{\varphi^+(k)}k \,|\nabla^a u| + \varphi^+(k) \Big). \end{split} \] Since the function on the right-hand side is integrable, we can choose $\delta_k>0$ such that
its integral over any measurable $A$ with $|A|<\delta_k$ is at least $-1$. Furthermore, since $\operatornamewithlimits{supp} D^su$ has measure zero, we can choose $\theta\in C^\infty_0(\Omega)$ as before to have support with Lebesgue measure at most $\delta_k$ and satisfy \[ \int_\Omega \theta w_\varepsilon \cdot dD^su > \frac1{2}\int_\Omega w_\varepsilon \cdot dD^su > \frac1{2\varepsilon}. \] Then \begin{align*}
&\sup_{w\in T^\varphi} \bigg[\int_\Omega \nabla^a u\cdot w - \varphi^*(x, |w|)\, dx + \int_\Omega w \cdot dD^su\bigg]\\ &\qquad \geqslant \int_\Omega \theta w_\varepsilon \cdot dD^su
+ \int_\Omega \nabla^a u\cdot (\theta w_\varepsilon) - \varphi^*(x, |\theta w_\varepsilon|)\, dx \geqslant \frac1{2\varepsilon} - 1. \end{align*} When $\varepsilon \to\infty$, the claim follows in this case also. \end{proof}
As a special case we obtain the following result in Orlicz spaces. Now the recession function is just a constant, either finite or infinite. As can be seen, we do not obtain any new spaces in this case, only the classical ${\rm BV}$-space or the regular Sobolev space.
\begin{cor}\label{cor:Orlicz} Let $\varphi\in \Phi_{\text{\rm c}}$ be independent of $x$ and satisfy \adec{}. If $u\in {\rm BV}(\Omega)$, then
$\varrho_{V,\varphi}(u) = \varrho_\varphi(|\nabla^a u|) + \varphi'_\infty\, |D^su|(\Omega)$ and so \begin{enumerate} \item ${\rm BV}^\varphi(\Omega)={\rm BV}(\Omega)$ if $\varphi'_\infty < \infty$; \item ${\rm BV}^\varphi(\Omega)=W^{1,\varphi}(\Omega)$ if $\varphi'_\infty = \infty$. \end{enumerate} \end{cor}
\section{Precise approximation and \texorpdfstring{$\Gamma$}{Gamma}-convergence} \label{sect:Gamma}
We can now prove a precise approximation lemma for the modular using the formula for $\varrho_{V, \varphi}$ from the previous section. In contrast to Lemma~\ref{lem:density} which provides only approximate equality of the limit we here obtain that the limit exactly equals $\varrho_{V,\varphi}(u)$, under appropriately stronger assumptions on $\varphi$. This is critical for $\Gamma$-convergence. A similar argument should also work for $V_\varphi$ with the same assumptions.
Note that we assume \hyperref[def:va1]{{\normalfont(VA1)}}{} for $\varphi^*$, not only its restricted version. This is used for Young's convolution inequality. In \cite[Lemma~4.1.7]{HarH19} it was shown that \hyperref[def:a1]{{\normalfont(A1)}}{} of $\varphi$ and $\varphi^*$ are equivalent provided \hyperref[def:a0]{{\normalfont(A0)}}{} holds; the corresponding statement is not known for \hyperref[def:va1]{{\normalfont(VA1)}}{}.
\begin{prop}[Modular approximation by smooth functions]\label{prop:modularDensity} Let $\varphi\in \Phi_{\text{\rm c}}(\Omega) \cap C(\Omega\times [0,\infty))$ satisfy \hyperref[def:a0]{{\normalfont(A0)}}{}, \adec{} and restricted \hyperref[def:va1]{{\normalfont(VA1)}}{} and assume that $\varphi^*$ satisfies \hyperref[def:va1]{{\normalfont(VA1)}}{}. For every $u \in L^\varphi(\Omega)$ there exist $u_k \in C^\infty(\Omega)$ such that \[ u_k \to u \text{ in }L^\varphi(\Omega) \quad\text{and}\quad
\varrho_{V,\varphi}(u) = \lim_{k \to \infty} \varrho_\varphi(|\nabla u_k|). \] If additionally $u \in L^2(\Omega)$, then the sequence can be chosen with $u_k \to u$ in $L^2(\Omega)$ as well. \end{prop} \begin{proof} Since the case $\varrho_{V,\varphi}(u)=\infty$ is trivial, we may assume that $\varrho_{V,\varphi}(u)<\infty$. Let $\varepsilon \in (0,1)$. We define $\xi_k$, $\eta_{\varepsilon_k}$, and $u_\varepsilon$ as in the proof of Lemma~\ref{lem:density} so that $V_\varphi(u_\varepsilon)\lesssim V_\varphi(u)$. It follows from \adec{q} that \[
\min\{ \|u\|_{\varrho_{V,\varphi}}, \|u\|_{\varrho_{V,\varphi}}^q\} \lesssim \varrho_{V,\varphi}(u) \lesssim
\max\{ \|u\|_{\varrho_{V,\varphi}}, \|u\|_{\varrho_{V,\varphi}}^q\}. \] Thus Lemma~\ref{lem:equivalence} and $V_\varphi(u_\varepsilon)\lesssim V_\varphi(u)$ imply that
$\varrho_\varphi(|\nabla u_\varepsilon|)\lesssim \varrho_{V,\varphi}(u)^q+1$. From Theorem~\ref{thm:exactFormula} we see that $U_1$ can be chosen so large (by choosing $m$ large in Lemma~\ref{lem:density}) that $\varrho_{V\setminus\overline{U_1},\varphi}(u)<\varepsilon$. Then $V_\varphi(u, \Omega\setminus \overline{U_1})\lesssim \varepsilon^{1/q}$.
Since $u_\varepsilon\in C^\infty(\Omega)$, $\nabla^a u_\varepsilon=\nabla u_\varepsilon$.
By the proof of Proposition~\ref{prop:exactFormulaAC} with $|\nabla u_\varepsilon|$ as $g$, there exists $w_\varepsilon\in C^1_0(\Omega; {\mathbb{R}^n})$ with
$\varphi^*(x, |w_\varepsilon|)\lesssim \varphi(x,|\nabla u_\varepsilon|)$ and \[
\int_\Omega \varphi(x,|\nabla u_\varepsilon|)\, dx \leqslant
(1+\varepsilon)\int_\Omega \nabla u_\varepsilon\cdot w_\varepsilon - \varphi^*(x,|w_\varepsilon|)\, dx. \] By \adec{} of $\varphi$, Theorem~\ref{thm:exactFormula} and the estimates above, \[
\varrho_{\varphi^*}(|w_\varepsilon|) \lesssim \varrho_\varphi(|\nabla u_\varepsilon|)
\lesssim \varrho_{V,\varphi}(u)^q+1. \]
Thus $\|w_\varepsilon\|_{\varphi^*}\leqslant c$. As in Lemma~\ref{lem:density}, we have \[ \begin{split} \int_\Omega \nabla u_\varepsilon \cdot w_\varepsilon \, dx = \underbrace{\sum_{k=1}^\infty \int_\Omega u \divop (\xi_k(\eta_{\varepsilon_k}* w_\varepsilon)) \, dx}_{=: I} - \underbrace{\sum_{k=1}^\infty \int_\Omega w_\varepsilon \cdot (\eta_{\varepsilon_k}*(u \nabla \xi_k) - (u\nabla \xi_k))\, dx}_{=: II} \end{split} \]
and the inequality $|II|\leqslant c\varepsilon$ again follows.
We divide the term $I$ into two parts. Let $\omega$ be from Corollary~\ref{cor:convolution}. Using the definition of $\varrho_{V,\varphi}$ to the first part of $I$, and and estimating the second part of $I$ as in Lemma~\ref{lem:density} but now with a test-function supported in $\Omega\setminus \overline{U_1}$, we find that \[ \begin{split}
|I| & =
\bigg| \int_\Omega u \divop (\xi_1(\eta_{\varepsilon_1}*w_\varepsilon)) \, dx +
\int_\Omega u \divop \bigg( \sum_{k=2}^\infty \xi_k(\eta_{\varepsilon_k}*w_\varepsilon)\bigg) \, dx \bigg|\\ &\leqslant
\varrho_{V,\varphi}\big((1+\omega(\varepsilon_1)) u\big) + \varrho_{\varphi^*}\Big( \frac{\xi_1|\eta_{\varepsilon_1}*w_\varepsilon|}{1+\omega(\varepsilon_1)}\Big) + cV_\varphi(u, \Omega\setminus \overline{U_1}) \\ &\leqslant
\varrho_{V,\varphi}\big((1+\omega(\varepsilon_1)) u\big) + \varrho_{\varphi^*}\Big( \frac{\eta_{\varepsilon_1}*|w_\varepsilon|}{1+\omega(\varepsilon_1)}\Big) + cV_\varphi(u, \Omega\setminus \overline{U_1}) \end{split} \] By Young's convolution inequality (Corollary~\ref{cor:convolution}), \[
\varrho_{\varphi^*}\Big( \frac{\eta_{\varepsilon_1}*|w_\varepsilon|}{1+\omega(\varepsilon_1)}\Big)
- \varrho_{\varphi^*}(|w_\varepsilon|)
\leqslant \varrho_{\varphi^*}( |w_\varepsilon|) + \omega(\varepsilon_1) -\varrho_{\varphi^*}(|w_\varepsilon|) \leqslant \omega(\varepsilon_1) \to 0 \] as $\varepsilon_1\to 0^+$. Combining the estimates, we obtain that \begin{align*}
\int_\Omega \varphi(x,|\nabla u_\varepsilon|)\, dx &\leqslant
(1+\varepsilon)\int_\Omega \nabla u_\varepsilon\cdot w_\varepsilon - \varphi^*(x,|w_\varepsilon|)\, dx\\ &\leqslant
(1+\varepsilon)(|I|-\varrho_{\varphi^*}(|w_\varepsilon|)) + c\varepsilon \\ &\leqslant (1+\varepsilon)\varrho_{V,\varphi}\big((1+\omega(\varepsilon_1)) u\big) +
c(|\Omega|\,\omega(\varepsilon_1) + \varepsilon^{1/q}). \end{align*} By \cite[Lemma~2.2.6]{HarH19}, there exists a constant $q_2$ depending on $q$ such that $\varrho_{V,\varphi}\big((1+\omega(\varepsilon_1)) u\big)\leqslant (1+\omega(\varepsilon_1))^{q_2}\varrho_{V,\varphi}(u)$. As $\varepsilon,\varepsilon_1\to 0^+$,
we obtain that $\limsup_{\varepsilon\to 0^+}\varrho_\varphi(|\nabla u_\varepsilon|)\leqslant \varrho_{V,\varphi}(u)$. The opposite inequality follows from Lemma~\ref{lem:sequence-in-BV} as in Lemma~\ref{lem:density}. \end{proof}
In \cite{EleHH_pp}, we introduced abstract ${\rm BV}^\varphi$-type spaces by a limit procedure.
We use here the version with a fidelity term which is most relevant for image processing. For $p >1$ and for a given $f \in L^2(\Omega)$, we defined functionals $F_p: L^2(\Omega) \to [0, \infty]$ by \begin{equation*}
F_p(u) := \begin{cases}
\displaystyle\int_\Omega \varphi(x, |\nabla u|)^p + |u-f|^2 \, dx & \text{when } u \in L^{1, \varphi^p}(\Omega); \\ \infty & \text{otherwise.} \end{cases} \end{equation*} and the limit functional $F: L^2 (\Omega) \to [0, \infty]$ by \begin{equation*} F(u) :=
\inf\bigg\{\liminf_{i\to\infty}\int_\Omega \varphi(x, |\nabla u_k|) + |u_k-f|^2\, dx \,\Big|\, u_k \in L^{1, \varphi} (\Omega) \cap L^2(\Omega)\text{ and } u_k\to u \text{ in } L^2(\Omega) \bigg\} \end{equation*}
Note that the energy in $F_p$ satisfies \ainc{} and \adec{} so it can be studied in a reflexive space and it is easy to prove existence of minimizers among other things \cite{HarHK16}.
We compare $F$ with the corresponding version of $\varrho_{V, \varphi}$ including the fidelity term, namely \[ \varrho_{V,\varphi}^f(u):= \varrho_{V,\varphi}(u)+\varrho_2(u-f) =
\sup\bigg\{ \int_\Omega u \divop w - \varphi^*(x, |w|)+|u-f|^2\, dx \,\Big|\, w \in C^1_0(\Omega; {\mathbb{R}^n})\bigg\}. \]
\begin{prop} \label{comparison-modular} Let $\varphi\in \Phi_{\text{\rm c}}(\Omega) \cap C(\Omega\times [0,\infty))$ satisfy \hyperref[def:a0]{{\normalfont(A0)}}{}, \adec{} and restricted \hyperref[def:va1]{{\normalfont(VA1)}}{} and assume that $\varphi^*$ satisfies \hyperref[def:va1]{{\normalfont(VA1)}}{}. Then $\varrho_{V,\varphi}^f(u) \leqslant F(u)$ for all $u \in L^2(\Omega)$ and $\varrho_{V,\varphi}^f(u) = F(u)$ for all $u \in L^2(\Omega)\cap L^\varphi(\Omega)$. \end{prop}
\begin{proof} Let us prove first that $\varrho_{V,\varphi}^f(u) \leqslant F(u)$. We may assume $F(u) < \infty$ and consider functions $u_k \in L^{1, \varphi}(\Omega) \cap L^2(\Omega)$ realizing the infimum from $F$ with $u_k \rightarrow u$ in $L^2(\Omega)$. Weak lower semicontinuity of $\varrho_{V,\varphi}$ (Lemma~\ref{lem:sequence-in-BV}) and in $L^2$ gives \[ \varrho_{V,\varphi}^f(u) \leqslant \liminf_{i\to \infty} \varrho_{V,\varphi}^f(u_k). \]
By Young's inequality, $\varrho_{V,\varphi}^f(u_k) \leqslant \varrho_{\varphi}(|\nabla u_k|) + \varrho_2(u_k - f)$ so that \[
\varrho_{V,\varphi}^f(u) \leqslant \liminf_{i\to \infty} \big(\varrho_{\varphi}(|\nabla u_k|) + \varrho_2(u_k - f) \big) = F(u). \] Thus the inequality is proved.
For the opposite inequality, $F(u) \leqslant \,\varrho_{V,\varphi}^f(u)$, we may assume that $\varrho_{V,\varphi}^f(u) < \infty$. By Proposition~\ref{prop:modularDensity}, there exist $u_k \in C^\infty(\Omega)$ such that $u_k \to u$ in $L^{\varphi}(\Omega)\cap L^2(\Omega)$ and \[
\varrho_{V,\varphi}(u) = \lim_{i \to \infty} \varrho_\varphi(|\nabla u_k|). \]
Since $\varrho_\varphi(|\nabla u_k|)<\infty$ and $u_k\in L^1(\Omega)$, we see that $u_k \in L^{1, \varphi}(\Omega)$, and so, by the definition of $F$, using the fact that the limit of the sum is the sum of the limits, we obtain that \[
F(u) \leqslant \liminf_{i \to \infty} \big(\varrho_\varphi(|\nabla u_k|) + \varrho_2(u_k - f) \big) = \varrho_{V,\varphi}^f(u). \qedhere \] \end{proof}
The concept of $\Gamma$-convergence was introduced by De Giorgi and Franzoni \cite{DeGF75}, see also \cite{Bra02, Dal93}. A family of functionals $F_p: L^2(\Omega) \to [0, \infty]$ is said to \textit{$\Gamma$-converge} to $F: L^2(\Omega) \to [0, \infty]$ in $L^2(\Omega)$ if the following hold for every sequence $(p_k)$ converging to one from above: \begin{enumerate}
\item[(a)] $\displaystyle F (u) \leqslant \liminf_{i \to \infty} F_{p_k} (u_{i})$ for every $u \in L^2(\Omega)$ and every sequence with $u_{i}\to u$ in $L^2(\Omega)$;
\item[(b)] $\displaystyle F (u) \geqslant \limsup_{i \to \infty} F_{p_k} (u_{i})$ for every $u \in L^2(\Omega)$ and some sequence with $u_{i}\to u$ in $L^2(\Omega)$. \end{enumerate}
We conclude by showing the $\Gamma$-convergence in the situation most relevant to image processing: convex planar domains. This allows us to simplify the assumptions.
\begin{cor} Let $\Omega\subset\mathbb{R}^2$ be convex and let $\varphi\in \Phi_{\text{\rm c}}(\Omega)$ satisfy \hyperref[def:a0]{{\normalfont(A0)}}{}, \adec{2} and \hyperref[def:va1]{{\normalfont(VA1)}}{} and assume that $\varphi^*$ satisfies \hyperref[def:va1]{{\normalfont(VA1)}}{}. Then $F_p$ $\Gamma$-converges to $\varrho_{V,\varphi}^f$ in $L^2(\Omega)$. \end{cor}
\begin{proof} To establish the necessary conditions we use some results from references without defining here all the terms. The references can be consulted if necessary. By \cite[Corollary~4.6]{Juu_pp}, $C^\infty(\overline{\Omega})$ is dense in $W^{1, \varphi}(\Omega)$ if $\Omega$ is an $(\varepsilon, \delta)$-domain and $\varphi$ satisfies \hyperref[def:a0]{{\normalfont(A0)}}{}, \hyperref[def:a1]{{\normalfont(A1)}}{} and (A2). We note that (A2) holds since $\Omega$ is bounded \cite[Lemma~4.2.3]{HarH19} and $\Omega$ is an $(\varepsilon, \delta)$-domain since it is convex.
Since $\varphi$ satisfies \adec{2}, $L^2(\Omega) \subset L^\varphi(\Omega)$ and thus $L^{1, \varphi}(\Omega) \cap L^2(\Omega) \hookrightarrow W^{1, \varphi}(\Omega)$. Since the dimension is $2$ we also have $W^{1, \varphi}(\Omega) \hookrightarrow W^{1, 1} (\Omega) \hookrightarrow L^2(\Omega)$. Thus $C^\infty(\overline{\Omega})$ is dense in $L^{1, \varphi}(\Omega) \cap L^2(\Omega)$ with respect to
the norm $u \mapsto \|u\|_2 + \|\nabla u\|_\varphi$. By density, \cite[Theorem~1.3(2)]{EleHH_pp} yields that $F_p$ $\Gamma$-converges to $F$ in $L^2(\Omega)$. Since $\varphi$ satisfies \hyperref[def:va1]{{\normalfont(VA1)}}{}, it belongs to $C(\Omega\times [0,\infty))$. Thus Proposition~\ref{comparison-modular} gives $F= \varrho_{V,\varphi}^f$ in $L^2(\Omega) = L^2(\Omega)\cap L^\varphi(\Omega)$. \end{proof}
\end{document} | arXiv |
\begin{document}
\title{Grover like Operator Using Only Single-Qubit Gates} \author{G.~Kato} \email{[email protected]} \affiliation{NTT Communication Science Laboratories,\\ NTT Corporation \\ 3-1, Morinosato Wakamiya, Atsugi-shi, Kanagawa Pref., 243-0198 Japan } \date{\today}
\begin{abstract} We propose a new quantum circuit for the quantum search problem. The
quantum circuit is superior to Grover's algorithm in some realistic cases. The reasons for the superiority are in short as follows: In the quantum circuit proposed in this paper, all the operators except for the oracle can be written as direct products of single-qubit gates. Such separable operators can be executed much faster than multi-particle operators, such as
c-NOT gates and
Toffoli gates, in many realistic systems. The idea of this quantum circuit is inspired by the Hamiltonian used in the adiabatic quantum computer. In addition, the scaling of the number of oracle calls for this circuit is the same as that for Grover's algorithm, i.e. $O\left(2^{n/2}\right)$. \end{abstract} \pacs{03.67.Lx}
\maketitle
\section{introduction} Since the concept of the quantum computer (QC) was proposed \cite{B82,D85,F85}, many quantum algorithms \cite{DJ92,S94,G97,TKM05} that are superior to classical algorithms have been proposed. These algorithms have inspired many researchers, and the number of the researchers investigating the QC has increased dramatically as a result.
Though many results are generated daily, there remains a serious problem. The generated quantum circuits utilize the properties of quantum mechanics effectively, but almost all of then are modifications or combinations of just three quantum circuits based on quantum Fourier transformation \cite{K95}, quantum amplitude amplification \cite{BHM+00} or discrete quantum random walk \cite{AAJ+01}. This indicates that it is very hard to design new quantum circuits that use the properties of quantum mechanics effectively.
Recently, some frameworks differing from the QC have been proposed,
such as the adiabatic quantum computer (AQC)\cite{FGG+00}, and the continuous random walk \cite{FG97}, and many results have been
forthcoming in this area. In this paper, we focus on the AQC, whose procedure is identified by a Hamiltonian. Recently, it was proved that the
calculation power of the AQC has the same as that of the QC \cite{ADK+04}. This means that the QC can be emulated using the AQC and vice versa with polynomial time and space with respect to the input size. On the other hand, the properties of the problems that the QC and AQC are good at are different. These two facts indicate that new concepts of quantum circuits must be given by the explicit modification from the Hamiltonians for the AQC into finite size quantum circuits for the QC. We think this is a good strategy for designing new quantum circuits that use the properties of quantum mechanics effectively.
In this paper, we propose a new quantum circuit modified from a Hamiltonian for the AQC. This is the first simple example of a quantum circuit obtained by following the above strategy. Here, we treat the well-investigated problem in the QC, i.e., the quantum search problem, in order to check the efficiency of the strategy. As a result, we get a new quantum circuit that is superior to the quantum circuit used in Grover's algorithm in some cases. The Hamiltonian just gives us some hints, and the new quantum circuit is intuitively generated using those hints. Thus, we can not show some explicit procedures for the modification.
Here, we have to mention that, from the past work \cite{L96}, quantum circuits for the QC can be easily modified from the Hamiltonians for the AQC, but the quantum circuits generated by the modification simply follow the time evolution of the AQC. Consequently, such quantum circuits are very redundant and inefficient for realistic calculations. The quantum circuits that we want to modify from the Hamiltonians are not such useless quantum circuits but practical ones.
To avoid any confusion, we should clarify that our circuit is superior in that it may be executed faster than Grover's algorithm in realistic systems since it uses only simple operators, each of which rotate just one-qubit, except for the oracle. However, the circuit does not offer reduced complexity. Actually, both it and Grover's algorithm have exactly the same complexity $O\left(2^{n/2}\right)$. For these reasons, the superiority of the new circuit will be meaningful mainly to experimentalists.
In Sec. \ref{sec:Grover_algorithm}, we briefly review Grover's algorithm to facilitate comparison between it and the expressions for the new quantum circuit.
In Sec. \ref{sec:G_algorithm}, we show the explicit form of the new quantum circuit and prove that the quantum circuit can execute quantum search efficiently. In Sec. \ref{sec:numerical_calculation}, we numerically simulate the new quantum circuit to show how well it executes quantum search. In Sec. \ref{sec:relation_AQC}, we show the relations between quantum circuits and Hamiltonians for the AQC. These relations are the hints for generating the new quantum circuit. The last section summarizes our conclusions. Technical details of a proof are in Appendix \ref{sec:proof_of_lemma}.
\section{Grover's algorithm} \label{sec:Grover_algorithm} By Grover's algorithm, the quantum search problem can be solved. This means that we can find integer $j$ from $0$ to $2^n-1$ using the oracle operator $\hat {O_r}$ such that \begin{eqnarray}
\hat{O_r}\left|m\right>\otimes\left|k\right> &:=&
\left|m\right>\otimes\left|k\oplus\delta\left(m,j\right)\right> \label{eq:def_oracle} \end{eqnarray} by the algorithm. The operator $\hat {O_r}$ acts on two registers: one is $2^n$-dimensional, corresponding to the search space, and the other is $2$-dimensional, corresponding to the output of the oracle.
Grover's algorithm can be expressed as follows. First, we generate the initial state \begin{eqnarray}
\left|\bar0\right>&:=&2^{-\frac n2}\sum_{m=0}^{2^n-1}\left|m\right>. \label{def:initial_state} \end{eqnarray} Next, we iterate the
two operations, which are identified by the following operator: \begin{eqnarray}
\hat G&:=&1-2\left|\bar 0\right>\left<\bar 0\right|,\\
\hat O&:=&1-2\left| j\right>\left< j\right|. \label{eq:def_oracle_Gr} \end{eqnarray} Note that, in general $\hat G\!\cdot\!\hat O$ is written by $G$, e.g., \cite{NC00}, and is called the Grover operator. The number of iterations is \begin{eqnarray} N&:=&\left[\frac{\pi}{4\arcsin 2^{-\frac n2}}\right], \end{eqnarray} where $[r]$ indicates the integer part of real number $r$. Note that the operator $\hat O$ (\ref{eq:def_oracle_Gr}) is outwardly different from the oracle operator $\hat{O_r}$ (\ref{eq:def_oracle}); however, $\hat O$ can be simulated from one use of $\hat{O_r}$
by using the second register as an ancilla prepared in state $\frac1{\sqrt2}\left|0\right>-\frac1{\sqrt2}\left|1\right>$.
Finally, we observe the state using the computational basis, i.e., $\left|0\right>$,
$\left|1\right>$,$\cdots$ ,$\left|2^n-1\right>$. The success probability of Grover's algorithm, i.e., the probability to detect the state $\left|j\right>$, goes to $1$ in the limit $n\rightarrow\infty$. This is equivalent to the following relation: \begin{eqnarray} \lim_{n\rightarrow\infty}
\left|\left<j\right|\left(\hat G\!\cdot\! \hat O\right)^N\left|\bar 0\right>\right|^2 &=&1. \label{eq:math_glover_algirism} \end{eqnarray} The scaling of the success probability versus $n$ is $1-O\left(2^{-n}\right)$. The relation (\ref{eq:math_glover_algirism}) can be easily proved as follows.
{\it Proof}:
The operator $\hat G\!\cdot\!\hat O$ modifies any vector in the space spanned by $\left|\bar 0\right>$ and $\left|j\right>$ into another vector in the same space. Then, we restrict the Hilbert space to the two dimensional space, i.e.,
$\left\{\left|\psi\right>=a\left|\bar0\right>+b\left|j\right>\right\}$ in this proof. Under this restriction, the operator $\hat G\!\cdot\!\hat O$ can be written as the following two dimensional matrix: \begin{eqnarray} \hat G\!\cdot\!\hat O &=& - \begin{pmatrix}
1-2^{1- n } & -2^{1-\frac n2}\sqrt{1-2^{-n}} \\
2^{1-\frac n2}\sqrt{1-2^{-n}}&1-2^{1- n } \end{pmatrix} \nonumber\\ &=& - \begin{pmatrix}
\cos2\theta&-\sin2\theta \\
\sin2\theta& \cos2\theta \end{pmatrix}, \\ \theta&:=&\arcsin2^{-\frac n2}\quad\quad 0<\theta\leq\frac\pi2. \label{def:theta} \end{eqnarray} Here, we use the basis
$\left\{\left|\bar 0\right>,\left|\bar1\right>:=\frac{\left|j\right>-2^{-\frac n2}\left|\bar 0\right>}{\sqrt{1-2^{-n}}}\right\}$. From this expression, it is easy to show that \begin{eqnarray}
&&\left|\left<j\right|\left(\hat G\!\cdot\!\hat O\right)^N\left|\bar 0\right>\right|^2 \nonumber\\ &=& \sin^2 \left(2N+1\right)\theta \nonumber\\ &=& \sin^2\left(2\left[\frac{\pi}{4\arcsin 2^{-\frac n2}}\right]+1\right) \arcsin2^{-\frac n2}. \end{eqnarray} From the last equation, it is clear that relation (\ref{eq:math_glover_algirism}) holds. $\square$
\section{Quantum search algorithm using a new quantum circuit} \label{sec:G_algorithm} \subsection{The case of one solution} We propose a new quantum circuit by which the Grover iteration can be replaced.
The outline of the algorithm is the same as Grover's algorithm, but in order to avoid misunderstanding we show whole algorithm below. First, we prepare the initial state $\left|\bar0\right>$, which is the same as the initial state of Grover's algorithm. Next, we iterate the
two operations, which are identified by the following operator: \begin{eqnarray} \hat G' &:=& \exp\left(\varphi\left(\omega\right) \sum_{\alpha=0}^{n-1}S_x^{(\alpha)} i\right), \label{eq:def_oracle_Go_G'}\\ \hat O' &:=&
\exp\left(\omega \left|j\right>\left<j\right| i\right). \label{eq:def_oracle_Go} \end{eqnarray} The number of iterations is \begin{eqnarray}
N'&:= &\left[\frac{\pi}{4 \sin\left|\frac\omega2\right|}2^{\frac n2}+\frac12\right]. \label{eq:suggestion_2} \end{eqnarray} The variable $\omega$ in the above definition can be chosen from the region $-\pi<\omega<\pi$ and is independent of $n$ and $j$. The operator $S_x^{(\alpha)}$ and the function $\varphi\left(\omega\right)$ are defined later. Finally, we observe the state using the computational basis. The success probability of this algorithm goes to $1$ in the limit $n\rightarrow \infty$. This is equivalent to the following relation: \begin{eqnarray} \lim_{n\rightarrow\infty}
\left|\left<j\right|\left(\hat G'\!\cdot\!\hat O'\right)^{N'}\left|\bar 0\right>\right|^2 &=&1. \label{eq:suggestion_1} \end{eqnarray} The scaling of the success probability versus $n$ is $1-O\left(n^{-1}\right).$ A proof of relation (\ref{eq:suggestion_1}) is located at the end of
this section.
Here, we have to note three things. First, the scaling of the number of oracle calls is $O\left(2^{n/2}\right)$ for any $-\pi<\omega<\pi$ when $\omega\neq0$. Here, we have to point out that the operator $\hat O'$ (\ref{eq:def_oracle_Go}) can actually be simulated by a constant number of calls to the oracle $\hat{O_r}$ (\ref{eq:def_oracle}), where the number depends on $\omega$. A method of simulation is as follows. We introduce a naturally generalized oracle as \begin{eqnarray}
\hat {O_r}'\left|m\right>\otimes\left|k\right>
&:=&\left|m\right>\otimes\left|k+\delta\left(m,j\right)\makebox{ mod }\omega_d\right> \label{eq:def_oracle_ge} \end{eqnarray} for arbitrary integer $\omega_d$. The operator $\hat {O_r}'$ (\ref{eq:def_oracle_ge}) acts on two registers:
one is $2^n$-dimensional and the other is $\omega_d$-dimensional. It is easy to show that this operator $\hat {O_r}'$
can be simulated by a constant number of calls to the oracle $\hat{O_r}$. Furthermore, the operator $\hat O'$ can be simulated from one use of $\hat{O_r}'$ by using the second register as ancillae prepared in state \begin{eqnarray}
\sum_{k=0}^{\omega_d-1}\exp\left(\frac{k\omega_c}{\omega_d}2\pi i\right) \left|k\right>. \end{eqnarray} In this definition, $\omega_d$ and $\omega_c$ are chosen so as to satisfy $\frac{\omega_c}{\omega_d}2\pi=\omega$. Then, the operator $\hat O'$ can be simulated by $\hat{O_r}$. Second,
the difference in execution time between $\hat {O_r}$
and $\hat O'$
probably will not depend on $n$ in most cases. This expectation comes form the following consideration. Once we know the explicit
circuit for $\hat {O_r}$, we will probably be able to make a circuit
corresponding to $\hat {O_r}'$ in such a way that the difference of
execution time of these two circuits does not depend on $n$. This expectation has no meaning from a computer science point of view,
since the oracle $\hat{O_r}$ is usually treated as a black-box. However, in case of actual calculations using a real system, it is important to
think in term of the execution time of operations. Third, when $\omega=\pm\pi$, the relation (\ref{eq:suggestion_1}) does not hold. This is related to the fact that the value $\omega$ influences not only the number of iterations $N'$ but also the speed of the convergence (\ref{eq:suggestion_1}).
For example, when $\omega$ approaches $\pm\pi$,
the speed of the convergence decreases.
On the other hand, when $\omega$ approaches $0$,
the speed of the convergence increases. Here, the change in the speed means the change in the constant factor of the scaling.
Here, we define the function $\varphi\left(\omega\right)$ and the operator $S_x^{(\alpha)}$ used in the above outline of the algorithm. First, $S_x^{(\alpha)}$ is the operator which acts only on the $\alpha$-th qubit, and the action on the qubit can be written as $\begin{pmatrix}0&\frac12 \\\frac12&0\end{pmatrix}$ using the computational basis. Therefore, we can write $S_x^{(\alpha)}$ as follows: \begin{eqnarray} S_x^{(\alpha)}&:=&Id\otimes \cdots\otimes \begin{pmatrix} 0&\frac12 \\\frac12&0 \end{pmatrix} \otimes\cdots\otimes Id. \end{eqnarray} Note that $2S_x^{(\alpha)}$ is simply Pauli operator $\sigma_x$ applied on qubit $\alpha$. Next, we define $\varphi\left(\omega\right)$ implicitly as follows: \begin{eqnarray} \cot\frac\omega2&=&\sum_{s=1}^{n}P_n\left(s\right)\cot\frac{s\varphi\left(\omega\right)}2, \nonumber \end{eqnarray}
\begin{equation} \makebox[1cm]{}-\frac{2\pi}{n}<\varphi\left(\omega\right)<\frac{2\pi}{n},\quad\operatorname{sgn}\left(\omega\right)=\operatorname{sgn}\left(\varphi\left(\omega\right)\right) \label{eq:definition_alpha_2} \end{equation} where \begin{eqnarray} P_n\left(s\right)&:=&\frac{n!2^{-n}}{s!\left(n-s\right)!}. \end{eqnarray} Recall that $2^n$ is the number of elements in the set from which item $j$ is selected and that $\omega$ is an arbitrary number in the region $-\pi<\omega<\pi$. Note that the function $\varphi\left(\omega\right)$ depends on $n$. \begin{figure*}
\caption{ A plot of the function $\varphi\left(\omega\right)$ defined by
(\ref{eq:definition_alpha_2}) for $n=10$ }
\label{fig:varphi}
\end{figure*} As an example, a plot of the function $\varphi\left(\omega\right)$ for $n=10$ is shown in Fig \ref{fig:varphi}.
The rest of this section is devoted to proving relation (\ref{eq:suggestion_1}).
{\it Proof:}
First of all, we show the main idea underlying this proof in order to provide
some insight into why it works. The idea consists of three parts. First, $\hat G'\hat O'$ leaves $\tilde S^2$ (\ref{eq:modified_total_spin})
eigenspaces invariant, and both $\left|\bar 0\right>$ and
$\left|j\right>$ lie in the same eigenspace, so we can restrict our study to this eigenspace. Second, $\left|\bar 0\right>$ and $\left|j\right>$ have most of their support on the $2$-dimensional subspace spanned by two particular eigenstates of $\hat G'\hat O'$, $\left|\psi_{\gamma_\pm}\right>$ whose eigenvalues are $\gamma_\pm$ (\ref{eq:inportant_eignevalue}), so we can even more restrict our study to this subspace.
Finally, due to the corresponding eigenvalues $\gamma_\pm$, we need to repeat $\hat G'\hat O'$ a certain number of times (\ref{eq:suggestion_2}) to rotate
$\left|\bar 0\right>$ to
$\left|j\right>$. Based on this idea, we obtain a strict proof as follows.
The operator $\hat G'\!\cdot\!\hat O'$ is a block diagonal matrix in the case of the computational basis and each block can be characterised by eigenvalues of the operator \begin{eqnarray} \tilde S^2:=
\left(\sum_{\alpha=0}^{n-1} S_x^{(\alpha)} \right)^2 \!\!+\!\left(\sum_{\alpha=0}^{n-1}\tilde S_y^{(\alpha)} \right)^2 \!\!+\!\left(\sum_{\alpha=0}^{n-1}\tilde S_z^{(\alpha)} \right)^2\!\!\!, \label{eq:modified_total_spin} \end{eqnarray} where \begin{eqnarray} &&{}\!\!\!\!\!\!\!\!\!\!\! \tilde S_y^{(\alpha)}:= \left(-\right)^{j^{(\alpha)}}\!\!\!\! Id\otimes \cdots\otimes \begin{pmatrix} 0&-\frac12 i \\\frac12 i&0 \end{pmatrix}
\otimes\cdots\otimes Id,\nonumber\\ &&{}\!\!\!\!\!\!\! \!\!\!\! \tilde S_z^{(\alpha)}:= \left(-\right)^{j^{(\alpha)}}\!\!\!\! Id\otimes \cdots\otimes \begin{pmatrix} \frac12&0 \\0&-\frac12 \end{pmatrix}
\otimes\cdots\otimes Id \label{eq:modified_Pauli} \end{eqnarray} and $j^{(\alpha)}$ is $0$ or $1$ such that \begin{eqnarray}
j&=&\sum_{\alpha=0}^{n-1}2^\alpha j^{(\alpha)}. \end{eqnarray} Note that the operators $2\tilde S_y^{(\alpha)}$ and $2\tilde S_z^{(\alpha)}$ defined by (\ref{eq:modified_Pauli}) reduce to the Pauli operators in the special case j=0. Otherwise, the operators $2\tilde S_y^{(\alpha)}$
and $2\tilde S_z^{(\alpha)}$ are equivalent to the Pauli operators up to an overall phase. The states $\left|\bar 0\right>$ and $\left|j\right>$ belong to the subspace whose eigenvalue for $\tilde S^2$ is $n\left(n+2\right)/4$. This subspace reduces to the maximal total spin subspace in the special case $j=0$.
In the rest of this section, we restrict the Hilbert space to this subspace and use the following two bases \begin{eqnarray} \left\{
\left|s_x\right>
\left|\sum_{\alpha=1}^n S_x^{(\alpha)}\left|s_x\right>=\left(-s+n/2\right)\left|s_x\right> \right. \right\},\\ \left\{
\left|s_z\right>
\left|\sum_{\alpha=1}^n\tilde S_z^{(\alpha)}\left|s_z\right>=\left(-s+n/2\right)\left|s_z\right> \right. \right\}. \end{eqnarray} Note that it is easy to see that
$\left|\bar0\right>\propto\left|0_x\right>$ and
$\left|j\right>\propto\left|0_z\right>$. Then, over all phases are defined in such a way that $\left<s_x|0_z\right>>0$,
$\left<0_x|s_z\right>>0$, $\left|\bar0\right>=\left|0_x\right>$ and
$\left|j\right>=\left|0_z\right>$.
The eigenvalues $ \exp\left(\gamma+\frac{n\varphi(\omega)}2\right) i$
and the corresponding eigenvectors $\left|\psi_\gamma\right>$ for $\hat G'\!\cdot\!\hat O'$ satisfy the relation \begin{eqnarray}
\frac{\left<s_x|\psi_\gamma\right>}{1-\exp\left(\omega i\right)}
&=&\frac{\left<s_x|0_z\right>\left<0_z|\psi_\gamma\right>}{1- \exp \left(\gamma+s\varphi\left(\omega\right) \right)i}. \label{eq:original_relation} \end{eqnarray} Then, the following two relations hold: \begin{eqnarray}
\frac{1}{1-\exp\left(\omega i\right)} &=&\sum_{s=0}^{n} \frac{P_n\left(s\right)}
{1- \exp\left(\gamma+s\varphi\left(\omega\right) \right)i}, \label{eq:definition_of_eigenvalue}\\ {}\!\!\!\!\!\!\!\!\frac{1}
{
\left|1-\exp\left(\omega i\right)\right|^2 } &=&\sum_{s=0}^{n}
\frac{P_n\left(s\right)\left|\left<0_z|\psi_\gamma\right>\right|^2}
{
\left|1-\exp\left(\gamma+s\varphi\left(\omega\right) \right)i\right|^2 }. \label{eq:definition_of_eigenvector} \end{eqnarray} In the derivation of the above two relations, we use the relation \begin{eqnarray}
\left|\left<s_x|0_z\right>\right|^2&=&P_n\left(s\right). \end{eqnarray}
Next, we show that there are two eigenvalue series $\exp\left(\gamma_\pm+\frac{n\varphi\left(\omega\right)}2\right)i$ for the operator $\hat G'\!\cdot\!\hat O'$ such that \begin{eqnarray}
\lim_{n\rightarrow\infty}2^{\frac n2}\gamma_\pm&=&\pm2\sin\frac\omega2 \label{eq:inportant_eignevalue}, \end{eqnarray} where we regard $\gamma_\pm$ as two series with respect to $n$ defined by $\omega$. In order to prove this relation, we use the following relation \begin{eqnarray}
\lim_{n\rightarrow \infty}\frac{n\varphi\left(\omega\right)}{2}&=&\omega. \label{eq:asymptotics_of_alpha} \end{eqnarray} Recall that the function $\varphi\left(\omega\right)$ is defined by (\ref{eq:definition_alpha_2}). This relation is derived from \begin{eqnarray}
\lim_{n\rightarrow\infty}\sum_{s=1}^{n}P_n\left(s\right)\cot\frac{sr}{2n} &=& \cot\frac r4, \label{eq:limit_of_definition_w_r_t_varphi} \end{eqnarray} where $-2\pi<r<2\pi$. Relation (\ref{eq:limit_of_definition_w_r_t_varphi}) is a special case of the following Lemma. \begin{itemize}
\item {\it Lemma}: \begin{eqnarray}
\lim_{n\rightarrow\infty}\sum_{s=1}^{n}P_n\left(s\right)f\left(\frac{s}{n}\right)&=&f\left(\frac12\right), \label{eq:limit_relation} \end{eqnarray} where $f\left(\zeta\in\mathbb{C}\right)$ is a meromorphic function in the region
$\left|\zeta-\frac12\right|<1/2+\delta$ and has only one pole at point $\zeta=0$. \end{itemize}
(A proof of this lemma is given in Appendix \ref{sec:proof_of_lemma}.) Then, relation (\ref{eq:asymptotics_of_alpha}) is proved. Now, we define two functions $g\left(n,\zeta\right)$ and $ g^{(q)}\left(\zeta\right)$ \begin{eqnarray} g\left(n,\zeta\right) &:=& \frac{1}{1-\exp\left(\omega i\right)} \nonumber\\&& -\sum_{s=0}^{n} \frac{P_n\left(s\right)}
{1- \exp\left(\zeta+s\varphi\left(\omega\right) \right)i}, \\
g^{(q)}\left(\zeta\right)&:=&\frac1{q!}\frac {d^q}{d{\tilde\zeta}^q} \left. \frac{1}
{1- \exp\left(\tilde\zeta+\zeta) \right)i}
\right|_{\tilde\zeta=0}. \end{eqnarray} It is clear that $g\left(n,\gamma\right)$ is equal to $0$ from condition (\ref{eq:definition_of_eigenvalue}).
Then, the sufficient condition of (\ref{eq:inportant_eignevalue}), \begin{eqnarray} &&\lim_{n\rightarrow\infty}g\left(n,\zeta2^{-\frac n2}\right)2^{\frac n2} \nonumber\\
&=& \frac1\zeta i-\lim_{n\rightarrow\infty} \sum_{q=1}^\infty \sum_{s=1}^{n}
P_n\left(s\right) g^{(q)}\left(s\varphi\left(\omega\right)\right) \zeta^q2^{-\frac {n\left(q-1\right)}2} \nonumber\\
&=& \frac1\zeta i-\sum_{q=1}^\infty \lim_{n\rightarrow\infty} \sum_{s=1}^{n}
P_n\left(s\right) g^{(q)}\left(s\varphi\left(\omega\right)\right) \zeta^q2^{-\frac {n\left(q-1\right)}2} \nonumber\\ &=&\frac1\zeta i-\frac \zeta{4\sin^2\frac\omega2}i \end{eqnarray}
\begin{equation} \nonumber \makebox[3cm]{} \zeta\in\mathbb{C},\quad \zeta\neq0, \end{equation} is derived by using relation (\ref{eq:asymptotics_of_alpha}) and lemma (\ref{eq:limit_relation}). In the first equality, we make the Laurent expansion at $\zeta=0$. In the second equality, we just exchange the order of the limit operations. In the last equality, we use relation (\ref{eq:asymptotics_of_alpha}) and lemma (\ref{eq:limit_relation}). Then, relation (\ref{eq:inportant_eignevalue}) is proved.
Next, we show the relations \begin{eqnarray} \lim_{n\rightarrow\infty}
\left<0_z|\psi_{\gamma_\pm}\right>\left<\psi_{\gamma_\pm}|0_x\right> &=&\pm \frac{\exp\left(-\frac\omega2i\right)}2, \label{eq:asymptotic_propaty_1} \end{eqnarray} \begin{equation}
\lim_{n\rightarrow \infty}\sum_{\gamma\neq\gamma_\pm}\left| \left<0_x|\psi_{\gamma}\right>\right|^2 =
\lim_{n\rightarrow \infty}\sum_{\gamma\neq\gamma_\pm}\left| \left<0_z|\psi_{\gamma}\right>\right|^2 =\:0, \label{eq:asymptotic_propaty_2} \end{equation} where $\sum_{\gamma\neq\gamma_\pm}$ means the summation with respect to all values $\gamma$ corresponding to eigenvalues of $\hat G'\!\cdot\!\hat O'$ except for $\gamma_\pm$. From relation (\ref{eq:definition_of_eigenvector}), \begin{eqnarray}
&&\lim_{n\rightarrow\infty}\left|\left<0_z|\psi_{\gamma_\pm}\right>\right|^{-2} \nonumber\\ &=& \lim_{n\rightarrow\infty} \sum_{s=0}^{n}
\frac{P_n\left(s\right)\left|1-\exp\left(\omega i\right)\right|^2}
{
\left|1-\exp\left(\gamma_\pm+s\varphi\left(\omega\right) \right)i\right|^2 } \nonumber\\ &=& 1+\lim_{n\rightarrow\infty} \sum_{s=1}^{n}
\frac{P_n\left(s\right)\left|1-\exp\left(\omega i\right)\right|^2}
{
\left|1-\exp\left(\gamma_\pm+s\varphi\left(\omega\right) \right)i\right|^2 } \nonumber\\ &=&2. \label{eq:asymptotic_propaty_3} \end{eqnarray} In the second equality, we use (\ref{eq:inportant_eignevalue}), and in the third equality, we use (\ref{eq:inportant_eignevalue}), (\ref{eq:asymptotics_of_alpha}) and (\ref{eq:limit_relation}). On the other hand, from relation (\ref{eq:original_relation}), \begin{eqnarray} \lim_{n\rightarrow\infty}
\frac{\left<0_z|\psi_{\gamma_\pm}\right>}
{\left<0_x|\psi_{\gamma_\pm}\right>} &=& \lim_{n\rightarrow\infty}\frac{1- \exp\left(\gamma_\pm i\right)}
{\left<0_x|0_z\right>\left(1-\exp\left(\omega i\right)\right)} \nonumber\\ &=&\pm \exp\left(-\frac\omega2i\right) \label{eq:asymptotic_propaty_4} \end{eqnarray} is derived. Using relations (\ref{eq:asymptotic_propaty_3}) and (\ref{eq:asymptotic_propaty_4}), relation (\ref{eq:asymptotic_propaty_1}) is proved. Furthermore, from (\ref{eq:asymptotic_propaty_3}) and (\ref{eq:asymptotic_propaty_4})
and the trivial relation \begin{eqnarray}
\sum_\gamma \left| \left<0_x|\psi_{\gamma}\right>\right|^2
\:=\:\sum_\gamma \left| \left<0_z|\psi_{\gamma}\right>\right|^2 \:=\:1, \end{eqnarray} (\ref{eq:asymptotic_propaty_2}) is derived.
Using some relations proved above, we obtain \begin{eqnarray} &&\lim_{n\rightarrow\infty}
\left|\left<0_z\right|\left(\hat G'\!\cdot\!\hat O'\right)^{N'}\left|0_x\right>\right| \nonumber\\ &=& \lim_{n\rightarrow\infty}
\left|\sum_{\gamma}\exp\left(N' \gamma i\right)
\left<0_z|\psi_\gamma\right>
\left<\psi_\gamma|0_x\right>
\right|\nonumber\\ &=& \lim_{n\rightarrow\infty}
\left|\exp\left(N'\gamma_+i\right)
\left<0_z|\psi_{\gamma_+}\right>
\left<\psi_{\gamma_+}|0_x\right> \right.\nonumber\\&&{}\left. + \exp\left(N' \gamma_-i\right)
\left<0_z|\psi_{\gamma_-}\right>
\left<\psi_{\gamma_-}|0_x\right>
\right|\nonumber\\ &=& \frac12\lim_{n\rightarrow\infty}
\left|\exp\left(N' \gamma_+i\right) - \exp\left(N' \gamma_- i\right)
\right|\nonumber\\ &=&1. \label{eq:end_of_proof_of_go} \end{eqnarray} In the second equality, we use relation (\ref{eq:asymptotic_propaty_2}), in the third equality we use relation (\ref{eq:asymptotic_propaty_1}), and in the last one we use (\ref{eq:inportant_eignevalue}) and (\ref{eq:suggestion_2}). Relation (\ref{eq:end_of_proof_of_go}) is exactly the same as (\ref{eq:suggestion_1}). $\square$
\subsection{The case of more than one solution} When there are two solutions, we also modify Grover's algorithm in the same way. However,
we have to know humming distance $d$ of the two solutions. This information is not used in Grover's algorithm. When we change the number of solutions, all we have to do is to change the definition of
$\varphi\left(\omega\right)$ and $N'$ as follows: \begin{eqnarray} \cot\frac\omega2&=&\sum_{s_1=0}^{n-d}\sum_{s_2=\delta\left(s_1,0\right)}^{d}\left(1+\left(-\right)^{s_2}\right)P_{n-d}\left(s_1\right)P_d\left(s_2\right) \nonumber\\&&\makebox[2cm]{}\times\cot\frac{\left(s_1+s_2\right)\varphi_2\left(\omega\right)}2, \nonumber \end{eqnarray}
\begin{equation} \makebox[.5cm]{}-\frac{2\pi}{n}<\varphi_2\left(\omega\right)<\frac{2\pi}{n},\quad \operatorname{sgn}\left(\omega\right)=\operatorname{sgn}\left(\varphi_2\left(\omega\right)\right), \label{eq:definition_alpha_2_second} \end{equation} \begin{eqnarray} N'_2&:= &\left[\frac{\pi}{4\sqrt2 \sin\frac\omega2}2^{\frac n2}+\frac12\right], \label{eq:suggestion_2_second} \end{eqnarray} where the subscript ``$2$'' of $\varphi\left(\omega\right)$ and $N'$ indicates just the number of solutions. Then, the operator $\hat G'$ and the oracle $\hat O'$ become \begin{eqnarray} \hat G'_2&:=&\exp\left(\varphi_2\left(\omega\right) \sum_{\alpha=0}^{n-1}S_x^{(\alpha)} i\right),\\ \hat O'_2&:=& \exp\left(\omega
\left( \left|j_1\right>\left<j_1\right|+ \left|j_2\right>\left<j_2\right|\right)
i\right). \end{eqnarray} The success probability goes to $1$ in the limit $n\rightarrow\infty$. This is equivalent to the following relation: \begin{eqnarray} \lim_{n\rightarrow\infty}
\sum_{\eta=1,2}\left|\left<j_\eta\right|\left(\hat G'_2\!\cdot\!\hat O'_2\right)^{N'_2}\left|\bar 0\right>\right|^2 &=&1. \label{eq:suggestion_1_second} \end{eqnarray} We can prove this relation in the same way as we have done in the one solution case, so we omit it. We believe that the same relations hold when there are more than two solutions, and we numerically checked this fact in several cases.
\section{numerical calculation} \label{sec:numerical_calculation} \begin{table*}
\begin{tabular}{|c||r|r|r|r|r|r|} \hline
\# of items, i.e., $2^n$ &Grover &$\omega=\frac\pi2$ &$\omega=\frac{2\pi}3$&$\omega=\frac{3\pi}4$&$\omega=\frac{4\pi}5$&$\omega=1$ \\ \hline \hline $2^{10}$ &25 &36 &29 &27&26&25 \\
&$5.4\times 10^{-4}$ &$2.2\times 10^{-1}$&$2.5\times 10^{-1}$ &$2.7\times 10^{-1}$ &$2.9\times 10^{-1}$&$6.8\times 10^{-1}$\\ \hline $2^{20}$ &804 &1137 &929 &871 &846&804 \\
&$2.4\times 10^{-7}$ &$8.5\times 10^{-2}$&$9.7\times 10^{-2}$ &$1.1\times 10^{-1}$ &$1.1\times 10^{-1}$&$6.2\times 10^{-1}$\\ \hline $2^{30}$ &25735 &36396 &29717 &27856 & 27060&25736\\
&$6.8\times 10^{-10}$&$5.0\times 10^{-2}$&$5.8\times 10^{-2}$ &$6.3\times 10^{-2}$ &$6.8\times 10^{-2}$&$6.1\times 10^{-1}$\\ \hline $2^{40}$ &823549 &1164675 &950953 &891404 &865931&823550\\
&$9.8\times 10^{-14}$&$3.5\times 10^{-2}$&$4.1\times 10^{-2}$ &$4.5\times 10^{-2}$ &$4.9\times 10^{-2}$&$6.0\times 10^{-1}$\\ \hline \end{tabular} \caption{\label{fig:wide} The upper integer in each cell indicates the optimal iteration number, i.e., $N$ or $N'$. The lower real number in each cell indicates the error rate, i.e.,
$1-\left|\left<j\right|\left(\hat G\!\cdot\!\hat O\right)^N\left|\bar0\right>\right|^2$
or
$1-\left|\left<j\right|\left(\hat G'\!\cdot\! \hat O'\right)^N\left|\bar0\right>\right|^2$. The optimal iteration number and error rate in the case of Grover's algorithm are in the leftmost column, and those for the algorithm using the proposed quantum circuit at $\omega=\frac12\pi,\frac23\pi,\frac34\pi,\frac45\pi,\pi$ are in the other columns. Note that, $N'$ and $\varphi\left(\omega\right)$ when $\omega=\pi$ are defined in the same way as the other four examples, i.e. (\ref{eq:suggestion_2}) and
(\ref{eq:definition_alpha_2}).
However, as pointed out in sec. \ref{sec:G_algorithm}, relation (\ref{eq:suggestion_1}) does not hold in that case.
All the values were calculated for the case of only one solution. } \end{table*} In order to check that the new quantum circuit works well, we numerically calculated the iteration number, i.e., $N'$ defined by (\ref{eq:suggestion_2}), and the error rate, i.e.,
$1-\left|\left<j\right|\left(\hat G'\!\cdot\!\hat O'\right)^{N'}\left|\bar0\right>\right|^2$, at $n=10,20,30,40$ and $\omega=\frac\pi2,\frac{2\pi}3,\frac{3\pi}4,\frac{4\pi}5,\pi$. The results are shown in Table \ref{fig:wide}. In order to compare the proposed quantum circuit with the quantum circuit used in Grover's algorithm, we also show the corresponding values for Grover's algorithm in the
table. Note that, $N'$ and $\varphi\left(\omega\right)$ when $\omega=\pi$ are defined in the same way as the other four example, i.e. (\ref{eq:suggestion_2}) and
(\ref{eq:definition_alpha_2}).
From the result when $\omega=\pi$, we predict that the relation \begin{eqnarray} \lim_{n_o\rightarrow\infty}\inf_{n_0<n}
\left|\left<j\right|\left(\hat G'\!\cdot\!\hat O'\right)^{N'}\left|\bar 0\right>\right|^2 &=&Const \nonumber\\ 0<Const<1\makebox[-1cm]{} \end{eqnarray} holds when case $\omega=\pi$. This relation may be proved in a way similar to that in the other $\omega$ case. This relation means that we can probably use the quantum circuit, i.e.
$\left(\hat G'\!\cdot\!\hat O'\right)^{N'}$, for the quantum search problem even when $\omega=\pi$, though the error rate for the circuit will be much bigger than that in other $\omega$ cases.
What we want to mention about the results for the cases $\omega=\frac\pi2,\frac{2\pi}3,\frac{3\pi}4,\frac{4\pi}5$ is that the error rate is sufficiently small for realistic $n$ cases.
On the other hand, it is fair to point out that with the algorithm using the new quantum circuit, the number of iterations
and the error rate are much higher than in Grover's algorithm. However, the results do not provide enough information for us to discuss the efficiency of the two algorithms. We remark that operator $\hat G$ is a really multi-particle operator, whereas operator $\hat G'$ is just a set of single-particle rotation, i.e., a direct product of single-qubit operators. The ``really multi-particle operators'' are those that
can not be expressed only by products of single-qubit operators.
Therefore, operator $\hat G'$ can be executed much faster than $\hat G$ in many realistic systems. Then, the average time to find solution $j$ by the algorithm using the new quantum circuit is shorter than that by Grover's algorithm in some cases on a realistic QC.
\section{relation between the proposed quantum circuit and the AQC} \label{sec:relation_AQC} The quantum circuit proposed in this paper is inspired by Farhi's Hamiltonian \cite{FGG+00} for the AQC. In this section, we briefly review the AQC, point out the simple relation between the
quantum circuit used in Grover's algorithm and Roland's Hamiltonian \cite{RC01} for quantum search on the AQC, and finally point out the similar relation between the proposed quantum circuit and Farhi's Hamiltonian for quantum search on the AQC. Recall that to generate a new quantum circuit, we assumed the existence of operator $\hat G'$ related to
Farhi's Hamiltonian as an analogy of the relation between the Grover operator and Roland's Hamiltonian. This relation is shown below. Then, we find the explicit expression of operator $\hat G'$, i.e., (\ref{eq:def_oracle_Go_G'}).
The AQC involve the following procedures. First, we define the parametrised hermitian matrix $\hat H\left(r\right)$ that has the following five properties. \begin{itemize}
\item The operator $\hat H\left(r\right)$ is continuously changed with respect to parameter $r\in\mathbb{R}$.
\item The ground state of $\hat H\left(0\right)$ is a simple general state.
\item The ground state of $\hat H\left(1\right)$ is an encoded solution of the problem.
\item At any $0\leq r\leq1$, the ground state of $\hat H\left(r\right)$ does not degenerate. \item The Hamiltonian can be easily defined using only the definition of the problem, i.e., the Hamiltonian can be defined without knowing the result. \end{itemize} Second, we prepare the initial state that is the ground state of $\hat H\left(0\right)$. Third, we make the time evolution of the state such that \begin{eqnarray}
i\frac\partial{\partial t}\left|\phi_T\left(t\right)\right> &=&
\hat H\left(\mu\left(\frac tT\right)\right)\left|\phi_T\left(t\right)\right> \nonumber \end{eqnarray}
\begin{equation} \makebox[2.1cm]{} \mu\left(0\right)=0\quad \mu\left(1\right)=1\quad \frac d{dr}\mu\left(r\right)>0. \end{equation} Note that $\mu\left(r\right)$ can be chosen arbitrarily until the above conditions are satisfied, but the choice affects the probability of success and the time for the calculation. Finally, we observe the state at time $t=T$. If $T$ is sufficiently large, the correct solution is obtained, i.e., \begin{eqnarray} \lim_{T\rightarrow\infty}
\left| \left<
\phi_g\left(r\right)|\phi_T\left(r T\right)
\right>\right| &=&1 \end{eqnarray}
where $\left|\phi_g\left(r\right)\right>$ is a ground state of the operator $\hat H\left(r\right)$. A suitable value of $T$ can be found from the adiabatic theorem. This is a rough sketch of the AQC.
Next, we show the relation between the quantum circuit used in Grover's algorithm and Roland's Hamiltonian for quantum search \cite{RC01} on the AQC. The Hamiltonian \begin{eqnarray} \hat H_R\left(r\right) &:=&
-\left(1-r\right)\left|\bar0\left>\right<\bar0\right|-r\left|j\right>\left<j\right| \label{eq:AQC_QC_1} \label{eq:hamiltonian_roland}\\ \mu_R\left(r\right) &:=&
\frac{\sin\left(\pi-2\theta\right)r}{\sin\left(\pi-2\theta\right)r+\sin\left(\left(\pi-2\theta\right)r+2\theta\right)} \label{eq:AQC_QC_2} \end{eqnarray}
executes quantum search, where $\left|\bar0\right>$ and $\theta$ mean the same state and value as those in the previous section, i.e., (\ref{def:initial_state}) and (\ref{def:theta}), and $j$ is the target of the search. The above function $\mu_R\left(r\right)$ is optimised so as to maximize the success probability. From this expression, it is readily known that \begin{eqnarray} \hat G&=&\exp\left(i\pi2\left(1-\mu_R^*\right)\hat H\left(0\right)\right)\nonumber\\ \hat O&=&\exp\left(i\pi2 \mu_R^* \hat H\left(1\right)\right), \label{rel:Grover_adiabatic_1} \end{eqnarray} where the operators $\hat G$ and $\hat O$ are defined by (\ref{eq:def_oracle_Gr}) and $\mu_R^*$ satisfies the condition that
the gap between the two lowest eigenvalues of $\hat H_R\left(r\right)$ becomes the minimum value at the point $r=\mu_R^*$. Furthermore, by some calculations, we can check that \begin{eqnarray}
\lim_{T\rightarrow\infty}\left|\left<\phi_T\left(\frac{4\theta T}{\pi-2\theta}m\right)\right|\left(\hat G\!\cdot\!\hat O\right)^{m}\left|\bar0\right>\right| &=&1 \label{rel:Grover_adiabatic_2} \end{eqnarray} where $0\leq m\leq\left[\frac\pi{4\theta}+\frac14\right]$ is an integer. This relation means that the optimal speed of an AQC using Roland's Hamiltonian is exactly the same as the speed of Grover's algorithm with respect to quantum search.
Next, we show the relation between the quantum circuit proposed in this paper and Farhi's Hamiltonian \cite{FGG+00} for quantum search on an AQC. The Hamiltonian \begin{eqnarray}
\hat H_F\left(r\right) &:=&
-\left(1-r\right)\sum_{\alpha=0}^{n-1}S^{(\alpha)}_x-r\left|j\right>\left<j\right| \label{eq:hamiltonian_farhi} \end{eqnarray} also executes quantum search. As is easily shown, the following relation holds \begin{eqnarray} \hat G'&=&\exp\left(i\pi\xi\left(1-\mu_F^*\right)\hat H_F\left(0\right)\right) \nonumber\\ \hat O'&=&\exp\left(i\pi\xi \mu_F^* \hat H_F\left(1\right)\right), \label{rel:Go_adiabatic_1} \end{eqnarray} where $\xi:=\omega/\mu_F^*$ is a real number. Relations (\ref{rel:Grover_adiabatic_1}) and (\ref{rel:Go_adiabatic_1}) are very similar. However, we can only check that the leading term of $\mu_F^*$ as a function of $n$ is the same as that of $\mu_F^{*\prime}$,
where at the point $r=\mu_F^{*'}$ the gap between the two lowest eigenvalues of $\hat H_F\left(r\right)$ becomes the minimum value.
Unfortunately, we have not yet found a relation like (\ref{rel:Grover_adiabatic_2}) in this case.
What we want to say in this section is that there are some relations between the quantum circuits for the QC and the Hamiltonians for the AQC, and these relations can be used to generate new quantum circuits. Some people may think that these relations are trivial or just accidental things. However, it is a truth that the proposed quantum circuit is found on the basis of the conviction that there must be an operator $\hat G'$ related to (\ref{eq:hamiltonian_farhi}) as an analogy of the relation between $\hat G$ and (\ref{eq:hamiltonian_roland}), i.e., (\ref{rel:Grover_adiabatic_1}) and (\ref{rel:Grover_adiabatic_2}). Accordingly, we believe that there are more hidden relations between
quantum circuits and Hamiltonians and that they would be powerful instruments for generating new quantum circuits and new Hamiltonians.
\section{conclusion} We have proposed a new quantum circuit for the quantum search problem. This quantum circuit is superior to the quantum circuit used in Grover's algorithm in some cases on a realistic quantum computer. The reasons for this superiority in short are as follows: In the quantum circuit proposed in this paper, all the operators except for the oracle are direct products of single-qubit gates. In the quantum circuit used in Grover's algorithm, there are the operators other than
the oracle, which are really multi-particle operators. On the other hand, it is a fact that the product of single-qubit gates can be executed much faster than multi-particle operators in many realistic systems. In addition, the scaling of the number of oracle calls for this circuit is the same as that for Grover's algorithm, i.e. $O\left(2^{n/2}\right)$.
The proposed circuit is found by a comparison of circuits for the quantum computer and Hamiltonian for the adiabatic quantum computer. This fact indicates that the comparison is probably one of the powerful instruments for finding efficient new quantum circuits.
One aspect of future work is to find a stricter relation between the quantum circuits for the quantum computer and the Hamiltonians for the adiabatic quantum computer that gives sufficient data for modification from the Hamiltonians into the
quantum circuits. Then, we will be able to automatically generate other efficient quantum circuits from Hamiltonians for the adiabatic quantum computer with respect to other problems that the adiabatic quantum computer is good at and discover new concepts for quantum circuits.
\section{Proof of Lemma (\ref{eq:limit_relation})} \label{sec:proof_of_lemma} Here, we prove lemma (\ref{eq:limit_relation}).
{\it Proof}:
The sufficient condition of (\ref{eq:limit_relation}) is the relation \begin{equation}
\lim_{n\rightarrow\infty} \sum_{s=1}^{n}P_n\left(s\right)\left(\frac sn\right)^q =2^{-q}\nonumber \label{eq:sufficient_condition} \end{equation} \begin{equation} q\in\mathbb{Z}, \end{equation} We can check this as follows: \begin{eqnarray} && \lim_{n\rightarrow\infty}\sum_{s=1}^{n}P_n\left(s\right)f\left(\frac sn\right)\nonumber\\ &=&
\lim_{n\rightarrow\infty}\sum_{s=1}^{n}P_n\left(s\right) \left( \sum_{q=-\alpha}^{-1}\left(\frac sn\right)^qf^{(q)} \right.\nonumber\\&&\left.{} \makebox[2cm]{}+\sum_{q=0}^{\infty}\left(\frac sn-\frac12\right)^qf^{(q)} \right) \nonumber\\ &=& \sum_{q=-\lambda}^{-1}f^{(q)}
\lim_{n\rightarrow\infty}\sum_{s=1}^{n}P_n\left(s\right) \left(\frac sn\right)^q \nonumber\\&&{} {}+\sum_{q=0}^{\infty}f^{(q)}
\lim_{n\rightarrow\infty}\sum_{s=1}^{n}P_n\left(s\right)\left(\frac sn-\frac12\right)^q \nonumber\\ &=& \sum_{q=-\lambda}^{-1}f^{(q)} 2^{-q}+f^{(0)} \nonumber\\ &=&f\left(\frac12\right), \end{eqnarray} where $f^{(q)}$ is defined as \begin{eqnarray}
f\left(\zeta\right)= \sum_{q=-\lambda}^{-1}\zeta^qf^{(q)} +\sum_{q=0}^{\infty}\left(\zeta-\frac12\right)^qf^{(q)}. \end{eqnarray} The (\ref{eq:sufficient_condition}) is used in the third equality. The other equalities are easily given from the above definition of $f^{(q)}$
In the rest of this appendix, we prove relation (\ref{eq:sufficient_condition}). We define some functions, \begin{eqnarray}
F\left(n,q\right)&:=&\sum_{s=1}^{n}P_n\left(s\right)\left(\frac sn\right)^q\\ n^q \tilde F\left(n,q\right)&:=&\sum_{s=\max\left(q,1\right)}^{n}\frac{s!}{\left(s-q\right)!}P_n\left(s\right), \\ &=& \left\{\begin{array}{l } \frac{n!2^{-q}}{\left(n-q\right)!} \makebox[1cm]{} \makebox{ in case of }q>0\\ \frac{n!2^{-q}}{\left(n-q\right)!}- \sum_{s=q}^{0}\frac{n!2^{-n}}{\left(s-q\right)!(n-s)!} \\ \makebox[2cm]{} \makebox{ in case of }q\leq 0.
\end{array}\right. \end{eqnarray} From these definitions, we can derive the relation \begin{eqnarray} &&\tilde F\left(n,q\right) \nonumber\\ &\leq& F\left(n,q\right) \nonumber\\ &\leq& \left(\frac{n}{n-4q}\right)^q\tilde F\left(n,q\right)+\sum_{s=1}^{\left[\frac n4\right]+1}P_n\left(s\right)\left(\frac sn\right)^q\!\!\!. \end{eqnarray} for $n>4q$. Using the following relation \begin{eqnarray} \lim_{n\rightarrow \infty} n!\frac{e^n}{n^{n+\frac12}\sqrt{2\pi}}&=&1, \end{eqnarray} we can see that both the upper bound and the lower bound of $F\left(n,q\right)$ goes to $2^{-q}$ in the limit $n\rightarrow\infty$. $\square$
\end{document} | arXiv |
\begin{document}
\title{Quantitative strong unique continuation for the Lam\'e system with less regular coefficients}
\begin{abstract} In this paper we prove a quantitative form of the strong unique continuation property for the Lam\'e system when the Lam\'e coefficients $\mu$ is Lipschitz and $\lambda$ is essentially bounded in dimension $n\ge 2$. This result is an improvement of our earlier result \cite{lin5} in which both $\mu$ and $\lambda$ were assumed to be Lipschitz. \end{abstract}
\section{Introduction}\label{sec1} \setcounter{equation}{0}
Assume that $\Omega$ is a connected open set containing $0$ in ${\mathbb R}^n$ for $n\geq 2$. Let $\mu(x)\in C^{0,1}(\Omega)$ and $\lambda(x),\rho(x)\in L^{\infty}(\Omega)$ satisfy \begin{equation}\label{1.1} \begin{cases} \mu(x)\geq\delta_0,\quad\quad \lambda(x)+2\mu(x)\geq\delta_0\quad\forall\ \text{a.e.}\ x\in\Omega,\\
\|\mu\|_{C^{0,1}(\Omega)}+\|\lambda\|_{L^{\infty}(\Omega)}\leq M_0,\quad \|\rho\|_{L^{\infty}(\Omega)}\le M_0 \end{cases} \end{equation} with positive constants $\delta_0, M_0$, where we define
$$\|f\|_{C^{0,1}(\Omega)}=\|f\|_{L^{\infty}(\Omega)}+\|\nabla f\|_{L^{\infty}(\Omega)}.$$ The isotropic elasticity system, which represents the displacement equation of equilibrium, is given by \begin{equation}\label{1.2} \text{div}(\mu(\nabla u+(\nabla u)^t))+\nabla(\lambda\text{div}u)+\rho u=0\quad\text{in}\ \Omega, \end{equation} where $u=(u_1,u_2,\cdots,u_n)^t$ is the displacement vector and $(\nabla u)_{jk}=\partial_ku_j$ for $j,k=1,2,\cdots,n$.
We are interested in the strong unique continuation property (SUCP) of \eqref{1.2}. More precisely, we would like to show that any nontrivial solution of \eqref{1.2} can only vanish of finite order at any point of $\Omega$. We also give an estimate of the vanishing order for $u$, which can be seen as a quantitative description of the SUCP for \eqref{1.2}. Here we list some of the known results on the SUCP for \eqref{1.2}:
\begin{itemize} \item $\lambda,\mu\in C^{1,1}$, $n\ge 2$ (quantitative): Alessandrini and Morassi \cite{almo}.
\item $\lambda,\mu\in C^{0,1}$, $n=2$ (qualitative): Lin and Wang \cite{lw05}.
\item $\lambda\in L^{\infty},\mu\in C^{0,1}$, $n=2$ (qualitative): Escauriaza \cite{es}.
\item $\lambda,\mu\in C^{0,1}$, $n\ge 2$ (quantitative): Lin, Nakamura, and Wang \cite{lin5}. \end{itemize}
In this paper, we relax the regularity assumption on $\lambda$ in \cite{lin5} to $\lambda\in L^{\infty}(\Omega)$. In view of counterexamples by Plis \cite{pl} or Miller \cite{mi}, this regularity assumption seems to be optimal. This improvement was inspired by our recent work on the Stokes system \cite{lin6}. We now state the main results of the paper. Assume that there exists $0<R_0\le 1$ such that $B_{R_0}\subset\Omega$. Hereafter $B_r$ denotes an open ball of radius $r>0$ centered at the origin. \begin{theorem}\rm{(Optimal three-ball inequalities)}\label{thm1.1} There exists a positive number $\tilde{R}<1$, depending only on $n,M_0,\delta_0$, such that if $\ 0<R_1<R_2<R_3\leq R_0$ and $R_1/R_3<R_2/R_3<\tilde{R}$, then \begin{equation}\label{1.7}
\int_{|x|<R_2}|u|^2dx\leq
{C}\left(\int_{|x|<R_1}|u|^2dx\right)^{\tau}\left(\int_{|x|<{R_3}}|u|^2dx\right)^{1-\tau} \end{equation} for $u\in H_{loc}^1({B}_{R_0})$ satisfying \eqref{1.2} in ${B}_{R_0}$, where the constant ${C}$ depends on $R_2/R_3$, $n$, $M_0,\delta_0$, and $0<\tau<1$ depends on $R_1/R_3$, $R_2/R_3$, $n,M_0,\delta_0$. Moreover, for fixed $R_2$ and $R_3$, the exponent $\tau$ behaves like $1/(-\log R_1)$ when $R_1$ is sufficiently small. \end{theorem}
\begin{theorem}\label{thm1.2} Let $u\in H^1_{loc}(\Omega)$ be a nontrivial solution of \eqref{1.2}, then there exist positive constants $K$ and $m$, depending on $n,M_0,\delta_0$ and $u$, such that \begin{equation}\label{1.8}
\int_{|x|<R}|u|^2 dx\ge KR^m \end{equation} for all $R$ sufficiently small. \end{theorem} \begin{remark}\label{rem1.2} Based on Theorem~\ref{thm1.1}, the constants $K$ and $m$ in \eqref{1.4} are explicitly given by $$
K=\int_{|x|<R_3}|u|^2dx $$ and $$
m=\tilde C+\log\Big{(}\frac{\int_{|x|<R_3}|u|^2dx}{\int_{|x|<R_2}|u|^2dx}\Big{)}, $$ where $\tilde C$ is a positive constant depending on $n,M_0,\delta_0$ and $R_2/R_3$. \end{remark}
\section{Reduced system and estimates}\label{sec2} \setcounter{equation}{0}
Here we want to find a reduced system from \eqref{1.2}. This is a crucial step in our approach. Let us write \eqref{1.2} into a non-divergence form: \begin{equation}\label{1.3} \mu\Delta u+\nabla((\lambda+\mu)\ {\rm div} u)+(\nabla u+(\nabla u)^t)\nabla\mu-{\rm div} u\nabla\mu+\rho u =0. \end{equation} Dividing \eqref{1.3} by $\mu$ yields \begin{eqnarray}\label{1.4} &&\Delta u+\frac{1}{\mu}\nabla((\lambda+\mu)\ {\rm div} u)+(\nabla u+(\nabla u)^t)\frac{\nabla\mu}{\mu}-{\rm div} u\frac{\nabla\mu}{\mu}+\frac{\rho}{\mu} u\notag\\ &=&\Delta u+\nabla(\frac{\lambda+\mu}{\mu}\ {\rm div} u)+(\nabla u+(\nabla u)^t)\frac{\nabla\mu}{\mu}-{\rm div} u(\frac{\nabla\mu}{\mu}+(\lambda+\mu)\nabla(\frac{1}{\mu}))\notag\\ &&+\frac{\rho}{\mu} u\notag\\ &=&\Delta u+\nabla(a(x)v)+G\notag\\ &=&0, \end{eqnarray} where $$a(x)=\frac{\lambda+\mu}{\lambda+2\mu}\in L^{\infty}(\Omega),\quad v=\frac{\lambda+2\mu}{\mu}\ {\rm div} u$$ and $$G=(\nabla u+(\nabla u)^t)\frac{\nabla\mu}{\mu}-{\rm div} u(\frac{\nabla\mu}{\mu}+(\lambda+\mu)\nabla(\frac{1}{\mu}))+\frac{\rho}{\mu} u.$$ Taking the divergence on \eqref{1.4} gives \begin{equation}\label{1.5} \Delta v+{\rm div} G=0. \end{equation} Our reduced system now consists of \eqref{1.4} and \eqref{1.5}. It follows easily from \eqref{1.5} that if $u\in H^1_{loc}(\Omega)$, then $v\in H^1_{loc}(\Omega)$.
To prove the main results, we rely on suitable Carleman estimates. Denote $\varphi_{\beta}=\varphi_{\beta}(x) =\exp (-\beta\tilde{\psi}(x))$, where $\beta>0$ and $\tilde{\psi}(x)=\log
|x|+\log((\log |x|)^2)$. Note that $\varphi_{\beta}$ is less singular than $|x|^{-\beta}$. We use the notation $X\lesssim Y$ or $X\gtrsim Y$ to mean that $X\le CY$ or $X\ge CY$ with some constant $C$ depending only on $n$.
\begin{lemma}{\rm\cite[Lemma 2.4]{lin5}}\label{lem2.1} There exist a sufficiently small number $r_1>0$ depending on $n$ and a sufficiently large number $\beta_1>3$ depending on $n$ such that for all $w\in U_{r_1}$ and $f=(f_1,\cdots,f_n)\in (U_{r_1})^{n}$, $\beta\geq \beta_1$, we have that \begin{eqnarray}\label{2.1}
&&\int\varphi^2_\beta (\log|x|)^2(\beta|x|^{4-n}|\nabla w|^2+\beta^3|x|^{2-n}|w|^2)dx\notag\\ &\lesssim& \int \varphi^2_\beta
(\log|x|)^{4}|x|^{2-n}[(|x|^{2}\Delta w+|x|{\rm div}
f)^2+\beta^2\|f\|^2]dx, \end{eqnarray} where $U_{r_1}=\{w\in C_0^{\infty}({\mathbb R}^n\setminus\{0\}): \mbox{\rm supp}(w)\subset B_{r_0}\}$. \end{lemma} Next, replacing $\beta$ by $\beta+1$ in \eqref{2.1}, we get another Carleman estimate. \begin{lemma}\label{lem2.2} There exist a sufficiently small number $r_1>0$ depending on $n$ and a sufficiently large number $\beta_1>2$ depending on $n$ such that for all $w\in U_{r_1}$ and $f=(f_1,\cdots,f_n)\in (U_{r_1})^{n}$, $\beta\geq \beta_1$, we have that \begin{eqnarray}\label{2.2}
&&\int\varphi^2_\beta (\log|x|)^{-2}(\beta|x|^{2-n}|\nabla w|^2+\beta^3|x|^{-n}|w|^2)dx\notag\\ &\lesssim& \int \varphi^2_\beta
|x|^{-n}[(|x|^{2}\Delta w+|x|{\rm div}
f)^2+\beta^2\|f\|^2]dx. \end{eqnarray} \end{lemma}
In addition to Carleman estimates, we also need the following Caccioppoli's type inequality. \begin{lemma}\label{lem3.1} Let $u\in (H^1_{loc}(\Omega))^{n}$ be a solution of \eqref{1.1}. Then for any $0<a_3<a_1<a_2<a_4$ such that
$B_{a_4r}\subset\Omega$ and $|a_4r|<1$, we have \begin{equation}\label{3.1}
\int_{a_1r<|x|<a_2r}|x|^{4}|\nabla v|^2+|x|^{2}|v|^2+|x|^{2}|\nabla u|^2dx\le C_0\int_{a_3r<|x|<a_4r}|u|^2dx \end{equation} where the constant $C_0$ is independent of $r$ and $u$. Here $v$ is defined in \eqref{1.4}. \end{lemma}
The proof of Lemma~\ref{lem3.1} will be given in the next section. Here we would like to outline how to proceed the proofs of main theorems. The detailed arguments can be found in \cite{lin5} or \cite{lin6}. Firstly, applying \eqref{2.2} to $w=u$, $f=|x|a(x)v$ and using \eqref{1.4}, we have that \begin{eqnarray}\label{2.3}
&&\int\varphi^2_\beta (\log|x|)^{-2}(\beta|x|^{2-n}|\nabla u|^2+\beta^3|x|^{-n}|u|^2)dx\notag\\
&\lesssim& \int \varphi^2_\beta|x|^{-n}[\big{(}|x|^{2}\Delta u+|x|{\rm div}(|x|a(x)v)\big{)}^2+\beta^2\||x|a(x)v\|^2]dx. \end{eqnarray}
Next, applying \eqref{2.1} to $w=v$, $f=|x|G$ and using \eqref{1.5}, we get that \begin{eqnarray}\label{2.4}
&&\int\varphi^2_\beta (\log|x|)^2(\beta|x|^{4-n}|\nabla v|^2+\beta^3|x|^{2-n}|v|^2)dx\notag\\ &\lesssim& \int \varphi^2_\beta
(\log|x|)^{4}|x|^{2-n}[\big{(}|x|^{2}\Delta v+|x|{\rm div}(|x|G)\big{)}^2+\beta^2\||x|G\|^2]dx.\notag\\ \end{eqnarray} Finally, adding $\beta\times$\eqref{2.3} and \eqref{2.4} together and using \eqref{3.1}, we can prove Theorem~\ref{thm1.1} and \ref{thm1.2} as in \cite{lin5} and \cite{lin6}.
\section{Proof of Lemma~\ref{lem3.1}}\label{sec3} \setcounter{equation}{0}
Define $b_1=(a_1+a_3)/2$ and $b_2=(a_2+a_4)/2$. Let $X=B_{a_4r}\backslash \bar{B}_{a_3r}$, $Y=B_{b_2r}\backslash \bar{B}_{b_1r}$ and $Z=B_{a_2r}\backslash \bar{B}_{a_1r}$. Let $\xi(x)\in C^{\infty}_0 ({\mathbb R}^n)$ satisfy $0\le\xi(x)\leq 1$ and \begin{eqnarray}\label{3.2} \xi (x)= \begin{cases} \begin{array}{l}
0,\quad |x|\leq a_3r,\\
1,\quad b_1r<|x|<b_2r,\\
0,\quad |x|\geq a_4r. \end{array} \end{cases} \end{eqnarray} From \eqref{1.2}, we have that \begin{eqnarray}\label{3.3} 0&=&-\int[\text{div}(\mu(\nabla u+(\nabla u)^t))+\nabla(\lambda\text{div}u)+\rho u]\cdot (\xi^2\bar{u})dx\notag\\
&=&\int\sum_{ijkl=1}^n[\lambda\delta_{ij}\delta_{kl}+\mu(\delta_{il}\delta_{jk}+\delta_{ik}\delta_{jl})]\partial_{x_l}u_k\partial_{x_j}(\xi^2\bar{u}_i)dx-\int\rho\xi^2|u|^2dx\notag\\ &=&\int\xi^2\sum_{ijkl=1}^n[\lambda\delta_{ij}\delta_{kl}+\mu(\delta_{il}\delta_{jk}+\delta_{ik}\delta_{jl})]\partial_{x_l}u_k\partial_{x_j}\bar{u}_idx\notag\\
&&+\int\sum_{ijkl=1}^n\partial_{x_j}(\xi^2)[\lambda\delta_{ij}\delta_{kl}+\mu(\delta_{il}\delta_{jk}+\delta_{ik}\delta_{jl})]\partial_{x_l}u_k\bar{u}_idx-\int\rho\xi^2|u|^2dx\notag\\ &=&I_1+I_2, \end{eqnarray} where $$I_1=\int\xi^2[\sum_{ij=1}^n\lambda\partial_{x_j}u_j\partial_{x_i}\bar{u}_i+\sum_{ij=1}^n\mu(\partial_{x_i}u_j\partial_{x_j}\bar{u}_i+\partial_{x_j}u_i\partial_{x_j}\bar{u}_i)]dx$$ and
$$I_2=\int\sum_{ijkl=1}^n\partial_{x_j}(\xi^2)[\lambda\delta_{ij}\delta_{kl}+\mu(\delta_{il}\delta_{jk}+\delta_{ik}\delta_{jl})]\partial_{x_l}u_k\bar{u}_idx-\int\rho\xi^2|u|^2dx.$$
Observe that \begin{eqnarray}\label{3.4} &&\int\xi^2(2\mu-\frac{\delta_0}{2}) \partial_{x_i}u_j\partial_{x_j}\bar{u}_idx\notag\\ &=& -\int\partial_{x_j}[\xi^2(2\mu-\frac{\delta_0}{2})]\partial_{x_i}u_j\bar{u}_idx-\int\xi^2(2\mu-\frac{\delta_0}{2}) \partial^2_{x_ix_j}u_j\bar{u}_idx\notag\\ &=& -\int\partial_{x_j}[\xi^2(2\mu-\frac{\delta_0}{2})]\partial_{x_i}u_j\bar{u}_idx+\int\partial_{x_i}[\xi^2(2\mu-\frac{\delta_0}{2})]\partial_{x_j}u_j\bar{u}_idx\notag\\ &&+\int\xi^2(2\mu-\frac{\delta_0}{2}) \partial_{x_j}u_j\partial_{x_i}\bar{u}_idx. \end{eqnarray} It follows from \eqref{3.4} that \begin{eqnarray}\label{3.5} I_1&=&\int\xi^2[\sum_{ij=1}^n\lambda\partial_{x_j}u_j\partial_{x_i}\bar{u}_i+\sum_{ij=1}^n(2\mu-\frac{\delta_0}{2})(\partial_{x_i}u_j\partial_{x_j}\bar{u}_i)]dx\notag\\ &&+\int\sum_{ij=1}^n\xi^2(\mu-\frac{\delta_0}{2})(\partial_{x_j}u_i\partial_{x_j}\bar{u}_i-\partial_{x_i}u_j\partial_{x_j}\bar{u}_i)dx\notag\\ &&+\frac{\delta_0}{2}\int\sum_{ij=1}^n\xi^2\partial_{x_j}u_i\partial_{x_j}\bar{u}_idx\notag\\ &=&\int(2\mu+\lambda-\frac{\delta_0}{2})\xi^2\sum_{ij=1}^n(\partial_{x_j}u_j\partial_{x_i}\bar{u}_i)dx\notag\\ &&+\int\sum_{ij=1}^n\xi^2(\mu-\frac{\delta_0}{2})(\partial_{x_j}u_i\partial_{x_j}\bar{u}_i-\partial_{x_i}u_j\partial_{x_j}\bar{u}_i)dx\notag\\ &&+\frac{\delta_0}{2}\int\sum_{ij=1}^n\xi^2\partial_{x_j}u_i\partial_{x_j}\bar{u}_idx+I_3, \end{eqnarray} where $$I_3=\sum_{ij=1}^n\int\partial_{x_i}[\xi^2(2\mu-\frac{\delta_0}{2})]\partial_{x_j}u_j\bar{u}_i-\partial_{x_j}[\xi^2(2\mu-\frac{\delta_0}{2})]\partial_{x_i}u_j\bar{u}_idx.$$ Since \begin{eqnarray*} &&\int\sum_{ij=1}^n\xi^2(\mu-\frac{\delta_0}{2})(\partial_{x_j}u_i\partial_{x_j}\bar{u}_i-\partial_{x_i}u_j\partial_{x_j}\bar{u}_i)dx\\
&=&\frac{1}{2}\int\sum_{ij=1}^n\xi^2(\mu-\frac{\delta_0}{2})|\partial_{x_j}u_i-\partial_{x_i}u_j|^2dx, \end{eqnarray*} we obtain that \begin{eqnarray}\label{3.7}
I_1\geq \frac{\delta_0}{2}\int|\xi\nabla u|^2dx+I_3. \end{eqnarray} Combining \eqref{3.3} and \eqref{3.7}, we have that \begin{equation*}
\int_{Y}|\nabla u|^2dx \le\int_{X} |\xi\nabla u|^2 dx\leq C_1\int_{X}|x|^{-2}|u|^2dx, \end{equation*} which implies \begin{equation}\label{3.8}
\int_{Y}|x|^{2}|\nabla u|^2dx\leq C_2\int_{X}|u|^2dx. \end{equation} Here and below all constants $C_1,C_2,\cdots$ depend on $\delta_0$, $M_0$.
To estimate $\nabla v$, we define $\chi(x)\in C^{\infty}_0 ({\mathbb R}^n)$ satisfy $0\le\chi(x)\leq 1$ and \begin{eqnarray*} \chi (x)= \begin{cases} \begin{array}{l}
0,\quad |x|\leq b_1r,\\
1,\quad a_1r<|x|<a_2r,\\
0,\quad |x|\geq b_2r. \end{array} \end{cases} \end{eqnarray*} By \eqref{1.5}, we derive that \begin{eqnarray}\label{3.10}
&& \int|\chi(x)\nabla v|^2dx\notag\\ &=&\int\nabla v\cdot\nabla(\chi^2 \bar{v})dx-2\int\chi\nabla v\cdot \bar{v}\nabla\chi dx\notag\\
&\le&|\int({\rm div} G) \chi^2\bar{v} dx|+2\int|\chi\nabla v\cdot \bar{v}\nabla\chi |dx\notag\\
&\le&|\int({\rm div} G) \chi^2\bar{v} dx|+\frac{1}{4}\int|\chi\nabla v|^2dx+C_3\int_{Y}|x|^{-2}|v|^2dx\notag\\
&\le&C_4\int_{Y} |\nabla u|^2 dx+C_4\int_{Y} |u|^2 dx+\frac{1}{2}\int|\chi\nabla v|^2dx+C_4\int_{Y}|x|^{-2}|v|^2dx\notag\\
&\le&C_5\int_{Y} |x|^{-2}|\nabla u|^2 dx+C_4\int_{Y} |u|^2 dx+\frac{1}{2}\int|\chi\nabla v|^2dx. \end{eqnarray} Therefore, we get from \eqref{3.10} that \begin{equation*}
\int_{Z}|\nabla v|^2dx\le2C_5\int_{Y} |x|^{-2}|\nabla u|^2 dx+2C_4\int_{Y} |u|^2 dx \end{equation*} and hence \begin{equation}\label{3.12}
\int_{Z}|x|^{4}|\nabla v|^2dx\le C_6\int_{Y} |x|^{2}|\nabla u|^2 dx+C_6\int_{Y}|x|^{4} |u|^2 dx. \end{equation}
Putting together $K\times$\eqref{3.8} and \eqref{3.12}, we have that \begin{eqnarray}\label{3.13}
&&K\int_{Y}|x|^{2}|\nabla u|^2dx+\int_{Z}|x|^{4}|\nabla v|^2dx\notag\\
&\le&KC_2\int_{X}|u|^2dx+C_6\int_{Y} |x|^{2}|\nabla u|^2 dx+C_6\int_{Y}|x|^{4} |u|^2 dx. \end{eqnarray} Choosing $K=2C_6$ in \eqref{3.13} yields \begin{eqnarray*}
&&\int_{Z}|x|^{2}|v|^2dx+\int_{Z}|x|^{2}|\nabla u|^2dx+\int_{Z}|x|^{4}|\nabla v|^2dx\notag\\
&\le&C_7\int_{Y}|x|^{2}|\nabla u|^2dx+C_7\int_{Z}|x|^{4}|\nabla v|^2dx\notag\\
&\le&C_8\int_{X}|u|^2dx, \end{eqnarray*} The proof is now complete.\eproof
\end{document} | arXiv |
\begin{document}
\title{\bf Regularity on abelian varieties II: basic results on linear series and defining equations}
\author[G. Pareschi]{Giuseppe Pareschi} \address{Dipartamento di Matematica, Universit\`a di Roma, Tor Vergata, V.le della Ricerca Scientifica, I-00133 Roma, Italy} \email{{\tt [email protected]}}
\author[M. Popa]{Mihnea Popa$^1$} \footnotetext[1]{The second author was partially supported by a Clay Mathematics Institute Liftoff Fellowship during the preparation of this paper.} \address{Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA} \email{{\tt [email protected]}}
\maketitle
\markboth{G. PARESCHI and M. POPA} {REGULARITY ON ABELIAN VARIETIES II}
\section*{\bf Abstract}
We apply the theory of M-regularity developed in \cite{us} to the study of linear series given by multiples of ample line bundles on abelian varieties. We define an invariant of a line bundle, called M-regularity index, which is seen to govern the higher order properties and (partly conjecturally) the defining equations of such embeddings. We prove a general result on the behavior of the defining equations and higher syzygies in embeddings given by multiples of ample bundles whose base locus has no fixed components, extending a conjecture of Lazarsfeld proved in \cite{pareschi}. This approach also unifies essentially all the previously known results in this area, and is based on Fourier-Mukai techniques rather than representations of theta groups.
\section{\bf Introduction}
This paper is mainly concerned with applying the theory of Mukai regularity (or $M$-regularity) introduced in \cite{us} to the study of linear series given by multiples of ample line bundles on abelian varieties. We show that this regularity notion allows one to define a new invariant of a line bundle, called $M$-regularity index, which will be seen to roughly measure how much better one can do, given a fixed line bundle, compared to the standard results of the theory. Based on the main result of \cite{us} ($M$-regularity criterion) and a related result proved here (W.I.T. regularity criterion), we show that all known results on such linear series can be recovered, and indeed generalized, under the same heading of $M$-regularity.
To make this precise, we start by recalling most of the basic results on ample line bundles existing in the literature. For simplicity we state them for powers of one line bundle, although most hold for suitable products of possibly distinct ones.
\begin{introtheorem} Let $A$ be an ample line bundle on an abelian variety $X$. The following hold: \newline \noindent (1) $A^2$ is globally generated. \newline \noindent (2)(Lefschetz Theorem) $A^3$ is very ample. \newline \noindent (3)(Ohbuchi's Theorem \cite{ohbuchi1}) If $A$ has no base divisor, then $A^2$ is very ample. \newline \noindent (4)(Bauer-Szemberg Theorem \cite{bauer}) $A^{k+2}$ is $k$-jet ample, and the same holds for $A^{k+1}$ if $A$ has no base divisor (extending (1), (2) and (3)). \newline \noindent (5)(Koizumi's Theorem \cite{koizumi}) $A^3$ gives a projectively normal embedding. \newline \noindent (6)(Ohbuchi's Theorem \cite{ohbuchi2}) $A^2$ gives a projectively normal embedding if and only if $0_X$ does not belong to a finite union of translates of the base locus of $A$ (cf. \S5 for the concrete statement). \newline \noindent (7)(Mumford's Theorem \cite{mumford1}, \cite{kempf1}) For $k\geq 4$, the ideal of $X$ in the embedding given be $A^k$ is generated by quadrics. In the embedding given by $A^3$ it is generated by quadrics and cubics. \newline \noindent (8)(Lazarsfeld's Conjecture \cite{pareschi}, extending results of Kempf \cite{kempf2}) $A^{p+3}$ satisfies property $N_p$ (extending (5) and (7)). \newline \noindent (9)(Khaled's Theorem \cite{khaled}) If $A$ is globally generated, then the ideal of $X$ in the embedding given by $A^2$ is generated by quadrics and cubics. \end{introtheorem}
These results turn out to be -- some quick while others non-trivial -- consequences of the general global generation criterion in \cite{us}, called the $M$-regularity criterion. Together with a more technical extension (the W.I.T. regularity criterion), described below, this approach yields new results and extensions as well. To introduce them, we first need some terminology.
Let $X$ be an abelian variety of dimension $g$ over an algebraically closed field, with dual abelian variety $\hat{X}$, and let ${\mathcal P}$ be a suitably normalized Poincar\'e line bundle on $X\times \hat X$. The Fourier-Mukai functor \cite{mukai} is the derived functor associated to the functor $\hat{\mathcal S}(\mathcal{F})={p_{\hat X}}_*(p_X^*\mathcal{F}\otimes {\mathcal P})$ from ${\rm Mod}(X)$ to ${\rm Mod}({\hat X})$. A sheaf $\mathcal{F}$ on $X$ is said to satisfy the Weak Index Theorem (W.I.T.) with index $i(\mathcal{F})=k$ if $R^i\hat{\mathcal S}(\mathcal{F})=0$ for all $i\ne k$, in which case $R^k\hat{\mathcal S}(\mathcal{F})$ is simply denoted $\hat \mathcal{F}$. A weaker condition, introduced in \cite{us}, is the following: $\mathcal{F}$ is called \emph{$M$-regular} if ${\rm codim}( {\rm Supp}~R^i\hat{\mathcal S}(\mathcal{F}))> i$ for all $i>0$. Moreover, we will consider the \emph{Fourier jump locus} of $\mathcal{F}$ to be the locus of $\xi\in \hat{X}$ where $h^0(F\otimes P_\xi)$ is different from the generic value (where $P_\xi$ is the line bundle on $X$ classified by $\xi$).
\noindent Given an ample line bundle $A$ on $X$, we define the \emph{$M$-regularity index} of $A$ to be
$$m(A):={\rm max}\{l~|~A\otimes m_{x_1}^{k_1}\otimes \ldots \otimes m_{x_p}^{k_p}~{\rm is ~}M{\rm -regular~ for ~all~distinct~}$$ $$x_1,\ldots,x_p\in X {\rm~with~} \Sigma k_i=l\}.$$ A first result is that this invariant governs the higher order properties of embeddings obtained from $A$.
\begin{introtheorem} If $A$ is an ample line bundle on $X$ and $k\geq m(A)$, then $A^{\otimes(k+2-m(A))}$ is $k$-jet ample. \end{introtheorem}
\noindent It is not hard to see that for example $m(A)\geq 1$ if and only if $A$ has no base divisor. The theorem thus recovers and extends the results of Lefschetz, Ohbuchi and Bauer-Szemberg mentioned above. Most interestingly though, this shows that results with seemingly unrelated proofs are simply steps in a hierarchy of regularity conditions. It is interesting to note also that the $M$-regularity indices are quite intimately related to the Seshadri constants measuring local positivity (cf. \cite{lazarsfeld2} for the case of abelian varieties); we will approach this in detail somewhere else.
By reversing the natural order in the body of the paper, the results presented in what follows suggest that it is quite natural to expect that a similar phenomenon governs the behavior of defining equations of $X$, and more generally higher syzygies, in embeddings of this kind.
\begin{introconjecture} Let $p\ge m$ be non-negative integers. If $A$ is ample and $m(A)\ge m$, then $A^k$ satisfies $N_p$ for any $k\ge p+3-m$. \end{introconjecture}
\noindent This extends Lazarsfeld's conjecture, which is the statement for $m=0$, meaning no conditions on $A$. That case has already been proved in \cite{pareschi}, by methods which are included in, and provide a basis for, the strategy adopted here.
The main result of this paper is a proof, and also a strenghtening, of the conjecture above for $m=1$, i.e. for line bundles whose base locus has no fixed components. We first recall the terminology introduced in \cite{green}: property $N_p$ for a very ample line bundle means that $I_{X,L}$, the homogeneous ideal of $X$ in the corresponding embedding, is generated by quadratic forms, and also that -- up to the $p$-th step -- the higher syzygies between these forms are generated in the lowest possible degree, i.e. by linear ones. Thus, in this language, the property that $I_{X,L}$ be generated by quadrics is condition $N_1$. Moreover, the property of "being off" by $r$ from $N_p$ was formalized in \cite{pareschi} into property $N_p^r$ (cf. \S6 for details). In a word, $N_p^0$ is equivalent to $N_p$, and $N_p^1$ means that $I_{X,L}$ is generated by quadrics and cubics.
\begin{introtheorem}\emph{(${\rm char}(k)$ does not divide $(p+1)$ and $(p+2)$.)} Let $A$ be an ample line bundle on $X$, with no base divisor. Then: \newline \noindent (a) If $k\ge p+2$ then $A^{k}$ satisfies property $N_p$. \newline \noindent (b) More generally, if $(r+1)(k-1)\ge p+1$ then $A^k$ satisfies property $N_p^r$. \end{introtheorem}
\noindent The first instance of this theorem, worth emphasizing individually, is the following:
\begin{introcorollary}(${\rm char}(k)\ne 2,3$.) Let $A$ be an ample line bundle on $X$, with no base divisor. Then: \newline \noindent (a) If $k\ge 3$ then $I_{X,A^{k}}$ is generated by quadrics. \newline \noindent (b) $I_{X,A^2}$ is generated by quadrics and cubics. \end{introcorollary} \noindent (Note in particular the improvement of Khaled's result above.)
For consistency reasons, we note that Ohbuchi's projective normality result (6) does not integrate in the discussion above, and does indeed suggest what happens in the cases left out by the above conjecture. However, it can still be obtained in a similar way, and in \S5 we sketch its proof as a toy version of that of the syzygy theorem.
As previously mentioned, all the proofs of the statements above are based on the basic $M$-regularity theorem, which we recall below.
\begin{introtheorem}{\bf ($M$-regularity criterion, \cite{us} Theorem 2.4.)} Let $\mathcal{F}$ be a coherent sheaf and $L$ an invertible sheaf supported on a subvariety $Y$ of the abelian variety $X$ (possibly $X$ itself). If both $\mathcal{F}$ and $L$ are $M$-regular as sheaves on $X$, then $\mathcal{F}\otimes L$ is globally generated. \end{introtheorem}
\noindent This has to be combined with a refined study, in a relative setting, of the notion of skew-Pontrjagin product introduced in \cite{pareschi}, and also with a different (but related) regularity criterion, which we prove here following a similar strategy. The new statement needs stronger hypotheses on the sheaf $\mathcal{F}$, but provides specific information about the loci where suitable tensor products are not globally generated.
\begin{introtheorem}{\bf (W.I.T. regularity criterion.)} Let $A$ be an ample line bundle on $X$. Let also $F$ be a locally free sheaf on $X$ such that \newline \noindent (1) the Fourier-jump locus $J(F)$ is finite. \newline \noindent (2) $F^\vee$ satisfies the W.I.T. with index $i(F^\vee)=g$. \newline \noindent (3) the torsion part of $\hat {F^\vee}$ is a sum of (possibly zero) skyscraper sheaves on the points of $J(F)$. \newline \noindent Then there is an inclusion of non-generation loci: $$B(F\otimes A)\subset \bigcup_{\xi \in J(F)}B(A\otimes P_\xi^\vee).$$ (For a sheaf $\mathcal{F}$, we denote by $B(\mathcal{F})$ the locus where $\mathcal{F}$ is not globally generated.) \end{introtheorem}
\noindent It is interesting to note that the W.I.T. regularity criterion applies to some sheaves for which the $M$-regularity criterion does not apply and conversely.
The underlying principle in this article is the use of vanishing theorems and Fourier-Mukai methods for vector bundles, or even arbitrary coherent sheaves, in the study of linear series. This completely bypasses methods based on theta-functions and representations of theta-groups originating in \cite{mumford1} (and employed in the original proofs of most of the previously known results) as it was somewhat hinted that it could be possible by earlier work of Kempf. Better still, the main advantage of the present methods is that they apply to a much wider spectrum of problems on abelian varieties, as it is described in \cite{us}.
The paper is organized as follows: in Section 2 we recall the main terminology and results from \cite{us}, and we introduce further notions of generation of sheaves. Section 3 contains the definition of the $M$-regularity index and the corresponding result on higher order properties of embeddings. Sections 4 and 5 are devoted to a rather long list of technical results needed in the study of defining equations. In the former we prove the W.I.T. regularity criterion, while in the latter we introduce the notion of relative skew Pontrjagin product and study its properties under various operations. Finally, Section 6 is devoted to the main results of this paper, on defining equations and syzygies of abelian varieties embedded with powers of line bundles whose base locus has no fixed components.
\noindent \textbf{Acknowledgements.} We would like to thank R. Lazarsfeld for some very useful conversations.
\section{\bf Background and preliminary results}
In what follows $X$ will be an abelian variety over an algebraically closed ground field $k$. Restrictions on ${\rm char}(k)$ will be specified along the paper. We denote by $\hat X$ the dual of $X$, which we identify with ${\rm Pic}^0(X)$. Given $\xi\in \hat X$, $P_\xi$ will denote the line bundle on $X$ classified by $\xi$. For a positive integer $n$, $X_n$ will denote the group of $n$-torsion points of $X$. When it appears in the text, we will always be in a situation where ${\rm char}(k)$ does not divide $n$.
\subsection*{\bf Various notions of generation of sheaves.} Let $\mathcal{F}$ be an arbitrary coherent sheaf on $X$. The support of the cokernel of the evaluation map $H^0(\mathcal{F})\otimes{\mathcal O}_X\rightarrow \mathcal{F}$ will be referred to as the \emph{non-generation locus} of $\mathcal{F}$ and denoted $B(\mathcal{F})$. The usual notions of global generation and generic global generation mean that $B(\mathcal{F})$ is empty or a proper subset respectively. On abelian varieties it is useful to consider weaker notions of generation, which can be in fact defined on any irregular variety: $\mathcal{F}$ is said to be \emph{continuously globally generated} (cf. \cite{us} \S2) if the map $$\bigoplus_{\xi\in U}H^0(\mathcal{F}\otimes P_\xi)\otimes P_\xi^\vee\rightarrow \mathcal{F}$$ is surjective for \emph{ any non-empty Zariski-open set $U\subset \hat X$}. For a line bundle $A$ this just means that $\bigcap_{\xi\in U}B(A\otimes P_\xi)$ is empty. In what follows we introduce an even weaker variant, needed in the sequel.
\begin{definition} Given a sheaf $\mathcal{F}$, we define its \emph{Fourier jump locus} as the locus $J(\mathcal{F})\subset \hat X$ consisting of $\xi\in \hat X$ where $h^0(\mathcal{F}\otimes P_\xi)$ jumps, i.e. it is different from its minimal value over ${\rm Pic}^0(X)$. $\mathcal{F}$ is said to be \emph{weakly continuously generated} if the map $$\bigoplus_{\xi\in U}H^0(\mathcal{F}\otimes P_\xi)\otimes P_\xi^\vee\rightarrow \mathcal{F}$$ is surjective for \emph{any non-empty Zariski-open set $U\subset \hat X$ containing $J(\mathcal{F})$}. Continuous global generation obviously implies weak continuous generation and the two notions are equivalent if the Fourier jump locus of $\mathcal{F}$ is empty. \end{definition}
\begin{remark}\label{generation} The following facts are easy to check (cf. also \cite{us} Remark 2.11): \item{(a)} If $\mathcal{F}$ is weakly continuously generated, then there exist $\xi_1,\dots,\xi_k\in \hat X$ such that the map $\oplus_{i=1}^kH^0(\mathcal{F}\otimes P_{\xi_i})\otimes P_{\xi_i}^\vee\rightarrow \mathcal{F}$ is surjective. \item{(b)} If $F$ is continuously globally generated then there is a positive integer $N$ such that \emph{for general $\xi_1,\dots ,\xi_N\in \hat X$, } the map $\oplus_{i=1}^NH^0(\mathcal{F}\otimes P_\xi)\otimes P_\xi^\vee\rightarrow \mathcal{F}$ is surjective. \item{(c)} If $\mathcal{F}$ is weakly continuously generated and $J(\mathcal{F})$ is finite, say $J(\mathcal{F})=\{\xi_1,\dots ,\xi_n\}$, then there is a positive integer $N$ such that \emph{for general $\xi_{n+1},\dots \xi_{n+N}\in \hat X$ } the map $\oplus_{i=1}^{n+N}H^0(\mathcal{F}\otimes P_\xi)\otimes P_\xi^\vee\rightarrow \mathcal{F}$ is surjective. \end{remark}
\noindent The following lemma, proved in \cite{us} Proposition 2.12, shows how to produce global generation from continuous global generation.
\begin{lemma} If $\mathcal{F}$ is a continuously globally generated sheaf on $X$ and $L$ is a continuously globally generated sheaf on $X$ which is everywhere of rank $1$ on its support, then $\mathcal{F}\otimes L$ is globally generated. \end{lemma} \noindent The proposition below is a variation of this result, relating the notions of weak continuous generation and generic global generation. At least if the Fourier jump locus of $\mathcal{F}$ is {\it finite}, one can describe the non-generation locus of $\mathcal{F}\otimes L$ in terms of $J(\mathcal{F})$.
\begin{proposition}\label{ggg} Let $\mathcal{F}$ be a weakly continuously generated sheaf on $X$. \newline \noindent (a) If $\mathcal{E}$ is a sheaf such that $\mathcal{E}\otimes P_\xi$ is generically globally generated for any $\xi\in \hat X$, then $\mathcal{F}\otimes \mathcal{E}$ is generically globally generated. \newline \noindent (b) If $\mathcal{E}\otimes P_\xi$ is globally generated for any $\xi\in \hat X$, then $\mathcal{F}\otimes \mathcal{E}$ is globally generated. \newline \noindent (c) Assume that the Fourier jump locus $J(\mathcal{F})$ is finite, and let $L$ be a continuously globally generated line bundle on $X$. Then $B(\mathcal{F}\otimes L)\subset \bigcup_{\xi\in J(\mathcal{F})}B(L\otimes P_\xi^\vee).$ \end{proposition} \begin{proof} (a) By Remark \ref{generation}(a), the map $\oplus_{i=1}^k H^0(\mathcal{F}\otimes P_{\xi_i})\otimes \mathcal{E}\otimes P_{\xi_i}^\vee\rightarrow \mathcal{F}\otimes \mathcal{E}$ is surjective. Therefore we have the inclusion of non-generation loci $B(\mathcal{F}\otimes \mathcal{E})\subset \bigcup_{i=1}^kB(\mathcal{E}\otimes P^\vee_{\xi_i})$. This proves the assertion since, by hypothesis, $B(\mathcal{E}\otimes P^\vee_{\xi_i})$ are proper subvarieties. This same argument also proves (b). \newline \noindent (c) If $J(\mathcal{F})=\{\xi_1,\dots ,\xi_N\}$, then the map $\oplus_{i=1}^{n+N} H^0(\mathcal{F}\otimes P_{\xi_i})\otimes L\otimes P_{\xi_i}^\vee\rightarrow \mathcal{F}\otimes L$ is surjective for general $\xi_{n+1},\dots \xi_N\in \hat X$ (cf. Remark \ref{generation}(c)). Therefore $B(\mathcal{F}\otimes L)$ is contained in the union of $\cup_{i=1}^n B(L\otimes P_{\xi_i})$ with the intersection -- for all $\xi_{n+1},\dots ,\xi_{n+N}$ general in $\hat X$ -- of $\cup_{i=n+1}^{n+N}B(L\otimes P_{\xi_i}^\vee)$. Since $L$ is a continuously globally generated \emph{line bundle}, this intersection is empty. This implies that $B(\mathcal{F}\otimes L)\subset \cup_{i=1}^nB(L\otimes P_{\xi_i}^\vee)$. \end{proof}
\subsection*{\bf Fourier-Mukai functor, index theorems and $M$-regularity.}
According to Mukai \cite{mukai}, one considers the left-exact functor $\hat{\mathcal S}$ from the category of ${\mathcal O}_X$-modules to the category of ${\mathcal O}_{\hat X}$-modules defined as $\hat{\mathcal S}(\mathcal{F})={p_{\hat X}}_*(p_X^*\mathcal{F}\otimes {\mathcal P})$. Mukai's main result \cite{mukai} Theorem 2.2 is that the derived functor ${\bf R}\hat {\mathcal S}$ establishes an equivalence of categories between ${\bf D}(X)$ and ${\bf D}(\hat X)$. A sheaf $\mathcal{F}$ on $X$ is said to satisfy the \emph{Index Theorem (I.T.) with index $i(\mathcal{F})=k$} if $H^j(F\otimes P_\xi)=0$ for any $\xi\in \hat X$ and any $j\ne k$. More generally, $\mathcal{F}$ is said to satisfy the \emph{Weak Index Theorem (W.I.T.) with index $i(F)=k$} if $R^j\hat{\mathcal S}(\mathcal{F})=0$ for $j\ne k$. In this case $R^{i(\mathcal{F})}\hat{\mathcal S}(\mathcal{F})$ is simply denoted $\hat \mathcal{F}$. A useful consequence of Mukai's theory is the following lemma (\cite{mukai}, Cor.2.5):
\begin{lemma}\label{exts} If $\mathcal{F}$ and $\mathcal{G}$ both satisfy W.I.T., then there is a natural isomorphism $$\phi:{\rm Ext}^j(\mathcal{F},\mathcal{G})\cong {\rm Ext}^{j+i(\mathcal{F})-i(\mathcal{G})}(\hat \mathcal{F},\hat \mathcal{G}).$$ \end{lemma}
\noindent We recall from \cite{us} the following weakening of the W.I.T. condition with index $0$, and the main regularity result proved there.
\begin{definition} A sheaf $\mathcal{F}$ on $X$ is said to be \emph{Mukai-regular} (or simply \emph{$M$-regular}) if $R^i\hat{\mathcal S}(\mathcal{F})$ is supported in {\rm codim}ension $> i$ for any $i>0$, where for $i=g$ this means that the support $R^g\hat{\mathcal S}(\mathcal{F})$ is empty. This happens in particular if the cohomological support loci
$$V^i(\mathcal{F}):=\{\xi~|~h^i(\mathcal{F}\otimes P_{\xi})\neq 0\}\subset {\rm Pic}^0(X)$$ have codimension $>i$ for all $i$. \end{definition}
\begin{theorem}{\bf ($M$-regularity criterion, \cite{us} Theorem 2.4 and Proposition 2.13.)}\label{F-reg} Let $\mathcal{F}$ be an $M$-regular sheaf on $X$, possibly supported on a subvariety $Y$ of $X$. Then the following hold: \newline \noindent (a) $F$ is continuously globally generated. \newline \noindent (b) Let also $A$ be a line bundle on $Y$, continuously globally generated as a sheaf on $X$. Then $F\otimes A$ is globally generated. \end{theorem}
\section{\bf The $M$-regularity index and properties of embeddings}
In this section we show that the concept of $M$-regularity is well-adapted to the study of linear series, and provides a uniform point of view on the study of (higher order) properties of embeddings. This will serve as an introduction to the deeper facts on defining equations treated in the subsequent sections via stronger regularity techniques. We first need to recall the notion of $k$-jet ampleness (cf. e.g. \cite{sommese} in general and \cite{bauer} in the context of abelian varieties).
\begin{definition} A line bundle $A$ is called $k$-\emph{jet ample}, $k\geq 0$, if the restriction map $$H^0(A)\longrightarrow H^0(A\otimes {\mathcal O}_X/ m_{x_1}^{k_1}\otimes \ldots \otimes m_{x_p}^{k_p})$$ is surjective for any distinct points $x_1,\ldots,x_p$ on $X$ such that $\Sigma k_i =k+1$. \end{definition}
\begin{remark} In particular $0$-jet ample means globally generated, $1$-jet ample means very ample. The notion of $k$-jet ampleness is stronger than a related notion of $k$-very ampleness, which takes into account $0$-dimensional subschemes of length equal to $k+1$. \end{remark}
\begin{lemma}\label{kva} For an ample line bundle $A$ on the abelian variety $X$, the following are equivalent: \newline \noindent (i) $A$ is $k$-jet ample. \newline \noindent (ii) $A\otimes m_{x_1}^{k_1}\otimes \ldots \otimes m_{x_p}^{k_p}$ satisfies I.T. with index $0$ for all $x_1,\ldots,x_{p}$ such that $\Sigma k_i = k+1$. \newline \noindent (iii) $A\otimes m_{x_1}^{k_1}\otimes \ldots \otimes m_{x_l}^{k_l}$ is globally generated for all $x_1,\ldots,x_{l}$ such that $\Sigma k_i= k$. \end{lemma} \begin{proof} This is based on the immediate fact that, since $h^1(A)=0$ as we are on an abelian variety, $k$-jet ampleness is equivalent to the vanishing $$H^1(A\otimes m_{x_1}^{k_1}\otimes \ldots \otimes m_{x_p}^{k_p})=0$$ for all $x_1,\ldots,x_{p}$ such that $\Sigma k_i = k+1$. If $A$ is $k$-jet ample, then so is any translate, thus (i) is equivalent to (ii) by the very definition. The equivalence with (iii) also follows quickly, since the required global generation is equivalent to the surjectivity of $$H^0(A\otimes m_{x_1}^{k_1}\otimes \ldots \otimes m_{x_l}^{k_l})\longrightarrow H^0(A\otimes m_{x_1}^{k_1}\otimes \ldots \otimes m_{x_l}^{k_l}\otimes {\mathcal O}_X/m_x) $$ for every $x\in X$. \end{proof}
\noindent The key definition is given below. We note that it is suggested naturally by Theorem \ref{F-reg} and Lemma \ref{kva}, and as a result the theorem which follows is almost tautological.
\begin{definition} The $M$-\emph{regularity index} of $A$ is defined as
$$m(A):={\rm max}\{l~|~A\otimes m_{x_1}^{k_1}\otimes \ldots \otimes m_{x_p}^{k_p}~{\rm is ~}M{\rm -regular~ for ~all~distinct~}$$ $$x_1,\ldots,x_p\in X {\rm~with~} \Sigma k_i=l\}.$$ \end{definition}
\noindent The following description provides more intuition for this definition.
\begin{proposition}\label{int} We say that a line bundle $A$ is $k$-jet ample in codimension $r$ if the set of points $x$ for which there exist $x_2,\ldots, x_p$ and $k_1,\ldots ,k_p$ with $\Sigma k_i=k+1$ such that $h^1(A\otimes m_x^{k_1} \otimes m_{x_2}^{k_2}\otimes \ldots \otimes m_{x_p}^{k_p})>0$ has codimension $r$ in $X$. We have that $m(A)\geq k+1$ if $A$ is $k$-jet ample in codimension $\geq 2$. \end{proposition} \begin{proof} If we assume that $m(A)<k+1$, then there exist $x_1,\ldots, x_p$ and $k_1,\ldots, k_p$ with $\Sigma k_i=k+1$ such that the set
$$\{y\in X~|~h^i(t_y^* A \otimes m_{x_1}^{k_1} \otimes m_{x_2}^{k_2}\otimes \ldots \otimes m_{x_p}^{k_p})>0\}$$ has codimension $\leq 1$. Since this is the same as the set
$$\{y\in X~|~h^i(A \otimes m_{x_1-y}^{k_1} \otimes m_{x_2 -y}^{k_2}\otimes \ldots \otimes m_{x_p -y}^{k_p})>0\},$$ the assertion follows immediately. \end{proof}
\begin{example}{\bf (Small values of $m(A)$.)}\label{base_div} If $A$ is an ample line bundle, then $m(A)\geq 1$ if and only if $A$ does not have a base divisor. Also, if $A$ gives a birational map which is an isomorphism outside a codimension $2$ subset, then $m(A)\geq 2$. Both assertions follow immediately from the proposition above. \end{example}
\begin{example}({\bf Abelian surfaces.}) (i) Let $A$ be a polarization of type $(1,2)$ on an abelian surface. It is an immediate consequence of the Decomposition Theorem (cf. \cite{lange} 4.3.1) that $A$ has a base divisor if and only if $X$ is a product of elliptic curves $E$ and $F$ and $A={\mathcal O}_X(E+2F)$. On the other hand, it is not hard to see that we always have $m(A)\leq 1$ (for example $A\otimes m_x^2$ is not $M$-regular, where $x$ is any point on $X$). Thus $m(A)=1$ exactly when the pair $(X,A)$ is not of the above form, while otherwise it is $0$. Polarizations of type $(1,3)$ are globally generated, and so $m(A)\geq 1$, but again an argument similar to Proposition \ref{int} shows that $m(A)=1$. \newline \noindent (ii) Let $A$ be a polarization of type $(1,4)$. If $X$ is general (cf. \cite{lange} Ch.10 \S5, or the original \cite{blvs}, for more precise conditions), then $A$ gives a birational morphism to ${\textbf P}^3$ which is not an embedding, and whose exceptional set is a curve (we thank the referee for pointing this out to us). However from the properties of this curve and the fact that the map separates tangent vectors outside a codimension $2$ subset, it follows easily that $m(A)\geq 2$, although one cannot directly use Proposition \ref{int}. On the other hand, the special such abelian variety is a cover of a product of elliptic curves, and in that case $m(A)=1$. \newline \noindent (iii) On a general abelian surface (more precisely, by Reider's Theorem \cite{lange} 10.4.1, one on which there are no elliptic curves $C$ such that $C\cdot A=2$), a polarization $A$ of type $(1,d)$ with $d\geq 5$ is very ample, and so $m(A)\geq 2$. \end{example}
Based on this definition we obtain the following theorem, which extends and places in a natural setting the basic results on embeddings given by multiples of ample line bundles existing in the literature.
\begin{theorem} If $A$ and $M_1,\ldots ,M_{k+1-m(A)}$ are ample line bundles on $X$, $k\geq m(A)$, then $A\otimes M_1\otimes \ldots\otimes M_{k+1-m(A)}$ is $k$-jet ample. In particular $A^{\otimes(k+2-m(A))}$ is $k$-jet ample. \end{theorem} \begin{proof} By definition, $A\otimes m_{x_1}^{k_1}\otimes \ldots \otimes m_{x_p}^{k_p}$ is $M$-regular for any $x_1,\ldots,x_p\in X$ as long as $\Sigma k_i = m(A)$. This in turn implies that $M_1\otimes A\otimes m_{x_1}^{k_1}\otimes \ldots \otimes m_{x_p}^{k_p}$ is globally generated, by Theorem \ref{F-reg}. Now, by Lemma \ref{kva}, this is the same as saying that $M_1\otimes A\otimes m_{x_1}^{k_1}\otimes \ldots \otimes m_{x_l}^{k_l}$ satisfies I.T. with index $0$ for all $x_1, \ldots, x_l$ and $\Sigma k_i=m(A)+1$. As this is a strong form of $M$-regularity, it allows us to continue the same procedure inductively to obtain the desired conclusion. \end{proof}
In particular, for small values of $m(A)$ (namely $0$ and $1$), this recovers as particular cases the theorems of Lefschetz, Ohbuchi, and more generally Bauer-Szemberg, mentioned in the introduction (cf. also Example \ref{base_div}).
We conclude by noting that a deeper reason for considering these invariants is that they seem to link in a natural way the geometry of the abelian variety in the embedding given by a line bundle with the equations, and more generally the syzygies, of that embedding. This will be explained in detail at the end of \S6.
\section{\bf Cohomological criteria for weak continuous generation}
In this section we provide a criterion, based on the weak index theorem, for the weak continuous generation of locally free sheaves on abelian varieties. To put the result into perspective, we recall that a key step in the proof of Lazarsfeld's conjecture \cite{pareschi} was based on the fact that if $F$ is a locally free sheaf on $X$ satisfying I.T. with index $0$, and $A$ is an ample line bundle, then $F\otimes A$ is globally generated. Theorem \ref{F-reg} above, proved in \cite{us}, provides a generalization of that criterion widely extending its range of applicability. For the purposes of this paper, we also need the following different generalization of the result mentioned above, based on even weaker hypotheses (but note the locally freeness assumption):
\begin{theorem}{\bf (W.I.T. regularity criterion)}\label{WIT} Let $F$ be a locally free sheaf on $X$ such that $F^\vee$ satisfies W.I.T. with index $g$ and the torsion part of $\widehat{F^\vee}$ is a torsion-free sheaf on a reduced subscheme of $X$. Then the following hold: \newline \noindent (a) $F$ is weakly continously generated. \newline \noindent (b) Let morever $A$ be a continuously globally generated line bundle on a subvariety of $X$. Then \begin{itemize} \item[(i)] $F\otimes A$ is generically globally generated. \item[(ii)] If the Fourier-jump locus $J(F)$ is finite then $B(F \otimes A)\subset \bigcup_{\xi\in J(F)}B(A\otimes P_\xi)$. \end{itemize} \end{theorem}
\begin{corollary}\label{bpf} Let $F$ and $A$ be a locally free sheaf, respectively an invertible sheaf on $X$. If $F$ satisfies the hypotheses of Theorem \ref{WIT} and $A$ is globally generated, then $F\otimes A$ is globally generated. \end{corollary}
\noindent The corollary follows immediately from Theorem \ref{WIT}(a) and Proposition \ref{ggg}(b). Turning to the proof of Theorem \ref{WIT}, note that, in view of Proposition \ref{ggg}(c), the only thing to prove is part(a). This in turn follows in a standard way (cf. \cite{pareschi} \S2(b) or \cite{us} \S2) from the following corresponding generalization of a result of Mumford-Kempf-Lazarsfeld type on multiplication maps (cf. \cite{us} Theorem 2.5).
\begin{lemma}\label{mult} Let $F$ be a locally free sheaf on X satisfying the hypotheses of Theorem \ref{WIT} and let $\mathcal{H}$ be a coherent sheaf on $X$ satisfying I.T. with index $0$. Then the sum of multiplication maps $$\mathcal{M}_U:\bigoplus_{\xi\in U}H^0(F\otimes P_\xi^\vee)\otimes H^0(\mathcal{H}\otimes P_\xi) \buildrel{\oplus m_\xi}\over\longrightarrow H^0(F\otimes \mathcal{H})$$ is surjective for any non-empty Zariski-open set $U\subset \hat X$ containing $J(F)$. \end{lemma}
\begin{proof} The argument follows the proof of \cite{us} Theorem 2.5 (although in fact the hypotheses allow us to avoid the use of derived categories). The statement is equivalent to proving the injectivity of the dual map (note that $F$ is locally free): $$\mathcal{M}_U^\vee:{\rm Ext}^g(\mathcal{H},F^\vee)\buildrel{\prod m_\xi^\vee}\over\longrightarrow \prod_{\xi\in U}{\rm Hom}(H^0(\mathcal{H}\otimes P_\xi),H^g(F^\vee\otimes P_\xi)), $$ where the maps $m_\xi^\vee$ are the co-multiplication maps taking an extension class $e\in {\rm Ext}^g(\mathcal{H},F^\vee)$ to its connecting map $H^0(\mathcal{H}\otimes P_\xi)\rightarrow H^g(F^\vee\otimes P_\xi)$. The index hypotheses allows us to write $\mathcal{M}_U^\vee$ as the composition of the map on global sections $\phi:{\rm Ext}^g(H,F^\vee)\rightarrow {\rm Hom}(\widehat{\mathcal{H}}, \widehat{F^{\vee}})$, followed by the evaluation map $$ev_U: H^0({\mathcal H}om(\widehat{\mathcal{H}}, \widehat{F^{\vee}}))\buildrel{\prod ev_\xi}\over\longrightarrow \prod_{\xi\in U}{\mathcal H}om(\widehat{\mathcal{H}},\widehat{F^{\vee}})(\xi),$$ where for a sheaf $\mathcal{E}$, we denote $\mathcal{E}(\xi):=\mathcal{E}\otimes k(\xi)$. In addition the hypotheses imply, by Lemma \ref{exts} above, that the map $\phi$ is an isomorphism.
On the other hand, if $U$ is a Zariski-open set containing $J(F)$, the map $ev_U$ is injective: note that by Nakayama's Lemma, given a non-zero global section $s$ of a sheaf $\mathcal{E}$ on $X$, we have that $s(x)\in \mathcal{E}(x)$ vanishes identically on a Zariski-open set $U$ only if either $U$ does not meet a component of the support of the torsion part of $\mathcal{E}$, or if the torsion part of $\mathcal{E}$ is a sheaf on a non-reduced subscheme of $X$. Taking $\mathcal{E}={\mathcal H}om(\widehat H,\widehat F^\vee)\cong (\widehat H)^\vee\otimes \widehat F^\vee$ in our case, we see that the torsion part $\tau({\mathcal H}om(\widehat H,\widehat F^\vee))$ is isomorphic to $(\widehat H)^\vee\otimes\tau(\widehat F^\vee)$. The hypothesis that $U$ contains the Fourier jump locus of $F$ excludes the first possibility (since the support of the torsion part of $\widehat F^\vee$ is certainly contained in $J(F)$). The second possibility is excluded by hypothesis. \end{proof}
\begin{remark} More generally, Lemma \ref{mult} and, consequently, Theorem \ref{WIT} continue to hold (with the same proof) under the hypotheses that $F$ is a locally free sheaf on an $m$-dimensional Cohen-Macaulay subvariety $Y$ of $X$, ${\mathcal H}om(F,\omega_Y)={\mathcal E}xt^{g-m}(F,{\mathcal O}_X)$ satisfies W.I.T. with index $m$, and the torsion part of the Fourier transform ${{\mathcal H}om(F,\omega_Y)}^{\widehat{}}$ is a torsion-free sheaf on a reduced subvariety of $X$. \end{remark}
\section{\bf Relative Pontrjagin products}
\subsection*{\bf Pontrjagin products, multiplication maps and relative Pontrjagin products.}
One of the key points emphasized in \cite{pareschi} is the relation between multiplication maps of sheaves on abelian varieties (which are in turn involved in the study of linear series) and (skew) Pontrjagin products (cf. Proposition 5.2 below). Here we develop an analogue relative setting for skew Pontrjagin products, required for our applications.
\begin{terminology/notation}{\bf ( Skew Pontrjagin product, P.I.T. and $~~~$ P.W.I.T.)}\label{PIT} Let us recall first that, given two sheaves $\mathcal{E}$ and $\mathcal{G}$ on $X$, their \emph{skew Pontrjagin product} (see \cite{pareschi} \S1) is defined as $$\mathcal{E}\hat * \mathcal{G}:={p_1}_*((p_1+p_2)^*(\mathcal{E})\otimes p_2^*(\mathcal{G})).$$ We will see, in the spirit of \cite{mukai} \S3, the skew Pontrjagin product as a bifunctor from ${\rm Mod}(X)\times {\rm Mod}(X)$ to ${\rm Mod}(X)$, and we denote by $\buildrel {\bf R}\over{\hat *}$ its derived functor. Moreover we adopt the following terminology: the pair $(\mathcal{E},\mathcal{G})$ \emph{satisfies the Pontrjagin Index Theorem (P.I.T.) with index $k=p(\mathcal{E},\mathcal{G})$} if $h^i((T_x^*\mathcal{E})\otimes \mathcal{G})=0$ for any $i\ne k$ and for any $x\in X$. If $R^i{p_1}_*((p_1+p_2)^*(\mathcal{E})\otimes {p_2}^*(\mathcal{G}))=0$ for $i\ne k$ we will say that $(\mathcal{E},\mathcal{G})$ \emph{satisfies the Weak Pontrjagin Index Theorem (P.W.I.T.) with index $k=p(\mathcal{E},\mathcal{G})$} and in this case (by abuse of notation) we will denote again $$\mathcal{E}\hat * \mathcal{G}=R^k{p_1}_*((p_1+p_2)^*(\mathcal{E})\otimes {p_2}^*(\mathcal{G})).$$ \end{terminology/notation}
We will also use the following notation: given two sheaves $\mathcal{E}$ and $\mathcal{G}$ on $X$, we denote by $\mathcal{M}(E,G)$ the locus of $x\in X$ where the multiplication map $$m_x:H^0(T_x^*\mathcal{E})\otimes H^0(\mathcal{G})\rightarrow H^0((T_x^*\mathcal{E})\otimes \mathcal{G})$$ is not surjective. The relationship between skew Pontrjagin products and multiplication maps is provided by the following:
\begin{proposition}\label{mult-pontr}(\cite{pareschi} Proposition 1.1) Let $\mathcal{E}$ and $\mathcal{G}$ be sheaves on $X$ such that $(\mathcal{E},\mathcal{G})$ satisfies P.I.T. with $p(E,G)=0$. Then $$\mathcal{M}(\mathcal{E},\mathcal{G})=B(\mathcal{E}\hat * \mathcal{G}).$$ \end{proposition}
\begin{remark} If $E$ and $G$ are locally free and $(E,G)$ satisfies P.I.T. with $p(E,G)=0$, then $(E^\vee,G^\vee)$ satisfies P.I.T. with $p(E^\vee,G^\vee)=g$ and, by relative Serre duality, $(E\hat * G)^\vee\cong E^\vee\hat * F^\vee$. In other words the dual of $E\hat * F$ is also a skew Pontrjagin product. \end{remark}
In view of Theorems \ref{F-reg} and \ref{WIT}, given a pair $(E,G)$ satisfying P.I.T. with $p(E,G)=0$ as in the remark above, in order to study the surjectivity of the multiplication map $m_0:H^0(F)\otimes H^0(G)\rightarrow H^0(F\otimes G)$ it is then natural to investigate whether there exists an ample line bundle $A$ on $X$ such that the "mixed" product $((E\hat * G)\otimes A^\vee)^\vee \cong (E^\vee\hat * F^\vee)\otimes A$ satisfies W.I.T. with $i((E^\vee\hat * F^\vee)\otimes A)=g$. Following \cite{pareschi}, an appropriate strategy turns out to be the following: first one establishes a suitable result of "exchange of Pontrjagin and tensor product under cohomology". Then, to prove the required vanishing, one uses the fact that, when the sheaves involved are line bundles algebraically equivalent to powers of a given one, say $A$, there is a suitable positive integer $n$ such that, pulling back via multiplication by $n$, the skew Pontrjagin product is a trivial bundle tensored by a suitable power of $A$. Here we generalize this technique to a relative setting: Proposition 5.5 below is the relative analogue of the exchange of Pontrjagin and tensor product under cohomology while Proposition 5.6 provides the formulas for the pullback via multiplication by an integer (both in the usual and the relative setting).
\begin{terminology/notation}{\bf (Pontrjagin product relative with respect to the second variable, relative P.I.T. and P.W.I.T.)} We denote by $p_i$ and $p_{ij}$ respectively, the projections and the intermediate projections of $X\times X\times \hat X$. Consider the bifunctor $$?~{\hat *}_{rel} ?={p_{13}}_*((p_1+p_2)^*?\otimes p_2^*?\otimes p_{23}^*{\mathcal P})$$ from ${\rm Mod}(X)\times {\rm Mod}(X)$ to ${\rm Mod}(X\times \hat X)$, and let ${\buildrel {\bf R}\over{\hat *}}_{rel}$ be its derived functor. As usual, we have corresponding notions of Index Theorem and Weak Index Theorem: e.g. relative P.I.T. with $p_{rel}(\mathcal{E},\mathcal{G})=k$ means that $H^i((T_x^*\mathcal{E})\otimes \mathcal{G}\otimes P_\xi)=0$ for any $x\in X$, $\xi \in \hat X$ and $i\ne k$. As above, if relative P.W.I.T. holds, we write ${\hat *}_{rel}$ rather than ${\buildrel {\bf R}\over{\hat *}}_{rel}$. We denote by $\Gamma$ the global sections functor. \end{terminology/notation}
\begin{proposition}\label{exchange}{\bf (Exchange of Pontrjagin and tensor product under (relative) cohomology.)} (a) Assume that $G$ and $H$ are locally free sheaves on $X$ and $?$ is either an object or a morphism in ${\bf D}(X)$. Then \newline \noindent (i) ${\bf R}\Gamma ((?\buildrel {\bf R}\over{\hat *}G)\otimes H)\cong {\bf R}\Gamma ((?{\buildrel {\bf R}\over{\hat *}}_{rel} H) \otimes p_X^*G)$. \newline \noindent (ii) ${\bf R}\hat S((?\buildrel {\bf R}\over{\hat *}G)\otimes H)\cong {\bf R}{p_{\hat X}}_*((?{\buildrel {\bf R}\over{\hat *}}_{rel} H) \otimes p_X^*G)$.
\noindent (b) Let in addition $\mathcal{E}$ be a sheaf on $X$ such that $(\mathcal{E},G)$ satisfies P.I.T. with $p(\mathcal{E},G)=k$. Then \newline \noindent (i) If $(\mathcal{E},H)$ satisfies P.I.T. with $p(\mathcal{E},H)=k$, then for any $i$, $$H^i((\mathcal{E}\hat * G)\otimes H)\cong H^i((\mathcal{E}{\hat *} H)\otimes G).$$ \newline \noindent (ii) If $(\mathcal{E},H)$ satisfies relative P.I.T. with $p_{rel}(\mathcal{E},H)=k$, then for any $i$, $$R^i\hat{\mathcal S}((\mathcal{E}\hat * G)\otimes H)\cong R^i{p_{\hat X}}_*((\mathcal{E}{\hat *}_{rel} H)\otimes p_X^*G).$$ \end{proposition}
\begin{proof} Part (b)(i) is precisely Lemma 3.2 of \cite{pareschi}. The rest of the proof follows the same argument in the derived setting. Note that we are suppressing part of the symbols showing that we are working with the derived functors, in order to simplify the notation. For the reader's convenience we prove (a)(ii) and (b)(ii). The left hand side of (a)(ii) can be written as \begin{eqnarray*} {\bf R}\hat{\mathcal S}((?\buildrel{\bf R}\over{\hat *} G)\otimes H)& = & {\bf R}{p_{\hat X}}_*({\mathcal P}\otimes p_X^*(H\otimes {\bf R}{p_1}_*((p_1+p_2)^*?\otimes p_2^*G))) \\ & \cong &{\bf R} {p_{\hat X}}_*({\mathcal P}\otimes p_X^*({\bf R}{p_1}_*(p_1^*H\otimes (p_1+p_2)^*?\otimes p_2^*G))) \\ &\cong & {\bf R}{p_{\hat X}}_*({\bf R}{p_{13}}_*(p_{13}^*\mathcal P\otimes p_1^*H\otimes (p_1+p_2)^*?\otimes p_2^*G)) \\ &\cong & {\bf R}{p_3}_* (p_{13}^*{\mathcal P}\otimes p_1^*H\otimes (p_1+p_2)^*?\otimes p_2^*G) \end{eqnarray*} On the other hand, working out the right hand side we have \begin{eqnarray*} {\bf R}{p_{\hat X}}_*((?{\buildrel {\bf R}\over{\hat *}}_{rel} H) \otimes p_X^*G) & = & {\bf R}{p_{\hat X}}_*({\bf R}{p_{13}}_*((p_1+p_2)^*?\otimes p_2^*H\otimes p_{23}^*{\mathcal P}_{23}) \otimes p_X^*G) \\ &\cong & {\bf R}{p_{\hat X}}_*({\bf R}{p_{13}}_*(p_1^*G\otimes (p_1+p_2)^*?\otimes p_2^*H\otimes p_{23}^*{\mathcal P}) \\ & \cong & {\bf R}{p_3}_*(p_1^*G\otimes (p_1+p_2)^*? \otimes p_2^*H\otimes p_{23}^*{\mathcal P}) \end{eqnarray*} The result follows via the automorphism $(x,y,\xi)\mapsto (y,x,\xi)$ of $X\times X\times \hat X$. As for (b)(ii), under the hypothesis at hand $(\mathcal{E}\buildrel {\bf R}\over{\hat *}G)\otimes H$ reduces to $(\mathcal{E}\hat * G)\otimes H[k]$ and $(\mathcal{E}{\buildrel {\bf R}\over{\hat *}}_{rel} H) \otimes p_X^*G$ reduces to $(\mathcal{E}{\hat *}_{rel} H)\otimes p_X^*G[k]$. Therefore the assertion follows from (a)(ii). \end{proof}
\subsection*{\bf Pulling back via multiplication by an integer.} When line bundles are involved, Pontrjagin products usually look simpler when pulled back via multiplication by an appropriate integer. The purpose of this subsection is to generalize the results of \cite{pareschi} \S3(b) to Pontrjagin products relative with respect to the second variable. Given a positive integer $n$, the map $x\mapsto nx$ will be denoted $n_X:X\rightarrow X$.
\begin{proposition}\label{calculations} (a) Let $n$ be a positive integer and let $L$ be a line bundle on~$X$. \newline \noindent (i) $n_X^*(L \buildrel{{\bf R}}\over {\hat *} ?)\cong (L^n \buildrel{{\bf R}}\over {\hat *} (?\otimes L^{-n+1}))\otimes n_X^*(L)\otimes p_X^*L^{-n}$ \newline \noindent (ii) $(n_X,1_{\hat X})^*(L{\buildrel{{\bf R}}\over {\hat *}}_{rel}?)\cong (L^n {\buildrel{{\bf R}}\over {\hat *}} (?\otimes L^{-n+1}))\otimes (n_X,1_X)^*(L)\otimes p_X^*L^{-n}$ \newline \noindent (b) Skew Pontrjagin products with the structure sheaf can be expressed as follows: \newline \noindent (i) $?{\buildrel{{\bf R}}\over {\hat *}} {\mathcal O}_X\cong {\bf R}\Gamma(?)\otimes {\mathcal O}_X$ \newline \noindent (ii) $?{\buildrel{{\bf R}}\over {\hat *}}_{rel} {\mathcal O}_X\cong p_{\hat X}^*({\bf R}{\mathcal S}(?))\otimes {\mathcal P}^\vee $ \newline \noindent (c) Let $A$ be an ample line bundle on $X$ and assume that $a$ and $a+b$ are positive integers. Then \newline \noindent (i) $(a+b)_X^*(A^a
{\hat *} (A^b\otimes P_\xi))\cong H^0(A^{a+b}\otimes P_\xi)\otimes (a+b)_X^*A^a\otimes a_X^*(A^{-a-b}) \otimes P_\xi^{-a}$ \item[] $(a+b)^*_X(A^{-a}\hat *(A^{-b}\otimes P_\xi))\cong H^g(A^{-a-b}\otimes P_\xi)\otimes (a+b)_X^*A^{-a}\otimes a_X^*(A^{a+b})\otimes P_\xi^{a} $ \newline \noindent (ii) $((a+b)_X \times 1_{\hat X})^*(A^a
{\hat *}_{rel} A^b)\cong p_{\hat X}^*((A^{a+b})^{\widehat{}}) \otimes p_X^*((a+b)_X^*A^a\otimes a_X^*A^{-a-b})\otimes {\mathcal P}^{-a}$ \item[] $((a+b)_X \times 1_{\hat X})^*(A^{-a} {\hat *}_{rel} A^{-b}) \cong p_{\hat X}^*((A^{-a-b})^{\widehat{}}) \otimes p_X^*((a+b)_X^*A^{-a}\otimes a_X^*A^{a+b})\otimes {\mathcal P}^{a}$ \end{proposition} \begin{proof} Note that (a)(i) is Proposition 3.4 of \cite{pareschi} and the proof of (a)(ii) identical. Furthermore, (b)(i) is Remark 3.5(b) of \cite{pareschi} and (b)(ii) is proved in the same way. Therefore all these proofs are omitted. The first isomorphism of (c)(i) is Proposition 3.6 of \cite{pareschi}, but note that in that paper there is a misprint: the last factor of the right hand side of \cite{pareschi} Prop. 3.6 should read $a_X^*(A^{-a-b}\otimes \alpha^\vee)$ instead of $a_X^*(A^{-a-b})\otimes \alpha^\vee$. The present formulation follows since $a_X^*P_\xi^\vee=P_\xi^{-a}$. The second isomorphism of (c)(i) follows by duality. Finally (c)(ii) is proved exactly in the same way, using Lemma \ref{standard}(a) below, and therefore its proof is omitted too. \end{proof}
\begin{lemma}\label{standard} (a)$(n_X, 1_{\hat X})^*{\mathcal P}= (1_X, n_{\hat X})^*{\mathcal P}={\mathcal P}^{\otimes n}$. \newline \noindent (b) $R^i{p_{\hat X}}_*({\mathcal P}^{\otimes n})= 0$ for $i< g$ and $={\mathcal O}_{{\hat X}_n}$ for $i=g$. \end{lemma} \begin{proof} (a) By double duality it is enough to prove the first equality. We prove it by induction on $n$. For $n=2$, we apply the Theorem of the Cube (\cite{mumford} Cor.2 p.58) with $f(x,\xi)=g(x,\xi)=(x,0)$, $h(x,\xi)=(0,\xi)$ (all maps $X\times \hat X\rightarrow X\times \hat X$)
and $L={\mathcal P}$. We get, using that ${\mathcal P}_{|\{0\}\times \hat X}
={\mathcal O}_{\hat X}$ and ${\mathcal P}_{|X\times \{0\}}={\mathcal O}_X$, that $(2_X, 1_{\hat X})^*{\mathcal P}={\mathcal P}^{\otimes 2}$. The general formula follows by induction, applying the same method with $f(x,y)=((n-1)_X,0)$, $g(x,y)=(x,0)$, $h(x,y)=(0,y)$. \newline \noindent (b) By flat base change, $R^i{p_{\hat X}}_*((1_X\times n_{\hat X})^*({\mathcal P})) =n_{\hat X}^*R^ip_{\hat X}^*{\mathcal P}= 0$ if $i<g$ and $n_{\hat X}^*{\mathcal O}_0={\mathcal O}_{{\hat X}_n}$ if $i=g$. \end{proof}
\subsection*{\bf A first application: theta-group-free proof of a theorem of Ohbuchi.} We end this section by presenting a theta-group-free proof of (part of) a classical theorem of Ohbuchi (\cite{ohbuchi2}, see also \cite{kempf3} Theorem 10.4, \cite{lange} Theorem 7.2.3 and \cite{khaled} for a proof working in ${\rm char}(k)\ne 2$) on the normal generation of a line bundle of the form $A^2$, where $A$ is an ample line bundle on an abelian variety. This is intended to be a toy version and an introduction to the new results on equations and syzygies in the next section, based on the techniques described above.
Within this framework, it is natural to state Ohbuchi's Theorem in a slightly more general way. (This can be however easily deduced from the usual statement: compare \cite{lange} Theorem 7.2.3 and Exercise 7.2.) Given an (ample) line bundle $A$ on $X$, let us denote $s(A):=A\otimes (-1)_X^*A^\vee$. The map $s:{\rm Pic}^{c_1(A)}(X)\rightarrow {\rm Pic}^0(X)$ is surjective and flat ($s(A)\in {\rm Pic}^0(X)$ classifies the "non-symmetry" of the line bundle $A$). Let also $t(A)$ denote a square root of $s(A)$.
\begin{theorem}\label{ohbuchi} ($char(k)\ne 2$) Let $A$ be an ample line bundle on $X$. Then $$\mathcal{M}(A^2,A^2)=\bigcup_{\xi\in{\hat X}_2}B(A\otimes P_{t(A)}\otimes P_\xi).$$ Hence $A^2$ is normally generated if and only if $0\not\in \bigcup_{\xi\in {\hat X}_2}B(A\otimes P_{t(A)}\otimes P_\xi)$. \end{theorem}
\begin{proof} We will prove only the "positive part" of the result, i.e. the inclusion $\mathcal{M}(A^2,A^2)\subset\cup_{\xi\in{\hat X}_2}B(A\otimes P_{t(A)}\otimes P_\xi)$, since this is the part to be generalized in the next section. The opposite inclusion can be proved similarly. We have that \begin{equation}\label{iso} R^i\hat{\mathcal S}( (A^{-2}\hat * A^{-2})\otimes A)\cong \left\{\begin{array}{ll} 0 & \textrm{if $i<g$} \\ \widehat{A^{-1}}\otimes{\mathcal O}_{{\hat X}_2+t(A)} &\textrm{if $i=g$,} \\ \end{array}\right. \end{equation}
where $\hat{X}_2+t(A)$ denotes the set $\{~\eta~|~\eta - t(A)\in \hat{X}_2\}$. Postponing the proof of (\ref{iso}) for a moment, let us show how it implies the statement. In fact (\ref{iso}) yields that the hypothesis of Theorem \ref{WIT} are fulfilled by $(A^2\hat * A^2)\otimes A^\vee$. Moreover (\ref{iso}) gives also that $J((A^2\hat * A^2)\otimes A^\vee)=\hat{X}_2+t(A)$ (this follows immediately from base change and Serre duality). Therefore, by Theorem \ref{WIT}, $B(A^2\hat *A^2)\subset \cup_{\xi\in{\hat X}_2}B(A\otimes P_{t(A)}\otimes P_\xi)$. On the other hand, by Proposition \ref{mult-pontr}, $\mathcal{M}(A^2,A^2)=B(A^2\hat * A^2)$, and the statement is proved.
\noindent \emph{Proof of (\ref{iso}):} \begin{eqnarray} R^i\hat{\mathcal S}( (A^{-2}\hat * A^{-2})\otimes A)\cong &&R^i{p_{\hat X}}_*((A^{-2}{\hat *}_{rel}A)\otimes p_X^*A^{-2}\\ \cong &&R^i{p_{\hat X}}_*(p_{\hat X}^*\widehat{A^{-1}}\otimes p_X^*(A^{-4}\otimes 2_X^*A)\otimes {\mathcal P}^{2})\\ \cong && R^i{p_{\hat X}}_*(p_{\hat X}^*\widehat{A^{-1}}\otimes p_X^*(A^{-1}\otimes (-1)_X^*A)\otimes {\mathcal P}^{2})\\ = && R^i{p_{\hat X}}_*(p_{\hat X}^*\widehat{A^{-1}}\otimes p_X^*P_{s(A)}^\vee\otimes {\mathcal P}^{2}) \\ \cong&&\left\{\begin{array}{ll} 0 & \textrm{if $i<g$} \\ \widehat{A^{-1}}\otimes{\mathcal O}_{{\hat X}_2+t(A)} &\textrm{if $i=g$}. \\ \end{array}\right. \end{eqnarray} In the sequence of congruences above, (2) follows by Proposition \ref{exchange}(b)(ii), (3) follows by Proposition \ref{calculations}(c)(ii) (second part) with $a=2$ and $b=-1$, (4) from the fact that $2_X^*A\cong A^3\otimes (-1)_X^*A$ and (6) from (a slight variant of) Lemma \ref{standard}(b) and the projection formula. \end{proof}
\section{\bf Equations defining abelian varieties and their syzygies}
Putting together the machinery of the previous paragraphs, in this section we adress the question of bounding the degrees of the generators (and their syzygies) of the homogeneous ideal $I_{X,L}$ of an
abelian variety $X$ embedded by a complete linear series $|L|$, where $L$ is a suitable power of an ample line bundle $A$. Our main result is:
\begin{theorem}\label{main} (${\rm char}(k)\ne 2,3$.) Let $A$ be an ample line bundle on $X$, with no base divisor. Then: \newline \noindent (a) If $k\ge 3$ then $I_{X,A^{k}}$ is generated by quadrics. \newline \noindent (b) $I_{X,A^2}$ is generated by quadrics and cubics. \end{theorem}
Theorem \ref{main} turns out to be a special case of a more general result, extending in the now well-known language of Green \cite{green} bounds on the degrees of generators of the ideal $I_{X,L}$
to a hierarchy of conditions about higher syzygies. Specifically, given a variety $X$ embedded in projective space by a complete linear series $|L|$, the line bundle $L$ is said to \emph{satisfy property $N_p$ } if the first $p$ steps of the minimal graded free resolution of the algebra $R_L=\oplus H^0(L^n)$ over the polynomial ring $S_L=\oplus {\rm Sym}^nH^0(L)$ are linear, i.e. of the form $$ S_L(-p-1)^{\oplus i_{p}}\rightarrow S_L(-p)^{\oplus i_{p-1}}\rightarrow\cdots\rightarrow S_L(-2)^{\oplus i_1}\rightarrow S_L\rightarrow R_L\rightarrow 0.$$ Thus $N_0$ means that the embedded variety is projectively normal (\emph{normal generation} in Mumford's terminology), $N_1$ means that the homogeneous ideal is generated by quadrics (\emph{normal presentation}), $N_2$ means that the relations among these quadrics are generated by \emph{linear} ones and so on.
More generally even (cf. \cite{pareschi}), one can define properties measuring how far the first $p$ steps of the resolution are from being linear. To do this, fix $p\ge 0$, and consider the first $p$ steps of the minimal free resolution of $R_L$ as an $S_L$-module: $$E_p\rightarrow E_{p-1}\rightarrow \cdots E_1\rightarrow E_0\rightarrow R_L\rightarrow 0,$$ where $E_0=S_L\oplus\bigoplus_jS_L(-a_{0j})$ with $a_{0j}\ge 2$ (since the linear series is complete), $E_1=\bigoplus_jS_L(-a_{1j})$ with $a_{1j}\ge 2$ (since the embedding is non-degenerate) and so on, up to $E_p=\bigoplus_jS_L(-a_{pj})$ with $a_{pj}\ge p+1$. Then $L$ is said to \emph{satisfy property $N_p^r$} if $a_{pj}\le p+1+r$. In particular $N_1^r$ means that $a_{1j}\le 2+r$, i.e. the ideal $I_{X,L}$ is generated by forms of degree $\le 2+r$, while property $N_p^0$ is the same as $N_p$.
\noindent With this terminology, the extension of Theorem \ref{main} to arbitrary syzygies is the following:
\begin{theorem}\label{syzygies} \emph{(${\rm char}(k)$ does not divide $(p+1)$ and $(p+2)$.)} Assume that $A$ has no base divisor. Then: \newline \noindent (a) If $k\ge p+2$ then $A^{k}$ satisfies property $N_p$. \newline \noindent (b) More generally, if $(r+1)(k-1)\ge p+1$ then $A^k$ satisfies property $N_p^r$. \end{theorem}
\noindent A word about the proofs. Although Theorem \ref{main} is subsumed in Theorem \ref{syzygies}, we prefer to start by proving it separately. The reason is that a substantially higher degree of technicality in the proof of Theorem \ref{syzygies} might potentially make the main idea less transparent -- with this separation, some of the similar arguments will not be repeated in the second proof.
\subsection*{\bf Background material.} We briefly recall some well-known facts about the relationship between condition $N_p$, or more generally $N_p^r$, and the surjectivity of suitable multiplication maps of vector bundles. For the facts surveyed here see e.g. \cite{lazarsfeld} and \cite{pareschi}. The main point is that condition $N_p^r$ is equivalent to the exactness in the middle of the piece of the Koszul complex (cf. \cite{green}): \begin{equation}\label{koszul} \Lambda^{p+1}H^0(L)\otimes H^0(L^h)\rightarrow \Lambda^p H^0(L)\otimes H^0(L^{h+1})\rightarrow \Lambda^{p-1}H^0(L)\otimes H^0(L^{h+2}) \end{equation} for all $h\geq r+1$. One can in turn express this as a vanishing condition for the cohomology of a suitable vector bundle. Specifically, for a globally generated line bundle $L$, let $M_L$ be the kernel of the evaluation map: \begin{equation}\label{ML} 0\rightarrow M_L\rightarrow H^0(L)\otimes {\mathcal O}_X\rightarrow L\rightarrow 0 \end{equation} It follows easily that the exactness of (\ref{koszul}) is equivalent to the surjectivity of the map $\Lambda^{p+1}H^0(L)\otimes H^0(L^h)\rightarrow H^0(\Lambda^p M_L\otimes L^{h+1})$ arising from the exact sequence (obtained by taking exterior powers in (\ref{ML})): $$0\rightarrow \Lambda^{p+1}M_L\otimes L^h\rightarrow \Lambda^{p+1}H^0(L)\otimes L^h\rightarrow \Lambda^pM_L\otimes L^{h+1}\rightarrow 0.$$ Therefore $N_p^r$ holds as soon as \begin{equation}\label{wedge} H^1(\Lambda^{p+1}M_L\otimes L^{h})=0, ~\forall~h\geq r+1. \end{equation} (On abelian varieties the converse is also true since $H^1(L^h)=0$ for $h\ge 1$.) This leads to:
\begin{proposition}\label{tensor} (a) \emph{$({\rm char}(k)$ does not divide $(p+1)$)} If $H^1(M_L^{\otimes (p+1)}\otimes L^h)=0$ for all $h\ge r+1$, then $L$ satisfies condition $N_p^r$. \newline \noindent (b) Assume that $H^1(M_L^{\otimes p}\otimes L^h)=0$. Then $H^1(M_L^{\otimes (p+1)}\otimes L^h)=0$ if and only if the multiplication map $$H^0(L)\otimes H^0(M_L^{\otimes p}\otimes L^h)\rightarrow H^0(M_L^{\otimes p}\otimes L^{h+1})$$ is surjective. \end{proposition} \begin{proof} Part (a) follows from (\ref{wedge}) since, under the assumption on the characteristic, $\Lambda^{p+1}M_L$ is a direct summand of $M_L^{\otimes (p+1)}$. Part (b) follows from the exact sequence \begin{equation}\label{sequence} 0\rightarrow M_L^{\otimes (p+1)}\otimes L^h\rightarrow H^0(L)\otimes M_L^{\otimes p}\otimes L^h\rightarrow M_L^{\otimes p}\otimes L^{h+1}\rightarrow 0. \end{equation} \end{proof}
\subsection*{\bf Proof of Theorem \ref{main}.} (a) The result is known for $k\ge 4$, so it is enough to prove it for $k=3$. By Proposition \ref{tensor}(a), it suffices to show that $$H^1(M_{A^3}^{\otimes 2}\otimes A^{3h})=0, ~\forall~h\geq 1.$$ Moreover, we know that $H^1(M_{A^3}\otimes A^{3h})=0$ for $h\ge 1$ -- by (\ref{sequence}) for $p=0$, this is equivalent to the \emph{normal generation} of $A^3$, i.e. Koizumi's Theorem. Therefore, by Proposition \ref{tensor}(b), it is enough to prove that the multiplication map $$H^0(A^3)\otimes H^0(M_{A^3}\otimes A^{3h})\rightarrow H^0(M_{A^3}\otimes A^{3(h+1)})$$ is surjective for $h\ge 1$. Again, this is well known for $h\ge 2$ -- it is equivalent to the fact that the homogeneous ideal of $X$ embedded by
$|A^3|$ is generated by forms of degree $2$ and $3$ (cf. \cite{kempf1},\cite{lange} or, in this interpretation, \cite{pareschi}). Therefore the only case to be examined is $h=1$.
We prove more generally that the locus $\mathcal{M}(A^3,M_{A^3}\otimes A^3)$ is empty (cf. Proposition \ref{mult-pontr}). By Proposition \ref{mult-pontr} we have $\mathcal{M}(A^3,M_{A^3\otimes A^3})= B(A^3\hat * (M_{A^3}\otimes A^3))$, since from the defining sequence \ref{ML} it is not hard to see that the pair $(A^3,M_{A^3}\otimes {A^3})$ satisfies P.I.T. with index $0$. To this end we make the following:
\noindent \emph{Claim. If $A$ has no base divisor, then $(A^3\hat * (M_{A^3}\otimes A^3))\otimes A^\vee$ is $M$-regular.}
\noindent By Theorem \ref{F-reg} this yields that $A^3\hat * (M_{A^3}\otimes A^3)$ is globally generated, and hence the theorem.
\noindent \emph{Proof of Claim.} Recall from \cite{us} \S3 that it is enough to prove that the cohomological support loci
$$V^i:=\{~\xi\in \hat X\> | \> h^i((A^3\hat *( M_{A^3}\otimes A^3))\otimes A^\vee\otimes P_\xi)>0\}$$ have {\rm codim}ension $> i$ for all $i>0$. By Proposition \ref{exchange}(b)(i) we have that
$$V^i=\{\xi\in \hat X \> |\> h^i((A^3\hat *(A^\vee \otimes P_\xi))\otimes M_{A^3}\otimes A^3)>0\}.$$ Let us consider the exact sequence obtained from \ref{ML} \begin{eqnarray*} &0\rightarrow (A^3\hat *(A^\vee \otimes P_\xi))\otimes M_{A^3}\otimes A^3\rightarrow H^0(A^3)\otimes (A^3\hat *(A^\vee \otimes P_\xi)) \otimes A^3\rightarrow&\\ &\rightarrow (A^3\hat *(A^\vee \otimes P_\xi)) \otimes A^6\rightarrow 0& \end{eqnarray*}
\vskip0.2truecm\noindent\emph{Subclaim. $h^i(((A^3\hat *(A^\vee \otimes P_\xi)) \otimes A^n)= 0$ for any $n\ge 3 $, $\xi\in \hat X$ and $i\ge 1$.}
\begin{proof} This is again a standard application of Proposition \ref{calculations}(c)(i): taking $a=3$ and $b=-1$ we have that $2_X^*(A^3\hat * (A^\vee\otimes P_\xi))\cong H^0(A^2\otimes P_\xi)\otimes 2_X^*A^3\otimes 3_X^*A^{-2}\otimes P_\xi^{-3}$. This implies that $$2_X^*(A^3\hat * (A^\vee\otimes P_\xi)\otimes A^n)\cong H^0(A^2\otimes P_\xi)\otimes 2_X^*A^3\otimes 3_X^*A^{-2}\otimes P_\xi^{-3}\otimes 2_X^*A^n,$$ which is isomorphic to a sum of copies of line bundles algebraically equivalent to $A^{(4n-6)}$, thus certainly ample for $n\ge 3$. As we are in characteristic $\ne 2$, $H^i((A^3\hat *(A^\vee \otimes P_\xi)) \otimes A^n)$ is a direct summand of $H^i(2_X^*((A^3\hat *(A^\vee \otimes P_\xi))\otimes A^n))$, which proves the subclaim. \end{proof}
\noindent Passing to cohomology in the exact sequence above, by the Subclaim we have that \begin{enumerate} \item[(i)]$V^i$ is empty for $i\ge 2$. \item[(ii)] $V^1$ coincides with the locus where the multiplication map $$H^0(A^3)\otimes H^0( (A^3\hat *(A^\vee \otimes P_\xi)) \otimes A^3)\rightarrow H^0((A^3\hat *(A^\vee \otimes P_\xi)) \otimes A^6)$$ is not surjective. \end{enumerate}
\noindent In view of (i), the Claim would be implied by the inequality
${\rm {\rm codim}}(V^1)>1$. Again by Proposition \ref{mult-pontr}, we have that \begin{equation}\label{S1}
V^1=\bigl\{\xi\in \hat X \>|\> 0_X\in B\bigl(A^3\hat * ((A^3\hat *(A^\vee \otimes P_\xi))\otimes A^3)\bigr)\bigr\}. \end{equation} We will approach this by the same trick of twisting with $A^\vee$ in order to try and apply Theorem \ref{WIT}. By relative duality we have $$((A^3\hat * (A^3\hat *(A^\vee \otimes P_\xi))\otimes A^3)\otimes A^\vee)^\vee\cong (A^{-3}\hat * (A^{-3}\hat *(A\otimes P_\xi^\vee))\otimes A^{-3})\otimes A.$$ By Proposition \ref{exchange}(b)(ii) \begin{equation}\label{strunz} R^i\hat{\mathcal S}((A^{-3}\hat * (A^{-3}\hat *(A\otimes P_\xi^\vee))\otimes A^{-3})\otimes A)\cong {R^ip_{\hat X}}_*((A^{-3}{\hat *}_{rel}A)\otimes p_X^*((A^{-3}\hat * (A\otimes P_\xi^\vee))\otimes A^{-3})). \end{equation} The key point that W.I.T. with index $g$ is satisfied goes through the following: \begin{eqnarray}\label{iso2} &R^i{p_{\hat X}}_*\bigl((2_X,1_{\hat X})^*[(A^{-3}{\hat *}_{rel}A)\otimes p_X^*((A^{-3}\hat * (A\otimes P_\xi^\vee))\otimes A^{-3})] \bigr)\cong&\nonumber \\ &\cong \left\{\begin{array}{ll}
0 &\textrm{if $i<g$} \\ H^g(A^{-2}\otimes P_\xi^\vee)\otimes \widehat{A^{-2}}\otimes{\mathcal O}_{{\hat X}_3 -s(A)-\xi}
&\textrm{if $i=g$} \end{array}\right.& \end{eqnarray} \noindent \emph{Proof of (\ref{iso2}).} By Proposition \ref{calculations}(c)(ii) with $a=3$ and $b=-1$: $$(2_X,1_{\hat X})_*( A^{-3}{\hat *}_{rel}A)\cong p_{\hat X}^*\widehat{A^{-2}}\otimes p_X^*(2_X^*A^{-3}\otimes 3_X^*A^2)\otimes {\mathcal P}^{3}.$$ In conclusion, using also Proposition \ref{calculations}(c)(i), \begin{eqnarray*} &(2_X,1_{\hat X})^*[(A^{-3}{\hat *}_{rel}A)\otimes p_X^*((A^{-3}\hat * (A\otimes P_\xi^\vee))\otimes A^{-3})]\cong &\\
& p_{\hat X}^*\widehat{A^{-2}}\otimes p_X^*(2_X^*A^{-3}\otimes 3_X^*A^2\otimes H^g(A^{-2}\otimes P_\xi^\vee)\otimes 2_X^*A^{-3}\otimes 3_X^*A^2 \otimes P_\xi^{-3}\otimes 2_X^*A^{-3})\otimes {\mathcal P}^{3} \cong & \\ &\cong p_X^*H^g(A^{-2}\otimes P_\xi^\vee)\otimes p_{\hat X}^*\widehat{A^{-2}}\otimes p_X^*(P_{s(A)}^\vee\otimes P_\xi^\vee)^3\otimes {\mathcal P}^{3} \end{eqnarray*} (Cf. the notation introduced before Theorem \ref{ohbuchi}.) Therefore (\ref{iso2}) follows from the projection formula and Lemma \ref{standard}.
We are now able to conclude the proof of the Claim. As we are in ${\rm char}(k)\ne 2$, $$R^i{p_{\hat X}}_*((A^{-3}{\hat *}_{rel}A)\otimes {p_X}^*((A^{-3}\hat * (A\otimes P_\xi^\vee))\otimes A^{-3}))$$ is a direct summand of $$R^i{p_{\hat X}}_*\bigl((2_X,1_{\hat X})^*[(A^{-3}{\hat *}_{rel}A)\otimes p_X^*((A^{-3}\hat * (A\otimes P_\xi^\vee))\otimes A^{-3})] \bigr).$$ According to (\ref{strunz}) and (\ref{iso2}), it follows that $(A^3\hat * (A^3\hat *(A^\vee \otimes P_\xi)\otimes A^3))\otimes A^\vee$ satisfies both hypotheses of Theorem \ref{WIT}. Moreover, by (\ref{iso2}) it also follows, using relative duality and base change, that there is an inclusion $$J((A^3\hat * (A^3\hat *(A^\vee \otimes P_\xi)\otimes A^3))\otimes A^\vee)\subset {\hat X}_3-s(A)-\xi.$$ Therefore Theorem \ref{WIT} implies that \begin{equation}\label{inclusion} B\bigl(A^3\hat * ((A^3\hat *(A^\vee \otimes P_\xi))\otimes A^3)\bigr) \subset \bigcup_{\eta\in {\hat X}_3}B(A\otimes P_{-s(A)-\xi}\otimes P_\eta). \end{equation} From (\ref{S1}) and (\ref{inclusion}) it follows that ${\rm codim}(S^1)={\rm codim}(B((A^3\hat *(A^\vee \otimes P_\xi))\otimes A^3))\ge {\rm codim}(B(A))$ and this proves the Claim.
\noindent (b) The proof is completely similar. This time one has to prove that the multiplication map $$H^0(A^2)\otimes H^0(M_{A^2}\otimes A^4)\rightarrow H^0(M_{A^2}\otimes A^6)$$ is surjective. As above, the result follows from the following statement, proved in the same way: \newline \noindent \emph{Claim. If $A$ has no base divisor then $(A^2\hat * (M_{A^2}\otimes A^4))\otimes A^\vee$ is $M$-regular.}
\subsection*{\bf Proof of Theorem \ref{syzygies}.} The argument is a combination between the proof of Lazarsfeld's conjecture in \cite{pareschi} and the idea of the proof of Theorem \ref{main}. We prove only (a), since the proof of (b) is completely analogous. First of all the theorem is known for $k\ge p+3$ and so we need to prove it only for $k=p+2$. For $L=A^{p+2}$, the exactness of the complex (\ref{koszul}) is known to hold for $h\ge 2$. (This means that the syzygies at the $p$-th step are generated at most in degree 2, i.e. condition $N_p^1$ in terminology of \cite{pareschi}, and so it follows from \cite{pareschi} Theorem 4.3.) Putting everything together, by Proposition \ref{tensor} it follows that it suffices to prove that the multiplication map $$H^0(A^{p+2})\otimes H^0(M_{A^p+2}^{\otimes p}\otimes A^p)\rightarrow H^0(M_{A^p+2}^{\otimes p}\otimes A^{2p}) $$ is surjective. Given $\xi\in\hat X$, we denote $$F_\xi^{(n,m)}:=A^n\hat * (A^m\otimes P_\xi)$$ We will prove the following:
\vskip0.2truecm\noindent\emph{Claim 1.} \emph{For every integer $k$ such that $1\le k\le p$ and every $\xi_1,\dots ,\xi_k\in \hat X$, the locally free sheaf $$A^{p+2}\hat * (M_{A^{p+2}}^{\otimes k} \otimes\bigotimes_{i=1}^{p-k} F_{\xi_i}^{(p+2,-1)} \otimes A^{p+2})$$ is globally generated.}
\vskip0.2truecm\noindent For $k=p$ this, together with Proposition \ref{mult-pontr}, proves the theorem (again, the fact that P.I.T. with index $0$ is verified follows from \cite{pareschi} Proposition 4.2).
\vskip0.2truecm\noindent\emph{Proof of Claim 1.} This goes by induction on $k$. Let us assume for a moment that we know the initial step $k=1$, and show that the statement for $k-1$ implies the statement for $k$, for all $k\geq 2$.
We fix $\xi_1, \dots ,\xi_k\in \hat X$. By Theorem \ref{F-reg}, it is enough to prove that the vector bundle $$(A^{p+2}\hat * (M_{A^{p+2}}^{\otimes k} \otimes\bigotimes_{i=1}^{p-k} F_{\xi_i}^{(p+2,-1)} \otimes A^{p+2}))\otimes A^\vee$$ is $M$-regular. In fact, for $k\geq 2$, it will even satisfy I.T. with index $0$, i.e. \begin{equation}\label{mah} H^i((A^{p+2}\hat * (M_{A^{p+2}}^{\otimes k} \otimes\bigotimes_{i=1}^{p-k} F_{\xi_i}^{(p+2,-1)} \otimes A^{p+2}))\otimes A^\vee\otimes P_\xi)=0 \end{equation} for any $i>0$ and any $\xi\in \hat X$. By Proposition \ref{exchange}(b)(i) we have that \begin{eqnarray*} &H^i((A^{p+2}\hat * (M_{A^{p+2}}^{\otimes k} \otimes\bigotimes_{i=1}^{p-k} F_{\xi_i}^{(p+2,-1)} \otimes A^{p+2}))\otimes A^\vee\otimes P_\xi)\cong &\\ &\cong H^i((A^{p+2}\hat * (A^\vee\otimes P_\xi))\otimes
M_{A^{p+2}}^{\otimes k}\otimes\bigotimes_{i=1}^{p-k} F_{\xi_i}^{(p+2,-1)} \otimes A^{p+2}).& \end{eqnarray*} (As usual, the hypotheses of Proposition \ref{exchange}(b)(ii) are fulfilled because of \cite{pareschi} Proposition 4.2.) Then, as in the proof of the previous theorem (and we won't go through all the details), the sequence $$0\rightarrow M_{A^{p+2}}^{\otimes k}\rightarrow H^0(A^{p+2})\otimes M_{A^{p+2}}^{\otimes k-1}\rightarrow M_{A^{p+2}}^{\otimes k-1}\otimes A^{p+2}\rightarrow 0$$ twisted by $(A^{p+2}\hat * (A^\vee\otimes P_\xi))\otimes \bigotimes_{i=1}^{p-k} F_{\xi_i}^{(p+2,-1)}\otimes A^{p+2}$ gives that the cohomology groups of (\ref{mah}) are zero, except for $H^1$ which vanishes if only if the multiplication map \begin{eqnarray*} &H^0(A^{p+2})\otimes H^0(M_{A^{p+2}}^{\otimes k-1}\otimes (A^{p+2}\hat * (A^\vee\otimes P_\xi))\otimes \bigotimes_{i=1}^{p-k} F_{\xi_i}^{(p+2,-1)}\otimes A^{p+2})\rightarrow&\\ &\rightarrow H^0(M_{A^{p+2}}^{\otimes k-1}\otimes (A^{p+2}\hat * (A^\vee\otimes P_\xi))\otimes \bigotimes_{i=1}^{p-k} F_{\xi_i}^{(p+2,-1)}\otimes A^{2(p+2)}) \end{eqnarray*} is surjective. But this follows from the inductive hypothesis and Proposition \ref{mult-pontr}.
We are left with proving Claim 1 for $k=1$. To this end we will apply the same reasoning as in the previous paragraph, only this time we use Theorem \ref{F-reg} with the weaker input that the sheaf in question is just $M$-regular.
\noindent \emph{Claim 2. The sheaf $A^{p+2}\hat * (M_{A^{p+2}} \otimes\bigotimes_{i=1}^{p-1} F_{\xi_i}^{(p+2,-1)} \otimes A^{p+2})$ is $M$-regular.}
\noindent \emph{Proof.} As before, the claim follows if we show that the locus of $\xi$ such that the multiplication map \begin{eqnarray*} &H^0(A^{p+2})\otimes H^0((A^{p+2}\hat * (A^\vee\otimes P_\xi))\otimes \bigotimes_{i=1}^{p-1} F_{\xi_i}^{(p+2,-1)}\otimes A^{p+2})\rightarrow&\\ &\rightarrow H^0((A^{p+2}\hat * (A^\vee\otimes P_\xi))\otimes \bigotimes_{i=1}^{p-1} F_{\xi_i}^{(p+2,-1)}\otimes A^{2(p+2)}) \end{eqnarray*} is surjective has codimension at least $2$. By Proposition \ref{mult-pontr}, this locus is precisely
$$\{~\xi~|~0_X\in B(A^{p+2}\hat * (F_\xi^{(p+2,-1)}\otimes \bigotimes_{i=1}^{p-1} F_{\xi_i}^{(p+2,-1)}\otimes A^{p+2}))\}.$$ By relative duality the dual of $$(A^{p+2}\hat * (F_\xi^{(p+2,-1)}\otimes \bigotimes_{i=1}^{p-1} F_{\xi_i}^{(p+2,-1)} \otimes A^{p+2}))\otimes A^\vee $$ is $$(A^{-p-2}\hat *( F_{-\xi}^{(-p-2,1)}\otimes \bigotimes_{i=1}^{p-1} F_{-\xi_i}^{(-p-2,1)} \otimes A^{-p-2}))\otimes A$$ By Proposition \ref{exchange} we have that \begin{eqnarray}\label{cagata} &R^i\hat{\mathcal S} ((A^{-p-2}\hat * (F_{-\xi}^{(-p-2,1)}\otimes \bigotimes_{i=1}^{p-1} F_{-\xi_i}^{(-p-2,1)} \otimes A^{-p-2}))\otimes A)\cong &\nonumber \\ &R^i{p_{\hat X}}_*((A^{-p-2}{\hat *}_{rel} A)\otimes p_X^*(F_{-\xi}^{(-p-2,1)}\otimes \bigotimes_{i=1}^{p-1} F_{-\xi_i}^{(-p-2,1)} \otimes A^{-p-2}))& \end{eqnarray} The key point, analogous to (\ref{iso2}) of the previous proof, is that \begin{eqnarray}\label{iso3} &R^i{p_{\hat X}}_*(((p+1)_X,1_{\hat X})^*((A^{-p-2}{\hat *}_{rel} A)\otimes F_{-\xi}^{(-p-2,1)}\otimes \bigotimes_{i=1}^{p-1} F_{-\xi_i}^{(-p-2,1)} \otimes A^{-p-2}))\cong&\nonumber\\ & \cong \left\{\begin{array}{ll}
0 &\textrm{if $i<g$} \\ V\otimes \widehat{A^{-p-1}}\otimes{\mathcal O}_{{\hat X}_{p+2}-\xi-\sum_{i=1}^{p-1}\xi_i-\frac{(p+1)(p+2)}{2}s(A)}
&\textrm{if $i=g$} \end{array}\right.& \end{eqnarray} where $V$ is a suitable vector space. \newline \noindent \emph{Proof of (\ref{iso3}).} By Proposition \ref{calculations}(c)(i) and (ii) we have that \begin{eqnarray*} &((p+1)_X,1_{\hat X})^*((A^{-p-2}{\hat *}_{rel} A))\otimes F_{-\xi}^{(-p-2,1)}\otimes \bigotimes_{i=1}^{p-1} F_{-\xi_i}^{(-p-2,1)} \otimes A^{-p-2})\cong &\\ &p_{\hat X}^*\widehat{A^{-p-1}}\otimes V\otimes p_X^*((p+1)_X^*A^{-(p+2)(p+1)}\otimes (p+2)_X^* A^{(p+1)^2}\otimes P_{-\xi}^{p+2}\otimes\bigotimes_{i=1}^{p-1}P_{-\xi_i}^{p+2})\otimes {\mathcal P}^{-p-2} &\\ \end{eqnarray*} where $V$ is the vector space $\bigotimes_{i=1}^{p-1}H^g(A^{-p-1}\otimes P_{{\xi}_i}^\vee)\otimes H^g(A^{-p-1}\otimes P_{\xi}^\vee)$. Therefore (\ref{iso3}) follows from Lemma \ref{standard}, noting that, by a standard calculation, $$(p+1)_X^*(A^{-(p+2)(p+1)}\otimes A^{-p-2})\otimes (p+2)_X^* A^{(p+1)^2} \cong P_{-s(A)}^{(p+1)^2(p+2)/2}.$$
\noindent The argument goes now as in the previous proof: we have first that $$R^i{p_{\hat X}}_*((A^{-p-2}{\hat *}_{rel} A) \otimes F_{-\xi}^{(-p-2,1)}\otimes \bigotimes_{i=1}^{p-1} F_{-\xi_i}^{(-p-2,1)} \otimes A^{-p-2})$$ is a direct summand of $$R^i{p_{\hat X}}_*((2_X,1_{\hat X})^*((A^{-p-2}{\hat *}_{rel} A) \otimes F_{-\xi}^{(-p-2,1)}\otimes \bigotimes_{i=1}^{p-1} F_{-\xi_i}^{(-p-2,1)} \otimes A^{-p-2})).$$
\noindent Summing up, the sheaf $$(A^{p+2}\hat * (F_\xi^{(p+2,-1)}\otimes \bigotimes_{i=1}^{p-1} F_{\xi_i}^{(p+2,-1)} \otimes A^{p+2}))\otimes A^\vee $$ satisfies the hypotheses of Theorem \ref{WIT}, and its Fourier jump locus is included in ${\hat X}_{p+2}-\xi-\sum_{i=1}^{p-1}\xi_i-\frac{(p+1)(p+2)}{2}s(A)$. Thus, by Theorem \ref{WIT}, we finally have that $$B(A^{p+2}\hat * (F_\xi^{(p+2,-1)}\otimes \bigotimes_{i=1}^{p-1} F_{\xi_i}^{(p+2,-1)}\otimes A^{p+2}))\subset \bigcup_{\eta\in {\hat X}_{p+2}}B(A\otimes P_{-\frac{(p+1)(p+2)}{2}s(A)-\sum\xi_i-\xi}\otimes P_\eta),$$ and the Claim follows since the base locus of $A$ is of codimension at least $2$.
\subsection*{\bf A conjecture based on the $M$-regularity index.}
As already mentioned in Section 3, Theorem \ref{syzygies} raises a natural question about a potentially general relationship between the equations and syzygies of $X$ in the embedding given by a power of a line bundle $A$, and the higher order properties of $A$, reflected in the $M$-regularity index $m(A)$ defined in \S3.
\begin{conjecture} \emph{ Let $p\ge m$ be non-negative integers. If $A$ is ample and $m(A)\ge m$, then $A^{\otimes k}$ satisfies $N_p$ for any $k\ge p+3-m$. } \end{conjecture}
This conjecture is a refinement of Lazarsfeld's conjecture, proved in \cite{pareschi}, which is the case $m(A)=0$, i.e. no conditions on $A$. Theorem \ref{syzygies} gives an affirmative answer to the conjecture for $m(A)=1$, which by Example \ref{base_div} happens precisely when $A$ has no base divisor. We remark though that the methods of this paper fail to apply for powers $A^k$ with $k\leq p+1$, so a new idea seems to be needed for the case of higher regularity indices.
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\end{document} | arXiv |
\begin{definition}[Definition:Upper Wythoff Sequence/Definition 2]
The '''upper Wythoff sequence''' is the Beatty sequence on the square $\phi^2$ of the golden section $\phi$.
It starts:
:$0, 2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, \ldots$
{{OEIS|A001950}}
\end{definition} | ProofWiki |
Area beneath $y=x$ from $-\infty$ to $\infty$
$$\int_{-\infty}^{\infty}x\,dx$$ According to my teacher, this improper integral diverges because "if one or both integrals diverge, the entire integral diverges." Evaluating it as a limit, however, it seems to cancel out and give $0$.
I understand this gives an indeterminate form, and that generally, it is incorrect to "cancel out infinity," but an indeterminate form doesn't mean that it can't be evaluated to diverge, as it seems to do in this case. Some intuition behind this conclusion also lies in the fact that either side is decreasing at the same rate, so it seems obvious that the area goes to $0$.
If it truly does diverge, then to what? it seems absurd to say that it blows up to $\pm\infty$.
calculus improper-integrals
edited Mar 7 at 5:36
let's have a breakdown
asked Mar 7 at 3:23
RoshanRoshan
$\begingroup$ odd function, it should be $0$ $\endgroup$ – Vasya Mar 7 at 3:31
$\begingroup$ You may find the answers here helpful. $\endgroup$ – jmerry Mar 7 at 3:36
$\begingroup$ Basically you have answered your own question: "generally, it is incorrect to cancel out infinity". Except for one thing: actually, it is always incorrect to cancel out infinity (when working in the real numbers), even though sometimes it will give you the right answer by accident. $\endgroup$ – David Mar 7 at 3:38
$\begingroup$ @Vasya: That's not the definition of the improper integral. An improper Riemann integral over the entire real line exists if and only if each of the integrals on $(-\infty,c]$ and on $[c,\infty)$ exist, for an arbitrary $c$, which requires two limits to exist. Here, neither of those limits exists. $\endgroup$ – Arturo Magidin Mar 7 at 3:40
$\begingroup$ Although the improper integral diverges, it has a Cauchy principal value of $0$. $\endgroup$ – Robert Israel Mar 7 at 3:59
The issue in assigning a value to $$ \int_{-\infty}^\infty x\,\mathrm{d}x $$ really involves what it means for a function $f$ to be integrable. Unfortunately, there is no short complete answer to this question. In the Lebesgue theory of integration, a function $f : \mathbb{R} \to \mathbb{R}$ is said to be integrable on $\mathbb{R}$ if $$ \int_{-\infty}^{\infty} \left\vert f(x)\right\vert\mathrm{d}x < \infty. $$ (Here we are ignoring any assumptions that are required for the above to make sense. In any case, these will always be satisfied for a continuous function and, in particular, your function $f(x) =x$.)
The requirement that $\int_{-\infty}^\infty |f|\mathrm{d}x < \infty$ is equivalent to asking that both $$ \int_{-\infty}^{\infty} f_+(x)\,\mathrm{d}x < \infty \quad \text{and} \quad \int_{-\infty}^{\infty} f_-(x)\,\mathrm{d}x < \infty $$ where $f_{+}(x) = \max(f(x), 0)$ and $f_-(x) = \max(-f(x),0)$. If $f$ is integrable according to the definition above, we then define \begin{equation}\label{eq:star}\tag{$\star$} \int_{-\infty}^\infty f(x)\,\mathrm{d}x = \int_{-\infty}^\infty f_+(x)\,\mathrm{d}x - \int_{-\infty}^\infty f_-(x)\,\mathrm{d}x \end{equation} which will be a finite number.
Clearly, the function $f(x) = x$ does not satisfy any of these hypothesis because \begin{align*} \int_{-\infty}^{\infty} f_+(x)\,\mathrm{d}x = \int_{0}^\infty x\,\mathrm{d}x = \infty. \end{align*}
Now, we go through all of this trouble to ensure that we never end up writing something along the lines of $\infty - \infty$ in \eqref{eq:star}, which cannot be made sense of.
However, as you have observed, something interesting happens with $f(x)=x$. For each $\alpha > 0$, you have shown that $$ \int_{-\alpha}^\alpha x\,\mathrm{d}x = \frac{\alpha^2 - \alpha^2}{2} = 0. $$ Hence, the limit $$ \lim_{\alpha \to \infty} \int_{-\alpha}^\alpha x\,\mathrm{d}x = 0 $$ exists and is well defined. This means that the number given by \begin{equation}\label{eq:dagger}\tag{$\dagger$} \int_{-\infty}^\infty x\,\mathrm{d}x \stackrel{?}{=} \lim_{\alpha \to \infty} \int_{-\alpha}^\alpha x\,\mathrm{d}x = 0 \end{equation} exists. Thus, the integral $\int_{-\infty}^\infty x\,\mathrm{d}{x}$ only exists in the improper sense (in this case, we are forced to use the Cauchy principle value as our definition of improper). In other words, $\int_{-\infty}^\infty x\,\mathrm{d}{x}$ should be interpreted as an improper integral (and even then, we need the Cauchy principle value). Although these do not make much sense in the Lebesgue sense (in which we require that $|f|$ be integrable), there are theories of integration that deal with these improper integrals (see the Gauge integral, for instance).
Short answer: Whether or not $\int_{-\infty}^\infty x\,\mathrm{d}x$ exists as an integral depends on the context. It does not exist as a Lebesgue (or Riemann) integral, but it does exist if you want to talk specifically about the value $$ \lim_{\alpha \to \infty} \int_{-\alpha}^\alpha x\,\mathrm{d}x $$
Edit: We also point out that the expression in \eqref{eq:dagger} would make a "bad" definition for an integral. To see why, consider first a (Lebesgue) integrable function $f : \mathbb{R} \to \mathbb{R}$. Then, $f$ will also be integrable on any interval $[a,b] \subset \mathbb{R}$. In fact, $f$ will be integrable on any interval of the form $(c,\infty)$. Moreover, the following additive rule would hold: $$ \int_{-\infty}^\infty f(x)\,\mathrm{d}x = \int_{-\infty}^c f(x)\,\mathrm{d}x + \int_{c}^\infty f(x)\,\mathrm{d}x. $$ Now, both of these properties are to be expected of an integral (after all, they are fundamental and very intuitive properties). However, despite existing as a limit, the "integral" $\int_{-\infty}^\infty x\,\mathrm{d}x$ fails both of these properties. Indeed, $$ \int_{c}^\infty x\,\mathrm{d}x = \infty \quad \text{and} \quad \int_{-\infty}^c x\,\mathrm{d}x = - \infty $$ for every $c \in \mathbb{R}$. Consequently, the additive rule $$ \int_{-\infty}^\infty x\,\mathrm{d}x \stackrel{?}{=} \int_{-\infty}^c x\,\mathrm{d}x + \int_{c}^\infty x\,\mathrm{d}x $$ also fails. In short, incorporating \eqref{eq:dagger} into our definition of the integral would cause us to lose many of the nice properties the integral satisfies. So, although we can partially avoid having $\infty - \infty$ in this case, we still end up breaking several familiar properties the integral should satisfy.
answered Mar 7 at 3:41
rolandcyprolandcyp
$\begingroup$ Okay, thanks, this makes sense, but I am still not completely understanding the purpose of these distinctions. Also, why do we at all "go through the trouble of avoiding $\infty - \infty$," in a case such as this where it is so easily simplifiable and really poses no problem at all and can indeed be made sense of? $\endgroup$ – Roshan Mar 7 at 7:14
$\begingroup$ Please see my edit. $\endgroup$ – rolandcyp Mar 7 at 15:26
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A low-latency communication protocol for target tracking in wireless sensor networks
Thu Ngo-Quynh1,
Vinh Tran-Quang2 &
Quan Nguyen-Trung1
Target tracking applications in wireless sensor networks need to achieve energy efficiency, tracking accuracy, and certain real-time constraints in response to fast-moving targets. From a layer view, an energy-efficient cross-layer communication protocol that consists of a medium access control layer and network routing layer is necessary for joint optimization. Due to the interference and contention over the wireless medium, the limited resources of battery-operated sensor nodes, and the dynamic topology of large-scale networks, this cross-layer design becomes a challenging task. In this research, we exploit a cluster routing algorithm over large-scale networks and propose a low-duty-cycle medium access control (MAC) algorithm to reduce collision, idle-listening, and overhearing. In addition, our work focuses on the joint optimization of routing and a MAC strategy for achieving a good trade-off between low delay, energy efficiency, and tracking accuracy. To deploy this protocol in a real tracking application, we also propose a clustering synchronization procedure that does not require distributing the global timing information over the complete network to achieve network-wide time synchronization. An analytical model and extensive simulations are proposed to evaluate and compare the performance of our work with existing protocols. Simulation and analysis results show that our approach achieves better communication delay and thus better tracking error while maintaining reasonable energy consumption compared to other cases.
Recent developments in sensor techniques have made wireless sensor networks (WSNs) available to many application domains. Most of these applications, such as battlefield surveillance and target tracking, address various types of real-time constraints in response to the physical world. For example, surveillance may require a sensor node to detect and classify a fast-moving target within 1 s before it moves out of the sensing range. Compared with the traditional distributed systems, achieving a low-latency guarantee for sensor networks is more challenging due to the following reasons. First, although the real-time performance is a key concern, it should be compatible with many other critical issues, such as energy efficiency due to the limited power of sensor nodes. For example, it is not efficient to activate the sensors all the time for only the benefit of a fast response. This naive approach severely reduces system lifetime. Second, a large-scale network of unreliable wireless links makes the tracking accuracy of a target not quite suitable for low-delay detection. Thus, the two most important objectives of tracking problems in WSNs are low delay and energy efficiency associated with tracking accuracy, which cannot be met concurrently. Therefore, the design of a WSN-based tracking system requires a trade-off between these considerations.
To provide a real-time guarantee, the authors in [1] present an analysis of end-to-end delay by giving a brief overview of tracking operations. Normally, after a target enters the area, nodes nearby are awakened to form a cluster to deliver aggregated reports to the base station (BS). More specifically, the end-to-end delay contains the following main phases. (1) Initial delay: Initial delay is the time required for the first node to start to sense the incoming target and confirm the detection. (2) Wake-up delay: After the initial delay, a cluster is formed to pass aggregated reports to the BS. To select a cluster with a reasonable size, nodes need to be awakened. The wake-up delay is the time required for an awakened node to wake up other sleeping nodes. (3) Aggregation delay: Each cluster is represented by a leader (called the cluster head—CH), which is responsible for collecting reports from cluster members. The CH, in turn, periodically transmits this report to the BS after the number of member reports exceeds a certain threshold. Aggregation delay is the time required to collect and process the detection reports from the member nodes. (4) Multi-hop delay: After cluster aggregation, the transmission from the CH to the BS causes a multi-hop transmission latency.
Among these delay components, the initial delay that consists of the hardware response, discrete sampling, etc. depends on the hardware structure of the sensor nodes, while three other elements (wake-up, aggregation, and multi-hop) depend mainly on the communication protocol implemented in the tracking system. From a layered view, these three latencies rely on the following components of the communication protocol. (1) Medium access control: The real-time property of a system depends on the effectiveness of the low-level medium access control (MAC) layer that is responsible for sharing the unreliable wireless medium. It plays a key role in determining the channel access delay and, thus, should provide a certain single-hop transmission time guarantee. (2) In addition, it has been observed that low-power sensors consume a significant amount of energy while idly listening in addition to the energy consumed during transmission and reception [2]. By controlling the fraction of time that sensor nodes are ACTIVE/INACTIVE, energy can be conserved. This technique is called a duty-cycle. However, duty-cycling the radio transceiver leads directly to the increase of the wake-up latency. (3) Even if a certain deadline can be provided in the MAC layer, the real-time property can still not be met if there is no guarantee in the network routing layer. The report of a tracked target's position from the CH transmitted to the BS over a large dynamic network should be bound by a certain multi-hop transmission time.
To guarantee low delay and improve energy efficiency for target tracking, it is important that all of these communication components (low-duty-cycle MAC and routing) in the protocol stack be optimized. There is some previous work in this area [1, 3–10]. However, this previous work for target tracking application almost exclusively focuses on separate components, and the issue of low latency is not fully solved nor is the trade-off between latency and energy efficiency sufficiently and explicitly addressed. In other words, a complete communication scheme based on cross-layer interaction providing energy efficiency and low delay for target tracking is still an open research area although the use of cross-layer techniques in WSN can help to achieve different objectives. Due to the interference and contention over the wireless medium, the limited resources of battery-operated sensor nodes, and the dynamic topology of large-scale networks, this cross-layer design is a challenging task. In this research, we exploit a cross-layer interaction involving cluster routing algorithm over large-scale networks and propose a low-duty-cycle MAC algorithm to reduce collision, idle-listening, and overhearing. In addition, our work focuses on the joint optimization of routing and MAC strategies to achieve a good trade-off between low delay, energy efficiency, and tracking accuracy.
Furthermore, the operation of a target tracking application often requires precise mapping of gathered sensor data with the time of the tracked target. To implement this application, the sensor nodes in the network need to have a common notion of time. Because the local clocks of the nodes operate independently and probably inaccurately [11], they need to be time-synchronized on a regular basis. Several time synchronization schemes have been extended for WSNs [12], taking into account some of the well-known constraints of WSNs. Normally, these schemes distribute global timing information over the entire network to achieve network-wide synchronization by using broadcast communication, but these solutions face severe challenges for target tracking. To overcome this problem in our cross-layer architecture, our cluster routing protocol is associated with a cluster working-cycle synchronization procedure that needs only to maintain the synchronized working cycles of nodes only within the clusters. This procedure helps to achieve precise mapping of gathered sensor data with the time of tracked target without implementing a network-wide time synchronization protocol. The contributions of our work are described as follows:
A new communication protocol CSP (Cluster-short Strobes-communication Protocol) consists of:
A cluster-based routing algorithm that reduces and balances the number of packets involved in communication. Thus, it minimizes the processing time (for searching the next-hop node towards the BS), energy consumed, and load differentiation between nodes. For this reason, this new approach decreases the multi-hop transmission time significantly compared to others of the same category while maintaining reasonable energy consumption.
A new low-duty-cycle MAC protocol: The data transmission of the above routing strategy produces high structured traffic of small packets towards the BS and only between the direct neighbors. This main characteristic reduces the packet overhearing and collision possibility and leads to the simple design of the CSP low-duty-cycle MAC protocol: an unbeaconed Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) without Request to Send/Clear to Send (RTS/CTS) associated with a short-preamble approach for waking up other nodes. These two main design features of CSP's low-duty-cycle MAC protocol help to avoid inefficient energy consumption and to minimize wake-up delay and also single-hop transmission delay.
A new cluster working-cycle synchronization procedure: the design of routing and low-duty-cycle strategies of the CSP leads to high structured traffic within clusters and from the CH towards the BS. In addition, the unbeaconed unslotted CSMA/CA algorithm associated with the low-duty-cycle scheme will cause a multi-hop asynchronous process of data transfer from the CH to the BS. Therefore, it is unnecessary to implement a complicated global timing procedure to achieve network-wide time synchronization. CSP, however, requires that the target distance estimation process realized by sensors be taken within a very short time interval to reduce "drifted" measurements due to target movements. To overcome this problem, we adopt a simple algorithm that maintains synchronization of working-cycles of nodes in a cluster without synchronizing the nodes' clocks.
We also present an analysis of delay produced by the CSP and compare to the other approaches of the same category (CSP using B-MAC [13] as an alternative low-duty-cycle algorithm and the CSP using Adaptive Routing Protocol with Energy Efficiency and Event Clustering for Wireless Sensor Networks (ARPEES) [14] as a routing strategy). The results from this analysis confirm that our CSP protocol produces the smallest delay among these schemes.
The performance of this communication protocol in terms of energy consumption, tracking delay, and trajectory accuracy is evaluated and compared with other similar protocols through simulation. The simulation results show that out new method achieves better communication delay and thus better tracking accuracy while maintaining reasonable energy consumption.
The remainder of the paper is organized as follows. Section 2 discusses the related work. Section 3 describes the system model. Section 4 presents the operation of the CSP protocol. Section 5 analyzes the delay produced by the CSP and compares it to others of the same category. Section 6 evaluates the performance of these communication protocols. The conclusion and future work are discussed in the last section.
Recently, communication protocols designed for tracking applications have almost exclusively focused on separate components (low-duty-cycle MAC and routing schemes) to provide energy efficiency. To the best of our knowledge, there are only a few complete communication schemes in the area that consider the importance of low latency associated with energy efficiency and tracking accuracy as the main design goals. In addition, time synchronization for target tracking is also rarely investigated.
EDAL [15] is an energy-efficient and delay-aware communication protocol that consists of a routing strategy associated with a compressing sensing algorithm. However, without low-duty-cycling mechanism and synchronization, it is difficult to achieve energy efficiency for uncompleted communication protocol EDA. RTSE [16] is a communication protocol that consists of data report and task execution for providing good performance by making trade-offs among delay, energy efficiency, and reliability when considering the characteristics of sensors and actuators, respectively. RTSE utilizes cluster-based routing strategy, and nodes are supposed to be synchronized using existed time synchronization schemes. The design goal of RTSE is similar to our work (energy efficiency, delay...), and RTSE also implements cluster-based routing schemes but RTSE has two disadvantages: it is difficult to implement RTSE for target tracking application in a network with hundreds synchronized nodes and without duty-cycling MAC, energy efficiency is hard to achieved in RTSE.
In [3, 4], the authors propose energy-efficient and low-latency target-tracking MAC (TT-MAC) and Distributed Time Division Multiple Access (D-TDMA) protocols based on TDMA that requires some authority to orchestrate activities within a network. This feature complicates the deployment of TT-MAC and D-TDMA in a multi-hop and large-scale network where nodes have limited resources. In addition, the real-time property of a MAC protocol based on a TDMA solution is difficult to achieve even using a tight scheduling scheme. Low-duty-cycle components of a communication protocol designed for tracking are proposed in [5, 6], called ELS Energy efficient low Latency Sleep Schedule and Minimal Contour Tracking Algorithm (MCTA). ELS differentiates two types of nodes, border and interior, each of which has a different sleeptime schedule. A border node is always on to communicate with interior nodes when the target appears. According to the network size, nodes are divided into different layers, and this differentiation makes this solution difficult to exploit in a large network of hundreds of nodes. MCTA conserves energy by letting only a minimum number of sensor nodes participate in communication by using the minimal tracking area based on the vehicular kinematics. Both ELS and MCTA are not concentrated on the design of low latency. Finally, t-tracking [17] is an interesting solution that achieves high tracking quality and energy efficiency. Consisting of a prediction scheme associated with duty-cycling and routing algorithms, t-tracking provides high quality tracking by utilizing mobile nodes and mobile sink.
Unlike classical approaches, Low Energy Self-Organizing Protocol (LESOP) [7] presents a cross-layer architecture of the application layer associated with the MAC layer and removes the transport and network layers. All the radio packets are simply broadcasted to the source node neighborhood wirelessly. LESOP controls the trade-off between the only tracking error and network energy consumption while not investigating the importance of system delay. Another cross-layer architecture [7] that consists of an extended 802.11 MAC and extended Dynamic Source Routing (DSR) algorithm. Unlike CSP, this cross-layer support does not consist of low-duty-cycle and synchronization schemes. Its main design feature is to reduce unnecessary routing maintenance. Therefore, this cross-layer DSR can not be implemented in the case of target tracking application. The multi-channel communication proposed in [18] is a special and interesting solution for reducing delay. The authors in this paper confirmed that decreased delay makes it possible to use higher sampling rates (or higher tracking performance) in network estimation applications and analyze only the dependencies between the communication protocol and the estimation parameters. The importance of delay required by certain real-time constraints in response to fast-moving targets is not evaluated.
Other solutions of routing protocol for target tracking are described in [9, 10, 19–21], called MRP-NEP: Non-Equal-Probability Multicast Routing. Protocol, HCTT: Hybrid Cluster-Based Target Tracking Protocol, OCO: Optimized Communication and Organization, PES: Prediction-based Energy Saving Scheme, and CTT&MAV: Mobile Target Tracking Scheme. The main purpose of these novel routing protocols is to reduce communication overhead by using different routing strategies. MRN-NEP utilizes non-equal-probabilistic forwarding. Sensor nodes forward packets with a probability that is mainly determined by how much the node's location deviates from the direction of target motion. HCTC, a dynamic clustering routing, constructs on-demand clusters at boundary regions. Nodes from different static clusters that detect the target can temporarily share information, and the tracking task can be handed over smoothly from one static cluster to another. PES is based on the fact that the movements of the tracked objects are sometimes predictable. This helps to reduce the transmission distance between the transmitter and receiver nodes and decrease the number of transmitted packets. OCO is an algorithm that ensures maximum accuracy of target tracking, efficient energy dissipation, and low computation overhead on the sensor nodes. CTT&MAV utilizes an energy-efficient clustering algorithm to form a Vonoroi-based diagram. However, these routing protocols have not focused on the importance of system latency.
Without a low-duty-cycle MAC and routing in a communication protocol, it is difficult for all previous work to assure low latency associated with energy efficiency and tracking accuracy. VigilNet [1] is one of the very few real-world tracking systems that simultaneously addresses energy efficiency, end-to-end real-time tracking, and accuracy by implementing a complete communication protocol of B-MAC [13] (as a low-duty-cycle MAC) associated with a Voronoi diagram [22] (as a routing algorithm). This system divides end-to-end delay into multiple sub-deadlines, each guaranteed by one system component. Wake-up, aggregation, and multi-hop delays are controlled by the B-MAC protocol and Voronoi diagram (that requires the employment of multiple BSs). Thus, the hard real-time property of the system is guaranteed at the expense of numerous BSs and increased price. However, VigilNet has not been evaluated regarding the performance of end-to-end delay by considering critical parameters of the communication protocol. In addition, the long-preamble approach of B-MAC still causes energy inefficiency and unnecessary delay compared to short-preamble approaches, especially when the hop count is high.
All these issues challenge us with the question: How can one design a low-delay communication protocol of low-duty-cycle MAC, routing, and synchronization schemes to provide good tracking accuracy while maintaining energy efficiency?
Measuring model
We assume that the BS has a fixed position at the edge of the network and infinite power. All the sensor nodes are identical, having fixed positions distributed uniformly over an area. Each sensor node is able to determine its position (geographical position or relative coordination in the concerned area) and the position of the BS during the deployment stage. Signals received from the target have the same original strength, which is known to the sensor. Sensors can estimate the distance to the target based on signal attenuation with some degree of error.
Sensing and network model
The delay from the beginning of the target localization mission until the time the result is returned is called senseDelay. The time period between two consecutive sensing processes is called sensePeriod. To track the position of a target precisely, we need several measurements that are taken in the same short time interval. Therefore, the sensing timers of sensor nodes need to be synchronized to provide enough measurements.
In this research, a simplified model of the physical layer imitating a typical transceiver is used to analyze the energy consumption and delay. Transceivers switch among IDLE, RX, and TX modes. The remaining energy will be updated when switching modes or by a periodical timer. Power consumption is calculated as follows:
$$ P=P_{RX}{\ast}~t_{RX}+P_{TX}{\ast}~t_{TX}+P_{IDLE}{\ast}~t_{IDLE} $$
where P R X , P T X , and P I D L E are the amounts of power consumed per unit of time when the node is in the corresponding state. t R X , t T X , and t I D L E are the durations of the node being in the RX, TX, and IDLE states, respectively.
CSP communication protocol
CSP routing scheme
After a target enters the area, nodes near the target become activated, form a cluster, and select a cluster head (CH) that is responsible for receiving aggregating sensed data from cluster members. The CH selects a next-hop node by a broadcasting process and delivers the aggregated report to this node. This routing process (called multi-hop) is repeated until the BS is reached, causing a high number of packets to be involved in broadcasting communication (or high energy consumption) and high multi-hop delay. To overcome these problems, we propose a cluster-based routing scheme (described in Fig. 1) to achieve low delay, energy efficiency, and good tracking accuracy. This scheme consists of three phases: initialization, cluster formation, and relaying of data to the BS with the main design features:
Data transmission towards the BS and pseudocode of CSP routing
Reducing multi-hop delay and energy consumption: During the initializing phase, two relay and backup nodes (RN and BN) of each sensor are selected according to a relay node function. After the cluster formation phase, a CH is selected according to a cluster head function. The selected CH utilizes this RN or BN as a next-hop candidate during the data-relaying phase. Without the broadcasting process, the CH can reduce the processing time for searching for the next hop. Thus, it minimizes multi-hop delay and energy consumption significantly.
Improving tracking accuracy: The data transmission from the CH to the RN or BN produces highly structured traffic of small packets towards the BS and between the only direct neighbors. This main design feature of the CSP cluster-based routing protocol reduces the packet overhearing and collision possibility and increases the tracking accuracy. In addition, each CH is always aware of its RN's and BN's energy levels and will replace them with new nodes if their energy falls below the critical level. Thus, it eliminates the chance that a node keeps sending data messages to its RN when the RN is already out of energy and hence improves the tracking accuracy.
Initializing phase
Each node in the network will choose a random time point in its initInterval (length of the initializing phase) to broadcast a RELAY_REQ {i D,E r e s (i),(x i ,y i )} packet containing its node identification iD, residual energy E r e s (i), and location (x i ,y i ) to its neighbors. When node i receives the packet, it chooses another random time point in the following interval of length waitRelayInfo to send back a RELAY_INFO {i s B S,E r e s (i),(x i ,y i ),d(i,B S}, where isBS is a flag specifying if node i is the BS or not and d(i,B S) is the distance from node i to the BS. It is assumed that all nodes know their own coordinates and those of the BS, and therefore each node can easily calculate the distance from it to the BS easily. At the end of its waitRelayInfo interval, nodei collects all its received RELAY_INFO packets from node j and assesses that information to choose a RN and a BN based on following relay node function:
$$ \begin{array}{l} {F_{{RN}}(i)=E_{{res}}(j)\ast\frac{1}{d(j,{{BS}})}\ast{cos}a_{j}} \\ \\ {{Max,\ SecondMax}F_{{RN}}(j)\mathop{\longrightarrow }\limits^{\forall i}_{{set\, as}} {RN,\ BN}} \end{array} $$
where a j is the angle created between j, i, and the BS. The c o s a j can be obtained by the following geometric calculation:
$$ {cos}a_{j}=\frac{{d(i,j)}^{2}+{d(i,{{BS}})}^{2}-{d(j,{{BS}})}^{2}}{2d\left(i,j\right)\ast d(i,{{BS}})} $$
According to this link cost function, two nodes that have relatively large residual energies, relatively small distances to the BS and relatively small angle values (or straight paths towards the BS) are selected as the RN and BN. Actually, the initInterval may not be identical for all nodes because the clocks of the nodes are not synchronized and it is hard to start the initializing phase in all nodes at the same time. However, as long as the initInterval is much longer than the startup time of the entire network, the initializing phase will provide an efficient routing topology.
Cluster forming
When a target enters the area, the measured signal of sensors exceeds the predefined threshold, and it will set a timer that has a duration of collectInterval with collectInterval <sensePeriod. This timer defines an interval for collecting measurements of neighbor nodes. At this time, a set of the nodes detecting the target forms a cluster and synchronizes their sensing cycles (described in the following section). After realizing this synchronization algorithm, the working cycles of the nodes are constructed by predefined consecutive working intervals with a fixed length. Therefore, the nodes in a synchronized cluster have synchronized working cycles. Each node i then selects a random point within collectInterval to broadcast a MEASUREMENT \(\{(x_{i},y_{i}),{\tilde {r}}_{i},E_{{res}}(i)\}\) packet containing its coordinates, the distance from it to the target \({\tilde {r}}_{i}\) and its current remaining energy. At the end of its collectInterval, each node will verify the number of collected measurements (including its own); if this number is at least three, the node checks the following cluster head function:
$$ F_{{{CH}}}=\frac{E_{{res}}(i)}{{\tilde{r}}_{i}} $$
The node then compares its own value to the others produced by the cluster head function; if its value is the largest, the node will promote itself to be the CH. At the end of collectInterval, the CH estimates the position of the target based on its collected measurements.
Relaying data to the BS
After the CH estimates the position of the target, it will relay these data to the BS by sending a packet to the RN or BN. More concretely, it creates a DATA_TO_BS packet and sends it to its RN and then sets a timer waitingRelayInfo to wait for E N E R G Y_I N F O{E r e s (R N)} from the RN. When a node receives a DATA_TO_BS, it checks its remaining energy and sends back this information to the CH in an ENERGY_INFO packet. At the end of the waitingRelayInfo timer, the CH performs the following route maintenance activities:
If the CH receives the ENERGY_INFO packet, it updates the energy information of its RN. If the remaining energy of the RN is less than the remaining energy of the BN by a predefined amount (switchingEnergy), the CH will switch the BN to the RN and the old RN to the BN.
If the CH cannot receive the ENERGY_INFO packet, it discards the current RN and chooses the BN as the new RN.
If both the BN and RN have remaining energies lower than a predefined threshold or experience an incident, the CH will request a new RN and BN the next time it needs to relay a packet.
After receiving the DATA_TO_BS packet, the RN, in turn, is served as the next CH and continues to relay this packet to its own RN. This process is repeated until reaching the BS.
CSP low-duty-cycle MAC protocol
The CSP low-duty cycle MAC protocol is designed to provide low delay, energy efficiency, and good tracking accuracy. As described in the previous section, the data transmission from the CH to the RN or the BN of the CSP cluster-based routing scheme leads to highly structured communication: towards the BS and only between direct neighbors. In addition, typical packets are small (approximately 100 bytes) because the in-network processing allows for reporting concise information instead of raw sensor readings. This traffic feature reduces the number of overhearing packets and the possibility of collision (and also tracking errors). Thus, we adopt the contention-based MAC protocol without the RTS/CTS mechanism and implement a simple back-off algorithm of unbeaconed and unslotted CSMA/CA. To minimize delay, CSP also implements a short strobe approach for waking up other nodes (based on the idea of X-MAC [23]). In conclusion, the CSP low-duty-cycle MAC protocol holds the following features with two working states: ACTIVE and INACTIVE.
In the ACTIVE state, the radio transceiver is switched to RX mode all the time (except when it is switched to TX mode in transmission), and the node can send or receive packets with minimal delay. To conserve energy, if a node has no activities (e.g., sending or receiving packets) for a predefined time interval, it will automatically change its working state to INACTIVE; this interval can be configured appropriately with the traffic load of the specific application.
In the INACTIVE state, the radio transceiver is switched to IDLE mode, which has very low power consumption; however, the transceiver cannot transmit or receive packets in this mode. To maintain connectivity with other nodes, a node in the INACTIVE state will periodically switch its transceiver to RX mode for a short interval. The interval in which the transceiver is switched to IDLE mode is called sleepInterval, and the interval in which the transceiver is switched to RX mode is called listenInterval. listenInterval enables the node to receive special control packets called strobes; therefore, strobes can be used by a node to wake up other nodes from the INACTIVE state.
When in the INACTIVE state, a node changes to the ACTIVE state in three cases:
It receives strobe packets addressed to it (in the short interval, its transceiver is switched to RX mode).
The upper layer wants to send a packet.
There is a demand from the upper layers to stay in the ACTIVE state for a specific amount of time or even forever (e.g., the BS does not need to ever be INACTIVE). In this case, the normal time-out for switching to the INACTIVE state is overridden. This feature helps to increase the availability of network nodes.
To wake up the receiver before sending a data packet, the sender transmits a sequence of short strobes containing the address of the sender and receiver, and each is sent during a time called strobeTime. When a node receives a strobe, it will check the receiver address. If the address is not its own or the broadcast address, it will ignore the strobe and not have to wake up unnecessarily. The time window between two consecutive strobes (called preservedInterval) is configured enough so that the sender can receive an ACK back from the receiver if the receiver successfully received the strobe. When the sender receives an ACK packet from the receiver, it stops sending the strobe sequence and sends the main packet immediately (Fig. 2). This mechanism reduces the per-hop latency and unnecessary energy spent waiting and transmitting.
Time line of sending strobes
The listenInterval of a node needs to be large enough so that a node can receives strobes from a sender. This time is calculated as follows:
$$ listenInterval=2~{\ast}~strobePeriod-preservedInterval $$
The maximum number of strobes in a sequence is fixed and calculated as follows:
$$ {nStrobe}_{{{max}}}=\left\lfloor \frac{sleepInterval}{strobePeriod}\right\rfloor $$
After sending all n S t r o b e m a x strobe packets, the sender will send the main packet even if it does not receive any ACK. (In most cases, this mechanism can wake up the receiver and deliver the main packet successfully).
CSP cluster synchronization algorithm
Within a cluster, each node is responsible for estimating the distance from it to the target and sending this information to the CH. If the clocks of these nodes are unsynchronized, the measurements of sensors will be "drifted" because of the target movement. To overcome this problem, we adopt a simple mechanism to synchronize the working cycles of nodes in a cluster. This mechanism is based on adjusting timers for triggering the sensing action of nodes at the same time by utilizing the broadcast nature of wireless communication and predicting the sensing cycles of nodes. It consists of two steps: synchronization during target detection and clustering and synchronization during target positioning at the CH and relaying data to the BS. After realizing this cluster synchronization algorithm, the working-cycles of nodes are constructed by predefined consecutive working intervals with fixed length. Therefore, nodes in a synchronized cluster have synchronized working cycles, and it is unnecessary to synchronize the clocks of nodes.
Synchronization during target detection and clustering
Initially, each node maintains its own sensing cycle and a SYNC flag set to FALSE. If a node detects the target for the first time, it changes its SYNC flag from FALSE to TRUE and broadcasts a SYNC_REQUEST packet to adjacent nodes. Its neighbor nodes, after receiving the SYNC_REQUEST packet, change their SYNC flag to TRUE and adjust their sensing cycles according to the cycle of the sender by calculating the following equation:
$$ diffTime=sensePeriod-senseDelay-txTime $$
where txTime is the calculated time for sending the SYNC_REQUEST packet. We can realize that the result of this equation (diffTime) is identical for all sensor nodes and can be calculated beforehand. In Fig. 3, node 2 broadcasts the SYNC_REQUEST packet. After receiving this packet, nodes 1 and 3 have to adjust their sensing cycles in two possible cases:
Sensing cycle adjustment (node 2 broadcasts SYNC_REQUEST)
Case 1: Node 1 has completed its sensing action when receiving the SYNC_REQUEST packet. To match its sensing cycle to the cycle of node 2, it calculates diffTime and adjusts its timer for the next sensing action.
Case 2: Node 3 has not completed its sensing action when receiving the SYNC_REQUEST packet; it also calculates diffTime and adjusts its timer for the next sensing cycle. However, in this case, the current cycle is shortened, and node 3 may not be able to complete the following steps. Therefore, node 3 discards the current sensing cycle and waits for the next cycle.
The SYNC flag of a node is reset to FALSE when the node cannot detect any target after receiving the sensing result from the sensor.
The synchronization step described previously provides an initial synchronized group for a node when it newly detects a target. However, this procedure is not enough to maintain a good number of sensor nodes to assure good tracking accuracy. The second synchronization step led by the CH is realized to solve this problem.
Synchronization during target positioning at the CH and relaying of data to the BS
The last action of the CH in a sensing cycle is maintaining the synchronization of the cluster. For re-synchronizing nodes in a cluster with the CH, the CH broadcasts a CH_BEACON packet at a specific moment near the end of each sensing cycle (called chBeaconTime). Similar to the previous step, nodes receiving this beacon will adjust their sensing cycle with the CH by calculating diffTimeaccording to the following equation:
$$ diffTime=sensePeriod - chBeaconTime - txTime $$
where txTime is calculated depending on the network configuration and may be optional. Because this beacon is broadcast at a moment near the end of a working cycle, the receiving nodes will cancel their remaining jobs of the current cycle if they have not finished (e.g., broadcast measurement, collecting measurements from others) and consider it to be the end of the cycle just like the CH. For example, if the sensePeriodis 0.5 s, CH_BEACON is broadcasted at 0.4 s into the cycle, and txTime is 0.001 s. If a node receives a CH_BEACON from the CH, whatever point in the current cycle it is, it will cancel its jobs of the current cycle and plan its next sensing action at 0.099 s after the current time.
Delay estimation
In this section, we evaluate the communication delay produced by CSP and utilize the denotations described in Table 1. These delay components can be calculated as:
$$ T_{{com-protocol}} = T_{{cluster}} + T_{{transmission}} $$
Table 1 Meaning of delay notifications
where T c l u s t e r is the time produced after the sensing signal until the end of the data exchange within the cluster. According to the routing strategy described before, this duration is constant and equals to collectInterval. We can have
$$ T_{{cluster}} = {collectInterval} $$
((10))
and T t r a n s m i s s i o n is the delay due to data transmission from the CH to the BS. This delay can be calculated as
$$ T_{{transmission}} = \sum^{n}_{i=1}{{t_{r}}_{i}} $$
where t r i is the delay produced at each hop for transmitting data to the next hop and n is the number of hops from the CH towards the BS. We now evaluate t r i for different communication protocols: CSP, CSP using B-MAC as an alternative low-duty-cycle scheme, and CSP using ARPEES [14] as an alternative cluster-based routing scheme. For simplicity, we assume here a perfect channel with no loss due to collision. In addition, no packets are lost due to buffer overflow at either sender or receiver.
\(t^{{CSP}}_{r_{i}}\) is caused at each hop by transmitting a DATA_TO_BS packet from the CH to the RN and receiving an ENERGY_INFO packet back from the RN. Before transmitting this packet, the CH needs to wake up the RN by sending a stream of nStrobe short strobes. This duration consists of the time for sending data and the time caused by all the elements of the frame sequence (back-off scheme, sending of an acknowledgement... of unbeaconed CSMA/CA IEEE 802.15.4 frame). Thus, we have:
$$ {\scriptsize\begin{aligned} t^{{CSP}}_{r_{i}} & = (nStrobe{-}1) * t_{strobePeriod} \\ & \quad+ t_{strobe\_TxTime} \\ & \quad+ t_{ACK\_TxTime} + t_{DATA\_TO\_BS\_TxTime} \\ & \quad+ t_{ENERGY\_INFO\_TxTime} \end{aligned}} $$
Similar to [24], \(t_{\textit {strobe\_TxTime}}\), \(t_{\textit {ACK\_TxTime}}\), \(t_{\textit {DATA\_TO\_BS\_TxTime}}\), and \(t_{\textit {ENERGY\_INFO\_TxTime}}\) can be calculated as follows:
$$ \begin{aligned} t_{strobe\_TxTime} = & \sum{t_{{BO}} + t_{{CCA}}} + t_{{frame}}({strobe})\\ & + + t_{{SIFS}} \end{aligned} $$
$$ \begin{aligned} t_{ACK\_TxTime} & = t_{{TA}} + \sum{t_{{BO}} + t_{{CCA}}} \\ & \quad+ t_{{frame}}({ACK}) + t_{{SIFS}} \end{aligned} $$
$$ \begin{aligned} t_{DATA\_TO\_BS\_TxTime} & = t_{{TA}} + \sum{t_{{BO}} + t_{{CCA}}} \\ & \quad+ t_{{frame}}(DATA\_TO\_BS) \\ & \quad+ t_{{LIFS}} \end{aligned} $$
$$ \begin{aligned} t_{ENERGY\_INFO\_TxTime} & = t_{{TA}} + \sum{t_{{BO}} + t_{{CCA}}} \\ & \quad+ t_{{frame}}(ENERGY\_INFO) \\ & \quad+ t_{{LIFS}} \end{aligned} $$
The back-off period is expressed as follows:
$$ T_{{BO}} = {BO}_{{slots}}~{\ast}~t_{{BO~slots}} $$
The number of back-off slots is a uniform random number in the interval (0,2BE−1) where BE is the back-off exponent, which has a minimum of 3. As we only assume one CH and a perfect channel, the BE will not change. In this case, the number of back-off slots can be represented as the mean of the interval (0,2BE−1)/2 or 3.5. Due to the characteristics of CSP, the transmission between the only CH and the RN is free, and the packet is transmitted after only one instance of back-off and CCA. In addition, we have the following relations in 802.15.4 where SP is the duration of one symbol [24]:
$$ \begin{array}{lcl} t_{{BO}} & = & 3.5~ {\ast}~ 20~ {\ast}~ {SP} \\ t_{{CCA}} & = & 8~ {\ast}~ {SP} \\ t_{{TA}} & = & 12~ {\ast}~ {SP} \\ t_{{SIFS}} & = & 12 ~{\ast}~ {SP} \\ t_{{LIFS}} & = & 40 ~{\ast}~ {SP} \end{array} $$
and the duration time for sending a packet is calculated as follows:
$$ t_{{frame}}({packet}) = \frac{{packet~size}}{{rate}} $$
With the different sizes of different packets given in Table 2 (strobe, ACK, D A T A_T O_B S, and E N E R G Y_I N F O) and the frequency band 2.4–2.4835 GHz with a symbol rate of 62.5 Kbaudps and bit rate of 250 Kbps (please refer to [24] for more information), Eqs. (13, 14, 15, 16) can be calculated, and the sum of all these equations is equal to 0.01232 s. Based on that, the average communication delay of our system for a wireless medium is
$$ \begin{aligned} t^{{CSP}}_{r_{i}} & = (nStrobe - 1)~ {\ast}~ {strobePeriod} \\ & \quad+ 0.01232~{s} \end{aligned} $$
Table 2 Simulation parameters
and the total communication delay produced by CSP is
$$ \begin{aligned} t^{{CSP}}_{com\_protocol} & = collectInterval + \\ & \quad+ \sum^{n}_{i=1}(nStrobe-1)~ {\ast}~ strobePeriod \\ & \quad+ 0.01232~{s} \end{aligned} $$
Finally, we estimate the delay of the communication protocol produced at each hop in the worst case, when the sender starts sending data after the receiver starts sleeping and after the maximum number of attempts of back-off (equal to 3). Similar to the previous case, we receive the following results:
$$ \begin{aligned} t_{r_{i}} & \leq (nStrobe_{{{max}}} - 1) * strobePeriod \\ & \quad+ 0.075664 ({s}) \end{aligned} $$
$$ t_{r_{i}} \leq sleepInterval + 0.075664 ({s}) $$
$$ \begin{aligned} t_{{com\_protocol}} & \leq collectInterval + \\ & \quad+ n(sleepInterval + 0.075664) ({s}) \end{aligned} $$
From the above equation, we realize that the total communication delay produced by CSP depends on collectInterval, sleepInterval, and also the number of hops towards the BS. Because the collectIntervalduration is constant and the number of hops towards the BS is a dynamic variable that cannot be controlled, we can adjust only sleepInterval to control the total communication delay of our communication protocol.
CSP using B-MAC as an alternative low-duty-cycle scheme
In this case, \(t^{{B-MAC}}_{r_{i}}\) is also caused by transmitting D A T A_T O_B S from the CH to the RN, but the B-MAC protocol produces different latency for the media access delay. Similar to the previous section, this delay can be evaluated as follows:
$$ \begin{aligned} t^{{B-MAC}}_{r_{i}} & = t^{{B-MAC}}_{LongPreamble\_TxTime} + \\ & \quad+ t_{ACK\_TxTime} + t_{DATA\_TO\_BS\_TxTime} \\ & \quad+ t_{ENERGY\_INFO\_TxTime} \end{aligned} $$
$$ \begin{aligned} t^{{B-MAC}}_{LongPreamble\_TxTime} & = \sum{t_{{BO}} + t_{{CCA}}} \\ & \quad+ t_{{frame}}({Long~preamble}) \\ & \quad+ t_{{LIFS}} \end{aligned} $$
Due to the routing strategy of CSP, the CH needs to wake up only the RN to transmit a D A T A_T O_B S packet. This characteristic helps to reduce the medium accessing time of a long preamble packet, or we can assume, in the ideal case, that a long preamble packet can successfully access the wireless medium after only one back-off. In this case, each hop delay produced by B-MAC is higher due to the large size of the long-preamble approach compared to the CSP protocol.
CSP using the ARPEES protocol as an alternative routing strategy
The cluster delay produced by ARPEES is similar to that of the CSP protocol. We investigate now the transmission delay produced at each hop by ARPEES. To search for the next relay node, the ARPEES protocol broadcasts a R E Q_R E L A Y message, waits to receive the A C K_R E L A Y from all its neighbors, and then evaluates its relay node function. Thus, different neighbors of the CH attempt to access the wireless medium to send their ACK messages. This feature leads to a significant increase of the delay produced at each hop compared to the CSP protocol.
$$ \begin{aligned} t^{{ARPEES}}_{r_{i}} & = t_{REQ\_RELAY\_TxTime} \\ & \quad+ t_{ACK\_TxTime} \\ & \quad+ t_{DATA\_TO\_BS\_TxTime} \end{aligned} $$
The main goal of the experimentation described in this article is to measure and compare the energy consumption, communication delay, and tracking accuracy of the proposed communication protocol as a function of varying the input parameters.
Experimental setups
We have developed an extensive computer simulation, implemented in the OMNET++ simulator [25], to evaluate performance of the three cases: CSP, CSP using B-MAC, and CSP using ARPEES. A total of 256 sensor nodes are evenly distributed over an area of (400×400)m2. A BS is positioned at the coordinates of (200,400) m. One target moves in the network area after 10 s from the start of the simulation; the target's speed ranges from 6 to 12 m/s. The sensing range of the target is 35 m, and the measurement error has a standard distribution with an expected value of 0 m and a standard deviation of 15 % of the sensing range. Each network node has a transmission range of 40 m (again, this representation is for simplicity). Sensor nodes work in sensing cycles with a length of 0.5 s. At the start of the simulation, every sensor node is provided with an identical amount of energy, 5 mWh.
Result analyses
In order to assure real-time property, the detection, localization, and report of target position need to be completed within each sampling interval. In other words, the end-to-end delay needs to be smaller than the sampling interval that equals to 0.5 s in our simulation. We perform experiments in three configurations (default CSP, CSP with B-MAC as duty-cycle scheme, and CSP with ARPEES as routing scheme) and plot variation of end-to-end delay, per hop delay, and total residual energy of these three cases in Figs. 4 and 5. The last figure (Fig. 6) presents tracking error and tracking trajectory. In each case, we simulate and select the simulation with the best tracking quality, and these three best cases of three configurations are utilized later for performance comparison.
a End-to-end delay, b delay per hop, and c total residual energy of CSP, CSP using B-MAC, and CSP using ARPEES
aEnd-to-end delay, b delay per hop, and c total residual energy of CSP according to sleepInterval
a End-to-end delay, b delay per hop, and c total residual energy of CSP according to sleepInterval
In Fig. 4, we realize that CSP achieves the best end-to-end delay performance where most of the packets arrive at the BS with end-to-end delay lower than 0.5 s. In addition, a small number of packets of CSP have high delay because the target moves to an area where sensor nodes have not been activated. When using B-MAC, end-to-end delay is higher but more stable compared to CSP. The last configuration, CSP using ARPEES causes the highest end-to-end delay. Unlike end-to-end delay, per-hop delay of the three cases varies differently: in CSP using B-MAC, packets have fixed per hop delay because of fixed size of long preamble, and therefore, the sum of per hop delay is stable and proportional to the number of hops towards BS. Note that in CSP using B-MAC, B-MAC works in heavy load by setting short checktime. In B-MAC [13], checktime is the duration between two consecutive CCA and similar to sleepInterval of CSP. In addition, the checktime of B-MAC is much shorter than the sleepInterval of the CSP case. In the worst case (when sender starts sending strobes right after receiver sleeps and receiver can only receive strobe at the end of sleepInterval), per hop delay produced by CSP is dominated by a full sleepInterval. In this worst case, default CSP works similarly as CSP using B-MAC and produces higher per hop delay because sleepInterval is higher than checktime (Fig. 4). However, this per-hop delay of CSP is generally much smaller than the per hop delay of the worst case due to two reasons.
Because of short preamble approach of CSP, the strobes train can be shortened by acknowledgment packets.
CSP efficient routing algorithm creates stable routing path in which, nodes stays in ACTIVE state and can reply immediately to the first few strobes. Therefore, most per hop delay will be the best case.
With CSP using ARPEES routing scheme, the process of finding relay node is repeated at each relay hop and causes too much overheads associated with high usage of wireless medium. After few hops, this process can cause delay higher than the period of traffic generation; therefore, congestion will occur.
We also track total energy of the network in the simulation in Fig. 4. By using duty-cycle protocol, the system achieves better energy efficiency compared to no duty-cycle solution (CSP with ARPEES). In the first 10 s of our simulations (initial phase of default configuration and configuration with B-MAC), the duty cycle is disabled for network initialization. During this interval, the power consumption is much higher than duty-cycle period. Result from this simulation shows that B-MAC has slightly better power consumption compared to two other cases. It is important to note that at any time, most of the nodes in network are in ACTIVE state and the total energy consumption of network is caused mainly by consumption of ACTIVE nodes. This consumption depends on the ratio of duration of transceiver in RX mode/duration of transceiver in IDLE mode. This ratio of CSP using B-MAC is very low because each node only needs to stay in RX mode to complete a CCA. In CSP, transceiver has to stay in RX mode to wait for a complete MAC packet. Therefore, the ratio of CSP using B-MAC is shorter than that of CSP.
As explained in Section 5, sleepInterval can be controlled in order to adjust the trade-off between communication delay and energy consumption of CSP. We repeat experiments when changing the sleepInterval, and the results (Fig. 5) consistently prove that the longer sleepInterval leads to the lower energy consumption and also to the higher delay. As stated above, increasing sleepInterval helps to decrease the ratio duration of transceiver in RX mode/duration of transceiver in IDLE mode and to reduce total power consumption. It is important to note that increasing sleepInterval does not influence the best case per hop delay, and it improves power consumption in the expense of increasing delay slightly. In addition, the worst case end-to-end delay causes additional packet loss by congestion at newly formed clusters and relay paths. Packet loss may lead to degraded tracking quality. Therefore, adjusting sleepInterval for controlling total residual energy must also depend on traffic load and tracking quality requirement.
Finally, the target tracking trajectories and tracking error of CSP, CSP using B-MAC, and CSP using ARPEES are plotted in Fig. 6. As described in the previous section, the characteristic of the low delay of CSP leads to better tracking error or accuracy, especially when system works under high load (due to high sampling rate and dense node distribution). The results from Fig. 6 also prove that CSP achieves the best performance of tracking accuracy, compared to CSP using B-MAC and CSP using ARPEES. Thus, low delay plays an important role of a target tracking system for providing good tracking accuracy.
The design of a low delay communication protocol plays an important role in providing good tracking accuracy and energy efficiency for a target tracking system in WSNs. In this work, we present a new communication protocol, CSP, that consists of a routing strategy associated with a low-duty-cycle MAC, and we focus on the joint optimization of these schemes to achieve low delay, low tracking error, and low energy consumption. Unlike other traditional tracking systems, CSP does not require global timing to achieve network-wide synchronization of the clocks of nodes. The design of a routing strategy and a low-duty-cycle MAC protocol of CSP leads to a simple cluster synchronization algorithm for synchronizing the working cycles of the nodes within a cluster. We present an analysis of the delay produced by CSP, CSP using B-MAC, and CSP using ARPEES and prove that CSP achieves the best delay performance. We also develop simulation experiments, and the results of such simulation show that CSP achieves the best delay performance, and thus the best tracking error, while maintaining reasonable energy consumption.
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This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.04-2012.06.
School of Information and Communication Technology, Hanoi University of Science and Technology, Hanoi, Vietnam
Thu Ngo-Quynh
& Quan Nguyen-Trung
School of Electronics and Telecommunications, Hanoi University of Science and Technology, Hanoi, Vietnam
Vinh Tran-Quang
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Correspondence to Thu Ngo-Quynh.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Ngo-Quynh, T., Tran-Quang, V. & Nguyen-Trung, Q. A low-latency communication protocol for target tracking in wireless sensor networks. J Wireless Com Network 2016, 33 (2016) doi:10.1186/s13638-016-0517-4
Target tracking | CommonCrawl |
\begin{document}
\title{Synthetic Aperture Imaging of Direction and Frequency Dependent
Reflectivities} \author{Liliana Borcea\footnotemark[1] \and Miguel
Moscoso \footnotemark[2] \and George Papanicolaou\footnotemark[3]
\and Chrysoula Tsogka\footnotemark[4]} \maketitle
\renewcommand{\arabic{footnote}}{\fnsymbol{footnote}} \footnotetext[1]{Department of Mathematics, University of Michigan,
Ann Arbor, MI 48109 {\tt [email protected]}} \footnotetext[2]{Gregorio Mill\'{a}n Institute, Universidad Carlos III
De Madrid, Madrid 28911, Spain {\tt [email protected]}} \footnotetext[3]{Stanford Mathematics Department, 450 Serra Mall Bldg.
380, Stanford CA 94305 {\tt [email protected]}} \footnotetext[4]{Mathematics and Applied Mathematics,
University of Crete and IACM/FORTH, GR-71409 Heraklion,
Greece {\tt [email protected]}} \renewcommand{\arabic{footnote}}{\arabic{footnote}}
\markboth{L. BORCEA, M. MOSCOSO, G. PAPANICOLAOU, AND
C. TSOGKA}{SYNTHETIC APERTURE IMAGING}
\begin{abstract} We introduce a synthetic aperture imaging framework that takes into consideration directional dependence of the reflectivity that is to be imaged, as well as its frequency dependence. We use an $\ell_1$ minimization approach that is coordinated with data segmentation so as to fuse information from multiple sub-apertures and frequency sub-bands. We analyze this approach from first principles and assess its performance with numerical simulations in an X-band radar regime. \end{abstract} \begin{keywords} synthetic aperture imaging, reflectivity, minimal support optimization. \end{keywords}
\section{Introduction}
{We introduce and analyze a novel algorithm for synthetic aperture radar (SAR) imaging, where a moving receive-transmit platform probes a remote region with signals $f(t)$ and records the scattered waves. The platform spans a large synthetic aperture so that high resolution images of the region may be obtained by processing the recorded data. A related application is inverse synthetic aperture radar (ISAR), where the receive-transmit antenna is stationary, and the synthetic aperture is due to the motion of an unknown scatterer. If this motion is known or can be estimated, the problem can be restated mathematically as SAR imaging of the scatterer, using the reference frame that moves with it.}
A schematic of the SAR imaging setup is in Figure \ref{fig:schematic}. The recordings $u(s,t)$ at the moving receive-transmit platform depend on two time variables: the slow time $s$ and the fast time $t$. The slow time parametrizes the trajectory of the platform, and it is discretized in uniform steps $h_s$, called the pulse repetition rate. At time $s$ the platform is at location ${\vec{\br}}(s)$. It emits the signal $f(t)$ and receives the backscattered returns $u(s,t)$. The fast time $t$ runs between consecutive signal emissions $t \in (0,h_s)$, and we assume a separation of time scales: The duration of $f(t)$ is smaller than the round trip travel time of the waves between the sensor and the imaging region, and the latter is smaller than $h_s$.
\begin{figure}
\caption{{Setup for imaging with a synthetic aperture.}}
\label{fig:schematic}
\end{figure}
In the usual synthetic aperture image formulation the reflectivity is modeled as a two dimensional function of location ${\vec{\mathbf{y}}}$ on a surface of known topography, say flat for simplicity. The assumption is that each point on the surface reflects the waves the same way in all directions, independent of the direction and frequency of the incident waves. This simplifies the imaging process and makes the inverse problem formally determined: the data are two-dimensional and so is the unknown reflectivity function.
The reflectivity can be reconstructed by the reverse time migration formula \cite{skolnik1970radar,Curlander,Jakowatz,cheney} \begin{equation}
{\mathcal{I}}({\vec{\mathbf{y}}}) = \sum_{j}\int dt\, u(s_j,t)
\overline{f\big(t-2\tau(s_j,{\vec{\mathbf{y}}})\big)}.
\label{eq:migration} \end{equation} Here $s_j$ are the slow time emission-recording instants, spaced by $h_s$, and the image is formed by superposing over the platform trajectory the data $u(s_j,t)$, match-filtered with the time reversed emitted signal $f(t)$, delayed by the roundtrip travel time $2\tau(s_j,{\vec{\mathbf{y}}}) = 2
|{\vec{\br}}(s_j)-{\vec{\mathbf{y}}}|/c$ between the platform location ${\vec{\br}}(s_j)$ and the imaging point ${\vec{\mathbf{y}}}$. The bar denotes complex conjugate and $c$ is the wave speed in the medium which is assumed homogeneous.
The assumption of an isotropic reflectivity may not always be justified in applications. Backscatter reflectivities are in general functions of five variables: the location ${\vec{\mathbf{y}}}$ on the known (flat) surface, the two angles of incidence and the frequency. Thus, the inverse problem is underdetermined and we cannot expect a reconstruction of the five dimensional reflectivity with a migration approach. Direct application of (\ref{eq:migration}) will produce low-resolution images of some effective, position-dependent reflectivity, and there will be no information about the directivity and frequency dependence of the actual reflectivity.
The reconstruction of frequency dependent reflectivities with synthetic aperture radar has been considered in \cite{cheney2013imaging}, where Doppler effects are shown to be useful in inversion, and in \cite{sotirelis2012study,elachi1990radar}, where data are segmented over frequency sub-bands, and then images are formed separately, for each data subset. Data segmentation is a natural idea, and we show here how to use it for reconstructing both frequency and direction dependent reflectivities.
The main result in this paper is the introduction and analysis of an algorithm for imaging direction and frequency dependent reflectivities of strong, localized scatterers. This algorithm is based on $\ell_1$ optimization. It reconstructs reflectivities of localized scatterers by seeking among all those that fit the data model the ones with minimal spatial support. Array imaging algorithms based on $\ell_1$ optimization are proposed and analyzed in \cite{baraniuk2007compressive,potter2010sparsity,chai2014imaging,
chai2013robust,
fannjiang2010compressed,fannjiang2012coherence,borcea2015resolution}. They consider only isotropic, frequency independent reflectivities.
A direct extension of $\ell_1$ optimization methods to imaging direction and frequency dependent reflectivities amounts to solving a grand optimization problem for a very long vector $\boldsymbol{\rho}$ of unknowns, the discretized reflectivity over spatial locations on the imaging grid, the angles of incidence/backscatter and the frequency. It has considerable computational complexity because of the high dimension of the space in which the discretized reflectivity vector lies. It also does not take into account the fact that many unknowns are tied to the same spatial location points within the discretized image window.
The synthetic aperture imaging algorithm introduced in this paper is designed to reconstruct efficiently direction and frequency dependent reflectivities by combining two main ideas: The first is to divide the data over carefully calibrated sub-apertures and frequency sub-bands, and solve an $\ell_1$ optimization problem to estimate the reflectivity for each data subset. {Data segmentation is useful assuming that the reflectivity changes continuously with the direction of probing and the frequency, so that we can approximate it by a piecewise constant function, pointwise in the imaging window. Over a sub-aperture of small enough linear size $a$, the platform receives scattered waves from a narrow cone with opening angle of the order $a/L$, where $L$ is the distance from the platform to the imaging window, and we can approximate the reflectivity by that at the center angle. Similarly, we can approximate the reflectivity by a constant over a small enough frequency sub-band. Then, we can use $\ell_1$ optimization to estimate the reflectivity as a function of location for each data subset.} The size of the sub-apertures and sub-bands determine the resolution of the reconstruction. The larger they are, the better the expected spatial resolution of the reflectivity. But the resolution is worse over direction and frequency dependence. The calibration of the data segmentation over sub-apertures and sub-bands reflects this trade-off. The second idea combines the $\ell_1$ optimizations by seeking reflectivities that have common spatial support. Instead of a single vector $\boldsymbol{\rho}$, the unknown is a matrix with columns of spatially discretized reflectivities. Each column corresponds to a direction of probing from a sub-aperture and a central frequency in a sub-band. The values of the entries in the columns are different, but they are zero (negligible) in the same rows. Moreover, the forward model, which is derived here from first principles, maps each column of the reflectivity matrix to the entries in the data subsets via one common reflectivity-to-data model matrix. The optimization can then be carried out within the multiple measurement vector (MMV) formalism described in \cite{malioutov05,cotter2005sparse,tropp2006algorithmsI,
tropp2006algorithmsII}.
The MMV formalism is used for solving matrix-matrix equations for an unknown matrix variable whose columns share the same support but have possibly different nonzero values. We show in this paper how to reduce the synthetic aperture imaging problem to an MMV format. The columns of the unknown matrix are associated with the discretized spatial reflectivities for different directions and frequencies. The solution of the MMV problem can be obtained with a matrix (2,1)-norm minimization where one seeks to minimize the $\ell_1$ norm of the vector formed by the $\ell_2$ norms of the rows of the unknown reflectivity matrix. The solutions obtained this way preserve the common support of the columns of the unknown matrix.
This paper is organized as follows. We begin in section \ref{sect:F} with the formulation of the imaging problem. We derive the data model, describe the complexity of the inverse problem, and motivate our imaging approach. The foundation of this approach is in section \ref{sect:MMVRed}, where we show how to reduce the imaging problem to an MMV format. The imaging algorithm is described in section \ref{sect:MMVAlg} and its performance is assessed with numerical simulations in section \ref{sect:NUM}. The presentation in sections \ref{sect:F}-\ref{sect:NUM} uses the so-called start stop approximation, which neglects the motion of the receive-transmit platform over the duration of the fast time data recording window. This is for simplicity and also because the approximation holds in the X-band radar regime used in the numerical simulations. However, the imaging algorithm can include Doppler effects due to the motion of the receive-transmit platform, as explained in section \ref{ap:Doppler}. We end with a summary in section \ref{sect:summary}.
\section{Formulation of the imaging problem} \label{sect:F} The data model is described in section \ref{sect:F1}. Then, we review briefly imaging of isotropic reflectivity functions via migration and $\ell_1$ optimization in section \ref{sect:F2}. The formulation of the problem for direction and frequency dependent reflectivities is in section \ref{sect:F3} \subsection{Synthetic aperture data model} \label{sect:F1} In synthetic aperture imaging we usually assume that the data $u(s,t)$, depending on the slow time $s$ and the fast time $t$, can be modeled with the single scattering approximation. For an isotropic and frequency independent reflectivity function $\rho = \rho({\vec{\mathbf{y}}})$ we have \begin{equation}
u(s,t) = \int \frac{d \omega}{2 \pi} \widehat u(s,\omega) e^{-i \omega t}, \label{eq:F1} \end{equation} with Fourier transform $\widehat u(s,\omega)$ given by \begin{equation}
\widehat u(s,\omega) \approx k^2 \widehat f(\omega) \int_{\Omega} d {\vec{\mathbf{y}}} \,
\rho({\vec{\mathbf{y}}}) \frac{\exp\big[2 i \omega \tau(s,{\vec{\mathbf{y}}})\big]}{(4
\pi|{\vec{\br}}(s)-{\vec{\mathbf{y}}}|)^2}.
\label{eq:F2} \end{equation} Here $k = \omega/c$ is the wavenumber and the integral is over points ${\vec{\mathbf{y}}}$ in $\Omega$, the support of $\rho$. The model (\ref{eq:F2}) uses the so-called start-stop approximation, where the platform is assumed stationary over the duration of the fast time recording window. We use this approximation throughout most of the paper for simplicity, and because it holds in the X-band radar regime considered in the numerical simulations. However, the results extend to other regimes, where Doppler effects may be important, as explained in section \ref{ap:Doppler}.
The inverse problem is to invert relation (\ref{eq:F2}) and thus estimate $\rho({\vec{\mathbf{y}}})$, given $u(s_j,t)$ at the slow time samples $s_j = (j-1)h_s$, for $j = 1, \ldots, N_s$. Here $h_s$ is the slow time sample spacing. The inversion is usually done by discretizing (\ref{eq:F2}), to obtain a linear system of equations for the unknown vector $\boldsymbol{\rho}$ of discretized reflectivities. The support $\Omega$ in (\ref{eq:F2}) is not known, so the inversion is done in a bounded search domain $\mathcal{Y}$ on the imaging surface, assumed flat. We call $\mathcal{Y}$ the image window. The reconstruction of $\boldsymbol{\rho}$ in $\mathcal{Y}$ is a solution of the linear system, as we review briefly in section \ref{sect:F2}.
The discretization of $\mathcal{Y}$ is adjusted so that it is commensurate with the expected resolution of the image in range and cross-range. The range direction is the projection on the imaging plane of the unit vector pointing from the imaging location ${\vec{\mathbf{y}}} \in \mathcal{Y}$ to the platform location. The cross-range direction is orthogonal to range. {It is well known in imaging that the range resolution is determined by the accuracy of travel time estimation, which in turn is determined by the temporal support of $f(t)$. Thus, it is useful to have a short pulse $f(t)$ whose support is of order $1/B$, where $B$ is the bandwidth. The range resolution with such pulses is of order $c/B$. The cross-range resolution is proportional to the central wavelength, which is why the emitted signals are typically modulated by high carrier frequencies $\omega_o/(2\pi)$. If $L$ is a typical distance between the platform and the imaging window and ${\mathcal{A}}$ is the length of the flight path, so that the platform receives waves within a cone of opening angle ${\mathcal{A}}/L$, the cross-range resolution is of the order $\lambda_o L/{\mathcal{A}}$, where $\lambda_o = 2 \pi c/\omega_o$ is the carrier wavelength. We assume that $\omega_o \gg B$, which is usually the case in radar. }
In synthetic aperture imaging applications like SAR, the platform emits relatively long signals $f(t)$ so as to carry sufficient energy to generate strong scatter returns, and thus high signal to noise ratios. Examples of such signals are chirps, whose frequency changes over time in an interval centered at the carrier frequency $\omega_o/(2\pi)$. To improve the precision of travel time estimation, and therefore range resolution, the returns $u(s_j,t)$ are compressed in time via match-filtering with the time reversed emitted signal \cite{skolnik1970radar}. Moreover, to remove the large phases and therefore avoid unnecessarily high sampling rates for the returns, the data are migrated via travel time delays calculated with respect to a reference point ${\vec{\mathbf{y}}}_o$ in the imaging window. The combination of these two data pre-processing steps is called down-ramping.
For the purposes of this paper it suffices to assume that $f(t)$
is a linear chirp, in which case the Fourier transform $|\widehat f(\omega)|^2$ of the compressed signal {has approximately the simple form} \begin{equation}
|\widehat f(\omega)| \approx |\widehat f(\omega_o)| 1_{[\omega_o-\pi B , \omega_o + \pi B]}(\omega),
\label{eq:L1.2} \end{equation} where $1_{[\omega_1,\omega_2]}(\omega)$ denotes the indicator function of the frequency interval $[\omega_1,\omega_2]$. The down-ramped returns are \begin{equation}
\int dt' \, u\Big(s,t-t' + 2 \tau(s,{\vec{\mathbf{y}}}_o)\Big) \overline{
f(-t')} = \int \frac{d \omega}{2 \pi} \, \overline{\widehat
f(\omega)}\widehat u(s,\omega) e^{-i \omega\big[t + 2\tau(s,{\vec{\mathbf{y}}}_o)\big]},
\label{eq:L1.0} \end{equation} and we let $\mathbf{d}$ be the vector of the samples of its Fourier transform \begin{equation}
\mathbf{d} = \left( d(s_j,\omega_l) \right)_{j = 1, \ldots N_s, l = 1, \ldots,
N_\omega}, \qquad d(s,\omega) = \overline{\widehat f(\omega)} \widehat
u(s,\omega)e^{-2 i \omega \tau(s,{\vec{\mathbf{y}}}_o)} .
\label{eq:L1.3} \end{equation} The size of the vector $\mathbf{d}$ is $N_s N_\omega$.
The linear relation between the unknown reflectivity vector $\boldsymbol{\rho}$ and the down-ramped data vector $\mathbf{d}$ follows from (\ref{eq:L1.3}) and (\ref{eq:F2}). We write it as \begin{equation}
\mathbf{A} \boldsymbol{\rho} = \mathbf{d},
\label{eq:L1.1} \end{equation} where the entries in $\boldsymbol{\rho} \in \mathbb{C}^{Q}$ are proportional to $\rho({\vec{\mathbf{y}}}_q)$, with ${\vec{\mathbf{y}}}_q$ the $Q$ discretization points of the image window $\mathcal{Y}$, and with the constant of proportionality taken to be the area of a grid cell. The reflectivity $\boldsymbol{\rho}$ is mapped by the reflectivity-to-data matrix $\mathbf{A} \in \mathbb{C}^{N_SN_\omega \times Q}$ to the data $\mathbf{d}$. The assumption of frequency independent reflectivity leads to a set of decoupled systems of equations $\mathbf{A}(\omega_l) \boldsymbol{\rho} = \mathbf{d}(\omega_l)$ indexed by the frequency $\omega_l$, where the entries of the $N_s \times Q$ matrices $\mathbf{A}(\omega_l)$ are \begin{equation}
\mathbf{A}_{j,q}(\omega_l) = \frac{k_l^2|\widehat f(\omega_l)|^2}{ (4 \pi
|{\vec{\br}}(s_j)-{\vec{\mathbf{y}}}_q|)^2} e^{2 i \omega_l \big[
\tau(s_j,{\vec{\mathbf{y}}}_q)-\tau(s_j,{\vec{\mathbf{y}}}_o)\big]}.
\label{eq:L1.4} \end{equation} Here $k_l = \omega_l/c$, $l = 1, \ldots, N_\omega$, $j = 1, \ldots, N_s,$ and $ q = 1, \ldots, Q.$
\subsection{Imaging isotropic reflectivities} \label{sect:F2} Imaging of the isotropic reflectivities amounts to inverting the linear system (\ref{eq:L1.1}). When this system is underdetermined, there are two frequently used choices for picking a solution: either minimize the Euclidian norm of $\boldsymbol{\rho}$ or its $\ell_1$ norm. The first choice gives \begin{equation}
\boldsymbol{\rho} = \mathbf{A}^\dagger \mathbf{d},
\label{eq:F3} \end{equation} where $\mathbf{A}^\dagger$ is the pseudo-inverse of $\mathbf{A}$. If $\mathbf{A}$ is full row rank, $ \mathbf{A}^\dagger = \mathbf{A}^* (\mathbf{A} \mathbf{A}^*)^{-1}$. The inversion formula (\ref{eq:F3}) also applies to overdetermined problems, where $\boldsymbol{\rho}$ is the least squares solution and $\mathbf{A}^\dagger = (\mathbf{A}^* \mathbf{A})^{-1} \mathbf{A}^*, $ for full column rank $\mathbf{A}$. The choice of the imaging window $\mathcal{Y}$ and its discretization is an essential part of the imaging process and, depending on the objectives and available prior information, we may be able to control whether the system (\ref{eq:L1.1}) is overdetermined or not. We explain in Appendix \ref{ap:discr} that by discretizing $\mathcal{Y}$ in steps commensurate with expected resolution limits we can make the columns of $\mathbf{A}$ nearly orthogonal. This means that in the overdetermined case $\mathbf{A}^* \mathbf{A}$ is close to a diagonal matrix. We also shown in Appendix \ref{ap:discr} that in the underdetermined case, for coarse enough sampling of the slow time $s$ and frequency $\omega$, the rows of $\mathbf{A}$ are nearly orthogonal, and therefore $\mathbf{A} \mathbf{A}^\star$ is close to a diagonal matrix. Thus, in both cases, $\mathbf{A}^\dagger$ is approximately $\mathbf{A}^\star$ up to multiplicative factors, and we can therefore image the support of $\boldsymbol{\rho}$ with $\mathbf{A}^*\mathbf{d}$. This is in fact the migration formula (\ref{eq:migration}) written in the Fourier domain, up to a geometrical factor, since the amplitude in (\ref{eq:L1.4}) is approximately constant for platform trajectories that are shorter than the imaging distance and for bandwidths $B \ll \omega_o$.
If we know that the imaging scene consists of a few strong, localized scatterers, as we assume here, a better estimate of $\boldsymbol{\rho}$ is given by the optimization \begin{equation}
\min \|\boldsymbol{\rho}\|_1 \quad \mbox{such that} ~ ~ \|\mathbf{A} \boldsymbol{\rho} -\mathbf{d}\|_2 \le
\epsilon.
\label{eq:L1.5} \end{equation}
Here $\epsilon$ is an error tolerance, commensurate with the noise level in the data, and $\|\cdot\|_1$ and $\|\cdot \|_2$ are the $\ell_1$ and the Euclidian norm, respectively. We refer to \cite{baraniuk2007compressive,potter2010sparsity,chai2013robust,
fannjiang2010compressed,fannjiang2012coherence} for studies of imaging with $\ell_1$ optimization. The main result in this context is that when there is no noise so that $\epsilon=0$, the reflectivities are recovered exactly provided that the inner products of the normalized columns of $\mathbf{A}$ are sufficiently small. An extension of the optimization to nonlinear data models that account for multiple scattering effects in $\mathcal{Y}$, is considered in \cite{chai2014imaging}. A resolution study of imaging with $\ell_1$ optimization is in \cite{borcea2015resolution}.
\subsection{Imaging direction and frequency dependent reflectivities} \label{sect:F3} In general, backscatter reflectivities are functions of five variables: the location ${\vec{\mathbf{y}}} \in \mathcal{Y}$, the unit direction vector ${\vec{\bf m}}$ and the frequency $\omega$. Hence, \begin{equation}
\rho = \rho({\vec{\mathbf{y}}},{\vec{\bf m}},\omega).
\label{eq:F4} \end{equation} This means that the down-ramped data model is more complicated than assumed in equations (\ref{eq:F2}) and (\ref{eq:L1.0}) or, equivalently, after discretization, in (\ref{eq:L1.3})-(\ref{eq:L1.4}). In integral form it is given by \begin{align}
d(s,\omega) &= \overline{\widehat f(\omega)} \widehat u(s,\omega) e^{-2 i \omega
\tau(s,{\vec{\mathbf{y}}}_o)} \nonumber \\ &= k^2 |\widehat f(\omega)|^2
\int_{\Omega} d {\vec{\mathbf{y}}} \, \rho(
{\vec{\mathbf{y}}},{\vec{\bf m}}(s,{\vec{\mathbf{y}}}),\omega) \frac{\exp \Big[ 2 i \omega
\big[ \tau(s,{\vec{\mathbf{y}}})-\tau(s,{\vec{\mathbf{y}}}_o) \big] \Big]}{\big(4 \pi
|{\vec{\br}}(s)-{\vec{\mathbf{y}}}|\big)^2},
\label{eq:F5} \end{align} where ${\vec{\bf m}}(s,{\vec{\mathbf{y}}})$ is the unit vector pointing from the platform location ${\vec{\br}}(s)$ to ${\vec{\mathbf{y}}} $ in the image window $\mathcal{Y}$. In discretized form we still have a linear system like (\ref{eq:L1.1}), except that now $\boldsymbol{\rho}$ is a vector of $Q N_s N_\omega$ unknowns, the discretized values of $\rho$ in the image window $\mathcal{Y}$.
Extending the inversion approaches described in the previous section to this model means inverting approximately the matrix $\mathbf{A}$ with a very large number of columns. We cannot expect the migration formula (\ref{eq:migration}) to give an accurate estimate of the reflectivity as a function of five variables, as pointed out in the introduction. The $\ell_1$ optimization approach works, but it becomes impractical for the large number $Q N_s N_\omega$ of unknowns. Moreover, it does not take into account the fact that the entries in $\boldsymbol{\rho}$ indexed by the slow time and frequency pairs $(j,l)$, with $j = 1, \ldots, N_s$ and $l = 1, \ldots, N_\omega$, refer to the same locations ${\vec{\mathbf{y}}}_q$ on the imaging grid.
The imaging approach introduced in this paper gives an efficient way of estimating direction and frequency dependent reflectivity functions of strong localized scatterers in $\mathcal{Y}$. It uses an approximation of the model (\ref{eq:F5}), motivated by the expectation that the backscatter reflectivity should not change dramatically from one platform location to the next and from one frequency to another. Instead of discretizing $\rho$ over all five variables at once, we discretize it only with respect to the location in the image window $\mathcal{Y}$, for one probing direction and frequency at a time. To do so, we separate the data over subsets defined by carefully calibrated sub-apertures and sub-bands, and freeze the direction and frequency dependence of the reflectivity for each subset. The grand optimization is divided this way into smaller optimizations for $Q$ unknowns, which are then coupled by requiring that the unknown vectors share the same spatial support in the imaging window $\mathcal{Y}$.
\section{Reduction to the Multiple Measurement Vector framework} \label{sect:MMVRed} \begin{figure}\label{fig:geometry}
\end{figure} We present here an analysis of how we can write the linear relation between the direction and frequency dependent reflectivity and the data as a linear matrix system \begin{equation}
\boldsymbol{\mathbb{A}} {\bf X} = {\bf D},
\label{eq:MMVEQ} \end{equation} where the unknown is the matrix ${\bf X}$ with $Q$ rows. The entries in the rows correspond to the discretization of this reflectivity at the $Q$ grid points ${\vec{\mathbf{y}}}_q$ in $\mathcal{Y}$. Each column of ${\bf X}$ depends on the reflectivity at the backscattered direction defined by the center of a sub-aperture and the center frequency of a sub-band. The data are segmented over ${\mathcal{N}}_\alpha$ sub-apertures and ${\mathcal{N}}_\beta$ sub-bands and are grouped in the matrix ${\bf D}$. The objective of this section is to describe the data segmentation and derive the linear system (\ref{eq:MMVEQ}), which can be inverted with the MMV approach as explained in section \ref{sect:MMVAlg}.
We begin in section \ref{sect:single} with a single sub-aperture and sub-band. We show in Lemma \ref{lem.1} that with proper calibration of the sub-aperture and sub-band size, the reflectivity-to-data matrix has a simple approximate form. Its entries have nearly constant amplitudes while the phases depend linearly on the slow time and frequency parametrizing the data subset. This simplification allows us to transform the linear system via coordinate rotation to a reference one, for all data subsets, as shown in section \ref{sect:MMVProb}. The matrix $\boldsymbol{\mathbb{A}}$ in (\ref{eq:MMVEQ}) corresponds to the reference sub-aperture and sub-band, and the statement of the result is in Proposition \ref{lem.2}.
\subsection{The sub-aperture and sub-band segmentation} \label{sect:setup} We enumerate the sub-apertures by $\alpha = 1, \ldots, {\mathcal{N}}_\alpha$, and denote by $s_\alpha^\star$ the slow time that corresponds to their center location ${\vec{\br}}(s_\alpha^\star)$. The choice of the sub-aperture size $a$ is important, and we address it in the next section. For now it suffices to say that it is small enough so that we can approximate it by a line segment, as illustrated in Figure \ref{fig:geometry}. The unit tangent vector along the trajectory, at the center of the sub-aperture, is denoted by ${\vec{\bf t}}_\alpha$, and the platform motion will be assumed uniform, at speed $V {\vec{\bf t}}_\alpha$. The unit vector from the reference location ${\vec{\mathbf{y}}}_o$ in the image window to ${\vec{\br}}(s_\alpha^\star)$ is ${\vec{\bf m}}_\alpha$. We call it the range vector for the $\alpha$ sub-aperture. The range (distance) to the imaging window is \begin{equation}
L_\alpha = |{\vec{\br}}(s_\alpha^\star)-{\vec{\mathbf{y}}}_o|. \end{equation} Each sub-aperture is parametrized by the slow time offset from $s_\alpha^\star$, denoted by \begin{equation}
\Delta s = s - s_\alpha^\star \in \Big[-\frac{a}{2V},\frac{a}{2V}
\Big].
\label{eq:M1} \end{equation} We do not index it by $\alpha$ because it belongs to the same interval for each sub-aperture. The discretization of $\Delta s$ is at the slow time sample spacing $h_s$, and there are \[n_s = \frac{a}{V h_s} + 1\] sample points, where $a/(V h_s)$ is rounded to an integer. Similarly, we divide the bandwidth in ${\mathcal{N}}_\beta$ sub-bands of support $b \le B$, centered at $\omega_\beta^\star$, and let $\Delta \omega$ be the frequency offset \begin{equation}
\Delta \omega = \omega - \omega_\beta^\star \in \Big[-\pi b,\pi b\Big].
\label{eq:M3} \end{equation} We sample the sub-band with $n_\omega$ points.
The reflectivity dependence on the direction and frequency is denoted by the superscript pair $(\alpha,\beta)$, and by discretizing it with the $Q$ points in $\mathcal{Y}$ we obtain the vector of unknowns $\boldsymbol{\rho}^{(\alpha,\beta)} \in \mathbb{C}^{Q}$. It is mapped to the data vector $\mathbf{d}^{(\alpha,\beta)}$ with entries given by the samples of $d(s_\alpha^\star + \Delta s, \omega_\beta^\star + \Delta \omega)$. The mapping is via the $n_s n_\omega \times Q$ reflectivity-to-data matrix $\mathbf{A}^{(\alpha,\beta)}$ described in Lemma \ref{lem.1}. \subsection{Reflectivity-to-data model for a single sub-aperture and sub-band} \label{sect:single} Here we explain how we can choose the size of the sub-apertures and frequency sub-bands so that we can simplify the reflectivity-to-data matrix. The calibration depends on the size of the imaging window $\mathcal{Y}$, which is quantified with two length scales \begin{equation}
{{Y}_\alpha} = \max_{q=1,\ldots Q} |({\vec{\mathbf{y}}}_q-{\vec{\mathbf{y}}}_o) \cdot {\vec{\bf m}}_\alpha|,
\label{eq:M4} \end{equation} and \begin{equation} {{Y}_\alpha^\perp} =
\max_{q=1,\ldots Q} |\mathbb{P}_\alpha({\vec{\mathbf{y}}}_q-{\vec{\mathbf{y}}}_o)|.
\label{eq:M4p} \end{equation} Here $\mathbb{P}_\alpha = I - {\vec{\bf m}}_\alpha {\vec{\bf m}}_\alpha^T$ is the projection on the cross-range plane orthogonal to ${\vec{\bf m}}_\alpha$, and $I$ is the identity matrix. The length scale ${{Y}_\alpha}$ gives the size of $\mathcal{Y}$ viewed from the range direction ${\vec{\bf m}}_\alpha$, and ${{Y}_\alpha^\perp}$ is the cross-range size.
The first constraints on the aperture $a$ and the cross-range size ${{Y}_\alpha^\perp}$ of the imaging window state that they are not too small, and thus imaging with adequate resolution can be done with the data subset. Explicitly, we ask that for all $\alpha = 1, \ldots, {\mathcal{N}}_\alpha$, \begin{align}
\label{eq:M5}
\frac{a^2}{\lambda_o L_\alpha} \gtrsim \frac{a {{Y}_\alpha^\perp}}{\lambda_o L_\alpha}
\gtrsim \frac{({{Y}_\alpha^\perp})^2}{\lambda_o L_\alpha} & \gtrsim 1. \end{align} The inequalities on the left involve { two Fresnel numbers $a^2/(\lambda_o L_\alpha)$ and $({{Y}_\alpha^\perp})^2/(\lambda_o L_\alpha)$, whose magnitudes define the imaging regime. If these numbers were small, we would be in a Fraunhofer diffraction regime, with approximately planar wavefronts on the scale of the sub-aperture and of the size of the imaging window. We consider a Fresnel diffraction regime, where these numbers are larger and we can get better resolution of images.} The cross-range resolution is $\lambda_o L_\alpha/a$, and naturally, the middle inequality in (\ref{eq:M5}) says that the image window is larger than the resolution limit. In the range direction we suppose that \begin{equation}
{{Y}_\alpha} \gtrsim \frac{c}{b} \gg \lambda_o,
\label{eq:M6} \end{equation} where $c/b$ is the range resolution for the sub-bands, and we used that $b \le B \ll \omega_o$.
{While we would like to have $a$ and $b$ large so as to get good spatial resolution of the unknown reflectivity, we recall that $\rho$ is frozen in our discretization in the small frequency sub-band and in the narrow cone of opening angle of the order $a/L_\alpha$, with axis defined by the center ${\vec{\br}}(s^\star_\alpha)$ of the sub-aperture and the reference point ${\vec{\mathbf{y}}}_o$. The larger $a$ and $b$ are, the coarser the estimation of the direction and frequency dependence of $\rho$. The more rapid the variation of $\rho$ with direction and frequency, the smaller $a$ and $b$ should be to represent it, at the expense of resolution. }
There is also a trade-off between resolution and the complexity of the inversion algorithm. By constraining $a$ and $b$ so that \begin{align}
\frac{b}{\omega_o} \frac{{{Y}_\alpha^\perp}}{\lambda_o L_\alpha/a} \ll 1, \label{eq:M8} \end{align} and \begin{align}
\frac{a^2{{Y}_\alpha}}{\lambda_o L_\alpha^2} \ll 1, \qquad \frac{a^2{{Y}_\alpha^\perp}}{\lambda_o
L_\alpha^2} \ll 1, \label{eq:M10} \end{align} we can simplify the mapping between the reflectivity and the data subset, as stated in Lemma \ref{lem.1}. This simplification allows us to use the efficient MMV framework to solve the large optimization problem for the entire data set, by considering jointly the smaller problems for the segmented data in an automatic way. The key observation here is that the unknown reflectivities for each data subset share the same spatial support. This is what the MMV formalism is designed to capture.
The next lemma gives the form of the reflectivity-to-data matrix in the linear system \begin{equation}
\mathbf{A}^{(\alpha,\beta)} \boldsymbol{\rho}^{\alpha,\beta} = \mathbf{d}^{(\alpha,\beta)},
\label{eq:lem.1p} \end{equation} for the $(\alpha,\beta)$ data subset. It is an approximation of the system (\ref{eq:L1.1}) restricted to the rows indexed by the $n_s$ slow times in the $\alpha-$aperture and the $n_\omega$ frequencies in the $\beta-$band. The expression of $\mathbf{A}^{(\alpha,\beta)}$ is derived in appendix \ref{ap:proofs}. \begin{lemma}
\label{lem.1}
Under the assumptions (\ref{eq:M5})-(\ref{eq:M10}), and with the
pulse model (\ref{eq:L1.2}), the matrix
$\mathbf{A}^{(\alpha,\beta)}$ consists of $n_\omega$ blocks
$\mathbf{A}^{(\alpha,\beta)}(\Delta \omega_l)$ indexed by the frequency offset
$\Delta \omega_l$, for $l = 1, \ldots, n_\omega$. Each block is an $n_s
\times Q$ matrix with entries defined by
\begin{align}
A^{(\alpha,\beta)}_{j,q}(\Delta \omega_l) = \frac{k_o^2 |\widehat
f(\omega_o)|^2}{(4 \pi L_\alpha)^2} \exp \Big\{ -2 i (k_\beta +
\Delta \omega_l/c) {\vec{\bf m}}_\alpha \cdot {\vec{\mathbf{y}}}_q \nonumber \\ -2 i k_\beta
V \Delta s_j \frac{ {\vec{\bf t}}_\alpha \cdot \mathbb{P}_\alpha \Delta
{\vec{\mathbf{y}}}_q}{L_\alpha} + i k_\beta \frac{\Delta {\vec{\mathbf{y}}}_q \cdot
\mathbb{P}_\alpha \Delta {\vec{\mathbf{y}}}_q}{L_\alpha} \Big\},
\label{eq:lem1}
\end{align}
where $k_\beta = \omega_\beta^\star/c$, and $\Delta {\vec{\mathbf{y}}}_q = {\vec{\mathbf{y}}}_q -
{\vec{\mathbf{y}}}_o$. \end{lemma}
\subsection{Multiple sub-aperture and sub-band model as an MMV system} \label{sect:MMVProb} It remains to show how to write equations (\ref{eq:lem.1p}) in the matrix form (\ref{eq:MMVEQ}) with a reflectivity-to-data matrix independent of the sub-apertures and sub-bands. This is accomplished via a rotation, that brings all the sub-apertures to a single reference sub-aperture. But to do this, we need to know that each data subset has a similar view of the image window. Mathematically, this is expressed by the following two additional constraints on $a$ and $b$ \begin{align}
\max_{1 \le \alpha \le {\mathcal{N}}_\alpha, 1 \le q \le Q} \frac{b}{c} \big|
({\vec{\bf m}}_\alpha - {\vec{\bf m}}_1) \cdot \Delta {\vec{\mathbf{y}}}_q \big| \ll 1,
\label{eq:lem2.1} \end{align} and \begin{align}
\max_{1 \le \alpha \le {\mathcal{N}}_\alpha, 1\le \beta \le {\mathcal{N}}_\beta, 1 \le q
\le Q} \Big| \Big(\frac{a k_\beta}{L_\alpha} {\vec{\bf t}}_\alpha \cdot
\mathbb{P}_\alpha -\frac{a k_1}{L_1} {\vec{\bf t}}_1 \cdot \mathbb{P}_1 \Big)
\Delta {\vec{\mathbf{y}}}_q\Big| \ll 1,
\label{eq:lem2.2} \end{align} The constraint (\ref{eq:lem2.1}) states that the imaging points remain within the range resolution limit $b/c$ for all the apertures. The constraint (\ref{eq:lem2.2}) states that the imaging points remain within the cross-range resolution limits, as well.
The derivation of the linear system (\ref{eq:MMVEQ}) is in appendix \ref{ap:proofs} and the result is stated in the next proposition. \begin{proposition}
\label{lem.2}
Under the same assumption as in Lemma \ref{lem.1} and in addition,
supposing that conditions (\ref{eq:lem2.1}) and (\ref{eq:lem2.2})
hold, we can combine the linear systems (\ref{eq:lem.1p}) in the
matrix equation (\ref{eq:MMVEQ}). The reference sub-aperture and
sub-band are indexed by $\alpha = 1$ and $\beta = 1$. The unknown
matrix ${\bf X}$ has $Q$ rows and ${\mathcal{N}}_\alpha
{\mathcal{N}}_\beta$ columns indexed by $(\alpha,\beta)$. Its entries are
\begin{equation}
X_q^{(\alpha,\beta)} = \rho_q^{(\alpha,\beta)} \exp \Big[
- 2 i k_\beta {\vec{\bf m}}_\alpha \cdot \Delta {\vec{\mathbf{y}}}_q + i k_\beta
\frac{\Delta {\vec{\mathbf{y}}}_q \cdot \mathbb{P}_\alpha \Delta
{\vec{\mathbf{y}}}_q}{L_\alpha} \Big],
\label{eq:lem2.4}
\end{equation}
where
\begin{equation}
\rho_q^{(\alpha,\beta)} =
\rho({\vec{\mathbf{y}}}_q,{\vec{\bf m}}_\alpha,
\omega_\beta^\star), \qquad
{\vec{\bf m}}_\alpha=
\frac{{\vec{\br}}(s_\alpha^\star)-{\vec{\mathbf{y}}}_o}{|{\vec{\br}}(s_\alpha^\star)-{\vec{\mathbf{y}}}_o|}.
\label{eq:lem2.5}
\end{equation}
The data matrix ${\bf D}$ has $n_s n_\omega$ rows and ${\mathcal{N}}_\alpha {\mathcal{N}}_\beta$
columns indexed by $(\alpha,\beta)$. We organize the equations in
blocks indexed by the frequency $\Delta \omega_l$, for $l = 1, \ldots,
n_\omega$. The entries of ${\bf D}$ are defined in terms of the down-ramped
data vectors $\mathbf{d}^{(\alpha,\beta)}$ as
\begin{equation}
D_j^{(\alpha,\beta)}(\Delta \omega_l) = \frac{(4 \pi
L_\alpha)^2}{k_o^2 |\widehat f(\omega_o)|^2} d^{(\alpha,\beta)}(\Delta
s_j,\Delta \omega_l),
\label{eq:lem2.6}
\end{equation}
where we recall that
\begin{equation}
d^{(\alpha,\beta)}(\Delta s_j,\Delta \omega_l) = d\big(s_\alpha^\star
+ \Delta s_j,\omega_\beta^\star + \Delta \omega_l\big),
\label{eq:lem2.7}
\end{equation}
and $d(s,\omega)$ is defined in (\ref{eq:L1.3}).
The reflectivity to data matrix $\boldsymbol{\mathbb{A}}$ has
$n_\omega$ blocks indexed by $\Delta \omega_l$, denoted by
$\boldsymbol{\mathbb{A}}(\Delta \omega_l)$. Each block is an $n_s
\times Q$ matrix with entries
\begin{equation}
\mathbb{A}_{j,q}(\Delta \omega_l) = \exp \Big[ -2 i \frac{\Delta
\omega_l}{c} {\vec{\bf m}}_1 \cdot \Delta {\vec{\mathbf{y}}}_q - 2 i k_1 \frac{V \Delta
s_j}{L_1} {\vec{\bf t}}_1 \cdot \mathbb{P}_1 \Delta {\vec{\mathbf{y}}}_q \Big].
\label{eq:lem2.8}
\end{equation} \end{proposition}
Note that the product of the reflectivity-to-data matrix $\boldsymbol{\mathbb{A}}$ with each column of ${\bf X}$ can be interpreted, up to a constant multiplicative factor, as a Fourier transform with respect to the range offset ${\vec{\bf m}}_1 \cdot \Delta {\vec{\mathbf{y}}}$ and cross-range offset ${\vec{\bf t}}_1 \cdot \mathbb{P}_1 \Delta {\vec{\mathbf{y}}}$ in $\mathcal{Y}$. Equation (\ref{eq:lem2.4}) shows that the columns of ${\bf X}$ differ from each other by a linear phase factor in $\Delta {\vec{\mathbf{y}}}$, which amounts to a rotation of the coordinate system of the $\alpha$ sub-aperture, and a quadratic factor which corrects for Fresnel diffraction effects. Thus, the linear system (\ref{eq:MMVEQ}) gives roughly the Fourier transform of the reflectivity $\rho$ for different range direction views, and the imaging problem is to invert it to estimate $\rho$.
\section{Inversion algorithm} \label{sect:MMVAlg} Here we describe the algorithm that estimates the reflectivity by inverting the linear system (\ref{eq:MMVEQ}). By construction, the columns of the $Q \times {\mathcal{N}}_\alpha {\mathcal{N}}_\beta$ unknown matrix ${\bf X}$ have the same spatial support, because they represent the same spatial reflectivity function. Thus, we formulate the inversion as a common support recovery problem for unknown matrices with relatively few nonzero rows \cite{rao1998sparse,chen06,cotter2005sparse,eldar10}. This Multiple Measurement Vector (MMV) formulation has been studied in \cite{eldar10,chen06,rao1998sparse} and has been used successfully for source localization with passive arrays of sensors in \cite{malioutov05} and for imaging strong scattering scenes, where multiple scattering effects cannot be neglected, in \cite{chai2014imaging}.
In the MMV framework the support of the unknown matrix ${\bf X}$ is quantified by the number of nonzero rows, that is the row-wise $\ell_0$ norm of ${\bf X}$. If we define the set \begin{equation}
\operatorname{rowsupp}({\bf X})= \{q = 1,\ldots, Q
~ ~\mbox{s.t.} ~ ~ \|{\bf e}_q^T
{\bf X}\|_{\ell_2}\neq0\},
\label{eq:rowsup} \end{equation} where ${\bf e}_q^T {\bf X}$ is the $q-$th row of ${\bf X}$ and ${\bf e}_q$ is the vector with entry $1$ in the $q-$th row and zeros elsewhere, then the row-wise $\ell_0$ norm of ${\bf X}$ is the cardinality of $\operatorname{rowsupp}({\bf X})$,
\[\Xi_0({\bf X})=|
\operatorname{rowsupp}({\bf X})|. \] To estimate ${\bf X}$ we must to solve the optimization problem \begin{equation}\label{MMV.NP}
\min\Xi_0({\bf X})\quad\text{s.t.}\quad
\boldsymbol{\mathbb{A}} {\bf X} = {\bf D}, \end{equation} but this is an NP hard problem. We solve instead the convex problem \begin{equation}\label{MMV21} \min J_{2,1}({\bf X})\quad\text{s.t.}\quad \boldsymbol{\mathbb{A}} {\bf X} = {\bf D}, \end{equation} which gives, under certain conditions on the model matrix $\boldsymbol{\mathbb{A}}$ \cite{eldar10,chai2014imaging}, the same solution as (\ref{MMV.NP}). In (\ref{MMV21}) $J_{2,1}$ denotes the $(2,1)$-norm \begin{equation}
J_{2,1}({\bf X})=\sum_{q=1}^m\|{\bf
e}_q^T{\bf X}\|_{\ell_2}, \label{eq:Jpq} \end{equation} which is the $\ell_1$ norm of the vector formed by the $\ell_2$ norms of the rows of ${\bf X}$. Furthermore, because data are noisy in practice, we replace the equality constraint in
(\ref{MMV21}) by $\|\boldsymbol{\mathbb{A}} {\bf X} -
{\bf D}\|_F < \epsilon$, where $\| \cdot \|_F$ is the Frobenius norm and $\epsilon$ is a tolerance commensurate with the noise level of the data.
There are different algorithms for solving \eqref{MMV21} or its reformulation for noisy data. We use an extension of an iterative shrinkage-thresholding algorithm, called GeLMA, proposed in \cite{moscoso12} for matrix-vector equations. This algorithm is very efficient for solving $\ell_1$-minimization problems, and has the advantage that the solution does not depend on the regularization parameter used to promote minimal support solutions, see \cite{moscoso12} for details. \begin{algorithm} \begin{algorithmic} \REQUIRE Set ${\bf X}=\vect0$, $\vect{\cal Z}=\vect 0$, and pick the step size $\mu$ and the regularization parameter $\gamma$. \REPEAT \STATE Compute the residual $\boldsymbol{\mathcal{E}} = {\bf D} - \boldsymbol{\mathbb{A}} {\bf X}$ \STATE ${\bf X}\Leftarrow{\bf X} + \mu \mathbf{A}^\ast(\vect{\cal Z} + \boldsymbol{\mathcal{E}})$ \STATE ${\bf
e}_q^T{\bf X} \Leftarrow\operatorname{sign}(\| {\bf
e}_q^T{\bf X}\|_{\ell_2}-\mu\gamma)\frac{\|{\bf
e}_q^T{\bf X} \|_{\ell_2}-\mu\gamma}{\|{\bf
e}_q^T{\bf X}\|_{\ell_2}}{\bf
e}_q^T{\bf X}$, $~ ~q=1,\ldots,Q$ \STATE $\vect{\cal Z}\Leftarrow\vect{\cal Z} + \gamma\boldsymbol{\mathcal{E}}$ \UNTIL{Convergence} \end{algorithmic} \caption{GeLMA-MMV} \label{algo} \end{algorithm}
After estimating ${\bf X}$ with Algorithm \ref{algo}, we recover the discretized direction and frequency dependent reflectivity using equation (\ref{eq:lem2.4}), \begin{equation}
\rho_q^{(\alpha,\beta)} = X_q^{(\alpha,\beta)} \exp \Big[
2 i k_\beta {\vec{\bf m}}_\alpha \cdot \Delta {\vec{\mathbf{y}}}_q - i k_\beta \frac{\Delta
{\vec{\mathbf{y}}}_q \cdot \mathbb{P}_\alpha \Delta {\vec{\mathbf{y}}}_q}{L_\alpha} \Big],
\label{eq:result} \end{equation} for the imaging points ${\vec{\mathbf{y}}}_q = {\vec{\mathbf{y}}}_o + \Delta {\vec{\mathbf{y}}}_q$ indexed by $q = 1, \ldots, Q$, the sub-apertures indexed by $\alpha = 1, \ldots, {\mathcal{N}}_\alpha$ and frequency sub-bands indexed by $\beta = 1, \ldots, {\mathcal{N}}_\beta$.
\section{Numerical simulations} \label{sect:NUM} We begin in section \ref{sect:NumSet} with the numerical setup, which is in the regime of the GOTCHA Volumetric data set \cite{gotcha} for X-band persistent surveillance SAR. Then we present in sections \ref{sect:results1D} and \ref{sect:results2D} the simulation results.
\subsection{Imaging in the X-band (GOTCHA) SAR regime} \label{sect:NumSet} The numerical simulations generate the data with the model (\ref{eq:F2}), for various scattering scenes. The regime of parameters is that of the GOTCHA data set, where the platform trajectory is circular at height $H=7.3$km, with radius $R=7.1$km and speed $V=70$m/s. The signal $f(t)$ is sent every $1.05$m along the trajectory, which gives a slow time spacing $h_s=0.015$s. The carrier frequency is $\omega_o/(2 \pi) = 9.6$GHz and the bandwidth is $B=622$MHz. The waves propagate at electromagnetic speed $c = 3 \cdot 10^8$m/s, so the wavelength is $\lambda_o = 3.12$cm. The image window $\mathcal{Y}$ is at the ground level, below the center of the flight trajectory, and the distance from the platform to its center ${\vec{\mathbf{y}}}_o$ is $L = 10.18$km. It is a square, with side length $Y = Y^\perp$ of the order of $40$m. The size of the sub-apertures is $a = 42$m and the width of each sub-band is $b = B/15$.
Given these parameters, the nominal resolution limits are \[
\lambda_o L/a =7.56\mbox{m}, \qquad c/b = 7.23\mbox{m}. \] The image window $\mathcal{Y}$ is discretized in uniform steps $h = 2$m in range and $h^\perp = 1$m in cross-range, and the reflectivity is modeled as piecewise constant on the imaging grid. {The image discretization affects the quality of the reconstruction with $\ell_1$ optimization. It must be coarse enough so that uniqueness of the $\ell_1$ minimizer holds, and yet fine enough so that modeling errors due to off-grid placement of the unknown are controlled. We refer to \cite{borcea2015resolution} for a study of this trade-off.}
{To illustrate the performance of the algorithm, we present in the next two sections results for various imaging scenes consisting of small scatterers supported on one pixel of the imaging grid, or over multiple adjacent pixels. The latter is for representing larger scatterers for which the direction dependent reflectivity can be motivated by Snell's law of reflection at their surface. }
The results presented in the next sections compare the images obtained with reverse time migration and the algorithm proposed in this paper, hereby referred to as the MMV algorithm. The migration image is computed with the formula \begin{equation}
{\mathcal{I}}({\vec{\mathbf{y}}}) = \frac{(4 \pi)^2}{k_o^2 |\widehat f(\omega_o)|^2 n_s n_\omega h
h^\perp} \sum_{j = 1}^{n_s} \sum_{l=1}^{n_\omega} d(s_j,\omega_l)
|{\vec{\br}}(s_j)-{\vec{\mathbf{y}}}|^2 e^{-2 i \omega_l
\big[\tau(s_j,{\vec{\mathbf{y}}})-\tau(s_j,{\vec{\mathbf{y}}}_o)\big]},
\label{eq:MIGN} \end{equation} which is a weighted version of (\ref{eq:migration}), where the weights are chosen so as to provide a quantitative estimate of the unknown $\rho$. That is to say, when we substitute the data model in (\ref{eq:MIGN}), under the assumption of an isotropic and frequency independent reflectivity we get that ${\mathcal{I}}({\vec{\mathbf{y}}})$ peaks at the true location of the scatterers and its value at the peaks equals the true reflectivity there.
Let us verify the assumptions (\ref{eq:M5})-(\ref{eq:M10}) with the GOTCHA parameters. The Fresnel numbers are larger than one, as stated in (\ref{eq:M5}), \[
\frac{a^2}{\lambda_o L} = 5.55 \quad \mbox{and} \quad
\frac{(Y^\perp)^2}{\lambda_o L} = 5.04. \] The size of the imaging region and the range resolution satisfy (\ref{eq:M6}). Moreover, \[ \frac{b}{\omega_o} \frac{Y^\perp}{\lambda_o L/a} = 0.0036, \] which is consistent with (\ref{eq:M8}), and (\ref{eq:M10}) is satisfied as well, \[ \frac{a^2 Y^\perp}{\lambda_o L^2} = \frac{a^2 Y}{\lambda_o L^2} = 0.022. \]
\begin{figure}
\caption{Estimation of an isotropic, frequency independent
reflectivity as a function of cross-range, using $N_\alpha = 8$
consecutive, non-overlapping apertures. The exact reflectivity is
shown with the full green line, the migration result with the blue
line and the MMV inversion result with the broken line. The abscissa
is cross-range in meters.}
\label{fig:isotropic}
\end{figure}
\subsection{Single frequency results} \label{sect:results1D} We begin with imaging results at the carrier frequency, where we assume we know the range of the scatterers and seek to reconstruct their reflectivity as a function of cross-range and direction. The image window extends over $120$m in cross-range, and it is sampled in steps $h^\perp = 1$m, where we recall that $\lambda_o L/a = 7.56$m.
\begin{figure}
\caption{Estimation of the reflectivity as a function of direction and
cross-range location for a scene with $6$ scatterers. The top plots
show the reflectivity as a function of cross-range (the abscissa in
meters), for the peak directions. The left plot is for noiseless
data and the right plot is for data contaminated with $10\%$
additive noise. The green line is the exact peak value and the broken
line the one obtained with MMV. The blue line is obtained with migration.
The bottom plots display the reflectivity of each
scatterer as a function of sub-aperture i.e., the slow time index
$\alpha = 1, \ldots, 10$, where $10$ is the number of sub-apertures.
The left plot is for the true reflectivity, the middle plot is for
the noiseless reconstruction and the right plot is for the noisy
reconstruction.}
\label{fig:1d_real1}
\end{figure}
The first result displayed in Figure \ref{fig:isotropic} is for an isotropic, frequency independent reflectivity of $11$ scatterers, ${\mathcal{N}}_\alpha = 8$ consecutive, non-overlapping apertures and noiseless data. We display in green the true reflectivity, in blue the reflectivity estimated with formula (\ref{eq:MIGN}), and with broken line the result of the MMV inversion algorithm. In the legend we abbreviate the migration formula result with the letters KM, standing for Kirchhoff Migration. The figure shows that the MMV algorithm reconstructs exactly the reflectivity, and that the weighted migration formula (\ref{eq:MIGN}) does indeed give quantitative estimates of the reflectivity. However, the migration estimates deteriorate when the reflectivity is anisotropic and frequency dependent, as illustrated next.
The results displayed in Figure \ref{fig:1d_real1} are obtained with ${\mathcal{N}}_\alpha = 10$ consecutive, non-overlapping apertures. The reflectivity depends on two variables: the cross-range location and the scattering direction, parameterized by the slow time $s_\alpha^\star$, for $\alpha = 1, \ldots, 10$. In discretized form it gives a matrix $\boldsymbol{\mathcal{R}}_{\mbox{true}}$ with row index corresponding to the pixel location in the image window, and column index corresponding to the sub-aperture.
The reconstruction of this matrix is denoted by ${\boldsymbol{\mathcal{R}}}$. The green and broken lines in the top plots in the figure display the true and reconstructed reflectivity at the peak direction, vs. cross-range. Explicitly, for each pixel in the image i.e., each row $q$ in $\boldsymbol{\mathcal{R}}_{\mbox{true}}$ or ${\boldsymbol{\mathcal{R}}}$, we display the maximal entry. The migration image of the reflectors is independent of the direction and is plotted with the blue line. The results show that we have $6$ small scatterers, which are well estimated by the MMV algorithm even for noisy data. The migration method identifies correctly the locations of the $6$ scatterers, but the reflectivity value is no longer accurate because only a few sub-apertures see each reflector, as we infer from the bottom plots described next. This also implies a deterioration in the cross-range resolution which is more visible in the next set of results in Figure \ref{fig:1dapertures_real1}. Naturally, the migration image gives no information about the direction dependence of the reflectivity.
In the bottom plots in Figure \ref{fig:1d_real1} we show the value of the reflectivity of each scatterer as a function of direction, parameterized by the slow time $s_\alpha^\star$. That is to say, we identify first the row indexes $q$ in $\boldsymbol{\mathcal{R}}_{\mbox{true}}$ or ${\boldsymbol{\mathcal{R}}}$ at which we have a strong scatterer (see top plots) and then display those rows. The left plot is for the true reflectivity, the middle is for the noiseless reconstruction, and the right is for the noisy reconstruction. We observe that the MMV method reconstructs the direction dependent reflectivity exactly in the noiseless case, and very well in the noisy case.
\begin{figure}\label{fig:1dapertures_real1}
\end{figure}
In Figure \ref{fig:1dapertures_real1} we illustrate the effect of the anisotropy of the reflectivity on the imaging process. We display the results the same way as in in the previous figure. The point is to notice that while the MMV method estimates accurately the direction dependent reflectivity in all cases, the migration method performs poorly when the anisotropy is strong, meaning that each scatterer is seen only by one sub-aperture at a time (top plots). The resolution is not that corresponding to the actual aperture of $10a = 420$m, but that for a single sub-aperture of $a = 42$m. The middle and bottom row plots show how migration images improve when the anisotropy of the reflectivity is weaker and more sub-apertures see each scatterer.
\subsection{Multiple frequency results} \label{sect:results2D} {Now we consider multiple frequency sub-bands and thus seek to estimate the reflectivity as a function of range, cross-range, direction and frequency.} We have ${\mathcal{N}}_\omega$ sub-bands of width $b$, and we sample each of them at $n_\omega = 15$ frequencies. The number of sub-apertures is ${\mathcal{N}}_\alpha = 8$. The imaging region is a square of side $40$m and it is sampled in cross-range in steps $h^\perp = 1$m and in range in steps $h = 2$m. We denote, as before, by $\boldsymbol{\mathcal{R}}_{\mbox{true}}$ the true matrix of discretized reflectivities and by $\boldsymbol{\mathcal{R}}$ the reconstructed ones. These are matrices of size $Q \times {\mathcal{N}}_\alpha {\mathcal{N}}_\omega$ and we display them in the image window $\mathcal{Y}$ as follows: For each pixel in the image window i.e., a row $q$ in $\boldsymbol{\mathcal{R}}_{\mbox{true}}$ or $\boldsymbol{\mathcal{R}}$, we display the maximum entry, the peak value of the reflectivity at point ${\vec{\mathbf{y}}}_q$ over directions and frequencies. Once we identify the location of the scatterers from these images, i.e., determine their associated rows, we display the entries in these rows, to illustrate the direction and frequency dependence of their reflectivity. These are the middle and right plots in the figures.
\begin{figure}\label{fig:2d_real2}
\end{figure} We begin in Figure \ref{fig:2d_real2} with a single frequency sub-band (${\mathcal{N}}_\omega = 1$), ${\mathcal{N}}_\alpha = 8$ consecutive, non-overlapping sub-apertures and data contaminated with $20\%$ additive noise. The anisotropic reflectivity model has four scatterers, as illustrated in the top plots. Each scatterer is seen by a single sub-aperture. The reconstructed reflectivity is shown in the bottom plots. On the left we show the migration image, which is blurry and is unable to locate the weaker scatterers. The MMV algorithm gives an excellent reconstruction as shown in the middle and right plots.
\begin{figure}\label{fig:2df_3}
\end{figure}
The results in Figures \ref{fig:2df_3} and \ref{fig:2df_4} are for ${\mathcal{N}}_\omega = 8$ consecutive, non-overlapping frequency bands and ${\mathcal{N}}_\alpha = 8$ consecutive, non-overlapping sub-apertures. The difference between the figures is the strength of the scatterers and their anisotropy. The results in Figure \ref{fig:2df_3} show that the MMV algorithm reconstructs well the location of the scatterers and the direction dependence of their reflectivity. The frequency dependence of the weaker scatterers is not that accurate, likely because the bandwidth is small and all frequencies are similar to the carrier. As in Figure \ref{fig:2d_real2}, the migration image is blurrier and does not locate the weak scatterers. Figure \ref{fig:2df_4} shows that the migration image improves when all scatterers are of approximately the same strength and they have weaker anisotropy.
\begin{figure}\label{fig:2df_4}
\end{figure}
\begin{figure}\label{fig:new2}
\end{figure}
{The last illustration considers a larger scatterer with direction dependent reflectivity, supported over four adjacent pixels, and a small isotropic scatterer with frequency dependent reflectivity. The data is contaminated with $20\%$ additive noise. We note that the migration method gives the correct location of the large scatterer, but not the value of its reflectivity. Moreover, it gives a blurry image of the small scatterer. The MMV algorithm determines well the support of both scatterers, as well as accurate estimates of the reflectivity as a function of direction and frequency.}
\section{Doppler effects} \label{ap:Doppler} All the results up to now use the start-stop approximation of the data model, which neglects the motion of the platform over the fast time recording window. Here we extend them to regimes where Doppler effects are important. We begin in section \ref{sect:Dop1} with the derivation of the generalized data model that includes Doppler effects, and an assessment of the validity of the start-stop approximation. Then we explain in section \ref{sect:Dop2} how to incorporate these effects in our imaging algorithm.
\subsection{Data model with Doppler effects} \label{sect:Dop1} For simplicity we first derive the data model for an isotropic reflectivity $\rho = \rho({\vec{\mathbf{y}}})$. Then we extend it in the obvious way to direction and frequency dependent reflectivities in a sub-aperture indexed by $\alpha$ and sub-band indexed by $\beta$, with reflectivity $\rho^{(\alpha,\beta)}({\vec{\mathbf{y}}})$.
The scattered wave $u(s,t)$ recorded at the transmit-receive platform is given by \begin{align}
u(s,t) &= - \int_{\Omega} d {\vec{\mathbf{y}}} \frac{\rho({\vec{\mathbf{y}}})}{c^2} \int_0^t dt_1 \int_0^{t_1}
d t_2 \, f''(t_2) G(t_1-t_2,{\vec{\br}}(s+t_2),{\vec{\mathbf{y}}})G(t-t_1,{\vec{\mathbf{y}}},{\vec{\br}}(s+t)), \nonumber
\\ &= - \frac{1}{c^2}\int_{\Omega} d {\vec{\mathbf{y}}} \rho({\vec{\mathbf{y}}}) \, \frac{f''\big(t_2(t)\big)}{(4 \pi)^2 |{\vec{\br}}\big(s+t_2(t)\big)-{\vec{\mathbf{y}}}|
|{\vec{\br}}(s+t)-{\vec{\mathbf{y}}}|} \label{eq:DM1} \end{align} where $t_2(t)$ is the solution of the equation \begin{equation}
t_2 + \frac{|{\vec{\br}}(s+t_2) - {\vec{\mathbf{y}}}|}{c} = t - \frac{|{\vec{\br}}(s+t)-{\vec{\mathbf{y}}}|}{c}, \label{eq:DM1p} \end{equation} and we used the expression of the Green's function of the wave equation \[
G(t,{\vec{\br}},{\vec{\mathbf{y}}}) = \frac{\delta \big[t-|{\vec{\br}}-{\vec{\mathbf{y}}}|/c\big]}{4 \pi |{\vec{\br}}-{\vec{\mathbf{y}}}|}, \] and the single scattering approximation. The expression (\ref{eq:DM1}) is simply the spherical wave emitted from ${\vec{\br}}(s+t_2)$, over the duration $t_2$ of the pulse, scattered isotropically at ${\vec{\mathbf{y}}}$, and then recorded at ${\vec{\br}}(t+s)$. Up to the single scattering approximation, this is an exact formula. Expanding with respect to $t$ the arguments in (\ref{eq:DM1}) and (\ref{eq:DM1p}) we obtain \begin{align}
u(s,t) &= - \frac{1}{c^2} \int_\Omega d {\vec{\mathbf{y}}} \rho({\vec{\mathbf{y}}})
\frac{1}{(4 \pi |{\vec{\br}}(s)-{\vec{\mathbf{y}}}|)^2\big(1 + O(Vt/L)\big)} \nonumber \times
\\
& \hspace{-0.2in}f''\Big[\Big( t \Big(1 - \gamma(s,{\vec{\mathbf{y}}}) +
O\Big(\frac{V}{c}\frac{Vt}{R}\Big)\Big) - 2 \tau(s,{\vec{\mathbf{y}}})\Big)/\Big(1+\gamma(s,{\vec{\mathbf{y}}})+ O\Big(\frac{V}{c}\frac{Vt}{R}\Big)\Big)\Big],
\label{eq:DM2p} \end{align} where we introduced the Doppler factor $\gamma$ defined by \begin{align}
\gamma(s,{\vec{\mathbf{y}}}) = \frac{{\vec{\br}}'(s)}{c} \cdot {\vec{\bf m}}(s,{\vec{\mathbf{y}}}), \qquad
{\vec{\bf m}}(s,{\vec{\mathbf{y}}}) = \frac{{\vec{\br}}(s)-{\vec{\mathbf{y}}}}{|{\vec{\br}}(s)-{\vec{\br}}(y)|}.
\label{eq:DM3} \end{align} We assume that the platform is moving at constant speed $V$ along a trajectory with unit tangent denoted by ${\vec{\bf t}}(s)$, and with radius of curvature $R$ assumed comparable to the range $L$. Thus, \[ \gamma(s,{\vec{\mathbf{y}}}) = O \Big(\frac{V}{c}\Big) \ll 1, \] because the platform speed is typically much smaller than $c$, the wave speed, and we can neglect the residual in (\ref{eq:DM2p}) which is even smaller than $\gamma$, because over the duration of the fast time window the platform travels a small distance compared with the radius of curvature $V t \ll R \sim L$. We have thus the data model \begin{equation} u(s,t) \approx - \frac{1}{c^2} \int_\Omega d {\vec{\mathbf{y}}}
\rho({\vec{\mathbf{y}}}) \frac{f''\Big[ t \big(1 - 2\gamma(s,{\vec{\mathbf{y}}})\big) - 2 \tau(s,{\vec{\mathbf{y}}})\big(1-\gamma(s,{\vec{\mathbf{y}}})\big)
\Big]}{(4 \pi |{\vec{\br}}(s)-{\vec{\mathbf{y}}}|)^2},
\label{eq:DM2}
\end{equation}
which includes first order Doppler effects.
The start-stop approximation is valid when the Doppler factor in the argument of $f''$ in (\ref{eq:DM2}) is negligible. Although $\gamma$ is small, $f''$ oscillates at the carrier frequency $\omega_o$ which is large and, depending on the scale of the fast time $t$, the Doppler factor may play a role. Recall that $t$ is limited by the slow time spacing $h_s$. In practice the duration of the fast time window may be much smaller than $h_s$, although it must be large enough so that the platform can receive the echoes delayed by the travel time, $2 \tau(s,{\vec{\mathbf{y}}})$. Explicitly, \[ t = O(L/c) + O(1/B), \] where $L/c$ is the scale of the travel time and $1/B$ is the scale of the duration of the signal.
We conclude that the start stop approximation holds when \[ \omega_o t \gamma(s,{\vec{\mathbf{y}}}) = O \Big(\frac{\omega_o L}{c} \frac{V}{c} \Big) + O \Big(\frac{\omega_o}{B} \frac{V}{c} \Big) \ll 1. \] In the GOTCHA regime, considered in the numerical simulations in section \ref{sect:NUM}, we have \[ \frac{\omega_o L}{c} \frac{V}{c} = 0.469, \qquad \frac{\omega_o}{B} \frac{V}{c} = 2.3 \cdot 10^{-5}, \] so $\omega_o t \gamma(s,{\vec{\mathbf{y}}})$ is slightly less than one. We may include it in the data model, but it amounts to a constant additive phase that has no effect in imaging. To see this, let us take the Fourier transform with respect to $t$ in (\ref{eq:DM2}) \begin{equation}
\widehat u(s,\omega) \approx k^2
\int_\Omega d {\vec{\mathbf{y}}} \rho({\vec{\mathbf{y}}}) \widehat f \Big[\omega \big(1 + 2 \gamma(s,{\vec{\mathbf{y}}})
\big) \Big] \frac{\exp \big[ 2 i \omega \big(1 + \gamma(s,{\vec{\mathbf{y}}}) \big)
\tau(s,{\vec{\mathbf{y}}})\big]}{(4 \pi |{\vec{\br}}(s)-{\vec{\mathbf{y}}}|)^2},
\label{eq:DMn2} \end{equation} and expand the arguments over the slow time $s$ and imaging point ${\vec{\mathbf{y}}}$. We use the approximation \begin{equation} \label{eq:DD3} {\vec{\br}}'(s) \approx V \Big[{\vec{\bf t}}(s^\star) - \vec{\bf n}(s^\star) \frac{V \Delta
s}{R}\Big], \end{equation} where $\Delta s$ is the slow time offset from the center $s^\star$ of the aperture, and ${\vec{\bf t}}(s^\star)$ is the unit tangent to the trajectory of the platform at the center point. The second term in (\ref{eq:DD3}) accounts for the curved platform trajectory, with unit vector $\vec{\bf
n}(s^\star)$ orthogonal to ${\vec{\bf t}}$, in the plane defined by ${\vec{\bf t}}$ and the center of curvature, and $R$ the radius of curvature. We also have \[
| {\vec{\bf m}}(s,{\vec{\mathbf{y}}}) -
{\vec{\bf m}}(s^\star,{\vec{\mathbf{y}}}_o)| = O\Big(\frac{V |\Delta s|}{L} \Big) +
O\Big(\frac{|Y^\perp|}{L} \Big), \] and \[
\omega_o \tau(s,{\vec{\mathbf{y}}}) = \omega_o \tau(s^\star,{\vec{\mathbf{y}}}_o) + O (k_o V\Delta s)
+ O(k_o \Delta y). \] {Substituting in (\ref{eq:DMn2}) and using the parameters of the GOTCHA regime, we see that, \[ \omega \gamma(s,{\vec{\mathbf{y}}}) \tau(s,{\vec{\mathbf{y}}}) \approx \omega_o \gamma(s^\star,{\vec{\mathbf{y}}}_o^) \tau(s^\star,{\vec{\mathbf{y}}}_o), \] so indeed, the Doppler effect amounts to a constant phase term.}
\subsection{Imaging algorithm with Doppler effects} \label{sect:Dop2} The model of the down-ramped data with the Doppler correction follows from (\ref{eq:DMn2}), \begin{align} d(s,\omega) &= \overline{\widehat f \big[ \omega \big( 1+2\gamma(s,{\vec{\mathbf{y}}}_o)\big)
\big]} \widehat u(s,\omega) \exp \big[-2 i \omega \big(1+\gamma(s,{\vec{\mathbf{y}}}_o)\big)
\tau (s,{\vec{\mathbf{y}}}_o) \big] \nonumber \\ &\approx k^2 \overline{\widehat f
\big[ \omega \big( 1+2\gamma(s,{\vec{\mathbf{y}}}_o) \big) \big]} \int_\Omega d {\vec{\mathbf{y}}} \, \widehat f \big[ \omega \big( 1+2\gamma(s,{\vec{\mathbf{y}}}) \big) \big]\rho({\vec{\mathbf{y}}}) \times \nonumber \\ &~ ~~~ \frac{\exp \big[ 2 i \omega \big(1+\gamma(s,{\vec{\mathbf{y}}})\big)
\tau(s,{\vec{\mathbf{y}}}) - 2 i \omega\big(1+\gamma(s,{\vec{\mathbf{y}}}_o)\big)
\tau(s,{\vec{\mathbf{y}}}_o)\big]}{\big(4 \pi |{\vec{\br}}(s)-{\vec{\mathbf{y}}}|\big)^2}. \label{eq:D1} \end{align} We are interested in direction and frequency dependent reflectivities, so to use formula (\ref{eq:D1}), we consider next the $\alpha-$th sub-aperture and the $\beta-$th sub-band, where we can replace $\rho$ by $\rho^{(\alpha,\beta)}({\vec{\mathbf{y}}})$. The data is denoted by $d^{(\alpha,\beta)}(\Delta s,\Delta \omega),$ where $\Delta s = s - s_\alpha^\star$ and $\Delta \omega = \omega - \omega_\beta^\star.$ The goal of the section is to include Doppler effects in the statements of Lemma \ref{lem.1} and Proposition \ref{lem.2}, which are the basis of our imaging algorithm.
We begin with the observation that \begin{equation} \omega \gamma(s,{\vec{\mathbf{y}}}) \tau(s,{\vec{\mathbf{y}}})= \frac{\omega}{c} \frac{{\vec{\br}}'(s)}{c} \cdot ({\vec{\br}}(s)-{\vec{\mathbf{y}}}) = \omega \gamma(s,{\vec{\mathbf{y}}}_o) \tau(s,{\vec{\mathbf{y}}}_o) - \frac{\omega}{c} \frac{{\vec{\br}}'(s)}{c} \cdot \Delta {\vec{\mathbf{y}}}, \end{equation} where $\Delta {\vec{\mathbf{y}}} = {\vec{\mathbf{y}}} - {\vec{\mathbf{y}}}_o$, and ${\vec{\br}}'(s)$ is given by (\ref{eq:DD3}), and assume henceforth that \begin{equation} \frac{V}{c} \frac{{{Y}_\alpha^\perp}}{L_\alpha} \ll \frac{b}{\omega_o} \ll 1. \label{eq:D4} \end{equation} This is consistent with our previous assumptions because ${{Y}_\alpha^\perp} \ll L_\alpha$ and $V \ll c$, and allows us to approximate the Doppler factor in the argument of the Fourier transform of the signal in (\ref{eq:D1}) by its value at the reference point. Then, using equation (\ref{eq:L1.2}) and noting also that \[
|{\vec{\br}}(s)-{\vec{\mathbf{y}}}| = L_\alpha \Big[ 1 + O \Big(\frac{a}{L_\alpha}\Big) +
O\Big(\frac{{{Y}_\alpha^\perp}}{L_\alpha}\Big)\Big], \qquad k = k_o \Big[ 1 +
O\Big( \frac{b}{\omega_o} \Big)\Big], \] we can simplify the amplitude factor in (\ref{eq:D1}) as \begin{align} \frac{k^2 \overline{ \widehat f \big[ \omega \big( 1+2\gamma(s,{\vec{\mathbf{y}}}_o) \big)
\big] }\widehat f \big[ \omega \big( 1+2\gamma(s,{\vec{\mathbf{y}}}) \big) \big]}{(4
\pi |{\vec{\br}}(s)-{\vec{\mathbf{y}}}|)^2} \approx \frac{k_o^2 |\widehat f(\omega_o)|^2}{(4 \pi
L_\alpha)^2} , \label{eq:DD4} \end{align} and obtain \begin{align}
d^{(\alpha,\beta)}(\Delta s, \Delta \omega) \approx \frac{k_o^2 |\widehat
f(\omega_o)|^2}{(4 \pi L_\alpha)^2} \sum_{q=1}^Q \rho^{(\alpha,\beta)}_q \exp \Big[ - 2 i ( k_\beta + \Delta k) \frac{{\vec{\br}}'(s_\alpha^\star+\Delta s)}{c} \cdot
\Delta {\vec{\mathbf{y}}}_q + \nonumber \\
2 i (\omega_\beta^\star + \Delta \omega)\big[\tau(s_\alpha^\star+\Delta s,{\vec{\mathbf{y}}}_o + \Delta {\vec{\mathbf{y}}}_q) - \tau(s_\alpha^\star+\Delta s
,{\vec{\mathbf{y}}}_o)\big] \Big].
\label{eq:datasp} \end{align} Here we have used that $k = k_\beta + \Delta k$, with center wavenumber $k_\beta = \omega_\beta^\star/c$ and offset $\Delta k = \Delta \omega/c$.
The difference between the travel times in the phase in (\ref{eq:datasp}) is approximated in the proof of Lemma \ref{lem.1} in appendix \ref{ap:proofs}. It remains to expand the first term in the phase, which is due to the Doppler factor. We use (\ref{eq:DD3}) and obtain \begin{align*} (k_\beta + \Delta k) \frac{{\vec{\br}}'(s_\alpha^\star + \Delta s)}{c} \cdot
\Delta {\vec{\mathbf{y}}}_q = k_\beta \frac{V}{c} \Big[{\vec{\bf t}}_\alpha \cdot \Delta
{\vec{\mathbf{y}}}_q - \frac{V \Delta s}{R} \vec{\bf n}_\alpha \cdot \Delta
{\vec{\mathbf{y}}}\Big] + \Delta k \frac{V}{c} {\vec{\bf t}}_\alpha \cdot \Delta {\vec{\mathbf{y}}}_q +
\\ O\Big(\frac{V}{c} \frac{a}{R} \frac{\vec{\bf n}_\alpha \cdot
\Delta {\vec{\mathbf{y}}}_q}{c/b} \Big) , \end{align*} with negligible residual under the assumption \begin{equation} \frac{V}{c} \frac{a}{R} \frac{{{Y}_\alpha}}{c/b} \ll 1. \label{eq:DD5} \end{equation} Recall that $c/b$ is the range resolution, and although we want ${{Y}_\alpha} \gg c/b$, the inequality (\ref{eq:DD5}) is easily satisfied because $ a \ll R \sim L_\alpha$ and $V \ll c$.
The generalization of the result in Lemma \ref{lem.1} is as follows. We have the linear system of equations \begin{equation} \mathbf{A}^{(\alpha,\beta)} \boldsymbol{\rho}^{(\alpha,\beta)} = \mathbf{d}^{(\alpha,\beta)}, \end{equation} where the reflectivity vector $\boldsymbol{\rho}^{(\alpha,\beta)}$ with entries $\rho_q^{(\alpha,\beta)}$ is mapped to the data vector $\mathbf{d}^{(\alpha,\beta)}$ with entries $d^{(\alpha,\beta)}(\Delta s_j,\Delta \omega_l)$ by the reflectivity-to-data matrix $\mathbf{A}^{(\alpha,\beta)}$. The entries of $\mathbf{A}^{(\alpha,\beta)}$ are given by \begin{align}
A_{j,q}^{(\alpha,\beta)}(\Delta \omega_l) = \frac{k_o^2 |\widehat
f(\omega_o)|^2}{(4 \pi L_\alpha)^2} \exp \Big\{ -2 i (k_\beta +\Delta \omega_l/c) \Big[ {\vec{\bf m}}_\alpha
\cdot \Delta {\vec{\mathbf{y}}}_q + \frac{V}{c} {\vec{\bf t}}_\alpha \cdot \Delta {\vec{\mathbf{y}}}_q \Big] \nonumber \\ \, -2 i \frac{k_\beta V
\Delta s}{L_\alpha} \Big[{\vec{\bf t}}_\alpha \cdot \mathbb{P}_\alpha \Delta {\vec{\mathbf{y}}}_q - \frac{L_\alpha}{R} \frac{V}{c}
{\vec{\bf n}}_\alpha \cdot \Delta {\vec{\mathbf{y}}}_q\Big] + i k_\beta \frac{\Delta {\vec{\mathbf{y}}}_q \cdot \mathbb{P}_\alpha \Delta
{\vec{\mathbf{y}}}_q}{L_\alpha} \Big\}. \end{align} The difference between this reflectivity-to-data matrix and the one given by (\ref{eq:lem1}) in Lemma \ref{lem.1} comes from the $V$ dependent terms in the square brackets in the phase, due to the Doppler effect.
We extend next the statement of Proposition \ref{lem.2}. We proceed as in appendix \ref{ap:proofs}, and show that the matrix-matrix equation (\ref{eq:MMVEQ}), $ \boldsymbol{\mathbb{A}} {\bf X} = {\bf D}$, still applies, with the same definition (\ref{eq:lem2.6}) of the data matrix ${\bf D}$,
\begin{equation}
D_j^{(\alpha,\beta)}(\Delta \omega_l) = \frac{(4 \pi
L_\alpha)^2}{k_o^2 |\widehat f(\omega_o)|^2} d^{(\alpha,\beta)}(\Delta
s_j,\Delta \omega_l),
\nonumber
\end{equation} and with the unknown matrix \begin{align} X_{q}^{(\alpha,\beta)} = \rho_q^{(\alpha,\beta)} \exp \Big\{-2 i k_\beta \Big[ \frac{V}{c} {\vec{\bf t}}_\alpha \cdot \Delta {\vec{\mathbf{y}}}_q + {\vec{\bf m}}_\alpha
\cdot \Delta {\vec{\mathbf{y}}}_q \Big] + i k_\beta \frac{\Delta {\vec{\mathbf{y}}}_q \cdot
\mathbb{P}_\alpha \Delta {\vec{\mathbf{y}}}_q}{L_\alpha} \Big\}. \label{eq:NewX} \end{align} This is under the assumptions that \begin{align} \max_{1 \le \alpha \le {\mathcal{N}}_\alpha, 1 \le q \le Q} \frac{V}{c}
\frac{b}{c} \Big|[{\vec{\bf t}}_\alpha - {\vec{\bf t}}_1] \cdot \Delta {\vec{\mathbf{y}}}_q\Big| &\ll 1, \\
\max_{1 \le \alpha \le {\mathcal{N}}_\alpha, 1 \le q \le Q} \frac{V}{c} \frac{a}{\lambda_o R} \big| ({\vec{\bf n}}_\alpha-{\vec{\bf n}}_1) \cdot \Delta {\vec{\mathbf{y}}}_q \big| &\ll1, \end{align} which are similar to (\ref{eq:lem2.1})-(\ref{eq:lem2.2}), and easier to satisfy for smaller $V$. The expression of the entries of the reflectivity-to-data matrix is a simple modification of that in equation (\ref{eq:lem2.8}), \begin{align} \label{eq:matrixdp} \mathbb{A}_{j,q}(\Delta \omega_l) = \exp \Big[ &-2 i \frac{\Delta
\omega_l}{c} \Big({\vec{\bf m}}_1 \cdot \Delta {\vec{\mathbf{y}}}_q + \frac{V}{c} {\vec{\bf t}}_1 \cdot
\Delta {\vec{\mathbf{y}}}_q \Big) \nonumber \\ &-2 i k_1 \frac{V \Delta s_j}{L_1} \Big({\vec{\bf t}}_1 \cdot
\mathbb{P}_1 \Delta {\vec{\mathbf{y}}}_q - \frac{L_1}{R}\frac{V}{c} {\vec{\bf n}}_1 \cdot \Delta {\vec{\mathbf{y}}}_q\Big) \Big]. \end{align}
Thus, the problem can be solved with the MMV approach, as described in section \ref{sect:MMVAlg}. The Doppler correction has two effects: It gives an extra rotation in the cross-range direction of the imaging window (the first phase term in (\ref{eq:NewX}), involving $V$), and two extra phase factors (involving $V$ inside the parentheses) in the reflectivity-to-data matrix $\mathbb{A}$ in (\ref{eq:matrixdp}).
\section{Summary} \label{sect:summary} We have introduced and analyzed from first principles a synthetic aperture imaging approach for reconstructing direction and frequency dependent reflectivities of localized scatterers. It is based on two main ideas: The first one is to segment the data over subsets defined by carefully calibrated sub-apertures and frequency sub-bands, and formulate the reflectivity reconstruction for each subset as an $\ell_1$ optimization problem. The direction and frequency dependence of the reconstructed reflectivity is frozen for each data subset but varies from one subset to another. The second idea is to fuse the sub-aperture and sub-band optimizations by seeking simultaneously from data subsets those reconstructions of the reflectivity that share the same spatial support in the image window. This is done with the multiple measurement vector (MMV) formalism, which leads to a matrix $\ell_1$ optimization problem. The main result of this paper is showing that synthetic aperture imaging of direction and frequency dependent reflectivities can be formulated and solved efficiently as an MMV problem.
Data segmentation is a natural idea that has been used before for synthetic aperture imaging of frequency dependent reflectivities \cite{sotirelis2012study,elachi1990radar}. Here we use it for estimating the direction dependence of the reflectivity, as well. We analyze how the size of the sub-apertures and frequency sub-bands in the data segmentation affects the resolution of the reconstructions as well as the computational complexity of the inversion. There is a trade-off in resolution in this approach: On one hand we want to have large sub-apertures and frequency sub-bands to get good spatial, range and cross-range, resolution of the reconstructed reflectivity. But on the other hand we also want to have small sub-apertures and frequency sub-bands to resolve well the direction and frequency dependence of the reflectivity. Small sub-apertures are also desirable so as to get images efficiently using Fourier transforms. The MMV formalism that we have introduced in this paper, and the associated algorithm for its implementation, deal well with these issues, as indicated by the numerical simulations.
Nearly all synthetic aperture imaging is done with reverse time migration algorithms, without regard to whether the reflectivities that are to be imaged are direction dependent or not. If the reflectivities are isotropic, then the spatial resolution of the reconstruction improves as the aperture increases. But this is not the case with direction dependent reflectivities as only part of the synthetic aperture will sense reflectivities from particular locations. This means that segmenting the data over sub-apertures is natural. The MMV-based imaging algorithm introduced in this paper handles automatically signals received by sub-apertures that are coming from directional reflectivities located in the image window.
\section{Derivation of the reflectivity to data model} \label{ap:proofs} Here we show that the expression of $A_{j,q}(\omega_l)$ in (\ref{eq:L1.4}) can be approximated by $A_{j,q}^{(\alpha,\beta)}(\Delta \omega_l)$ given in Lemma \ref{lem.1}, for $\omega_l = \omega_\beta^\star + \Delta \omega_l$ and $s_j = s_\alpha^\star + \Delta s_j$. For simplicity of notation we drop the indexes $j$ and $l$ of the frequency and slow time.
It is easy to see from (\ref{eq:L1.2}) and the assumptions $\omega_o \gg b$ and $L_\alpha \gg a \gtrsim {{Y}_\alpha}$ that \begin{equation}
\frac{k^2 |\widehat f(\omega)|^2}{\big(4 \pi |{\vec{\br}}(s)-{\vec{\mathbf{y}}}|\big)^2} \approx
\frac{k_o^2 |\widehat f(\omega_o)|^2}{(4 \pi L_\alpha)^2},
\label{eq:AP1} \end{equation} for $k = \omega/c$ and $k_o = \omega_o/c$. It remains to show the phase approximation \begin{align}
2 \omega \big[\tau (s,{\vec{\mathbf{y}}})-\tau(s,{\vec{\mathbf{y}}}_o)\big] \approx -2 k
{\vec{\bf m}}_\alpha \cdot {\vec{\mathbf{y}}} - 2 k_\beta V \Delta s \frac{{\vec{\bf t}}_\alpha \cdot
\mathbb{P}_\alpha \Delta {\vec{\mathbf{y}}}}{L_\alpha} + k_\beta \frac{\Delta
{\vec{\mathbf{y}}} \cdot \mathbb{P}_\alpha \Delta {\vec{\mathbf{y}}}}{L_\alpha},
\label{eq:AP2} \end{align} where $ \omega = \omega_\beta^\star + \Delta \omega$ lies in the frequency sub-band of width $b$, $s = s_\alpha^\star+\Delta s$ is in the sub-aperture of size $a$ and ${\vec{\mathbf{y}}} = {\vec{\mathbf{y}}}_o + \Delta {\vec{\mathbf{y}}}$ is in $\mathcal{Y}$.
We begin by expanding the travel time in $\Delta {\vec{\mathbf{y}}}$, \begin{align*}
\Phi &= 2 \omega \big[\tau (s,{\vec{\mathbf{y}}})-\tau(s,{\vec{\mathbf{y}}}_o)\big] \\&= - 2 k
{\vec{\bf m}}(s,{\vec{\mathbf{y}}}_o) \cdot \Delta {\vec{\mathbf{y}}} + \frac{k}{|{\vec{\br}}(s)-{\vec{\mathbf{y}}}_o|} \Delta
{\vec{\mathbf{y}}} \cdot \big[ I - {\vec{\bf m}}(s,{\vec{\mathbf{y}}}_o) {\vec{\bf m}}^T(s,{\vec{\mathbf{y}}}_o)\big] \Delta {\vec{\mathbf{y}}} +
\mathcal{E}_1, \end{align*} with small residual \[ \mathcal{E}_1 = O \Big( \frac{{{Y}_\alpha^\perp}^2 {{Y}_\alpha}}{\lambda_o L_\alpha^2} \Big) \ll 1, \] by assumption (\ref{eq:M10}) and ${{Y}_\alpha^\perp} \lesssim a$, inferred from (\ref{eq:M5}). Here we used the expression of the gradient \[
\nabla_{\vec{\mathbf{y}}} |{\vec{\br}}(s)-{\vec{\mathbf{y}}}| = - \frac{{\vec{\br}}(s)-{\vec{\mathbf{y}}}}{|{\vec{\br}}(s)-{\vec{\mathbf{y}}}|} = - {\vec{\bf m}}(s,{\vec{\mathbf{y}}}), \] the Hessian \[
\nabla_{\vec{\mathbf{y}}} \otimes \nabla_{\vec{\mathbf{y}}} |{\vec{\br}}(s)-{\vec{\mathbf{y}}}| = \frac{1}{|{\vec{\br}}(s)-{\vec{\mathbf{y}}}|} \Big[ I - {\vec{\bf m}}(s,{\vec{\mathbf{y}}}){\vec{\bf m}}^T(s,{\vec{\mathbf{y}}})\Big], \] and \begin{align*} \sum_{i,j,q=1}^3 \Delta y_i \Delta y_j \Delta y_q
\partial^3_{y_i,y_j,y_q} |{\vec{\br}}(s)-{\vec{\mathbf{y}}}| &= \frac{3 {\vec{\bf m}}_\alpha \cdot
\Delta {\vec{\mathbf{y}}}}{|{\vec{\br}}(s)-{\vec{\mathbf{y}}}|^2} \big[ |\Delta {\vec{\mathbf{y}}}|^2 - \big({\vec{\bf m}}_\alpha
\cdot \Delta {\vec{\mathbf{y}}} \big)^2 \big]. \end{align*} Next, we expand in $\Delta \omega = \omega - \omega_\beta^\star$ and obtain \begin{align*}
\Phi &= - 2(k_\beta + \Delta k) {\vec{\bf m}}(s,{\vec{\mathbf{y}}}_o) \cdot \Delta {\vec{\mathbf{y}}} +
\frac{k_\beta}{|{\vec{\br}}(s)-{\vec{\mathbf{y}}}_o|} \Delta
{\vec{\mathbf{y}}} \cdot \big[ I - {\vec{\bf m}}(s,{\vec{\mathbf{y}}}_o) {\vec{\bf m}}^T(s,{\vec{\mathbf{y}}}_o)\big] \Delta {\vec{\mathbf{y}}} +
\mathcal{E}_2, \end{align*} where $\Delta k = \Delta \omega/c$ and \[ \mathcal{E}_2 = \mathcal{E}_1 + O \Big(\frac{b}{\omega_o} \frac{{{Y}_\alpha^\perp}^2}{\lambda_o L_\alpha}\Big) \ll 1. \] The last estimate is by assumption (\ref{eq:M8}). Finally, we expand in $\Delta s = s - s_\alpha^\star$, and recalling the notation in section \ref{sect:setup}, we get \begin{align}
\Phi = -2(k_\beta + \Delta k) {\vec{\bf m}}_\alpha \cdot \Delta {\vec{\mathbf{y}}} - 2
k_\beta \frac{V \Delta s}{L_\alpha} {\vec{\bf t}}_\alpha \cdot
\mathbb{P}_\alpha \Delta {\vec{\mathbf{y}}} + \nonumber \\ k_\beta \frac{\Delta {\vec{\mathbf{y}}}
\cdot \mathbb{P}_\alpha \Delta {\vec{\mathbf{y}}}}{L_\alpha} +
\mathcal{E}. \label{eq:Phi} \end{align} The residual is the sum of four terms \[ \mathcal{E} = \mathcal{E}_2 + \mathcal{E}_3 + \mathcal{E}_4 + \mathcal{E}_5, \] with $\mathcal{E}_2$ given above. The term $\mathcal{E}_3$ comes from the quadratic part of the expansion of $k_\beta {\vec{\bf m}}(s,{\vec{\mathbf{y}}}_o) \cdot \Delta {\vec{\mathbf{y}}}$, \begin{align*} \mathcal{E}_3 \sim k_\beta (V \Delta s)^2 \Big[ \frac{{\vec{\bf n}}_\alpha \cdot
\mathbb{P}_\alpha \Delta {\vec{\mathbf{y}}} }{R L_\alpha} + \frac{ {\vec{\bf t}}_\alpha
\cdot \big[ {\vec{\bf m}}'_\alpha {\vec{\bf m}}_\alpha^T + {\vec{\bf m}}_\alpha
({\vec{\bf m}}'_\alpha)^T\big] \Delta {\vec{\mathbf{y}}}}{V L_\alpha} +
\frac{\big({\vec{\bf t}}_\alpha\cdot\mathbb{P}_\alpha \Delta {\vec{\mathbf{y}}}\big)
\big({\vec{\bf t}}_\alpha \cdot {\vec{\bf m}}_\alpha\big)}{L_\alpha^3} \Big]. \end{align*} Here $\sim$ denotes order of magnitude, and the primes denote derivative with respect to $s$. The unit vector ${\vec{\bf n}}_\alpha $ is normal to ${\vec{\bf t}}_\alpha$, in the plane defined by ${\vec{\bf t}}_\alpha$ and the center of curvature of the trajectory of the platform. It enters the definition \begin{equation} \label{eq:defTp} {\vec{\bf t}}'_\alpha = -\frac{V {\vec{\bf n}}_\alpha}{R}, \end{equation}
where $R \sim L_\alpha$ is the radius of curvature. Moreover \begin{equation} {\vec{\bf m}}'_\alpha = \frac{V}{L_\alpha} \mathbb{P}_\alpha {\vec{\bf t}}_\alpha. \end{equation} We conclude that \[ \mathcal{E}_3 = O \Big( \frac{a^2 {{Y}_\alpha^\perp}}{\lambda_o L_\alpha^2} \Big) + O \Big( \frac{a^2 {{Y}_\alpha}}{\lambda_o L_\alpha^2} \Big) \ll 1, \] where the inequality is by assumption (\ref{eq:M10}).
The term $\mathcal{E}_4$ in the residual is \begin{align*}
\mathcal{E}_4 \sim \frac{\Delta \omega}{c} \frac{V \Delta s}{L_\alpha}
{\vec{\bf t}}_\alpha \cdot \mathbb{P}_\alpha \Delta {\vec{\mathbf{y}}} = O
\Big(\frac{b}{\omega_o} \frac{a {{Y}_\alpha^\perp}}{\lambda_o L_\alpha} \Big) \ll 1, \end{align*} by assumption (\ref{eq:M8}), and the last term $\mathcal{E}_5$ comes from the expansion of the quadratic term in $\Delta {\vec{\mathbf{y}}}$ in the expression of $\Phi$. We estimate it as \begin{align*}
\mathcal{E}_5 = O \Big(\frac{a {{Y}_\alpha} {{Y}_\alpha^\perp}}{\lambda_o L_\alpha^2} \Big) \ll 1, \end{align*} where we used assumption (\ref{eq:M10}). The statement of Lemma \ref{lem.1} follows from (\ref{eq:AP1}) and (\ref{eq:Phi}). $\Box$
Proposition \ref{lem.2} follows easily from the expression (\ref{lem.1}) of $A_{j,q}^{(\alpha,\beta)}$ and assumptions (\ref{eq:lem2.1}) and (\ref{eq:lem2.2}). Writing the linear system (\ref{eq:lem.1p}) component-wise we get \begin{align*}
\sum_{q=1}^Q X_q^{(\alpha,\beta)} \exp \Big[ -2 i
\frac{\Delta \omega_l}{c} {\vec{\bf m}}_\alpha \cdot {\vec{\mathbf{y}}}_q - 2 i k_\beta V
\Delta s_j \frac{ {\vec{\bf t}}_\alpha \cdot \mathbb{P}_\alpha \Delta
{\vec{\mathbf{y}}}_q}{L_\alpha} \Big] = D_j^{(\alpha,\beta)}(\Delta \omega_l), \end{align*} with $X_q^{(\alpha,\beta)}$ given in (\ref{eq:lem2.4}) and $D_j^{(\alpha,\beta)}$ defined in (\ref{eq:lem2.6}). The result (\ref{eq:lem2.8}) follows from this equation and assumptions (\ref{eq:lem2.1}) and (\ref{eq:lem2.2}). $\Box$
\section{Inner products for rows and columns of the reflectivity-to-data matrix} \label{ap:discr}
Here we analyze the relation between the discretization of the imaging window $\mathcal{Y}$ and the linear independence of the columns of the reflectivity to data matrix. This is done by computing inner products of of normalized rows and columns of the reflectivity-to-data matrix. If the column inner products multiplied by the number of elements in the support of the reflectivities are below a threshold then the MMV algorithm will give an exact reconstruction, in the noiseless case \cite{chai2014imaging}.
We consider the restriction to a data subset, defined by a sub-aperture and frequency sub-band satisfying the assumptions in section \ref{sect:MMVRed}. Thus, we work with matrices $\mathbf{A}^{(\alpha,\beta)}$, but to simplify notation we drop the indexes $(\alpha,\beta)$.
Let us denote by ${\bf a}_q$ the $q-$th column of matrix $\mathbf{A}$ and calculate the inner product \[ \left< {\bf a}_q, {\bf a}_{q'} \right> = \Big(\mathbf{A}^\star \mathbf{A}\Big)_{q,q'} = \sum_{j = 1}^{n_s} \sum_{l=1}^{n_\omega} \overline{A_{j,q}(\Delta \omega_l)} A_{j,q'}(\Delta \omega_l). \] Using Lemma \ref{lem.1} we get \begin{align*}
\left< \frac{{\bf a}_q}{\|{\bf a}_q\|}, \frac{{\bf a}_{q'}}{\|{\bf
a}_{q'}\|} \right> = \exp \Big[ -2 i k_\beta {\vec{\bf m}}_\alpha \cdot
({\vec{\mathbf{y}}}_{q'}-{\vec{\mathbf{y}}}_q) + \frac{i k_\beta\Big( \Delta {\vec{\mathbf{y}}}_{q'}
\mathbb{P}_\alpha \Delta {\vec{\mathbf{y}}}_{q'} - \Delta {\vec{\mathbf{y}}}_{q}
\mathbb{P}_\alpha \Delta {\vec{\mathbf{y}}}_{q}\Big)}{L_\alpha} \Big] \times \\\frac{1}{n_s n_\omega}\sum_{j = 1}^{n_s} \sum_{l=1}^{n_\omega} \exp \Big[-\frac{2 i \Delta \omega_l}{c} {\vec{\bf m}}_\alpha \cdot ({\vec{\mathbf{y}}}_{q'}-{\vec{\mathbf{y}}}_q) -
\frac{2 i k_\beta V \Delta s_j}{L_\alpha} {\vec{\bf t}}_\alpha \cdot
\mathbb{P}_\alpha ({\vec{\mathbf{y}}}_{q'}-{\vec{\mathbf{y}}}_q)\Big], \end{align*} where we normalized the columns by their Euclidian norm. The sums can be approximated by integrals over the frequency band and aperture, as long as they are sampled at intervals $h_\omega$ and $h_s$ satisfying \[
\frac{h_\omega}{b} \frac{|{\vec{\bf m}}_\alpha \cdot ({\vec{\mathbf{y}}}_q-{\vec{\mathbf{y}}}_{q'})|}{c/b} \ll 1,
\qquad \frac{V h_s}{a} \frac{|{\vec{\bf t}}_\alpha \cdot \mathbb{P}_\alpha
({\vec{\mathbf{y}}}_{q'}-{\vec{\mathbf{y}}}_q)|}{\lambda_o L/a} \ll 1. \] We obtain after taking absolute values that \begin{align}
\left|\left< \frac{{\bf a}_q}{\|{\bf a}_q\|}, \frac{{\bf
a}_{q'}}{\|{\bf a}_{q'}\|} \right>\right| \approx
\left|\mbox{sinc} \Big( \frac{b}{c} {\vec{\bf m}}_\alpha \cdot
({\vec{\mathbf{y}}}_{q'}-{\vec{\mathbf{y}}}_q) \Big) \mbox{sinc}\Big( \frac{k_o a}{L_\alpha}
{\vec{\bf t}}_\alpha \cdot \mathbb{P}_\alpha ({\vec{\mathbf{y}}}_{q'}-{\vec{\mathbf{y}}}_q)\Big)\right|. \end{align} This is small for $q \ne q'$ when we sample the imaging window $\mathcal{Y}$ in steps that are larger than the resolution limits $c/b$ in range and $\lambda_o L/a$ in cross-range.
A similar calculation can be done for the rows of $\mathbf{A}$, denoted by ${\bf a}_{(j,l)}$. We have \[
\left< {\bf a}_{(j',l')},{\bf a}_{(j,l)} \right> = \Big(\mathbf{A} \mathbf{A}^*
\Big)_{(j,l),(j',l')} = \sum_{q=1}^Q \overline{A_{j',q}(\Delta
\omega_{l'})} A_{j,q}(\Delta \omega_l), \] and using Lemma \ref{lem.1} we get \begin{align*}
\left|\left< \frac{ {\bf a}_{(j',l')}}{\|{\bf a}_{(j',l')}\|},
\frac{{\bf a}_{(j',l')}}{\|{\bf a}_{(j',l')}\|} \right> \right|
= \frac{1}{Q} \sum_{q=1}^Q \exp \Big[ \frac{2i
(\omega_{l'}-\omega_{l}){\vec{\bf m}}_\alpha \cdot \Delta {\vec{\mathbf{y}}}_q}{c} + \\\frac{2 i
k_\beta V (s_{j'}-s_{j}) {\vec{\bf t}}_\alpha \cdot \mathbb{P}_\alpha \Delta
{\vec{\mathbf{y}}}_q}{L_\alpha} \Big]. \end{align*} Furthermore, for discretizations of the imaging window in steps $h$ in range and $h^\perp$ in cross-range, satisfying \[
\frac{|\omega_{l'}-\omega_l|}{b} \frac{h}{c/b} \ll 1, \qquad
\frac{V |s_{j}-s_{j'}|}{a} \frac{h^\perp}{\lambda_o L_\alpha/a} \ll 1, \] we can approximate the sum over $q$ by an integral over the imaging window and obtain \begin{align*}
\left|\left< \frac{ {\bf a}_{(j',l')}}{\|{\bf a}_{(j',l')}\|},
\frac{{\bf a}_{(j',l')}}{\|{\bf a}_{(j',l')}\|} \right> \right|
\approx \left| \mbox{sinc} \Big(\frac{(\omega_{l'}-\omega_l) {{Y}_\alpha}}{c}\Big)
\mbox{sinc} \Big(\frac{k_o V (s_{j'}-s_j) {{Y}_\alpha^\perp}}{L_\alpha}\Big) \right|. \end{align*} This result shows that the inner product of the rows is small when the frequency is sampled in steps larger than ${{Y}_\alpha}/c$ and the slow time is sampled in steps larger than $(1/V)/(\lambda_o {{Y}_\alpha^\perp}/L_\alpha)$.
\end{document} | arXiv |
Memorize: A set is a well-defined collection of objects called elements or members of the set. If $x$ is a member of the set $S$, we write $x \in S$, and if $x$ is a not member of the set $S$, we write $x \notin S$
Here, well-defined means that any given object must either be an element of the set, or not be an element of the set.
Memorize: We say sets $A$ and $B$ are equal, and write $A=B$ if $x \in A \Leftrightarrow x \in B$ (that is, have exactly the same elements).
Here are three ways of specifying a set:
1. Explicit listing: list its elements between brackets, as in $\{2,3,5,7\}$.
2. Implicit listing: list enough of its elements to establish a pattern and use an elipsis (...). At least two elements must be listed to establish the pattern, sometimes more are needed. As examples, consider $\{\ldots-3,-1,1,3, \ldots\}$ and $\{0,2,4, \ldots, 120\}$, the set of odd integers and the set of non-negative even integers less than or equal to 120 , respectively.
3. Set builder notation: specify the set as the set of all $x$ (say) that make some propositional function true, as in $\{x:(x$ is prime $) \wedge(x<10)\}$.
Note that these are all ways of describing the set, but the set itself does not depend on the description. It just exists, how you describe it is a choice. In particular, for each object, what matters is whether or not it belongs to the set. This is why $\{1,2,2,3\},\{1,2,3,3\}$ and $\{1,2,3\}$ all describe the same set.
Memorize: We say that a set $A$ is a subset of a set $B$ if every element of $A$ is an element of $B$ (i.e., $x \in A \Rightarrow x \in B$ ). If $A$ is a subset of $B$ we write $A \subseteq B$, and otherwise we write $A \nsubseteq B$.
Memorize: The empty set is the set that contains no elements. It is denoted by $\emptyset$ or \{\} .
For every set $A$, we have $A \subseteq A$ and $\emptyset \subseteq A$. Both statements follow from the definition of subset. The second statement is true because the condition $x \in \emptyset$ is never true. (You should be able to explain this if asked.)
Memorize: We say that $A$ is a proper subset of $B$, and write $A \subset B$, if $A \subseteq B$ and $A \neq B$.
That is, $A$ is a proper subset of $B$ if $A \subseteq B$ and there is an element of $B$ which is not an element of $A$. This is consistent with the general use of the word "proper" in mathematics - roughly spreaking it is used for "not equal to the whole thing".
Notice that two sets $A$ and $B$ are equal if $x \in A \Leftrightarrow x \in B$. This is the same as $x \in A \Rightarrow x \in B$ and $x \in B \Rightarrow x \in A$. That is $A=B$ is the same as $A \subseteq B$ and $B \subseteq A$.
How to prove two sets $A$ and $B$ are equal. Here are two ways.
1. Showing that each is a subset of the other. A proof like this has two parts. First you show $A \subseteq B$ by starting with "Assume $x \in A$ " and then arguing that $x \in B$, and then you show $B \subseteq A$ by starting with "Assume $x \in B$ " and then arguing that $x \in A$. The argument will usually have to make use of other information you know (and/or are given).
2. Using set buider notation to demonstrate that the sets can be described by logically equivalent propositional functions.
You must be able to distinguish between $\in$ and $\subseteq$. The first one makes the assertion that a particular object belongs to a set; the second one says that every element of one set belongs to another set. The confusion usually creeps in when the sets in question contain other sets as elements.
Memorize: The power set of a set $X$ is the set $\mathcal{P}(X)$ whose elements are the subsets of $X$.
You need to keep the following facts straight:
- $\mathcal{P}(X)$ is a set.
- the elements of $\mathcal{P}(X)$ are sets (too). - $A \in \mathcal{P}(X) \Leftrightarrow A \subseteq X$ (this is the definition).
- In particular, $\emptyset \in \mathcal{P}(X)$ and $X \in \mathcal{P}(X)$.
We always assume our sets are subsets of some (large) set called the universe (or universal set), and denoted by $\mathcal{U}$.
Memorize: Let $A$ and $B$ be sets:
- The union of $A$ and $B$ is the set $A \cup B=\{x: x \in A \vee x \in B\}$.
- The intersection of $A$ and $B$ is the set $A \cap B=\{x: x \in A \wedge x \in B\}$.
- The difference of $A$ and $B$ is the set $A-B=\{x: x \in A \wedge x \notin B\}$.
- The complement of $A$ is the set $\bar{A}=\{x: x \notin A\}=\mathcal{U}-A$.
- The symmetric difference of $A$ and $B$ is the set $A \Delta B=(A-B) \cup(B-A)$.
Note that $A-B$ is, in general, not equal to $B-A$.
Set identities. These arise from using set builder notation and the logical equivalences from before (that is, they can all be proved that way). You should memorize them.
- $A \cap \mathcal{U}=A, \quad A \cap \emptyset=\emptyset$
- $A \cup \mathcal{U}=\mathcal{U}, \quad A \cup \emptyset=A$
- $A \cup A=A, \quad A \cap A=A$
- $A \cup B=B \cup A, \quad A \cap B=B \cap A$
- $(A \cap B) \cap C=A \cap(B \cap C), \quad(A \cup B) \cup C=A \cup(B \cup C)$
- Law of Double Complement: $\overline{\bar{A}}=A$
- Distributive Laws: $A \cup(B \cap C)=(A \cup B) \cap(A \cup C), \quad A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$
- DeMorgan's Laws: $\overline{A \cup B}=\bar{A} \cap \bar{B}, \quad \overline{A \cap B}=\bar{A} \cup \bar{B}$
You should be able to prove each of the above in two ways (set builder notation and showing that each side is a subset of the other).
Venn diagrams. These are a pictorial representation of sets and a good way to get intuition about (possible) set equalities. You should be able to use Venn diagrams to investigate whether two sets are equal. If they are equal, you should be able to prove this using one of the methods discussed before (a Venn diagram does not suffice as a proof). If the sets are not equal, you should be able to use the Venn diagram to get a particular example showing they are not equal.
| Textbooks |
\begin{document}
\title{The saturation number of $K_{3,3} \begin{abstract}
A graph $G$ is called $F$-saturated if $G$ does not contain $F$ as a subgraph (not necessarily induced) but the addition of any missing edge to $G$ creates a copy of $F$. The saturation number of $F$, denoted by $sat(n,F)$, is the minimum number of edges in an $n$-vertex $F$-saturated graph. Determining the saturation number of complete bipartite graphs is one of the most important problems in the study of the saturation number. The value of $sat(n,K_{2,2})$ was shown to be $\lfloor\frac{3n-5}{2}\rfloor$ by Ollmann, and a shorter proof was later given by Tuza. For $K_{2,3}$, there has been a series of study aiming to determine $sat(n,K_{2,3})$ over the years. This was finally achieved by Chen who confirmed a conjecture of Bohman, Fonoberova, and Pikhurko that $sat(n, K_{2,3})= 2n-3$ for all $n\geq 5$.
Pikhurko and Schmitt conjectured that $sat(n, K_{3,3})= (3+o(1))n$. In this paper, for $n\geq 9$, we give an upper bound of $3n-9$ for $sat(n, K_{3,3})$, and prove that $3n-9$ is also a lower bound when the minimum degree of the $K_{3,3}$-saturated graphs is $2$ or $5$, where it is trivial when the minimum degree is greater than $5$.
\\
\noindent\textbf{Keywords:} saturation number; complete bipartite graph; minimum degree\\ \end{abstract}
\section{Introduction} \baselineskip 16pt
All graphs in this paper are finite and simple. Throughout the paper we use the terminology and notation of \cite{W2001}. Given a graph $G$, we use $|G|$, $e(G)$, $\delta(G)$, and $\Delta(G)$ to denote the number of vertices, the number of edges, the minimum degree and the maximum degree of $G$, respectively. Let $\overline{G}$ denote the complement graph of $G$. For any $v\in V(G)$, let $d_G(v)$ and $N_G(v)$ denote the degree and neighborhood of $v$ in $G$, respectively, and let $N_G[v]=N_G(v)\cup\{v\}$. We shall omit the subscript $G$ when the context is clear. For $A,B\subseteq V(G)$ with $A\cap B=\emptyset$, let $A\sim B$ denote that each vertex in $A$ is adjacent to each vertex in $B$ and $G[A, B]$ be the subgraph with vertex set $A\cup B$ and edge set $E(G[A, B])=\{xy\in E(G): x\in A, y\in B\}$. For $S\subseteq V(G)$, we denote by $G[S]$ the subgraph of $G$ induced by $S$. Let $n$ be a positive integer. For positive integer $k$, we let $[k]=\{1,2,\ldots,k\}$. We denote a path, a cycle, a star, and a complete graph with $n$ vertices by $P_n$, $C_n$, $S_n$, and $K_n$, respectively. For $r\geq2$ and positive integers $s_1,\ldots,s_r$, let $K_{s_1,\ldots,s_r}$ denote the complete $r$-partite graph with part sizes $s_1,\ldots,s_r$. Let $G$ and $H$ be two disjoint graphs.
Denote by $G\cup H$ the union of $G$ and $H$. The {\it join} $G\vee H$ is the graph obtained from $G\cup H$ by joining each vertex of $G$ to each vertex of $H$.
Given a family of graphs $\mathcal{F}$, a graph $G$ is \dfn{$\mathcal{F}$-saturated} if no member of $\mathcal{F}$ is a subgraph of $G$, but for any $e\in E(\overline{G})$, some member of $\mathcal{F}$ is a subgraph of $G+e$. The \dfn{saturation number} of $\mathcal{F}$, denoted by $sat(n,\mathcal{F})$, is the minimum number of edges in an $n$-vertex $\mathcal{F}$-saturated graph. Define $sat_\delta(n,\mathcal{F})$ to be the minimum number of edges in a graph with $n$ vertices and minimum degree $\delta$ that is $\mathcal{F}$-saturated. If $\mathcal{F}=\{F\}$, we also write $sat(n,\{F\})$ and $sat_\delta(n,\{F\})$ as $sat(n,F)$ and $sat_\delta(n,F)$, respectively.
Saturation numbers were first studied in 1964 by Erd\H{o}s, Hajnal, and Moon \cite{EHM1964}, who proved that $sat(n,K_{k+1})=(k-1)n-{{k}\choose{2}}$. Furthermore, they proved that equality holds only for the graph $K_{k-1}\vee \overline{K_{n-k+1}}$. In 1986, K\'{a}szonyi and Tuza in \cite{KT1986} determined $sat(n,F)$ for $F\in\{S_k, kK_2, P_k\}$, and they proved that $sat(n,\mathcal{F})=O(n)$ for any family $\mathcal{F}$ of graphs. Since then, there has been extensive research on saturation numbers for various graph families $\mathcal{F}$.
We now mention some results for complete multipartite graphs. When all but at most one parts have size $1$, Pikhurko \cite{P1999} and Chen, Faudree, and Gould \cite{CFG2008} independently determined the saturation number of complete multipartite graphs with sufficiently large order. When there are at least two parts of size at least 2, the exact values were only known for $K_{2,2}$ and $K_{2,3}$. The exact value for $K_{2,2}$ was first determined by Ollmann \cite{O1972}. Later on, a shorter proof was given by Tuza \cite{T1989}. For $K_{2,3}$, there have been several papers aiming to determine $sat(n,K_{2,3})$ over the years. This was finally achieved by Chen \cite{C2014} who confirmed a conjecture of Bohman, Fonoberova, and Pikhurko \cite{BFP2010} that $sat(n, K_{2,3})= 2n-3$ for all $n\geq 5$. For the case where the graph has $r$ parts and all parts have size 2, Gould and Schmitt \cite{GS2007} conjectured that $sat(n, K_{2,\ldots,2})=\lceil((4r-5)n-4r^2+6r-1)/2\rceil$, and they proved the conjecture when the minimum degree of the $K_{2,\ldots,2}$-saturated graphs is $2r-3$.
For general complete multipartite graphs $K_{s_1,\ldots,s_r}$ with $s_r\geq \cdots \geq s_1\geq1$, Bohman, Fonoberova, and Pikhurko \cite{BFP2010} determined the asymptotic bound on $sat(n, K_{s_1,\ldots,s_r})$ as $n\rightarrow \infty$. \begin{thm}[\cite{BFP2010}]\label{Ks1sr}
Let $r\geq2$ and $s_r\geq \cdots \geq s_1\geq1$. Define $p=s_1+\cdots+s_{r-1}-1$. Then, for all large $n$, \begin{align*} (p+\frac{s_r-1}{2})n-O(n^{3/4})\leq sat(n, K_{s_1,\ldots,s_r})\leq {p\choose 2}+p(n-p)+\big\lceil\frac{(s_r-1)(n-p)}{2}-\frac {s_r^2}{8}\big\rceil. \end{align*}
In particular, $sat(n, K_{s_1,\ldots,s_r})=(s_1+\ldots+s_{r-1}+0.5s_r-1.5)n + O(n^{3/4}).$ \end{thm}
We continue to study the saturation number for complete multipartite graphs. In light of the known results, studying $sat(n,K_{3,3})$ is the natural next step. In 2008, Pikhurko and Schmitt \cite{PS2008} conjectured that $sat(n, K_{3,3})=(3+o(1))n$.
In this paper, we give an upper bound on $sat(n, K_{3,3})$. Moreover, we consider its lower bound. In particular, we determine the exact value of $sat(n, K_{3,3})$ for $6\le n\le 8$ and provide a lower bound on $sat(n,K_{3,3})$ when the minimum degree of the $K_{3,3}$-saturated graphs is $2$ or $5$.
The main results are the following theorems.
\begin{thm}\label{main1} Let $n$ be a positive integer and $n\ge 6$. Then
$sat(n, K_{3,3})\leq\begin{cases}
2n, & 6\leq n \leq 8,\\[2mm]
3n-9, & n \geq 9.\\
\end{cases}$ \end{thm}
\begin{thm}\label{main23}
\begin{description}
\item[(i)] For $6\le n\le 8$, $sat(n,K_{3,3})=2n$.
\item[(ii)] For $n\ge 9$, $sat_2(n,K_{3,3})= 3n-9$ and $sat_5(n,K_{3,3})\ge 3n-9$.
\end{description}
\end{thm}
Let $G$ be a $K_{3,3}$-saturated graph with $n$ vertices and $n\geq 9$. If $\delta(G)\ge 6$, then $e(G)\ge 3n\geq 3n-9$. Hence, for $n\geq9$, to determine the exact value of $sat(n,K_{3,3})$, we only need to consider $K_{3,3}$-saturated graphs with the minimum degree at most $5$.
An outline of this paper is as follows. To prove Theorem \ref{main1}, we construct an $n$-vertex $K_{3,3}$-saturated graph with $2n$ edges when $ 6\leq n \leq 8$ and $3n-9$ edges when $n\geq 9$ in Section \ref{upper}. In Section \ref{689}, we first prove that $sat(n,K_{3,3})\geq 2n$ when $6\le n\le 8$ in Section \ref{6n8}, then we prove $sat_{\delta}(n,K_{3,3})\ge 3n-9$ when $\delta\in\{2,5\}$ in Section \ref{n9}.
\section{Proof of Theorem \ref{main1}}\label{upper} In this section, for $n\geq6$, we construct an $n$-vertex $K_{3,3}$-saturated graph $G_n$ with $2n$ edges when $ 6\leq n \leq 8$, and $3n-9$ edges when $n\geq 9$. Let $G_{11}$ be a graph as depicted in Figure \ref{fig1}. Then $G_n=G_{11}[\{v_1,\ldots,v_n\}]$ for $6\leq n\leq 11$. \begin{figure}
\caption{The graph $G_{11}$.}
\label{fig1}
\end{figure} \begin{prop}\label{construction1} For $6\leq n\leq 11$, the graph $G_n$ is $K_{3,3}$-saturated and \begin{equation*} e(G_n)=\begin{cases} 2n, & 6\le n\le 8,\\ 3n-9, & 9\le n \le 11.\\ \end{cases} \end{equation*} \end{prop}
\noindent {\emph{Proof.}}~~ It is easy to verify that $e(G_n)=2n$ when $6\leq n \leq 8$, and $e(G_n)=3n-9$ when $9\leq n \leq 11$. Next we show that $G_n$ contains no copy of $K_{3,3}$ for $6\leq n\leq 11$. Suppose $R$ is a copy of $K_{3,3}$ of $G_{11}$. Then $v_9\notin V(R)$ because $d_{G_{11}}(v_9)=2$. For $u\in\{v_7,v_8,v_{10},v_{11}\}$, since $d_{G_{11}}(u)=3$ and there exists $v\in N_{G_{11}}(u)$ such that $d_{G_{11}}(v)=3$ and $|N_{G_{11}}(u)\cap N_{G_{11}}(v)|=2$, we have $u\notin V(R)$. Thus $R\subseteq G_6$. Since $v_1v_2\notin E(G_6)$, $v_1$ and $v_2$ lie in the same part of $R$. Then $R[\{v_3,v_4,v_5,v_6\}]$ contains a copy of $K_{1,3}$, a contradiction. So $G_{11}$ contains no copy of $K_{3,3}$. Note that $G_n$ ($6\leq n\leq 10$) is a subgraph of $G_{11}$. Hence $G_n$ contains no copy of $K_{3,3}$ for any $6\leq n\leq 11$.
Let $xy$ be an edge in the complement of $G_n$. It remains to show that the graph $G_n'$ obtained by adding $xy$ to $G_n$ has a copy of $K_{3,3}$. We consider the following cases. \begin{description}
\item[(a)] If $\{x,y\}\cap\{v_1,v_2\}\neq \emptyset$ or $x,y\in\{v_7,v_8,v_{10},v_{11}\}$, then the subgraph of $G_n'$ induced by $\{x,y\}\cup \{v_3,v_5\}\cup\{v_4,v_6\}$ contains a copy of $K_{3,3}$.
\item[(b)] If $\{x,y\}\cap\{v_3,v_5\}\neq \emptyset$ or $x=v_9$, $y\in\{v_8, v_{11}\}$, then the subgraph of $G_n'$ induced by $\{x,y\}\cup \{v_1,v_2\}\cup\{v_4,v_6\}$ contains a copy of $K_{3,3}$.
\item[(c)] If $\{x,y\}\cap\{v_4,v_6\}\neq \emptyset$ or $x=v_9$, $y\in\{v_7, v_{10}\}$, then the subgraph of $G_n'$ induced by $\{x,y\}\cup \{v_1,v_2\}\cup\{v_3,v_5\}$ contains a copy of $K_{3,3}$. \end{description} For $6\leq n\leq 11$, in all cases, $G_n'$ contains a copy of $K_{3,3}$, hence $G_n$ is $K_{3,3}$-saturated.
\vrule height3pt width6pt depth2pt
\begin{defn}\label{Gn} For $n\geq12$, let $H=\overline{K}_{2}\vee(C_4\cup C_{n-9}\cup K_1)$, where $V(\overline{K}_{2})=\{v_1,v_2\}$, $C_4=v_3v_4v_5v_6v_3$, $C_{n-9}=v_7v_8\ldots v_{n-3}v_7$, $V(K_1)=\{v_{n-2}\}$. Let $G_n$ be the graph obtained from $H$ by adding new vertices $\{v_{n-1},v_n\}$ and new edges $\{v_{n-1}v_3,v_{n-1}v_5,v_nv_4,v_nv_6\}$. \end{defn}
\begin{prop}\label{construction} For $n\geq12$, the graph $G_n$ defined in Definition~\ref{Gn} is $K_{3,3}$-saturated and has $3n-9$ edges. \end{prop}
\noindent {\emph{Proof.}}~~ Clearly, $e(G)=2(n-4)+(n-5)+4=3n-9$. Firstly, We show that $G_n$ has no subgraph isomorphic to $K_{3,3}$. Suppose $R$ is a copy of $K_{3,3}$ of $G_n$. From the structure of $G_n$, we see that $d(v_{n-1})=d(v_n)=2$ and hence $v_{n-1}, v_n\notin V(R)$. Thus $R\subseteq H$. Since each vertex of $C_4\cup C_{n-9}\cup K_1$ has at most two neighbors in $C_4\cup C_{n-9}\cup K_1$, $v_1,v_2\in V(R)$ and they lie in different parts of $R$. This contradicts $v_1v_2\notin E(G_n)$. So $G_n$ contains no copy of $K_{3,3}$.
Let $xy$ be an edge in the complement of $G_n$. It remains to show that the graph $G''$ obtained by adding $xy$ to $G_n$ has a copy of $K_{3,3}$. We consider the following cases. \begin{description}
\item[(a)] If $x,y\in\{v_1,v_2,v_{n-1},v_{n}\}$, then the subgraph of $G''$ induced by $\{x,y\}\cup\{v_3,v_5\}\cup\{v_4,v_6\}$ contains a copy of $K_{3,3}$.
\item[(b)] If $x=v_{n-1}$, $y\in\{v_4,v_6,v_7,\ldots,v_{n-2}\}$ or $x=v_4$, $y=v_6$ or $x\in\{v_4,v_6\}$, $y\in\{v_7,\ldots,v_{n-2}\}$, then the subgraph of $G''$ induced by $\{x,v_1,v_2\}\cup\{y,v_3,v_5\}$ contains a copy of $K_{3,3}$.
\item[(c)] If $x=v_{n}$, $y\in\{v_3,v_5,v_7,\ldots,v_{n-2}\}$ or $x=v_3$, $y=v_5$ or $x\in\{v_3,v_5\}$, $y\in\{v_7,\ldots,v_{n-2}\}$, then the subgraph of $G''$ induced by $\{x,v_1,v_2\}\cup\{y,v_4,v_6\}$ contains a copy of $K_{3,3}$.
\item[(d)]If $x,y\in \{v_7,\ldots,v_{n-2}\}$ and $x\neq v_{n-2}$, let $N(x)\cap\{v_7,\ldots,v_{n-3}\}=\{x',x''\}$, then the subgraph of $G''$ induced by $\{x,v_1,v_2\}\cup\{y,x',x''\}$ contains a copy of $K_{3,3}$. \end{description} In all cases, $G''$ contains a copy of $K_{3,3}$. Hence $G_n$ is $K_{3,3}$-saturated.
\vrule height3pt width6pt depth2pt
By Proposition \ref{construction1} and Proposition \ref{construction}, we complete the proof of Theorem \ref{main1}.
\section{Proof of Theorem \ref{main23}}\label{689}
In the rest of the paper, we consider the lower bound on $sat(n, K_{3,3})$. Let $G=(V,E)$ be a $K_{3,3}$-saturated graph. We firstly choose a vertex $a$ such that $d(a)=\delta(G)$ and $e(G[N(a)])$ is as small as possible.
We partition $V$ into four parts $V_1$, $V_2$, $V_3$ and $V_4$, where $V_1=N[a]$, $V_2=\{x \in V \backslash V_1: |N(x) \cap N(a)| \geq 2 $\},
$V_3=\{y \in V \backslash (V_1 \cup V_2): |N(y) \cap N(a)|=1$\} and $V_4=V\backslash (V_1 \cup V_2 \cup V_3 )$. Let $N_G(a)=\{a_1,a_2,\ldots,a_{d(a)}\}$.
For $i_1,i_2,\ldots, i_s\in [d(a)]$, let $V_{i_1i_2\ldots i_s}=\{x \in V_2: N(x) \cap V_1=\{a_{i_1},a_{i_2},\ldots, a_{i_s}\}\}$.
In the following, we will first describe some useful properties of the $K_{3,3}$-saturated graph $G$.
\begin{prop}\label{k22}
The following statements hold. \begin{description} \item[(i)] For any $x,y\in V$, if $x y \notin E$, then there are $\{x_1, x_2\}\subseteq N(x) $ and $\{y_1, y_2\}\subseteq N(y)$ such that $\{x_1,x_2\}\sim\{y_1, y_2\}$. (We usually say there is a copy of $K_{2,2}$ between $N(x)$ and $N(y)$.)
\item[(ii)] For any $x \in V \setminus V_1$ , we have $|N(x) \cap N(a_i) \cap N(a_j)|\leq2$ for any $i,j\in[d(a)]$ with $i\neq j$, and there exist $i,j\in[d(a)]$ with $i\neq j$ such that $|N(x) \cap N(a_i) \cap N(a_j)|=2$.
\item[(iii)] For any $x \in V_3$, we have $|N(x) \cap V_2| \geq 1$. For any $x \in V_4$, we have $|N(x) \cap V_2| \geq 2$.
\item[(iv)] When $G[V_1 \backslash \{a\}]$ contains no copy of $K_{1,2}$, we have $|N(x) \cap V_2|\geq2$ for any $x \in V \setminus V_1$, and $|V_2|\geq3$. When $G[V_1\setminus\{a\}]$ contains no copy of $K_{2,2}$, we have $|N(x)\cap V_2|\ge 1$ for any $x\in V_2$, and $|V_2|\ge 2$. \end{description} \end{prop}
\noindent {\emph{Proof.}}~~ Suppose $x y \notin E$. Then there is a copy of $K_{3,3}$ in $G+xy$, and (i) follows. For any $x \in V \setminus V_1$, if there is a vertex $x \in V \setminus V_1$ such that $|N(x) \cap N(a_i) \cap N(a_j)|\geq3$ for some $i,j\in[d(a)]$ with $i\neq j$, then we would obtain a copy of $K_{3,3}$ of $G$, a contradiction. So $|N(x) \cap N(a_i) \cap N(a_j)|\leq2$ for any $x \in V \setminus V_1$ and $i,j\in[d(a)]$ with $i\neq j$. Since $ax\notin E$ for any $x\in V\backslash V_1$, there exist $i,j\in[d(a)]$ such that $|N(x) \cap N(a_i) \cap N(a_j)|=2$ by (i). This proves (ii). Let $x \in V \setminus V_1$ and $i,j\in[d(a)]$ with $i\neq j$ such that $|N(x) \cap N(a_i) \cap N(a_j)|=2$, we say $\{u,v\}=N(x) \cap N(a_i) \cap N(a_j)$. Then $u,v\in (V_1\cup V_2)\setminus\{a\}$. If $x \in V_3$, then we have $|N(x) \cap V_2| \geq 1$ by the definition of $V_3$. If $x \in V_4$, then we have $|N(x) \cap V_2| \geq 2$ by the definition of $V_4$. This proves (iii). Suppose $G[V_1 \backslash \{a\}]$ contains no copy of $K_{1,2}$. Then $u,v\in V_2$. Hence we have $|N(x) \cap V_2|\geq2$ for any $x \in V \setminus V_1$, and $|V_2|\geq3$. Suppose $G[V_1 \backslash \{a\}]$ contains no copy of $K_{2,2}$. Then $\{u,v\}\cap V_2\ne \emptyset$. Hence we have $|N(x) \cap V_2|\geq1$ for each $x \in V_2$, and $|V_2|\geq2$.
This proves (iv).
\vrule height3pt width6pt depth2pt
Proposition \ref{k22}(i) implies $\delta(G)\ge 2$ for each $K_{3,3}$-saturated graph $G$. Thus we consider $\delta(G)\ge 2$.
\subsection{Proof of Theorem \ref{main23}(i)}\label{6n8} By Theorem \ref{main1}, to prove $sat(n, K_{3,3})= 2n$ for $6\leq n\leq 8$, it suffices to prove $sat(n, K_{3,3}) \geq 2n$.
We consider the minimum degree of $G$. If $\delta(G) \geq 4$, then we have $e(G) \geq 2n$. So we assume that $2\leq \delta(G)\leq 3$. For $i\in \{2,3,4\}$ and $x\in V_i$, we define $f(x)=|N(x) \cap (V_1 \cup \dots \cup V_{i-1} )|+0.5|N(x) \cap V_i|-2$. Let $ s_i=\sum_{x \in V_i} f(x)$, where $i \in \{2,3,4\}$.
We first observe that one can relate the number of edges to $s_2$, $s_3$ and $s_4$ in the following way: \begin{align}\label{2ef} e(G)=\notag&~e(G[V_1])+e(G[V_2])+e(G[V_1,V_2])+e(G[V_3])+e(G[V_1,V_3])+e(G[V_2,V_3])+e(G[V_4])\\\notag&+e(G[V_4,V_2\cup V_3])\\\notag
=&~e(G[V_1])+2(|V_2|+|V_3|+|V_4|)+s_2+s_3+s_4\\
=&~e(G[V_1])+2(n-|V_1|)+s_2+s_3+s_4. \end{align}
\begin{lem}\label{n678} For $6\le n\le 8$, \begin{description}
\item[(i)] if $\delta(G)=2$, then $s_2+s_3+s_4\geq |V_2|+|V_3|$.
\item[(ii)] if $\delta(G)=3$, then $s_2+s_3+s_4 \geq |V_2|+|V_3|+|V_4|$ when $e(G[V_1 \backslash \{a\}])\leq 1$ and $s_2+s_3+s_4 \geq 0.5(|V_2|+|V_3|+|V_4|)$ when $e(G[V_1 \backslash \{a\}])\geq2$. \end{description} \end{lem}
\noindent {\emph{Proof.}}~~ Suppose that $\delta(G)=2$. Then $G[V_1 \backslash \{a\}]$ contains no $K_{1,2}$. Thus $f(x) \geq 1$ for each $x\in V_2\cup V_3$ and $f(x) \geq 0$ for each $x\in V_4$ by Proposition \ref{k22} (iii). So $s_2+s_3+s_4 \geq |V_2|+|V_3|$. Suppose that $\delta(G)=3$. If $e(G[V_1 \backslash \{a\}])\leq 1$, then $|V_4|\leq1$ because $n\leq8$ and $|V_2|\ge 3$ by Proposition \ref{k22}(iv). Thus $f(x) \geq 1$ for each $x\in V\setminus V_1$ by Proposition \ref{k22} (iii). So $s_2+s_3+s_4 \geq |V_2|+|V_3|+|V_4|$. If $e(G[V_1 \backslash \{a\}])\geq2$, then we have $|N(x) \cap V_2|\geq1$ for each $x \in V_2\cup V_3$ and $|N(x) \cap V_2|\geq2$ for each $x \in V_4$ by Proposition \ref{k22} (iii). Thus for $x \in V_2$, $f(x) \geq 0.5$; for $y \in V_3$, $f(y) \geq 0.5$ or $f(y)=0$ and there exists a vertex $z \in V_4$ such that $f(z)=1$; for $z \in V_4$, $f(z) \geq 0.5$. Proposition \ref{k22}(iv) implies $|V_2|\ge 2$ and so
$|V_3\cup V_4|\leq 2$, we have $s_2+s_3+s_4 \geq 0.5(|V_2|+|V_3|+|V_4|)$.
\vrule height3pt width6pt depth2pt
Suppose that $\delta(G)=2$. If $a_1a_2 \in E$, then $e(G)\geq2n+|V_2|+|V_3|-3$ by Lemma \ref{n678}(i) and (\ref{2ef}). By Proposition \ref{k22}(iii), we have $|V_2| \geq 3$. So $e(G) \geq 2n$. If $a_1a_2 \notin E(G)$, then $e(G)\geq2n+|V_2|+|V_3|-4$ by Lemma \ref{n678}(i) and (\ref{2ef}). Proposition \ref{k22}(i) implies that there is a copy of $K_{2,2}$ between $N(a_1)$ and $N(a_2)$, we have $|V_2\cup V_3| \geq 4$. So $e(G) \geq 2n$.
Suppose that $\delta(G)=3$. If $n=6$, then $|V_2|=2$, $|V_3|=|V_4|=0$ and $e(V_1)=6$ by Proposition \ref{k22}(i). Otherwise, $a_ia_j\notin E$ where $i,j\in [3]$ with $i\ne j$, Proposition \ref{k22}(i) implies that there is a copy of $K_{2,2}$ between $N(a_i)$ and $N(a_j)$, which contradicts the fact that $|V_2\cup V_3|=2$.
Let $V_2=\{x_1, x_2\}$. Proposition \ref{k22}(iv) implies $x_1x_2\in E$. If $x_1a_i\notin E$ for some $i\in [3]$, then $x_2\in V_{123}$ by Proposition \ref{k22} (i). Thus $e(G)\ge 12=2n$.
If $n=7$ and $e(G[V_1])\le 4$, then $G[V_1\setminus\{a\}]$ contains no copy of $K_{1,2}$. Proposition \ref{k22}(iv) implies
$|V_2|=3$
and $e(G[V_2])=3$. Since $a_ia_j\notin E$ for some $i,j\in[3]$, Proposition \ref{k22}(i) implies there is a copy of $K_{2,2}$ between $N(a_i)$ and $N(a_j)$. Since $|V_2\cup V_3|=|V_2|= 3$, $e(G[V_1])\ge 4$. We see $|V_{123}|\le 1$, else $G$ contains a copy of $K_{3,3}$. There exists a vertex $x$ such that $|N(x)\cap V_1|=2$ and $xa_k\notin E$ for some $k\in [3]$. Proposition \ref{k22}(i) implies that there is a copy of $K_{2,2}$ between $N(x)$ and $N(a_k)$, say $\{x_1, x_2\}\sim\{a_{k1},a_{k2}\}$. When $\{a_{k1},a_{k2}\}\subseteq V_2$, then $\{x_1,x_2\}\subseteq V_1$ and $\{a_{k1},a_{k2}\}\subseteq V_{123}$, a contradiction. When $\{a_{k1},a_{k2}\}\cap V_1\ne \emptyset$, since $e(G[V_1])\le 4$, $|\{a_{k1},a_{k2}\}\cap\{a_1,a_2,a_3\}|\le 1$. If $a_{k1}\in \{a_1,a_2,a_3\}$, then $a_{k2}\in V_2$. By $|V_2|=3$, $\{x_{k1},x_{k2}\}\cap V_1\ne \emptyset$, which contradicts $e(G[V_1])\le 4$. If $a\in \{a_{k1},a_{k2}\}$, say $a_{k1}=a$, then $\{x_{1},x_2\}\subseteq V_1$, $a_{k2}\in V_2$ and $a_{k2}\in V_{123}$. Then $e(G)=e(G[V_1])+e(G[V_2])+e(G[V_1,V_2])\ge 4+3+7=14=2n$.
If $n=7$ and $e(G[V_1])=6$, by Lemma \ref{n678}(ii), then $e(G)\ge 2n-0.5$, that is $e(G)\ge 2n$. Suppose $n=7$ and $e(G[V_1])=5$. Let $E(G[V_1\setminus\{a\}])=\{a_1a_2,a_1a_3\}$.
If $|V_2|=2$, then let $V_2=\{x_1,x_2\}$. Applying Proposition \ref{k22}(i) to $ax_1\notin E$ ($ax_2\notin E$), we have the $K_{2,2}$ between $N(a)$ and $N(x_1)$ ($N(x_2)$) is $\{a_2,a_3\}\sim\{a_1, x_2\}(\{a_1, x_1\})$.
Then $\{x_1,x_2\}\subseteq V_{123}$, and so $\{a_1,a_2,a_3\}\sim\{a, x_1, x_2\}$ is a copy of $K_{3,3}$ of $G$, a contradiction.
If $|V_2|\ge 3$, then $|V_2|=3$ by $n=7$. Let $V_2=\{x_1,x_2, x_3\}$. Note that $f(x_i)\ge 0.5$ for each $i\in [3]$. If there exists a vertex $x_i\in V_{123}$ or there are two vertices $x_i,x_j\in V_2$ such that $f(x_i)\geq1$ and $f(x_j)\geq1$, then $e(G)\ge 2n-0.5$ by (\ref{2ef}), and so $e(G)\ge 2n$. Thus we may assume $V_{123}=\emptyset$ and there is at most one vertex $x_i\in V_2$ such that $f(x_i)\geq1$.
Since there is a copy of $K_{2,2}$ between $N(x)$ and $N(a)$ for each $x\in V_2$, there is some vertex $x_i\in V_2$ with $f(x_i)=1$, say $x_1$. Then $x_1\in V_{23}$ and $\{x_2,x_3\}\subseteq V_{1i}$ for some $i\in \{2,3\}$, say $i=2$. Then $N(a_3)=\{a,a_1,x_1\}$, but $e(G[N(a_3)])\le 1$, which contradicts the minimality of $e(G[N(a)])$. So $e(G)\ge 2n$.
If $n=8$, then $e(G) \geq 2n$ when $e(G[V_1 \backslash \{a\}])=1$ or $3$ by Lemma \ref{n678}(ii). Suppose $n=8$ and $e(G[V_1 \backslash \{a\}])=0$, then $ e(G)=2n+s_2+s_3+s_4-5$. So we need to show $s_2+s_3+s_4 \geq 4.5$.
If $|V_{123}|\geq1$, then $f(x) \geq 2$ for each $x\in V_{123}$. So $s_2+s_3+s_4 \geq |V_2|+|V_3|+|V_4|+1 \geq 5$ by the proof of Lemma \ref{n678}(ii). Now we consider $|V_{123}|=0$.
Since $a_1a_2\notin E$, Proposition \ref{k22}(i) implies that there is a copy of $K_{2,2}$ between $N(a_1)$ and $N(a_2)$, say $\{x_1,x_2\}\sim \{x_3,x_4\}$. Then $\{x_1,x_2,x_3,x_4\}\subseteq V_2\cup V_3$. Since $n=8$, $|V_2\cup V_3|=4$. If $x_1\in V_3$, then we can not find a copy of $K_{2,2}$ between $N(a_2)$ and $N(a_3)$ because $|(N(a_2)\cup N(a_3))\cap (V_2\cup V_3)|\leq 3$, a contradiction. By symmetry, we have $\{x_1,x_2,x_3,x_4\}\subseteq V_2$.
If there exists $i\in[4]$ such that $|N(x_i)\cap V_2|\ge 3$, then $e(G)= e(G[V_1])+e(G[V_2])+e(G[V_1,V_2])\ge3+5+8=16=2n$. If $|N(x_i)\cap V_2|=2$ for each $i\in [4]$, then $E(G[V_2])=\{x_ix_j|i\in\{1,2\}, j\in\{3,4\}\}$. Since $x_1x_2\notin E$, Proposition \ref{k22}(i) implies that there is a copy of $K_{2,2}$ between $N(x_1)$ and $N(x_2)$. Note that $N(x_1)\cup N(x_2)\subseteq\{a_1,a_2,a_3,x_3,x_4\}$, $e(G[\{a_1,a_2,a_3\}])=0$ and $x_3x_4\notin E$. So the $K_{2,2}$ between $N(x_1)$ and $N(x_2)$ must be $\{a_1,a_2\}\sim \{x_3,x_4\}$. Then $d(a_3)=1$, this contradicts $\delta(G)\geq2$.
Suppose $n=8$ and $e(G[V_1 \backslash \{a\}])=2$. Then $ e(G)=2n+s_2+s_3+s_4-3$. So we need to show $s_2+s_3+s_4 \geq 2.5$. If $f(x)\geq1$ for some $x\in V_2$, then $s_2+s_3+s_4 \geq 2.5$ by the proof of Lemma \ref{n678}(ii). If $f(x)=0.5$ for some $x\in V_2$, then $f(x')\geq 1$ where $\{x'\}=N(x)\cap V_2$. So $s_2+s_3+s_4 \geq 2.5$.
This completes the proof of the lower bound on $sat(n,K_{3,3})$ for $6\leq n\leq 8$.
\vrule height3pt width6pt depth2pt
\subsection{Proof of Theorem \ref{main23}(ii)}\label{n9} Note that for $n\geq9$, the minimum degree of the $K_{3,3}$-saturation graph we constructed in Section \ref{upper} with $3n-9$ edges is $2$. Thus $sat_2(n,K_{3,3})\le 3n-9$ for $n\ge 9$. Hence, to prove $sat_2(n,K_{3,3})=3n-9$, it suffices to prove $sat_2(n,K_{3,3})\ge 3n-9$ for $n\ge 9$. In this section, we give the lower bound of $3n-9$ for $sat_{\delta}(n, K_{3,3})$ for $\delta\in\{2,5\}$ and $n\geq9$. We first consider the case where the minimum degree of $G$ is $2$. \subsubsection{ $\delta(G)=2$}
We prove $sat_2(n, K_{3,3})\ge 3n-9$ for $n\geq 9$ in this part. According to the partition of $V$, we define $h(x)=|N(x) \cap (V_1 \cup \ldots\cup V_{i-1} )|+0.5|N(x) \cap V_i|-3$ for each $ x \in V_i $ and $q_i=\sum_{x \in V_i} h(x)$ where $i\in \{2,3,4\}$. For each $x\in V$, we say that the $h$-value of $x$ is $k$ if $h(x)=k$.
\begin{align}\label{eG2}
e(G)=&~e(G[V_1])+e(G[V_2])+e(G[V_1,V_2])+e(G[V_3])+e(G[V_1,V_3])+e(G[V_2,V_3])+e(G[V_4])\notag\\&+e(G[V_4,V_2\cup V_3])\notag\\
=&~e(G[V_1])+3(|V_2|+|V_3|+|V_4|)+q_2+q_3+q_4\notag\\
=&~e(G[V_1])+3(n-|V_1|)+q_2+q_3+q_4.
\end{align}
By (\ref{eG2}), we have $e(G) \geq 3n-7+q_2+q_3+q_4$. Therefore, it suffices to prove \begin{align}\label{-2.5} q_2+q_3+q_4 \geq -2.5. \end{align}
By Proposition \ref{k22}(iii), we have $|N(x)\cap V_2|=2$ for each $x\in V\setminus V_1$. So $h(z) \geq 0$ for each $z \in V_2\cup V_3 $ and $ h(z)\geq -1$ for each $z \in V_4$. Thus, $q_2 \geq 0$ and $q_3 \geq 0$. Therefore, to prove (\ref{-2.5}), it suffices to show $q_4 \geq -2.5$.
Let $V_4^{-}=\{z\in V_4:h(x)<0\}=\{z_1,z_2,\ldots,z_{|V_4^{-}|}\}$ and $n^-_4(x)=|N(x)\cap V^-_4|$ for each $x\in V$. By Proposition \ref{k22}(iii), each vertex $z \in V_4^{-}$ has exactly two neighbors in $V_2$, so we let $N(z_i)\cap V_2=\{x_{i1},x_{i2}\}$. Note that if $h(z_i)=-1$, then $N(z_i)=\{x_{i1},x_{i2}\}$ and so $z_i$ has no neighbor in $V_4^{-}$, and if $h(z_i)=-0.5$, then $d(z_i)=3$ and $z_i$ has one neighbor in $V_4$, saying $N_4(z_i)=\{c_i\}$.
For each $z_i, z_j\in V_4^{-}$ with $z_iz_j\notin E$, there is a $K_{2,2}$ between $N(z_i)$ and $N(z_j)$ by Proposition \ref{k22}(i), we define four different types of $K_{2,2}$ as follows.
\noindent {\bf Type 1} : $\{x_{i1}, x_{i2}\}\sim\{x_{j1}, x_{j2}\}$;
\noindent {\bf Type 2} : $\{x_{i1}, x_{i2}\}\sim\{x_{jt}, c_j\}$, where $t\in\{1,2\}$;
\noindent {\bf Type 3} : $\{x_{is}, c_i\}\sim\{x_{j1}, x_{j2}\}$, where $s\in\{1,2\}$;
\noindent {\bf Type 4} : $\{x_{is}, c_i\}\sim\{x_{jt}, c_j\}$, where $s,t\in\{1,2\}$.
If there are three vertices in $V_4$ with an $h$-value of $-1$, then there are six distinct vertices $x_1,x_2,\ldots, x_6 \in V_2$
such that $\{x_1, x_2\} \sim \{x_3, x_4\}$, $\{x_3, x_4\} \sim \{x_5, x_6\}$ and $\{x_1, x_2\} \sim \{x_5, x_6\}$. Thus $G$ contains a copy of $K_{3,3}$ as $\{a_1, a_2, x_1\} \sim \{x_3, x_4, x_5\}$, a contradiction. So there are at most two vertices in $V_4$ with an $h$-value of $-1$. Thus $q_4 \geq -2.5$ when $|V_4^{-}|\leq 3$. In the following, we assume that $|V_4^{-}|\geq 4$.
\begin{claim}\label{three -1} There is at most one vertex in $V_4^{-}$ with an $h$-value of $-1$. \end{claim}
\noindent {\emph{Proof.}}~~ Suppose that, by contradiction, there are exactly two vertices with an $h$-value $-1$, say $z_1$ and $z_2$. Then $z_1 z_2 \notin E$ and the $K_{2,2}$ between $N(z_1)$ and $N(z_2)$ is Type 1. Since $|V_4^{-}|\geq 4$, there exists a vertex, say $z_3$, such that $d(z_3)=3$ and $z_1z_3\notin E$, $z_2z_3\notin E$. Applying Proposition \ref{k22}(i) to $z_1z_3\notin E$ and $z_2z_3\notin E$, we obtain that there exists $b \in N(z_3)$ such that $b \sim \{x_{11}, x_{12}, x_{21}, x_{22}\}$. Then $G$ contains a copy of $K_{3,3}$ as $\{a_1, a_2, b\} \sim \{x_{11}, x_{12}, x_{21}\}$, a contradiction. Hence there is at most one vertex in $V_4^{-}$ with an $h$-value of $-1$.
\vrule height3pt width6pt depth2pt
By Claim \ref{three -1}, if $|V_4^{-}| \leq 4$, then $q_4\geq -2.5$. So we assume that $|V_4^{-}| \geq 5$ in the following.
\begin{claim}\label{one -1}
If there exists a vertex in $ V_4^{-}$ with an $h$-value of $-1$, then $q_2+q_3+q_4 \geq -2.5$.
\end{claim}
\noindent {\emph{Proof.}}~~ Without loss of generality, we assume that $h(z_1)=-1$. For each $z_i\in V_4^{-}\setminus\{z_1\}$, since $z_1z_i\notin E$, $\{x_{11},x_{12}\}\not\subseteq N(z_i)$.
We first prove that there is at most one vertex $z_i\in V_4^{-}$ such that $\{x_{11}, x_{12}\}\sim \{x_{i1}, x_{i2}\}$. Suppose not. Then there exist two vertices, say $z_2$ and $z_3$, such that $\{x_{11}, x_{12}\} \sim \{x_{t1}, x_{t2}\}$ for each $t\in \{2,3\}$. Since $|N(x) \cap V_2| =2$ for each $x \in V \setminus V_1$, $\{x_{21}, x_{22}\} =\{x_{31}, x_{32}\}$. Note that $z_2z_3\in E$ for otherwise the non-edge $z_2z_3$ contradicts Proposition \ref{k22}(i). Since $|V_4^{-}| \geq 5$, there exists a vertex, say $z_4$, such that $z_4 z_p\notin E$ for each $p \in [3]$. By applying Proposition \ref{k22}(i) to $z_1z_4$, we have $\{x_{4i},c_4\}\sim \{x_{11}, x_{12}\}$ for some $i\in \{1,2\}$ and thus $x_{4i} \in \{x_{21}, x_{22}\}$. Since $c_2=z_3$, there is no $K_{2,2}$ between $N(z_2)$ and $N(z_4)$, contradicting Proposition \ref{k22}(i). This proves the statement.
Now for $i\in \{3,4,\ldots,|V_4^{-}|\}$, since $z_1z_i\notin E$, we have $\{x_{11},x_{12}\}\sim \{x_{ij}, c_i\}$, where $j\in[2]$.
Applying Proposition \ref{k22}(i) to $c_iz_1\notin E$,
we have $|N(c_i)\cap V_4|\ge 3$ and thus $h(c_i)\ge 0.5$.
Now we show that $c_i\neq c_j$ for $i,j\in \{3,4,\ldots,|V_4^{-}|\}$ with $i\neq j$. Since $c_i\notin V_4^{-}$, we have $z_iz_j\notin E$. By Proposition \ref{k22} (i), there is a $K_{2,2}$ between $N(z_i)$ and $N(z_j)$. By considering the $K_{2,2}$ between $N(z_k)$ and $N(z_1)$ for $k\in \{i,j\}$, we see $N(c_k)\cap V_2=\{x_{11},x_{12}\}$. It follows that the $K_{2,2}$ between $N(z_i)$ and $N(z_j)$ must be Type 4. So $c_i\neq c_j$. Now we have \[
q_4\ge h(z_1)+h(z_2)+\sum_{i=3}^{|V_4^{-}|}(h(z_i)+h(c_i))\geq -1.5. \] This completes the proof.
\vrule height3pt width6pt depth2pt
By Claim \ref{one -1}, we assume $h(z) =-0.5$ for each vertex $z \in V_4^{-}$.
If $|V_4^{-}| \leq 5$, then $q_4 \geq -2.5$. So we assume $|V_4^{-}| \geq 6$ in the following.
\begin{claim}\label{type1} If $h(z) =-0.5$ for each vertex $z \in V_4^{-}$ and there exist two non-adjacent vertices in $ V_4^{-}$ satisfying the $K_{2,2}$ between their neighborhood is Type 1, then $q_2+q_3+q_4 \geq -2.5$.
\end{claim}
\noindent {\emph{Proof.}}~~ Suppose $z_1z_2\notin E$ and the $K_{2,2}$ between $N(z_1)$ and $N(z_2)$ is Type 1. Let $U=\{z \in V_4^{-}\setminus\{z_1,z_2\}$ with $zz_1, zz_2\notin E\}$. Since $|V_4^{-}| \geq 6$, we have $|U|\ge 2$.
Let $z_i\in U$. By applying Proposition \ref{k22} (i) to $z_i z_1 \notin E$, there is a copy of $K_{2,2}$ between $N(z_1)$ and $N(z_i)$. Note that $|N(v)\cap V_2|=2$ for each $v\in V\setminus V_1$. If the $K_{2,2}$ is Type 1 or Type 3, then $\{x_{i1},x_{i2}\}=\{x_{21},x_{22}\}$. If the $K_{2,2}$ is Type 2, then $N(c_3)\cap V_2=\{x_{11},x_{12}\}$ and $x_{is}\in\{x_{21},x_{22}\}$ for some $s\in[2]$. In each case, we cannot find a $K_{2,2}$ between $N(z_2)$ and $N(z_i)$. So the $K_{2,2}$ between $N(z_1)$ and $N(z_i)$ is Type 4. Similarly, the $K_{2,2}$ between $N(z_2)$ and $N(z_i)$ is Type 4. So we have $x_{is}\in\{x_{11},x_{12}\}$ and $x_{it}\in\{x_{21},x_{22}\}$, where $\{s,t\}=[2]$, and $c_1,c_2,c_i\notin V_4^{-}$.
Hence for each $z_i, z_j \in U$, the $K_{2,2}$ between $N(z_i)$ and $N(z_j)$ is Type 4. So $c_i \neq c_j$. This means that for each $z\in U$, its unique neighbor $c\in V_4$ has at least 3 neighbors in $V_4\setminus V^-_4$, so $h(z)+h(c)\ge 0$. And for any $z_i, z_j \in U$, $c_i \neq c_j$, so $q_4\ge -2$.
\vrule height3pt width6pt depth2pt
By Claim \ref{type1}, we suppose there are no two vertices $z_i,z_j\in V_4^-$ with $z_iz_j\notin E$ such that the $K_{2,2}$ between $N(z_i)$ and $N(z_j)$ is Type 1. Suppose that $c \in V_4^{-}$ for each $z \in V_4^{-}$. Let $z_i,z_j \in V_4^{-}$ with $z_iz_j\notin E$. By Proposition \ref{k22}(i), Claim \ref{type1}, and $c_i, c_j \in V_4^{-}$, we may assume the $K_{2,2}$ between $N(z_i)$ and $N(z_j)$ is Type $2$. Then there is no copy of $K_{2,2}$ between $N(z_i)$ and $N(c_j)$, a contradiction. So we choose $z \in V_4^{-}$ with $c \notin V_4^{-}$ as $z_1$. Let $A_0= \emptyset$. Let $A_\ell=\{z|z\in V_4^{-} \setminus (A_0 \cup \ldots \cup A_{\ell-1}) \text{ and the $K_{2,2}$ between $N(z_1)$ and $N(z)$ is Type $\ell$ } \}$ and $C_\ell=\{c_i:z_i\in A_\ell\}$ for $\ell \in [4]$. By Claim \ref{type1}, we have $A_1=C_1=\emptyset$. Thus $|A_2|+|A_3|+|A_4|=|V_4^{-}|-1$. Let $C=\{c_1\}\cup C_2\cup C_3 \cup C_4$ and $C'=\{c_1\}\cup C_2\cup C_4$. Note that $C_j$ and $C_k$ may intersect when $j \neq k$ and $j,k\in[4]$.
For any $z\in A_2$, we have $c \notin V_4^{-}$ for otherwise there is no copy of $K_{2, 2}$ between $N(z_1)$ and $N(c)$. Thus for each $z_i,z_j\in A_2$, we have $z_iz_j\notin E$. Since $z_i,z_j\in A_2$, we have $N(c_i)\cap V_2=N(c_j)\cap V_2=\{x_{11},x_{12}\}$ and there exist $s,t\in[2]$ such that $x_{is}\notin\{x_{11},x_{12}\}$ and $x_{jt}\notin\{x_{11},x_{12}\}$. If the $K_{2,2}$ between $N(z_i)$ and $N(z_j)$ is Type 2 or Type 3, then $N(c_j)\cap V_2=\{x_{i1},x_{i2}\}$ or $N(c_i)\cap V_2=\{x_{j1},x_{j2}\}$, a contradiction. So the $K_{2,2}$ between $N(z_i)$ and $N(z_j)$ is Type 4. This implies that $C_2$ is a clique.
For each two vertices $z_i,z_j\in A_3$, we have $N(z_i)\cap V_2=N(z_j)\cap V_2$ since $|N(c_1)\cap V_2|=2$. If $z_iz_j\notin E$, then the $K_{2,2}$ between $N(z_i)$ and $N(z_j)$ is Type $4$. If $z_iz_j\in E$, then $c_ic_j\in E$. This implies that $C_3$ is a clique. Thus if $|A_3| \geq 3$, then for each $z\in A_3$, we have $c \notin V_4^{-}$.
Let $|C_2|=p$, $|C_3\setminus C_2|=q$ and $|C_4\setminus(C_3\cup C_2)|=r$. Note that $|C|\le p+q+r+1$ and the equation $|C|= p+q+r+1$ implies that $c_1\notin C_2\cup C_3$. Note that \begin{align}\label{q4=-2.5}
q_4\ge&\sum_{v\in C\setminus V_4^-}h(v)+\sum_{v\in V_4^-}h(v)=\sum_{v\in C\setminus V_4^-}h(v)-0.5|V_4^-|. \end{align}
To prove $q_4\ge -2.5$, it suffices to prove $\sum_{v\in C\setminus V_4^-}h(v)\ge0.5|V_4^-|-2.5$ by (\ref{q4=-2.5}).
Recall that $C_2$ and $C_3$ are two cliques of $G$, $(C_2\cup C_4)\cap V_4^-=\emptyset$ and $C_3\cap V_4^-=\emptyset$ if $|A_3|\geq3$.
\noindent {\bf Case 1:} $|C_3|=|A_3|\geq 3$.
In this case, we have $(C_2\cup C_3\cup C_4)\cap V_4^-=\emptyset$. Thus $\sum_{v\in C\setminus V_4^-}h(v)=\sum_{v\in C}h(v)$.
If $C_2\cap C_3\ne \emptyset$, then \begin{align*}
\sum_{v\in C}h(v)\ge& 2|C|+e(G[C])+0.5e(G[C,V_4^-])-3|C|\\
\ge&2|C|+{{p}\choose{2}}+{{q}\choose{2}}+q+r+0.5|V_4^-|-3|C|\\
=&{{p}\choose{2}}+{{q}\choose{2}}+q+r+0.5|V_4^-|-|C|\\
\ge &max\{0,p-1\}+max\{0,q-1\}+p+r+0.5|V_4^-|-(p+q+r+1)\\
\ge & 0.5|V_4^-|-2. \end{align*}
If $C_2\cap C_3=\emptyset$, then $q\geq3$ and \begin{align*}
\sum_{v\in C}h(v)\ge&\notag 2|C|+e(G[C])+0.5e(G[C,V_4^-])-3|C|\\\notag
\ge &2|C|+{{p}\choose{2}}+{{q}\choose{2}}+r+0.5|V_4^-|-3|C|\\
=&{{p}\choose{2}}+{{q}\choose{2}}+r+0.5|V_4^-|-(p+q+r+1)\\
\ge& p-1+q+r+0.5|V_4^-|-(p+q+r+1)\\
=& 0.5|V_4^-|-2. \end{align*}
\noindent {\bf Case 2:} $|A_3|\le 2$ and $|A_2|=p\geq 3$. \begin{align*}
\sum_{v\in C\setminus V_4^-}h(v)\ge&\sum_{v\in C'}h(v)\ge 2|C'|+e(G[C'])+0.5e(G[C',V_4^-])-3|C'|\\
\ge &2|C'|+{{p}\choose{2}}+|C_4\setminus C_2|+0.5(|V_4^-|-2)-3|C'|\\
\ge&{{p}\choose{2}}+|C_4\setminus C_2|+0.5|V_4^-|-1-(p+|C_4\setminus C_2|+1)\\
\ge& {{p}\choose{2}}-p+0.5|V_4^-|-2\\
\ge& 0.5|V_4^-|-2. \end{align*}
\noindent {\bf Case 3:} $|A_2|\le 2$ and $|A_3|\le 2$.
Note that $(\{c_1\}\cup C_4)\cap (\{z_1\}\cup A_4)=\emptyset$. We have
\begin{align}\label{A4-1}
\sum_{v\in \{c_1\}\cup C_4}h(v)\ge &2(|C_4|+1)+e(G[\{c_1\}\cup C_4])+0.5e(G[\{c_1\}\cup C_4,\{z_1\}\cup A_4])-3(|C_4|+1)\notag\\
\ge &2(|C_4|+1)+ |C_4|+0.5(|A_4|+1)-3(|C_4|+1)=0.5(|A_4|-1).
\end{align} Then \begin{align*} q_4\ge & \sum_{v\in \{c_1\}\cup C_4}h(v)+\sum_{v\in V_4^-}h(v)\\
\ge &0.5(|A_4|-1)-0.5(|A_2|+|A_3|+|A_4|+1)=-0.5(|A_2|+|A_3|)-1. \end{align*}
Observe that $q_4\ge -2.5$ when $|A_2|+|A_3|\le 3$. Thus we just need to consider the case $|A_2|=|A_3|=2$.
Note that $C'\cap V_4^-=\emptyset$. Suppose $C_2\cap (\{c_1\}\cup C_4)\ne \emptyset$. Then $G[C']$ is a connected graph, and so $e(G[C'])\ge |C'|-1$. We see \begin{align*}
\sum_{v\in C\setminus V^-_4}h(v)\ge\sum_{v\in C'}h(v)\ge& 2|C'|+e(G[C'])+0.5e(G[C',V_4^-\setminus A_3])-3|C'|\\
\ge & e(G[C'])-|C'|+0.5(|V_4^-|-2)\\
\ge & |C'|-1-|C'|+0.5|V_4^-|-1\\
\ge & 0.5|V_4^-|-2. \end{align*}
Suppose $C_2\cap (\{c_1\}\cup C_4)=\emptyset$. Let $C_2=\{c_2, c_3\}$. If $h(c_2)>0$ or $h(c_3)>0$, by (\ref{A4-1}), then \begin{align*} q_4\ge&\sum_{v\in \{c_1\}\cup C_4}h(v)+\sum_{v\in C_2}h(v)+\sum_{v\in V_4^-}h(v)\\
\ge& 0.5(|A_4|-1)+0.5-0.5(1+4+|A_4|)=-2.5. \end{align*}
If $h(c_2)=h(c_3)=0$, then $N(c_2)=\{x_{11}, x_{12}, c_3, z_2\}$ and $N(c_3)=\{x_{11}, x_{12}, c_2, z_3\}$. Since $z_1c_2\notin E$, the $K_{2,2}$ between $N(z_1)$ and $N(c_2)$ must be Type 4, which contradicts $c_3c_1\notin E$.
In a conclusion, $q_4\ge -2.5$ and so $e(G)\ge 3n-9$. This completes the proof of the lower bound on $sat_2(n,K_{3,3})$ for $\geq 9$.
\vrule height3pt width6pt depth2pt
\subsubsection{$\delta(G)=5$} We prove $sat_5(n, K_{3,3})\ge 3n-9$ for $n\geq 9$ in this part. Since $\delta(G)=5$, we have $e(G)\ge 2.5n$. Then $e(G)\ge 3n-9$ when $n\le 19$. Thus we assume $n\ge 20$ in the following.
We define a new function $g$ as follows.
\begin{itemize}
\item For $x\in V_2$, $g(x)=|N(x)\cap V_1|+0.5|N(x)\cap (V_2\cup V_3)|+0.25|N(x)\cap V_4|-3$.
\item For $x\in V_3$, $g(x)=|N(x)\cap V_1|+0.5|N(x)\cap (V_2\cup V_3\cup V_4)|-3$.
\item For $x\in V_4$, $g(x)=0.75|N(x)\cap V_2|+0.5|N(x)\cap (V_3\cup V_4)|-3$.
\end{itemize} Observe that \begin{align}\label{eg} e(G)=\notag&~e(G[V_1])+e(G[V_2])+e(G[V_1,V_2])+e(G[V_3])+e(G[V_1,V_3])+e(G[V_2,V_3])+e(G[V_4])\\&+e(G[V_4,V_2\cup V_3])\notag\\
=&~e(G[V_1])+3(|V_2|+|V_3|+|V_4|)+\sum_{x\in V\setminus V_1}g(x)\notag\\
=&~e(G[V_1])+3(n-|V_1|)+\sum_{x\in V\setminus V_1}g(x). \end{align}
Note that $\delta(G)=5$. Then $g(x)\ge -0.25$ for each $x\in V_2$ because $|N(x)\cap V_1|\ge 2$; $g(x)\ge 0$ for each $x\in V_3$ because $|N(x)\cap V_1|=1$; $g(x)\ge 0$ for each $x\in V_4$ because $|N(x)\cap V_2|\ge 2$. If there exists a vertex $x_0\in V_2$ such that $g(x_0)<0$, then $g(x_0)=-0.25$, $d(x_0)=5$, $N(x_0)\cap (V_2\cup V_3)=\emptyset$, $|N(x_0)\cap V_1|=2$ and $|N(x_0)\cap V_4|=3$. We may assume that $N(x_0)=\{a_i,a_j,z_1,z_2,z_3\}$, where $i,j \in [5]$, $i\ne j$ and $\{z_1,z_2,z_3\}\subseteq V_4$. Since $ax_0\notin E(G)$, Proposition \ref{k22}(ii) implies that there is a copy of $K_{2,2}$ in $G[V_1\setminus\{a\}]$. Let $s=1$ if $a_ia_j\in E$ and $s=0$ if $a_ia_j\notin E$. Thus $e(G[V_1\setminus\{a\}])\ge 4+s$. But $e(G[N(x_0)])\le 3+s$ because $N(z_i)\cap V_1=\emptyset$ for each $i\in [3]$, which contradicts the minimality of $e(G[N(a)])$. Hence, $g(x)\ge 0$ for each $x\in V\setminus V_1$ and so $\sum_{x\in V\setminus V_1}g(x)\ge 0$. When $e(G[V_1])\ge 9$, by (\ref{eg}), we have $e(G)\ge 3n-9$ .
Thus we next consider $e(G[V_1])\leq8$. Note that $|N(x)\cap V_2|\geq 1$ for each $x\in V_2$ when $e(G[V_1])\leq8$. The following discussion is split into three cases below.
\noindent {\bf Case 1:} $e(G[V_1])=8$.
If $\sum_{x\in V\setminus V_1}g(x)>0$, then $e(G)=3n-10+\sum_{x\in V\setminus V_1}g(x)>3n-10$ by (\ref{eg}) and so $e(G)\ge 3n-9$ because $e(G)$ is an integer. Next we prove $\sum_{x\in V\setminus V_1}g(x)>0$. If there exists a vertex $x\in V_2$ with $|N(x)\cap V_1|\ge 3$, then $g(x)>0$ and so $\sum_{x\in V\setminus V_1}g(x)>0$. So we may assume that $|N(x)\cap V_1|=2$ for each $x\in V_2$. Since $e(G[V_1\setminus\{a\}])=3$, there is a vertex $a_i$ such that $N(a_i)\cap N(a)=\emptyset$ or $N(a_i)\cap N(a)=\{a_j\}$ with $N(a_j)\cap N(a)=\{a_i\}$, where $i,j\in [5]$ and $i\neq j$. We denote such a vertex by $a_1$. There is a vertex $a_k$ such that $a_1a_k\notin E$ for $k\in [5]$ and $k\ne 1$. Since $a_1a_k\notin E$, by Proposition \ref{k22}(i), $N(a_1)\cap(V_2\cup V_3)\ne \emptyset$. Let $x\in N(a_1)\cap(V_2\cup V_3)$ and $x_1\in N(x)\cap V_2$. If $x\in V_3$, then $|N(x_1)\cap (V_2\cup V_3)|\ge 2$. If $x\in V_2$, by the choice of $a_1$, then we have $|N(x_1)\cap V_2|\ge 2$, else there is no $K_{2,2}$ between $N(x_1)$ and $N(a)$. So $g(x_1)\ge 0.25$, which implies $\sum_{x\in V\setminus V_1}g(x)>0$. Hence $e(G)\ge 3n-9$.
\noindent {\bf Case 2:} $e(G[V_1])=7$ and there is a copy of $K_{1,2}$ in $G[V_1\setminus\{a\}]$.
We may assume that $E(G[V_1\setminus\{a\}])=\{a_1a_2, a_1a_3\}$. If $\sum_{x\in V\setminus V_1}g(x)>1$, by (\ref{eg}), then $$e(G)=e(G[V_1])+3(n-|V_1|)+\sum_{x\in V\setminus V_1}g(x)>7+3(n-6)+1=3n-10.$$ Since $e(G)$ is an integer, $e(G)\ge 3n-9$. Thus we just need to prove $\sum_{x\in V\setminus V_1}g(x)>1$.
Let $V_2^1=\{x\in V_2: |N(x)\cap V_2|=1 \}$ and $V_2^2=\{x\in V_2: |N(x)\cap V_2|\ge 2 \}$. Let $x\in V_2^1$ and $xx_1\in E(G[V_2])$. Applying Proposition \ref{k22}(i) to $ax\notin E(G)$, we have $x\in N(a_1)$ and $x_1\in N(a_2)\cap N(a_3)$. If $x_1\in V_2^1$, then $x_1\in N(a_1)$ and $x\in N(a_2)\cap N(a_3)$ by $x_1a\notin E(G)$. Thus $\{a_1, a_2, a_3\}\subseteq (N(x)\cap V_1)\cap (N(x_1)\cap V_1$). There is a copy of $K_{3,3}$ in $G$, that is $\{a,x,x_1\}\sim\{a_1, a_2, a_3\}$, a contradiction.
This implies that
$e(G[V_2^1])=0$, $V_2^2\ne \emptyset$ and $|V_2|\ge 3$.
Since $a_4a_5\notin E$, there is a copy of $K_{2,2}$ between $N(a_4)$ and $N(a_5)$, say $\{x_{41},x_{42}\}\sim \{x_{51}, x_{52}\}$. Notices that $N(a_4)\cap V_1=N(a_5)\cap V_1=\{a\}$. Thus $\{x_{41},x_{42},x_{51}, x_{52}\}\subseteq V_2\cup V_3$. For each $y\in \{x_{41},x_{42},x_{51}, x_{52}\}\cap V_3$, by Proposition \ref{k22}(i), then $|N(y)\cap V_2|\ge 2$. By the definition of $g$-function, for each $x\in V_2$, we have \begin{align*}
g(x)=&|N(x)\cap V_1|+0.25|N(x)\cap (V_2\cup V_3\cup V_4)|+0.25|N(x)\cap (V_2\cup V_3)|-3\\
=& |N(x)\cap V_1|+0.25|N(x)\cap (V_2\cup V_3\cup V_4)|+0.25|N(x)\cap V_2|-3+0.25|N(x)\cap V_3|. \end{align*}
If $x\in V_2^1$, then \begin{align*}
g(x)\geq 2+0.25\times3+0.25\times1-3+0.25|N(x)\cap V_3|
= 0.25|N(x)\cap V_3|. \end{align*} If $x\in V_2^2$, then \begin{align*}
g(x)\geq 2+0.25\times3+0.25\times2-3+0.25|N(x)\cap V_3|
= 0.25+0.25|N(x)\cap V_3|. \end{align*}
If $|N(x)\cap V_1|\geq 3$, then \begin{align*}
g(x)\geq 3+0.25\times2+0.25\times1-3+0.25|N(x)\cap V_3|
= 0.75+0.25|N(x)\cap V_3|. \end{align*}
Suppose $|\{x_{41},x_{42},x_{51}, x_{52}\}\cap V_3|\ge 2$.
Then $e(G[V_2,V_3])\ge2|\{x_{41},x_{42},x_{51}, x_{52}\}\cap V_3| \ge 4$. Note that $V_2^2\ne \emptyset$. Thus $$\sum_{x\in V\setminus V_1}g(x)\ge \sum_{x\in V_2}g(x)\ge 0.25+\sum_{x\in V_2}0.25|N(x)\cap V_3|=0.25+0.25e(G[V_2,V_3])\ge 1.25.$$
Suppose $|\{x_{41},x_{42},x_{51}, x_{52}\}\cap V_3|= 1$, say $x_{41}\in V_3$. Then $\{x_{42}, x_{51}, x_{52}\}\subseteq V_2$ and $x_{42}\in V_2^2$. We see $\{x_{51},x_{52}\}\subseteq V_2^2$ or $x_{42}\in N(a_2)\cap N(a_3)$, that is $|N(x_{42})\cap V_1|\ge 3$. Thus $$\sum_{x\in V\setminus V_1}g(x)\ge \sum_{x\in V_2}g(x)\ge 0.75+\sum_{x\in V_2}0.25|N(x)\cap V_3|=0.75+0.25e(G[V_2,V_3])\ge 1.25.$$
It remains to consider the case $ \{x_{41},x_{42},x_{51}, x_{52}\} \subseteq V_2$, that is $ \{x_{41},x_{42},x_{51}, x_{52}\} \subseteq V_2^2$.
If $V_3\ne \emptyset$, then $e(G[V_2,V_3])\ge 1$ and
$$\sum_{x\in V\setminus V_1}g(x)\ge \sum_{x\in V_2}g(x)\ge 0.25|V_2^2|+\sum_{x\in V_2}0.25|N(x)\cap V_3|
\ge 1+0.25e(G[V_2,V_3])\ge
1.25.$$
If $|N(x)\cap V_1|\ge 3$ for some $x\in V_2$, then $$\sum_{x\in V\setminus V_1}g(x)\ge \sum_{x\in V_2}g(x)\ge 0.75+0.25(|V_2^2|-1)+\sum_{x\in V_2}0.25|N(x)\cap V_3|
\ge1.5.$$
Next we assume that $|N(x)\cap V_1|=2$ for each $x\in V_2$ and $|V_3|=0$. Note that for each $x\in V_2^1$, let $xx_1\in E(G[V_2])$, we have $x_1\in N(a_2)\cap N(a_3)$. Thus $x_1\notin \{x_{41},x_{42},x_{51}, x_{52}\}$. If $|V_2|\ge 5$, then $V_2^2\setminus\{x_{41},x_{42},x_{51}, x_{52}\}\ne \emptyset$. Thus $|V_2^2|\ge 5$ and $\sum_{x\in V\setminus V_1}g(x)\ge 1.25$. If $|V_2|\le 4$, that is $V_2=\{x_{41},x_{42},x_{51}, x_{52}\}$, then we have $|V_4|\ge n-|V_2|-|V_3|-6=n-10$ because $|V_3|=0$. Note that $n\geq 20$. Thus \begin{align*} e(G)=&e(G[V_1])+e(G[V_2])+e(G[V_1\cup V_4,V_2])+e(G[V_4])\\
\ge & 7+4+8+2|V_4|+\frac{3|V_4|}{2}>3n-9 \end{align*}
\noindent {\bf Case 3:} $e(G[V_1])=7$ and there is no copy of $K_{1,2}$ in $G[V_1\setminus\{a\}]$ or $5\leq e(G[V_1])\leq6$.
In this case, we define a new function $g'$ as follows. \begin{itemize}
\item For $x\in V_2$, $g'(x)=|N(x)\cap V_1|+0.5|N(x)\cap V_2|-3$.
\item For $x\in V_3\cup V_4$, $g'(x)=|N(x)\cap (V_1\cup V_2)|+0.5|N(x)\cap ( V_3\cup V_4)|-3$. \end{itemize} We see \begin{align}\label{egprime} e(G)=&\notag~e(G[V_1])+e(G[V_2])+e(G[V_1,V_2])+e(G[V_3])+e(G[V_1,V_3])+e(G[V_2,V_3])+e(G[V_4])\\\notag &+e(G[V_4,V_2\cup V_3])\\\notag
=&~e(G[V_1])+3(|V_2|+|V_3|+|V_4|)+\sum_{x\in V\setminus V_1}g'(x)\\
=&~e(G[V_1])+3(n-|V_1|)+\sum_{x\in V\setminus V_1}g'(x). \end{align}
For each $x\in V_2$, by Proposition \ref{k22}(iv), $|N(x)\cap V_2|\ge 2$. Thus $g(x)\ge0.25$ because $d(x)\ge 5$. It follows that $\sum_{x\in V\setminus V_1}g(x)\ge 0.25|V_2|$. It suffices to consider the following two subcases.
\noindent {\bf Subcase 3.1:} $|V_2|\ge 13$ or $|V_3\cup V_4|\geq 7$
Suppose $|V_2|\ge 13$. Then $$e(G)=e(G[V_1])+3(n-|V_1|)+\sum_{x\in V\setminus V_1}g(x)\ge 5+3n-18+0.25|V_2|\ge 3n-9.75$$ and so $e(G)\ge 3n-9$ because $e(G)$ is an integer.
Suppose $|V_3\cup V_4|\geq 7$. By Proposition \ref{k22}(iv), $|N(x)\cap V_2|\ge 2$ for each $x\in V\setminus V_1$. Thus $g'(x)\ge 0$ for each $x\in V_2$, $g'(x)\ge 1$ for each $x\in V_3$, and $g'(x)\ge 0.5$ for each $x\in V_4$.
It follows that $$e(G)=e(G[V_1])+3(n-|V_1|)+\sum_{x\in V\setminus V_1}g'(x)\ge 5+3n-18+0.5|V_3\cup V_4|\ge 3n-9.5.$$ Since $e(G)$ is an integer, $e(G)\ge 3n-9$.
\noindent {\bf Subcase 3.2:} $|V_2|\le 12$ or $|V_3\cup V_4|\leq 6$
Since $n\geq12$, $|V_3\cup V_4|\ge 2$. We first prove the following claim. \begin{claim}\label{v34}
If there is no copy of $K_{1,2}$ in $G[V_1\setminus\{a\}]$ and $|V_3\cup V_4|\ge 2$, then $\sum_{x\in V_3\cup V_4}g'(x)\ge 2$. In particular, if $|V_3|\ge 1$ or $|N(z)\cap V_2|\ge 3$ for some $z\in V_4$, then $\sum_{x\in V_3\cup V_4}g'(x)\ge 3$. \end{claim}
\noindent {\emph{Proof.}}~~ By the definition of $g'$-function and $\delta(G)= 5$, we have for each $x\in V_3$, $g'(x)\ge 1$ and for each $x\in V_4$, $g'(x)\ge 0.5$. When $|V_3\cup V_4|\ge 4$, $\sum_{x\in V_3\cup V_4}g'(x)\ge 2$. When $2\le |V_3\cup V_4|\le 3$, for each $x\in V_3\cup V_4$, we have $|N(x)\cap (V_1\cup V_2)|=5-(|V_3\cup V_4|-1)$. Thus $$g'(x)\ge (6-|V_3\cup V_4|)+0.5(|V_3\cup V_4|-1)-3=2.5-0.5(|V_3\cup V_4|$$ and $$\sum_{x\in V_3\cup V_4}g'(x)\ge (2.5-0.5(|V_3\cup V_4|))|V_3\cup V_4|\ge 3.$$
Next we assume that $|V_3|\ge 1$ or $|N(z)\cap V_2|\ge 3$ for some $z\in V_4$. To prove $\sum_{x\in V_3\cup V_4}g'(x)\ge 3$, it suffices to consider the case $|V_3\cup V_4|\ge 4$ by the above discussion. If $|V_3\cup V_4|\ge 5$ or $|V_3|\ge 2$, then $\sum_{x\in V_3\cup V_4}g'(x)\ge 3$. Suppose $|V_3\cup V_4|= 4$ and $|V_3|\leq 1$. Let $V_3\cup V_4=\{y_1,y_2,y_3,y_4\}$ and $\{y_1,y_2,y_3\}\subseteq V_4$. Let $y_4\in V_3$ or $|N(y_4)\cap V_2|\ge 3$ when $y_4\in V_4$. If $g'(y_i)\geq 1$ for some $i\in [3]$, then $\sum_{x\in V_3\cup V_4}g'(x)\ge 3$. So we assume $g'(y_i)=0.5$ for each $i\in [3]$, then we have $|N(y_i)\cap (V_3\cup V_4)|=3$. Thus $G[\{y_1,y_2,y_3,y_4\}]$ is a clique. It follows that $g'(y_4)\ge 1.5$ and $\sum_{x\in V_3\cup V_4}g'(x)\ge 3$.
\vrule height3pt width6pt depth2pt
Since $|V_3\cup V_4|\ge 2$, $\sum_{v\in V_3\cup V_4}g'(v)\ge 2$ by Claim \ref{v34}. When $e(G[V_1])\ge 7$, by inequality (\ref{egprime}), $e(G)\ge 3n-9$. Now we consider the case $e(G[V_1])= 6$.
If we can show $\sum_{v\in V_2}g'(v)>0$ or $\sum_{v\in V_3\cup V_4}g'(v)>2$, by Claim \ref{v34} and (\ref{egprime}), then $e(G)>3n-10$ and so $e(G)\ge 3n-9$.
If there exists a vertex $u\in V_2$ such that $|N(u)\cap V_1|\ge 3$, then $g'(u)\ge 1$ and so $\sum_{v\in V_2}g'(v)>0$. If $V_3\ne \emptyset$, then $\sum_{v\in V_3\cup V_4}g'(v)\ge 3$ by Claim \ref{v34}. Thus we may assume that $|N(v)\cap V_1|=2$ for each $v\in V_2$ and $V_3=\emptyset$. We choose a vertex $x\in V_2$. Without loss generality, suppose $x\in N(a_1)\cap N(a_2)$. Since $xa_i\notin E$ for each $i\in \{3,4,5\}$, there is a copy of $K_{2,2}$ between $N(a_i)$ and $N(x)$, say $\{a_{i1}, a_{i2}\}\sim \{x_{i1}, x_{i2}\}$. Note that there is no copy of $K_{1,2}$ in $G[V_1\setminus\{a\}]$. Thus $\{a_{i1},a_{i2}\}\cap V_2\ne \emptyset$ for each $i\in \{3,4,5\}$. Recall that $|N(v)\cap V_1|=2$ for each $v\in V_2$ and $V_3=\emptyset$. We have $\{x_{i1},x_{i2}\}\cap (V_2\cup V_4)\ne \emptyset$ for each $i\in \{3,4,5\}$. By Proposition \ref{k22}(ii), we have $|N(w)\cap V_2|\ge 3$ for each $w\in (\bigcup_{i\in \{3,4,5\}}\{x_{i1},x_{i2}\})\cap (V_2\cup V_4)$. Thus $\sum_{v\in V_3\cup V_4}g'(v)\ge 3$ by Claim \ref{v34} or $\sum_{v\in V_2}g'(v)\ge 0.5$.
Next we consider $e(G[V_1])=5$. If $\sum_{v\in V\setminus V_1}g'(v)>3$, by (\ref{egprime}), then $e(G)>3n-10$ and so $e(G)\ge 3n-9$. Thus we prove $\sum_{v\in V\setminus V_1}g'(v)>3$ in the following.
Recall $\sum_{v\in V_3\cup V_4}g'(v)\ge 2$. If there is a vertex $x\in V_2$ with $|N(x)\cap V_1|\ge 4$, then $g'(x)\ge 2$ and $$\sum_{v\in V\setminus V_1}g'(v)\geq g'(x)+\sum_{v\in V_3\cup V_4}g'(v)\geq 4.$$ If there are two different vertices $x,y\in V_2$ with $|N(x)\cap V_1|=|N(y)\cap V_1|=3$, then $g'(x)\ge 1$, $g'(y)\ge 1$ and $$\sum_{v\in V\setminus V_1}g'(v)\geq g'(x)+g'(y)+\sum_{v\in V_3\cup V_4}g'(v)\geq 4.$$
Suppose $x\in V_2$ with $|N(x)\cap V_1|=3$, and $|N(v)\cap V_1|=2$ for each $v\in V_2\setminus\{x\}$. Let $N(x)\cap V_1=\{a_1, a_2, a_3\}$. Since $xa_4\notin E$, there is a copy of $K_{2,2}$ between $N(x)$ and $N(a_4)$, say $\{x_{11},x_{12}\}\sim \{a_{41},a_{42}\}$. If $V_3\ne \emptyset$, by Claim \ref{v34}, then $\sum_{v\in V_3\cup V_4}g'(v)\ge 3$. Thus $$\sum_{v\in V\setminus V_1}g'(v)\ge g'(x)+\sum_{v\in V_3\cup V_4}g'(v)\geq4.$$
So we may assume that $V_3=\emptyset$. Then $\{a_{41},a_{42}\}\subseteq V_2$ and $\{x_{11},x_{12}\}\cap (V_2\cup V_4)\ne \emptyset$. Let $w\in \{x_{11},x_{12}\}\cap (V_2\cup V_4)$. By Proposition \ref{k22}(ii), $|N(w)\cap V_2|\ge3$. Thus $g'(w)\ge 0.5$. If $w\in V_2$,
then $$\sum_{v\in V\setminus V_1}g'(v)\ge g'(x)+g'(w)+ \sum_{v\in V_3\cup V_4}g'(v)\geq 3.5.$$ If $w\in V_4$, by Claim \ref{v34}, then $\sum_{v\in V\setminus V_1}g'(v)\ge 4$.
Suppose $|N(v)\cap V_1|=2$ for each $v\in V_2$.
Since $|V_3\cup V_4|\le 6$ and $n\geq 20$, $|V_2|\ge 8$. Recall the definition of $g$-function, for each $v\in V_2$, we have $g(v)\ge 0.25$ and if $g(v)>0.25$, then $g(v)\ge 0.5$. We see there exists a vertex $x\in V_2$ such that $g(x)=0.25$, otherwise, $g(v)\ge 0.5$ for each $v\in V_2$ and so $\sum_{v\in V_2}g(v)\ge 0.5|V_2|\ge 4$. By (\ref{eg}), $e(G)\ge 5+3(n-6)+4=3n-9$. We choose such a vertex $x\in V_2$ such that $g(x)=0.25$. Then $d(x)=5$ and let $N(x)=\{a_1,a_2,x_{11},x_{12},z\}$, where $\{a_1, a_2\}\subseteq V_1$, $\{x_{11},x_{12}\}\subseteq V_2$ and $z\in V_4$. Note that $xa_j\notin E$ for each $j\in \{3,4,5\}$. By Proposition \ref{k22}(i), there is a copy of $K_{2,2}$ between $N(x)$ and $N(a_j)$, say $\{x_{j1},x_{j2}\}\sim \{a_{j1},a_{j2}\}$. We see $\{a_{j1},a_{j2}\}\subseteq V_2\cup V_3$ for each $j\in \{3,4,5\}$. Since $|N(v)\cap V_1|=2$ for each $v\in V_2$, $\{x_{j1},x_{j2}\}\nsubseteq V_1$ for each $j\in \{3,4,5\}$. Otherwise, $\{x_{j1},x_{j2},a_j\}\subseteq N(a_{j1})\cap V_1$, a contradiction.
Suppose $V_3\neq \emptyset$. Then we have $\sum_{v\in V_3\cup V_4}g'(v)\ge 3$ by Claim \ref{v34}.
We have $|V_3|\le 3$, otherwise $\sum_{v\in V_3}g'(v)\ge 4$ and we are done. Note that $\{a_{j1},a_{j2}\} \subseteq V_2\cup V_3$ for any $j\in \{3,4,5\}$. When $|\bigcup_{j\in \{3,4,5\}}\{a_{j1},a_{j2}\}|\leq 5$, we may assume $a_{31}=a_{41}$. When $|\bigcup_{j\in \{3,4,5\}}\{a_{j1},a_{j2}\}|=6$, we have $|\bigcup_{j\in \{3,4,5\}}\{a_{j1},a_{j2}\}\cap V_2|\geq 3$ because $|V_3|\le 3$, so we may assume $\{a_{31}, a_{41}\}\subseteq \bigcup_{j\in \{3,4,5\}}\{a_{j1},a_{j2}\}\cap V_2$. In two cases, we have $\{a_{31}, a_{41}\}\subseteq V_2$. Let $k\in\{3,4\}$. If $\{x_{k1},x_{k2}\}\cap V_2\ne \emptyset$, let $w\in \{x_{k1},x_{k2}\}\cap V_2$, then
$\{x,a_{k1}\}\subseteq N(w)\cap V_2$, Proposition \ref{k22}(ii) implies that $|N(w)\cap V_2|\ge 3$ and so $g'(w)\ge 0.5$. Thus $$\sum_{v\in V\setminus V_1}g'(v)\ge g'(w)+ \sum_{v\in V_3\cup V_4}g'(v)\geq3.5.$$
So we assume $\{x_{k1},x_{k2}\}\cap V_2=\emptyset$. Note that $\{x_{k1},x_{k2}\}\nsubseteq V_1$. Since $d(x)=5$,
$\{x_{k1},x_{k2}\}=\{a_1,z\}$ or $\{a_2,z\}$, and so $N(a_{k1})\cap N(a_{k2})\cap V_1=\{a_k,a_{\ell_k}\}$ for some $\ell_k\in [2]$. Then $\{x,a_{31},a_{32},a_{41},a_{42}\} \subseteq N(z)\cap V_2$. Note that $|N(v)\cap V_1|=2$ for any $v\in V_2$. Since
$x\in V_{12}$, $\{a_{31},a_{32}\}\subseteq V_{1\ell_3}$ and $\{a_{41},a_{42}\}\subseteq V_{1\ell_4}$, $|\{x,a_{31},a_{32},a_{41},a_{42}\}|= 5$, which follows that
$g'(z)\ge 2$. Note that $V_3\ne \emptyset$ and $g'(y)\ge 1$ for each $y\in V_3$ and $g'(v)\ge 0.5$ for each $v\in V_4$. Recall $z\in V_4$ and $|V_3\cup V_4|\ge 2$.
So $\sum_{v\in V_3\cup V_4}g'(v)>3$ when $|V_3\cup V_4|\geq3$. If $|V_3\cup V_4|=2$, then $g'(y)>1$ for $y\in V_3$ because $d(y)\ge 5$. Therefore $\sum_{v\in V_3\cup V_4}g'(v)>3$.
It remains to consider $V_3=\emptyset$. Then $\{a_{j1},a_{j2}\}\subseteq V_2$ for any $j\in \{3,4,5\}$. When $\{x_{j1},x_{j2}\}\cap V_2=\emptyset$ for any $j\in \{3,4,5\}$, then $\{x_{j1},x_{j2}\}=\{a_1,z\}$ or $\{a_2,z\}$. Note that $|N(v)\cap V_1|=2$ for each $v\in V_2$. Since $\{a_{j1},a_{j2}\}\subseteq V_{j\ell_j}$ for $\ell_j\in [2]$, $|(\bigcup_{j\in \{3,4,5\}}\{a_{j1},a_{j2}\})\cup \{x\}|=7$. Thus $|N(z)\cap V_2|\ge 7$, which implies that $\sum_{v\in V_2}g'(v)\ge 4$. When there exists $j\in \{3,4,5\}$ such that $\{x_{j1},x_{j2}\}\cap V_2\ne \emptyset$, then $g'(w)\ge 0.5$ for $w\in \{x_{j1},x_{j2}\}\cap V_2$ because $|N(w)\cap V_2|\ge 3$ by Proposition \ref{k22}(ii). In this case, we have $z\notin \{x_{j1},x_{j2}\}\cap V_4$. Otherwise, Proposition \ref{k22}(ii) implies $|N(z)\cap V_2|\ge 3$. By
Claim \ref{v34}, $$\sum_{v\in V\setminus V_1}g'(v)\ge g'(w)+\sum_{v\in V_3\cup V_4}g'(v)\geq 3.5.$$ Thus we are done. If $|N(x_{11})\cap V_2|+|N(x_{12})\cap V_2|\ge 7$, then we have $$g'(x_{11})+g'(x_{12})= e(G[\{x_{11},x_{12}\},V_1])+0.5(e(G[\{x_{11}\},V_2])+e(G[\{x_{12}\},V_2]))-6\ge 4+3.5-6=1.5.$$ Thus $\sum_{v\in V\setminus V_1}g'(v)\ge 3.5$, and we are done. So it suffices to prove $|N(x_{11})\cap V_2|+|N(x_{12})\cap V_2|\ge 7$ in the following. Since $z\notin \{x_{j1},x_{j2}\}$, we have $\{x_{j1},x_{j2}\}\cap V_2\ne \emptyset$ for any $j\in \{3,4,5\}$. Recall $N(x)=\{a_1, a_2, x_{11}, x_{12}, z\}$ and $x\in N(x_{11})\cap N(x_{12})\cap V_{12}$. Then $\{a_{31},a_{32},a_{41},a_{42},a_{51},a_{52}\}\subseteq N(x_{11})\cup N(x_{12})$.
If $|\{a_{31},a_{32},a_{41},a_{42},a_{51},a_{52}\}|\ge 5$, then
$$|N(x_{11})\cap V_2|+|N(x_{12})\cap V_2|=|(N(x_{11})\cup N(x_{12}))\cap V_2|+|(N(x_{11})\cap N(x_{12}))\cap V_2| \ge 7.$$
Suppose that $|\{a_{31},a_{32},a_{41},a_{42},a_{51},a_{52}\}|\le 4$. Note that $|N(x)\cap V_1|=2$ for each $x\in V_2$. We obtain $|\{a_{31},a_{32},a_{41},a_{42},a_{51},a_{52}\}|\ge 3$. When $\{x_{31},x_{32}\}\cap V_1\ne \emptyset$, say $a_\ell\in \{x_{31},x_{32}\}$ for some $\ell\in [2]$, then $\{a_{31},a_{32}\} \subseteq V_{3\ell}$
and $\{a_{31},a_{32}\}\cap \{a_{k1},a_{k2}\}=\emptyset$ for each $k\in \{4,5\}$. Since $|\{a_{31},a_{32},a_{41},a_{42},a_{51},a_{52}\}|\le 4$, we have $\{x_{k1},x_{k2}\}=\{x_{11},x_{12}\}$ for each $k\in \{4,5\}$, that is $|N(x_{11})\cap N(x_{12}) \cap \bigcup_{j\in \{3,4,5\}}\{a_{j1},a_{j2}\}|\ge 2$. Thus \begin{align*}
|N(x_{11})\cap V_2|+|N(x_{12})\cap V_2|=&\notag|(N(x_{11})\cup N(x_{12}))\cap V_2|+|(N(x_{11})\cap N(x_{12}))\cap V_2|\\\notag
\ge&|\bigcup\nolimits_{j\in \{3,4,5\}}\{a_{j1},a_{j2}\}\cup \{x\}|+3 \geq 7. \end{align*}
When $\{x_{j1},x_{j2}\}\cap V_1= \emptyset$ for each $j\in\{3,4,5\}$, then $\{x_{j1},x_{j2}\}=\{x_{11},x_{12}\}$ for each $j\in \{3,4,5\}$ and $\bigcup_{j\in \{3,4,5\}}\{a_{j1},a_{j2}\}\subseteq N(x_{11})\cap N(x_{12})$. By $|\bigcup_{j\in \{3,4,5\}}\{a_{i1},a_{i2}\}|\ge 3$ and $x\notin\bigcup_{j\in \{3,4,5\}}\{a_{i1},a_{i2}\}$, we have $|N(x_{11})\cap V_2|+|N(x_{12})\cap V_2|\geq 8$.
As a result, we have $e(G)\ge 3n-9$ for $n\ge 9$ in each case and so $sat_5(n,K_{3,3})\ge 3n-9$.
\vrule height3pt width6pt depth2pt
This completes the proof of Theorem \ref{main23}. \section{Conclusion}
Based on above results, we make the following conjecture, which gives an exact value for $sat(n,K_{3,3})$.
\begin{conj}\label{k33conj}
For $n\ge 9$, $sat(n,K_{3,3})= 3n-9$. \end{conj}
By Theorem \ref{main1}, $sat(n,K_{3,3})\le 3n-9$ for $n\ge 9$. To confirm Conjecture \ref{k33conj}, it suffices to prove $sat(n,K_{3,3})\ge 3n-9$ for $n\ge 9$. Let $G$ be a $K_{3,3}$-saturated graph with $n$ vertices and $n\geq9$. Proposition \ref{k22}(i) implies $\delta(G)\ge 2$. If $\delta(G)\ge 6$, then $e(G)\ge 3n\geq 3n-9$. Thus we only need to consider $2\le \delta(G)\le 5$. We have proved $sat_{\delta}(n,K_{3,3})\ge 3n-9$ when $\delta\in\{2,5\}$. Actually, for $\delta\in \{3,4\}$, we can also apply the method in this paper, but it is more complex and there are quite a few cases to consider.
{\bf Acknowledgments.} Shi and Zhang are partially supported by the National Natural Science Foundation of China (Nos. 11922112, 12161141006), Natural Science Foundation of Tianjin (Nos. 20JCZDJC00840, 20JCJQJC00090). Lei was partially supported by the National Natural Science Foundation of China (No. 12001296), Natural Science Foundation of Tianjin (No. 21JCQNJC00060).
\end{document} | arXiv |
$A,B,C,D$ and $E$ are five persons who are to be seated around a circular table such that $A$ and $B$ must sit together and $C$ and $D$ must never sit together. In how many ways can they be seated?
First we make $(AB)$ and $E$ sit which can be done in $2$ ways since $A$ and $B$ can arrange themselves in $2!=2$ ways.
One of $C$ and $D$ can be put into gaps between $E$ and $A$ and the other can be put into the gap between $B$ and $E$ for which there are obviously $2$ ways.
To obtain total number of ways it appears that we should multiply number of ways in Step 1($=2$) and number of ways in Step 2($=2$) ways, i.e. total number of ways $=2\times 2=4$.
But it is easy to notice that in these $4$ ways two of the arrangements are rotation of the other two.
So should the answer be $4$ or should it be $2$.
In general what should be the approach?
Say we have $10$ persons sitting around a circular table with $3$ of them wanting to sit together only whereas $4$ other persons do not want to sit next to each other?
Should the rotation of a particular arrangement be construed as same or different?
$A$, $B$, $C$, $D$ and $E$ are five persons who are to be seated around a circular table such that $A$ and $B$ must sit together and $C$ and $D$ must never sit together. In how many ways can they be seated?
There are four possible seating arrangements.
Seat E. Since A and B sit together and C and D are separated, C and D must both be adjacent to E. Therefore, choosing whether C or D sits to E's immediate left also determines who sits to E's immediate right and choosing whether A or B sits two seats to E's left also determines who sits two seats to E's right. Hence, there are $2 \cdot 2 = 4$ permissible seating arrangements, as shown below.
Notice that none of these seating arrangements can be obtained from another by rotation.
Should the rotation of a particular arrangement be construed as the same or different?
By convention, a rotation of a particular arrangement is considered to be the same unless the seats are labeled or we are given a particular reference point (such as a special chair or the north end of the table).
Notice that we have already accounted for rotational invariance by measuring our seating arrangements relative to the position of E.
Say we have $10$ persons sitting around a circular table, with $3$ of them wanting to sit together and $4$ other persons who wish to be separated? In how many ways can they be seated?
We use the block of three people who wish to sit together as our reference point. Say the people are $A$, $B$, and $C$. In how many ways can they be arranged within the block?
Suppose the four people who wish to be separated are $D$, $E$, $F$, and $G$. Since there are only seven seats left at the table, they must be seated in the seats that are $1$, $3$, $5$, and $7$ positions to the left of the block. In how many ways can they be seated?
Let's call the remaining three people $G$, $H$, and $I$. In how many ways, can they be seated in the remaining three chairs?
Counting the arrangements of 8 people around a square table?
Circular permutation - Arranging 4 persons around a circular table where 8 seats are there.
In how many ways can 2 adults, 2 girls, and 2 boys be seated around a circular table? | CommonCrawl |
\begin{document}
\title[Binary digits of squarefree numbers and quadratic residues] {Prescribing the binary digits of squarefree numbers and quadratic residues}
\author[R. Dietmann] {Rainer Dietmann} \address{Department of Mathematics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, United Kingdom} \email{[email protected]}
\author[C. Elsholtz]{Christian Elsholtz} \address{Institute of Analysis and Computational Number Theory, Technische Universit\"at Graz, A-8010 Graz, Austria }
\email{[email protected]}
\author[I. E. Shparlinski]{Igor E. Shparlinski} \address{Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia} \email{[email protected]} \begin{abstract}
We study the equidistribution of multiplicatively defined sets, such as the squarefree integers, quadratic non-residues or primitive roots, in sets which are described in an additive way, such as sumsets or Hilbert cubes. In particular, we show that if one fixes any proportion less than $40\%$ of the digits of all numbers of a given binary bit length, then the remaining set still has the asymptotically expected number of squarefree integers. Next, we investigate the distribution of primitive roots modulo a large prime $p$, establishing a new upper bound on the largest dimension of a Hilbert cube in the set of primitive roots, improving on a previous result of the authors. Finally, we study sumsets in finite fields and asymptotically find the expected number of quadratic residues and non-residues in such sumsets, given their cardinalities are big enough. This significantly improves on a recent result by Dartyge, Mauduit and S{\'a}rk{\"o}zy. Our approach introduces several new ideas, combining a variety of methods, such as bounds of exponential and character sums, geometry of numbers and additive combinatorics. \end{abstract}
\subjclass[2010]{Primary 11A63, 11B30, 11N25; Secondary 11H06, 11L40, 11P70, 11T30} \keywords{Digital problems, square-free numbers, non-residues, finite fields, Hilbert cubes}
\maketitle
\section{Introduction}
\subsection{Motivation} Given an integer $n \ge 2$, we denote by ${\mathcal D}_{n}$ the set of vectors $\vec{d} = (\delta_0, \ldots, \delta_{n-1})$ where $\delta_i \in \{\ast, 0, 1\}$, $i =0, \ldots, n-1$, and for $\vec{d} \in {\mathcal D}_n$ we consider the set $$ {\mathcal N}_n(\vec{d}) = \left\{\sum_{i=0}^{n-1} d_i 2^i~:~ d_i \in \{0, 1\}\ \text{if}\ \delta_i=\ast, \ d_i =\delta_i\ \text{otherwise}\right\}. $$
Furthermore, for integers $n >k \ge 1$ we denote by ${\mathcal D}_{k,n}$ the subset of $n$-dimensional vectors $\vec{d} \in {\mathcal D}_n$ with exactly $k$ components of $\vec{d}$ that are fixed as either $0$ or $1$ and $n-k$ components that are $\ast$. We also use ${\mathcal D}_{k,n}^*$ to denote the set of vectors $\vec{d} \in {\mathcal D}_{k,n}$ with $\delta_0 = 1$. In particular, for $\vec{d} \in {\mathcal D}_{k,n}^*$ all elements of ${\mathcal N}_n(\vec{d})$ are odd.
Various arithmetic properties of elements from ${\mathcal N}_n(\vec{d})$ as well as of other integers with restricted digits have been studied in a number of works.
We first recall that Bourgain~\cite{Bour1,Bour2} has recently obtained several very strong results about prime numbers with prescribed binary digits, see also~\cite{HaKa}. For example, the result of~\cite{Bour2} gives an asymptotic formula for the number of primes $p \in {\mathcal N}_n(\vec{d})$ for very dense vectors $\vec{d} \in {\mathcal D}_{k,n}^*$, more precisely, when $k \le cn$ for some absolute constant $c>0$, which is a dramatic improvement over the previous results of~\cite{Bour1,HaKa}. Mauduit and Rivat~\cite{MauRiv} have recently settled a problem of Gelfond about the distribution of primes with the sums of digits in a prescribed arithmetic progression, see also~\cite{DMR}. Partially motivated by some cryptographic applications, the distribution and construction of RSA moduli and smooth numbers with some binary digits prescribed have been studied in~\cite{GraShp,Shp2}. Various results on prime divisors and other arithmetic properties of numbers with very few non-zero binary digits can be found in~\cite{BaSh1,Bour0,EMS,Luca,Shp1,Shp3}. For a diverse variety of results on integers with various restrictions on their digits (for example, palindromes) see~\cite{BaSh2,Col1,Col2,DrMa,Kon1,KMS,MaSha,MoShk} and references therein.
\subsection{Our results and methods} Here we combine a variety of methods, such as bounds on exponential and character sums, geometry of numbers, additive combinatorics, to derive new results about the arithmetic structure of elements of ${\mathcal N}_n(\vec{d})$. Throughout, our goal is to treat $\vec{d} \in {\mathcal D}_{k,n}$ with the ratio $k/n$ as large as possible (that is, for thin sets of integers with as large as possible proportion of pre-assigned digits). We believe that our ideas and results can find application to several other problems of this type.
More precisely, in Section~\ref{sec:sqfr} we first study the distribution of squarefree numbers in sets ${\mathcal N}_n(\vec{d})$.
Using some combinatorial arguments, the theory of lattice minima and a result of Bourgain~\cite[Lemma~4]{Bour2} we obtain an asymptotic formula for the number of squarefree integers $s \in {\mathcal N}_n(\vec{d})$ for $\vec{d} \in {\mathcal D}_{k,n}^*$ provided that $k\le (2/5- \varepsilon) n$ for any fixed $\varepsilon>0$. In Section~\ref{sec:Euler} we also give an asymptotic for the sums of values of the Euler function in essentially full range $k\le (1- \varepsilon) n$.
Furthermore, we also estimate multiplicative character sums and in Section~\ref{sec:nonres} obtain results about the distribution of quadratic non-residues and primitive roots modulo $p$ among the elements of ${\mathcal N}_n(\vec{d})$ for $\vec{d} \in {\mathcal D}_{k,n}$ provided that $k\le (1/2 - \varepsilon) n$ for any fixed $\varepsilon>0$. This result complements those of~\cite{BaCoSh,DES1,DES2,OstShp} where similar questions are considered for integers with various restrictions on their binary digits (and also digits in other bases).
Finally, in Section~\ref{sec:Hilb}, we consider a related question about quadratic residues and primitive roots in {\it Hilbert cubes\/}. For a prime power $r = p^n$, let $\mathbb{F}_r$ denote the finite field of $r$ elements. For $a_0, a_1, \ldots, a_d \in \mathbb{F}_p$ we define the {\it Hilbert cube\/} as \begin{equation} \label{eq:hilbert_cube} {\mathcal H}(a_0; a_1, \ldots, a_d) = \left\{ a_0 + \sum_{i=1}^d \vartheta_i a_i~:~ \vartheta_i \in \{0,1\}\right\}. \end{equation}
We define $f(p)$ as the largest $d$ such that there are $a_0, a_1, \ldots, a_d \in \mathbb{F}_p$ with pairwise distinct $a_1, \ldots, a_d$ such that ${\mathcal H}(a_0; a_1, \ldots, a_d)$ does not contain a quadratic non-residue modulo $p$. Furthermore, we define $F(p)$ as the largest $d$
such that there are $a_0, a_1, \ldots, a_d \in \mathbb{F}_p$ with pairwise distinct $a_1, \ldots, a_d$ such that ${\mathcal H}(a_0; a_1, \ldots, a_d)$ does not contain a primitive root modulo $p$. Clearly $$
f(p) \le F(p). $$
Hegyv\'{a}ri and S\'{a}rk\"{o}zy~\cite[Theorem~2]{HS} give the bound
$f(p) < 12 p^{1/4}$ for sufficiently large $p$, which has been improved to $$ F(p) \le p^{1/5+o(1)} $$ as $p\to \infty$, by Dietmann, Elsholtz and Shparlinski~\cite[Theorem~1.3]{DES1}. Here we improve this further to $$ F(p) \le p^{3/19+o(1)} $$ and recall that reducing the exponent below $1/8$ immediately implies an improvement of the Burgess bound on the least primitive root (note that $3/19 -1/8 = 0.0328\ldots$).
As a further application of our method of Section~\ref{sec:nonres}, in Section~\ref{sec:rest dig}, we outline a substantial improvement of one of the results of Dartyge, Mauduit and S{\'a}rk{\"o}zy~\cite{DMS}. Namely, given a basis $\omega_1, \ldots, \omega_n$ of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$, and a collection of $n$ sets ${\mathfrak A} = \{{\mathcal A}_i\subseteq \mathbb{F}_p~:~i=1, \ldots, n\}$, we consider the set \begin{equation} \label{eq:Set WA} {\mathcal W}_{\mathfrak A} = \left\{a_1 \omega_1+\ldots+a_n \omega_n~:~ a_i \in {\mathcal A}_i, \ i=1, \ldots, n\right\}, \end{equation} which has a natural interpretation of elements in $\mathbb{F}_{p^n}$ ``with restricted digits''. Dartyge, Mauduit and S{\'a}rk{\"o}zy~\cite[Theorem~2.1]{DMS} show that if for some fixed $\varepsilon > 0$ the lower bound \begin{equation} \label{eq:DMS Cond} \min_{1 \le i \le r}\# {\mathcal A}_i \ge \(\frac{ \sqrt{5}-1}{2} + \varepsilon\)p \end{equation} holds, then, as $p\to \infty$, the set ${\mathcal W}_{\mathfrak A}$ contains asymptotically equal proportions of quadratic residues and non-residues (note that in~\cite{DMS} only the case of ${\mathcal A}_1 =\ldots ={\mathcal A}_r$ is considered but the proof immediately extends to different sets).
Here, in Section~\ref{sec:rest dig} we prove a similar asymptotic equidistribution of quadratic residues and non-residues under a much more relaxed condition than~\eqref{eq:DMS Cond}. Namely, for our result we only assume that for some fixed $\varepsilon > 0$ we have $$ \prod_{1 \le i \le n}\# {\mathcal A}_i \ge p^{(1/2 + \varepsilon)n^2/(n-1)} \qquad \mbox{and} \qquad \min_{1 \le i \le n}\# {\mathcal A}_i \ge p^\varepsilon. $$ For $n \geq 3$ this is a much wider range of parameters than the earlier restriction~\eqref{eq:DMS Cond} that is linear in $p$.
\subsection{Notation} Throughout the paper the implied constants in the symbols ``$O$'' and ``$\ll$'' may depend on the real parameter $\varepsilon > 0$ and
an integer parameter $\nu \ge 1$. We recall that the expressions $A \ll B$ and $A=O(B)$ are each equivalent to the statement that $|A|\le cB$ for some constant $c$.
As usual, $\log z$ denotes the natural logarithm of $z$.
The letter $p$ always denotes a prime.
As we have mentioned, we use $\mathbb{F}_r$ to denote the finite field of $r$ elements.
\section{Preparations} \subsection{Bounds of some exponential and character sums} \label{sec:exp char}
We need the following result of Bourgain~\cite[Lemma~4]{Bour2}.
\begin{lemma} \label{lem:ExpSums} Let $n > k \ge 1$ and $\vec{d} \in {\mathcal D}_{k,n}^*$. Then for any integers $a$ and $q$ with $\gcd(2a,q) = 1$ and
$3 \le q \le n^{1/10 \kappa}$, where $\kappa = k/n$, we have
$$
\left|\sum_{s \in {\mathcal N}_n(\vec{d})} \exp(2 \pi i as/q)\right|
< \#{\mathcal N}_n(\vec{d}) 2^{-\sqrt{n}}. $$ \end{lemma}
We need the following bound of a double character sum due to Karatsuba~\cite{Kar1}, see also~\cite[Chapter~VIII, Problem~9]{Kar2}, which in turn follows from the Weil bound (see~\cite[Corollary~11.24]{IwKow}) and the H{\"o}lder inequality.
We present it in the settings of arbitrary finite fields.
\begin{lemma} \label{lem:DoubleSums} For any integer $\nu \ge 1$, any sets ${\mathcal A}, {\mathcal B} \subseteq \mathbb{F}_r$ and any non-trivial multiplicative character $\chi$ of $\mathbb{F}_r$, we have $$ \sum_{a\in {\mathcal A}}\sum_{b \in {\mathcal B}} \chi(a+b) \ll (\# {\mathcal A})^{1-1/2\nu} \#{\mathcal B} r^{1/4\nu} + (\# {\mathcal A})^{1-1/2\nu} (\#{\mathcal B})^{1/2} r^{1/2\nu} , $$ where the implied constant depends only on $\nu$. \end{lemma}
In particular, taking $\nu =\rf{\varepsilon^{-1}}$ for a fixed $\varepsilon>0$ we derive from Lemma~\ref{lem:DoubleSums}
\begin{cor} \label{cor:DoubleSums} For any $\eta>0$ there exists $\delta >0$ such that for any sets ${\mathcal A}, {\mathcal B} \subseteq \mathbb{F}_r$ with $\# {\mathcal A} \ge r^{1/2 + \eta}$ and $\# {\mathcal B} \ge r^{\eta}$ and any non-trivial multiplicative character $\chi$ of $\mathbb{F}_r$, we have $$ \sum_{a\in {\mathcal A}}\sum_{b \in {\mathcal B}} \chi(a+b) \ll \# {\mathcal A} \#{\mathcal B} r^{-\delta}, $$ where the implied constant depends only on $\eta$. \end{cor}
We make use of the following special case of a result of Shao~\cite{Shao}.
\begin{lemma} \label{lem:MomentCharSums} Let $\nu\ge 1$ be a fixed integer. Let $0 \le w_1 < \ldots < w_J< p$ be $J\ge 1$ arbitrary integers with $$ w_{j+1} - w_j \ge H, \qquad j = 1, \ldots, J-1, $$ for some real $H\ge p^{1/2\nu}$. Then for any non-principal multiplicative character $\chi$ modulo $p$, we have $$
\sum_{j=1}^{J-1} \max_{h \le H} \left|\sum_{i=1}^{h} \chi\(i+w_j\)\right|^{2\nu} \ll p^{1/2 + 1/2\nu + o(1)}H^{2\nu - 2}. $$ \end{lemma}
\subsection{Bounds of the number of solutions of some congruences} \label{sec:cong}
For $\vec{d} \in {\mathcal D}_n$ and an integer $q \ge 2$ we consider the set $$ {\mathcal N}_n(\vec{d}, q) = \left\{s \in {\mathcal N}_n(\vec{d})~:~
s \equiv 0 \pmod q\right\}. $$
\begin{lemma} \label{lem:Cong Small q} Let $n > k \ge 1$ and $\vec{d} \in {\mathcal D}_{k,n}^*$. Then for any odd $q$ with
$1 < q \le n^{1/10 \kappa}$, where $\kappa = k/n$, we have
$$ \# {\mathcal N}_n(\vec{d}, q) = \frac{1}{q} \#{\mathcal N}_n(\vec{d}) + O\( \#{\mathcal N}_n(\vec{d}) 2^{-\sqrt{n}}\). $$ \end{lemma}
\begin{proof} Using the orthogonality of exponential functions we write $$ \# {\mathcal N}_n(\vec{d}, q) = \sum_{s \in {\mathcal N}_n(\vec{d}) } \frac{1}{q} \sum_{a =0}^{q-1} \exp(2 \pi i as/q) = \frac{1}{q} \sum_{a =0}^{q-1} \sum_{s \in {\mathcal N}_n(\vec{d}) } \exp(2 \pi i as/q). $$ The term corresponding to $a = 0$ is equal to $ \#{\mathcal N}_n(\vec{d})/q$ while it is easy to see that Lemma~\ref{lem:ExpSums} also applies to exponential sums with denominators $q/\gcd(a,q)\ge 3$ instead of $q$. \end{proof}
For larger values of $q$ we only have an upper bound on $\# {\mathcal N}_n(\vec{d}, q)$.
For real positive $\kappa$ and $\varrho$ we define \begin{equation} \label{eq:tau} \tau(\kappa, \varrho) = \frac{1+\varrho - \sqrt{(1-\varrho)^2+4\varrho\kappa}}{2} \end{equation} as the root of the equation $$ \tau^2 - \tau(1+\varrho) +\varrho(1-\kappa) = 0 $$ which belongs to the interval $[0, \varrho]$. We now set \begin{equation} \label{eq:theta} \vartheta(\kappa, \varrho) =\frac{\tau(\kappa, \varrho)}{\varrho}. \end{equation}
\begin{lemma} \label{lem:Cong Med q} Let $\varepsilon > 0$ be fixed. Let $$n(1-\varepsilon) > k \ge 1 \qquad \mbox{and} \qquad 2^{n(1-\varepsilon)} \ge q \ge 1. $$
Then for any $\vec{d} \in {\mathcal D}_{k,n}^*$ we have $$ \# {\mathcal N}_n(\vec{d}, q) \ll \#{\mathcal N}_n(\vec{d}) q^{-\vartheta(\kappa, \varrho)}, $$ where the implied constant is absolute, $\kappa$ and $\varrho$ are defined by $$ \kappa = k/n \qquad \mbox{and} \qquad q = 2^{\varrho n}, $$ and $\vartheta(\kappa, \varrho)$ is given by~\eqref{eq:theta}. \end{lemma}
\begin{proof} We refer to the digits of $s \in {\mathcal N}_n(\vec{d})$ on positions $j$ with $\delta_j=\ast$ as to {\it free positions\/} and we refer to other digits as to {\it fixed positions\/}.
We set $$ r = \rf{\frac{\log q}{\log 2}} -1. $$ Let $\vec{d} = (\delta_0, \ldots, \delta_{n-1})$.
We set $\delta_i=0$ for all integers $i \not \in [0, n-1]$.
Now, for $j \in \mathbb{Z}$, we denote by
$w_j$ the number of free positions amongst the positions $j, \ldots,j+r-1$ and let $\chi^{\ast}$ be the characteristic function of the symbol `$\ast$' defined on the set $\{\ast, 0, 1\}$. Then $$ \sum_{j=-r+1}^{n-1} w_j = \sum_{j=-r+1}^{n-1} \sum_{i=0}^{r-1} \chi^{\ast}(\delta_{i+j}) = r(n-k). $$
We now set $t = \fl{\tau(\kappa, \varrho) n}$, where $\tau(\kappa, \varrho)$ is given by~\eqref{eq:tau}. We count the total number $W$ of free positions which appear in each of the $n+r -2t+1$ blocks of width $r$ starting at the points $j = -r+t, \ldots, n-t$. Then we have \begin{equation} \label{eq:W lower} W = \sum_{j=-r+t}^{n-t} w_j = r(n-k) - \sum_{j=-r+1}^{-r+t-1} w_j - \sum_{j=n-t+1}^{n-1} w_j . \end{equation}
We now note that for $j < 0$ we have $w_j \le r-|j|$ and for $j <r$ we have $w_{n-j} \le j$. Hence, $$ \sum_{j=-r+1}^{-r+t-1} w_j +\sum_{j=n-t+1}^{n-1} w_j \le 2\sum_{i=1}^{t-1} i = t^2 + O(t). $$ Therefore, we conclude from~\eqref{eq:W lower} that $$ W \ge r(n-k) - t^2 + O(t). $$
Hence, for some $h \in [-r+t, n-t]$ we have \begin{equation} \label{eq:large w_h} \begin{split} w_h & \ge \frac{W}{n+r-2t+1}\ge \frac{r(n-k) - t^2 + O(t)}{n+r-2t+1}\\ & = n\frac{\varrho(1-\kappa) - \tau(\kappa, \varrho)^2}{1+\varrho-2\tau(\kappa, \varrho)} + O(1)\\ & = n\tau(\kappa, \varrho) + O(1) = n \varrho \vartheta(\kappa, \varrho) + O(1), \end{split} \end{equation} where the implied constant is absolute.
Now fixing the digits on the remaining $n-k - w_h$ free positions $j \not \in [h,h+r-1]$ of the numbers $$ \sum_{i=0}^{n-1} d_i 2^i \in {\mathcal N}_n(\vec{d}) $$ and recalling that $2^r < q$, we see that the number $$ s = \sum_{i=h}^{h+r-1} d_i 2^{i-h} $$ belongs to a prescribed residue class modulo $q$ and since $0 \le s < 2^r < q$, $s$ is uniquely defined. Hence, using~\eqref{eq:large w_h}, we obtain
$$ \# {\mathcal N}_n(\vec{d}, q) \le 2^{n-k -w_h} \le \# {\mathcal N}_n(\vec{d}, q) 2^{-w_h} \ll
\# {\mathcal N}_n(\vec{d}, q) q^{-\vartheta(\kappa, \varrho)}, $$ and the result follows. \end{proof}
\begin{lemma} \label{lem:Cong Aver q} Let $\varepsilon>0$ be sufficiently small and \begin{equation} \label{eq:rd12}
\frac{1}{2} \ge \varrho \ge \frac{1}{4}. \end{equation} Moreover, let \begin{equation} \label{kappa37}
\kappa=\frac{k}{n}<\frac{3}{7}- 2 \varepsilon \end{equation} and $$
2^{\varrho n} \ll A \ll 2^{\varrho n}, $$ and suppose that \begin{equation} \label{eq:cond1} (3+4\varepsilon)\varrho \le 2(1-\kappa) \end{equation} and \begin{equation} \label{eq:cond2}
\varrho(1+5\varepsilon)<4(1-\kappa)-2. \end{equation} Then \begin{equation} \label{goal}
\sum_{A<q \le 2A} \# {\mathcal N}_n(\mathbf{d}, q^2) \ll
\# {\mathcal N}_n(\mathbf{d}) A^{-\varepsilon/2}. \end{equation} \end{lemma}
\begin{proof} We follow the definition of free and fixed positions as in the proof of Lemma~\ref{lem:Cong Med q}.
Let us divide the set of all $n$ positions into three blocks $W_1$, $W_2$, $W_3$ of consecutive positions (from the left to the right) in the following way: The number of positions in $W_1 \cup W_2$ as well as in $W_2 \cup W_3$ is $2\varrho n+O(1)$. This is certainly possible since we have~\eqref{eq:rd12}
(more explicitly, $W_1$ and $W_3$ contain $n(1-2 \varrho)+O(1)$ positions and $W_2$ contains $n(4\varrho-1)+O(1)$ positions). Let $w_i$ be the number of free positions in block $W_i$, $i=1,2,3$. Since the total number of free positions is $(1-\kappa)n$, we obtain \begin{equation} \label{eq:rd1}
w_1+w_2+w_3 = (1-\kappa)n. \end{equation} Now let $\alpha$ be the number of free positions in $W_1$ and $W_2$ together, that is, $\alpha=w_1+w_2$, and analogously let $\beta=w_1+w_3$ and $\gamma=w_2+w_3$. Then \eqref{eq:rd1} implies that \begin{equation} \label{eq:rd8}
\alpha+\beta+\gamma = 2(1-\kappa)n. \end{equation} Now regarding the neighbouring blocks $W_1$ and $W_2$ as one block with $\alpha$ free positions, as in the proof of Lemma \ref{lem:Cong Med q} we obtain \begin{equation} \label{eq:N alpha} \#{\mathcal N}_n(\mathbf{d}, q^2) \ll \#{\mathcal N}_n(\mathbf{d}) 2^{-\alpha} \end{equation} whenever $A<q \le 2A$. Note that here we use the fact that $A \gg 2^{\varrho n}$, whence $q^2 \gg 2^{2\varrho n}$, so a congruence modulo $q^2$ fixes all the free positions in the block composed of $W_1$ and $W_2$. Analogously, we obtain the alternative bound \begin{equation} \label{eq:N gamma} \#{\mathcal N}_n(\mathbf{d}, q^2) \ll \#{\mathcal N}_n(\mathbf{d}) 2^{-\gamma}, \end{equation} using the block composed of $W_2$ and $W_3$. Our first observation is that we can assume that $\alpha < (1+\varepsilon) \varrho n$, as otherwise~\eqref{eq:N alpha} implies $$ \#{\mathcal N}_n(\mathbf{d}, q^2) \ll \#{\mathcal N}_n(\mathbf{d}) A^{-1-\varepsilon} $$ and the result follows. Similarly, using~\eqref{eq:N gamma}, we can assume that
$\gamma < (1+\varepsilon) \varrho n$. Hence, by~\eqref{eq:rd8} we can also assume that \begin{equation} \label{eq:beta large}
\beta \ge 2n(1-\kappa-(1+\varepsilon)\varrho). \end{equation} Note that by \eqref{eq:cond1}, this implies that \begin{equation} \label{betanote}
2^{-\beta} \ll A^{-1-\varepsilon}. \end{equation} Moreover, as trivially $\beta \le (1-\kappa)n$ where $(1-\kappa)n$ is the total number of free positions, we obtain $$
\varrho \ge \frac{1-\kappa}{2(1+\varepsilon)}. $$ By \eqref{kappa37}, for sufficiently small $\varepsilon>0$ this implies that $$
\varrho>\frac{2}{3}\kappa +\varepsilon, $$ whence \begin{equation} \label{fix}
\varrho n - \beta + (2-6\varrho)n \le
n (2\kappa-3\varrho+2\varepsilon \varrho) \le -\varepsilon n \end{equation} for sufficiently small $\varepsilon>0$.
Working with $W_1$ and $W_3$ is more difficult, as we are no longer dealing with one, but rather with two intervals. Writing $r$ for the bit position at the right of $W_1$, and $s$ for the position at the right of $W_2$, we are now considering congruences of the form \begin{equation} \label{eq:rd2}
2^ra+2^s b+c \equiv 0 \pmod {q^2}. \end{equation} Note that from the construction of $W_1$, $W_2$, $W_3$ it follows that $$
r \ge 2\varrho n+O(1). $$ Once $b$, corresponding to $W_2$, has been fixed, the solution set of \eqref{eq:rd2} is of the form \begin{equation} \label{eq:ac} (a,c) = (a_0, c_0) + (a_1, c_1), \end{equation} where $(a_0, c_0) \in \mathbb{Z}^2$ is a fixed solution of~\eqref{eq:rd2} and $(a_1, c_1)$ runs over all solutions of the homogeneous congruence \begin{equation} \label{eq:rd3}
2^ra_1 + c_1 \equiv 0 \pmod {q^2}. \end{equation} By construction of the $W_i$, we see that $a$ and $c$ are non-negative integers with $a, c \ll 2^{(1-2\varrho)n}$, whence also
$|a_1|, |c_1| \ll 2^{(1-2\varrho)n}$. Moreover, the congruence~\eqref{eq:rd3} describes a two-dimensional lattice with a basis $\{(1, -2^r), (0, q^2)\}$ and of determinant $q^2$. Let $2^{\lambda_1(q)}, 2^{\lambda_2(q)}$ be its successive minima, where $\lambda_1(q) \le \lambda_2(q)$. For the general background on lattices we refer to~\cite{GrLoSch}.
Then $$
q^2 \ll 2^{\lambda_1(q)+\lambda_2(q)} \ll q^2. $$ Let us first discuss the case that $$
\lambda_2(q) \le (1-2\varrho)n. $$ Then the number of solutions to~\eqref{eq:rd3} with
$|a_1|, |c_1| \ll 2^{(1-2\varrho)n}$ can be estimated as $$ O\( \(2^{(1-2\varrho)n- \lambda_1(q)}+1\) \(2^{(1-2\varrho)n- \lambda_2(q)}+1\) \) = O\(2^{2(1-2\varrho)n}q^{-2}\) $$ (note that $q^2 \ll 2^{\lambda_1(q) + \lambda_2(q)} \le 2^{2 \lambda_2(q)} \le 2^{2(1-2\varrho)n}$). Furthermore, since $q^2\ge A^2 \gg 2^{2 \varrho n}$, we obtain the bound $O\(2^{(2-6\varrho)n}\)$ for the number of solutions to~\eqref{eq:rd3}.
Considering all the possible $$ 2^{w_2} = 2^{n- k-\beta} = \# {\mathcal N}_n(\mathbf{d}) 2^{-\beta} $$ choices for $b$,
we therefore obtain $$ \# {\mathcal N}_n(\mathbf{d}, q^2) \ll \# {\mathcal N}_n(\mathbf{d}) 2^{(2-6\varrho)n-\beta}. $$ By~\eqref{fix}, this contribution, when summed over $A<q \le 2A$, is negligible with respect to~\eqref{goal}.
We may therefore without loss of generality assume that $$
\lambda_2(q)>(1-2\varrho)n. $$
Again, from~\eqref{eq:ac} we conclude that the number of solutions of~\eqref{eq:rd2} with $|a_1|, |c_1| \ll 2^{(1-2\varrho)n}$ is $$ O\( \(2^{(1-2\varrho)n- \lambda_1(q)}+1\) \(2^{(1-2\varrho)n- \lambda_2(q)}+1\) \) = O\(2^{(1-2\varrho)n-\lambda_1(q)} + 1\), $$ and the number of possible choices for $b$ is bounded by $\# {\mathcal N}_n(\mathbf{d}) 2^{-\beta}$, so \begin{equation} \label{eq:N beta} \# {\mathcal N}_n(\mathbf{d}, q^2) \ll \# {\mathcal N}_n(\mathbf{d}) 2^{-\beta} (2^{(1-2\varrho)n-\lambda_1(q)}+1). \end{equation} Let us define a real parameter \begin{equation} \label{eq:lambda} \lambda = (1-2\varrho)n-2(1-\kappa)n+3(1+\varepsilon)\varrho n. \end{equation} If $\lambda_1(q)> \lambda$, then~\eqref{eq:beta large} and~\eqref{betanote} give $$
2^{-\beta}\(2^{(1-2\varrho)n-\lambda_1(q)} +1\) \le 2^{-(1+\varepsilon)\varrho n} + 2^{-\beta} \ll A^{-1-\varepsilon}, $$ so $$
\sum_{\substack{A<q \le 2A:\\ \lambda_1(q) \ge \lambda}}
\#{\mathcal N}_n(\mathbf{d}, q^2) \ll \# {\mathcal N}_n(\mathbf{d}) A^{-\varepsilon}. $$
It now remains to estimate the contribution from $q$ with $ \lambda_1(q) \le \lambda$. Furthermore, it is enough to show that for any real positive $\mu< \lambda$ we have \begin{equation} \label{eq:rd13}
\sum_{\substack{A<q \le 2A:\\ \mu \le \lambda_1(q) < \mu+1}}
\#{\mathcal N}_n(\mathbf{d}, q^2) \ll \# {\mathcal N}_n(\mathbf{d}) A^{-\varepsilon}. \end{equation}
Now $\lambda_1(q) \le \mu+1$ means that there exist $a_1, c_1 \in \mathbb{Z}$, not both zero, such that $|a_1|, |c_1| \ll 2^\mu$
and~\eqref{eq:rd3} holds true. Note that $2^r a_1 + c_1 = 0$ is impossible, as it implies that $2^r \mid c_1$, so $|c_1| \ge 2^r \gg 2^{2\varrho n}$, contradicting $|c_1| \ll 2^\mu$ as by~\eqref{eq:rd12} and~\eqref{eq:cond1} we have $$\mu<\lambda \le (1-2\varrho)n- \varepsilon\varrho n \le 2\varrho n- \varepsilon\varrho n. $$
Therefore, $2^r a_1 + b_1 \ne 0$, so by~\eqref{eq:rd3} for each fixed pair $(a_1, c_1)$ there are only $2^{o(n)}$ possibilities for $q$ that are integer divisors of $2^r a_1 + b_1 = O(2^n)$, see~\cite[Theorem~317]{HardyWright}. The number of possible $(a_1, c_1)$ can be bounded by $O(2^{2\mu})$, and $\# N_n(\mathbf{d}, q^2)$, by~\eqref{eq:N beta}, is at most of order of magnitude $$ \#{\mathcal N}_n(\mathbf{d})(2^{-\beta+(1-2\varrho)n-\mu}+2^{-\beta}) \ll \#{\mathcal N}_n(\mathbf{d})(2^{-\beta+(1-2\varrho)n-\mu}+A^{-1-\varepsilon}). $$ We therefore obtain \begin{align*}
\sum_{\substack{A<q \le 2A:\\ \mu \le \lambda_1(q) < \mu+1}}
\#{\mathcal N}_n(\mathbf{d}, q^2) & \ll \# {\mathcal N}_n(\mathbf{d})
(2^{-\beta+(1-2\varrho)n+\mu+n\varepsilon} + A^{-\varepsilon}) \\
& \ll \#{\mathcal N}_n(\mathbf{d})
(2^{-\beta+(1-2\varrho)n+\lambda+o(n)}+A^{-\varepsilon}). \end{align*} Now by \eqref{eq:cond2}, \eqref{eq:beta large} and~\eqref{eq:lambda}, we have $$
-\beta+(1-2\varrho)n+ \lambda < 0, $$ completing the proof. \end{proof}
In particular, covering an interval $[A,B]$ by dyadic intervals, we derive from Lemma~\ref{lem:Cong Aver q} the following result suitable for our applications.
\begin{cor} \label{cor:Cong Aver q} Let $\varepsilon>0$ and $$
\kappa=\frac{k}{n}<\frac{3}{7}-2\varepsilon $$ Suppose that $$ \frac{1}{2} \ge \zeta \ge \xi \ge \frac{1}{4}, \qquad (3+4\varepsilon)\zeta \le 2(1-\kappa), \qquad
\zeta(1+5\varepsilon) < 4(1-\kappa)-2. $$ Then for any $A$ and $B$ with $$
2^{\xi n} \ll A \le B \ll 2^{\zeta n} $$ we have $$
\sum_{A<q \le B} \# {\mathcal N}_n(\mathbf{d}, q^2) \ll
\# {\mathcal N}_n(\mathbf{d}) A^{-\varepsilon/2} \log B. $$ \end{cor}
\begin{lemma} \label{two_windows} Keeping the notation of Lemma \ref{lem:Cong Med q}, suppose that $$
\frac{\kappa}{2} \le \varrho \le \frac{1}{2}. $$ Then $$ \# {\mathcal N}_n(\vec{d}, q) \ll \#{\mathcal N}_n(\vec{d}) q^{-1+\kappa/2\varrho}. $$ \end{lemma} \begin{proof} We use a similar, but simpler argument as in the proof of Lemma \ref{lem:Cong Aver q}. As $\varrho \le \frac{1}{2}$, we can divide the $n$ bits into three blocks $W_1, W_2, W_3$ (from the left to the right), such that $W_1$ and $W_3$ have size $\varrho n + O(1)$, each, and $W_2$ has size $(1-2\varrho)n+O(1)$. Then in one of $W_1$ and $W_3$, say $W_1$, there must be at least $$
\frac{1}{2}\left((1-\kappa)n-(1-2\varrho)n\right) + O(1) =
n(\varrho-\kappa/2)+O(1) $$ many free positions. Once all the bits outside $W_1$ have been chosen, a congruence modulo $q$ fixes all $\gg n(\varrho-\kappa/2)$ remaining free positions in $W_1$, whence $$ \# {\mathcal N}_n(\vec{d}, q) \ll \#{\mathcal N}_n(\vec{d}) 2^{-n(\varrho-\kappa/2)} \ll \#{\mathcal N}_n(\vec{d}) q^{-1+\kappa/2\varrho}, $$ which concludes the proof. \end{proof}
To apply Lemma~\ref{lem:Cong Med q} we also need the following technical statement.
\begin{lemma} \label{lem:theta} For $1> \kappa> 0$ the function $\vartheta(\kappa, \varrho)$ given by~\eqref{eq:theta} is monotonically decreasing as $\varrho$ is increasing. \end{lemma}
\begin{proof} The result follows from the observation that the derivative $$ \frac{{\partial } \vartheta}{{\partial } \varrho}= \frac{1 + (-1 + 2 \kappa) \varrho - \sqrt{1 + (-2 + 4 \kappa) \varrho + \varrho^2}}{ 2 \varrho^2 \sqrt{ 1 + (-2 + 4 \kappa) \varrho + \varrho^2}} $$
is negative, as $1+2c\varrho+ \varrho^2>0$ and $1+c\varrho -\sqrt{1+2c\varrho+ \varrho^2}<0$ when $|c|<1$. \end{proof}
\subsection{Some results from additive combinatorics} \label{sec:addcomb}
We now recall a recent result by Schoen~\cite[Theorem~3.3]{Sch} in additive combinatorics. As in~\cite{DES1}, we note that~\cite[Theorem~3.3]{Sch} is only stated for subset sums but can be easily extended to Hilbert cubes.
\begin{lemma} \label{lem:hilbcube} For any $a_0 \in \mathbb{F}_p$ and pairwise distinct $a_1, \ldots, a_d \in \mathbb{F}_p$ such that $d \ge 8(p/\log p)^{1/D}$, where $D$ is an integer satisfying $$ 0<D\le \sqrt{\frac{\log p}{2 \log \log p}}, $$ the Hilbert cube~\eqref{eq:hilbert_cube} contains an arithmetic progression of length $L$ where $$
L \ge 2^{-10} (d/\log p)^{1+1/(D-1)}. $$ \end{lemma}
For a set ${\mathcal S} \subseteq \mathbb{F}_p$ we use $\Sigma_k({\mathcal S})$ to denote the set of all $k$-elements subset sums of ${\mathcal S}$, that is, $$ \Sigma_k({\mathcal S}) = \left\{\sum_{t \in {\mathcal T}} t~:~ {\mathcal T} \subseteq {\mathcal S}, \ \#{\mathcal T}=k\right\}. $$ We make use of the following result of Dias da Silva and Hamidoune~\cite[Theorem~4.1]{DiDaSiHa}.
\begin{lemma} \label{lem:subset} For a set ${\mathcal S} \subseteq \mathbb{F}_p$ and an integer $k \ge 1$, we have $$ \# \Sigma_k({\mathcal S}) \ge \min\{p, k\#{\mathcal S} -k^2 +1\}. $$ \end{lemma}
We now define $$ \Sigma_*({\mathcal S}) = \bigcup_{k=0}^{\# {\mathcal S}} \Sigma_k({\mathcal S}). $$
Taking $k = \fl{\#{\mathcal S}/2}$ in Lemma~\ref{lem:subset} we immediately derive:
\begin{cor} \label{cor:subset} For a set ${\mathcal S} \subseteq \mathbb{F}_p$ and an integer $k \ge 1$ we have $$ \# \Sigma_*({\mathcal S}) \gg \min\{p, (\#{\mathcal S})^2 +1\}. $$ \end{cor}
\section{Main Results}
\subsection{Squarefree integers with fixed digits} \label{sec:sqfr} Let $S_n(\vec{d})$ be the number of squarefree integers $s \in {\mathcal N}_n(\vec{d})$.
\begin{theorem} \label{thm:SF} For any $\varepsilon > 0$, uniformly over integer $k <(2/5 - \varepsilon) n$ and $\vec{d} \in {\mathcal D}_{k,n}^*$, we have $$ S_n(\vec{d})= \(\frac{8}{\pi^2} +o(1)\) \#{\mathcal N}_n(\vec{d}). $$ \end{theorem}
\begin{proof} The inclusion-exclusion principle yields $$ S_n(\vec{d}) = \sum_{q=1}^\infty \mu(q) \# {\mathcal N}_n(\vec{d}, q^2), $$ where $\mu(q)$ is the M{\"o}bius function, see~\cite[Section~16.3]{HardyWright}.
As before, we define $\kappa = k/n$ and we also use the function $\vartheta(\kappa, \varrho)$ that is given by~\eqref{eq:theta}.
For $\kappa<2/5-\varepsilon$ we have \begin{equation} \label{eq:25}
\vartheta(\kappa, 2/5)>1/2 \end{equation} as $\vartheta(2/5, 2/5)=1/2$ and for fixed $\varrho$, the function $\vartheta(\kappa, \varrho)$ given by \eqref{eq:theta} is obviously decreasing in $\kappa$. Choose $\zeta>2/5$ such that \begin{equation} \label{eq:zeta}
(3+4\varepsilon) \zeta \le 2(1-\kappa), \qquad
\zeta (1+5 \varepsilon) < 4(1-\kappa)-2, \end{equation} which for sufficiently small $\varepsilon>0$ is possible since $\kappa< 2/5 - \varepsilon$. Note that in particular, $\zeta>\kappa$.
We set $$ T = n^{1/20 \kappa}, \qquad U=2^{n/5}, \qquad V = 2^{n/4}, \qquad W = 2^{\zeta n}, $$ and write \begin{equation} \label{eq:Si} S_n(\vec{d}) = S_1 + S_2 + S_3 + S_4 + S_5, \end{equation} where \begin{equation*} \begin{split} S_1 & =\sum_{q \le T} \mu(q) \# {\mathcal N}_n(\vec{d}, q^2),\\ S_2 & =\sum_{T< q \le U} \mu(q) \# {\mathcal N}_n(\vec{d}, q^2),\\ S_3 & =\sum_{U< q \le V} \mu(q) \# {\mathcal N}_n(\vec{d}, q^2),\\ S_4 & =\sum_{V < q \le W} \mu(q) \# {\mathcal N}_n(\vec{d}, q^2),\\ S_5 & =\sum_{q > W} \mu(q) \# {\mathcal N}_n(\vec{d}, q^2). \end{split} \end{equation*} We use Lemma~\ref{lem:Cong Small q} for $q \le T$, getting the main term \begin{equation*} \begin{split} S_1 &= \#{\mathcal N}_n(\vec{d}) \sum_{\substack{q \le T\\ q~\text{odd}}} \frac{\mu(q)}{q^2} + O\( \#{\mathcal N}_n(\vec{d}) T 2^{-\sqrt{n}}\) \\ & = \#{\mathcal N}_n(\vec{d}) \sum_{q~\text{odd}} \frac{\mu(q)}{q^2} + O\( \#{\mathcal N}_n(\vec{d}) T 2^{-\sqrt{n}} + \#{\mathcal N}_n(\vec{d}) T^{-1} \) \\ & = \#{\mathcal N}_n(\vec{d}) \prod_{\substack{\ell \ge 3\\ \ell~\text{prime}}} \(1 - \frac{1}{\ell^2}\) + O\( \#{\mathcal N}_n(\vec{d}) T 2^{-\sqrt{n}} + \#{\mathcal N}_n(\vec{d}) T^{-1}\)\\ & = \frac{4}{3}\#{\mathcal N}_n(\vec{d}) \prod_{\ell~\text{prime}} \(1 - \frac{1}{\ell^2}\) + O\( \#{\mathcal N}_n(\vec{d}) T 2^{-\sqrt{n}} + \#{\mathcal N}_n(\vec{d}) T^{-1}\). \end{split} \end{equation*} So we now obtain the main term \begin{equation} \label{eq:S1} S_1 = \(\frac{8}{\pi^2} + o(1)\)\#{\mathcal N}_n(\vec{d}), \end{equation} see~\cite[Theorem~280]{HardyWright}.
To estimate $S_2$, we use Lemma~\ref{lem:Cong Med q}. First we note that by Lemma~\ref{lem:theta}, for $T < q \le U$, we have $$ \vartheta(\kappa, 2\varrho) \ge \vartheta(\kappa, 2/5), $$ where, in analogy to Lemma~\ref{lem:Cong Med q}, $\varrho$ is defined by $q = 2^{\varrho n}$. Hence in this range we have $$ \# {\mathcal N}_n(\vec{d}, q^2) \ll
\#{\mathcal N}_n(\vec{d}) q^{-2\vartheta(\kappa, 2\varrho)} \ll \#{\mathcal N}_n(\vec{d}) q^{-2\vartheta(\kappa, 2/5)}. $$ Since by~\eqref{eq:25} we have $2\vartheta(\kappa, 2/5)> 1$, we now derive \begin{equation} \label{eq:S2} S_2 \ll \#{\mathcal N}_n(\vec{d}) T^{1-2\vartheta(\kappa, 2/5)} = o\(\#{\mathcal N}_n(\vec{d})\). \end{equation}
For $S_3$ we use a similar argument as for $S_2$, this time with Lemma~\ref{two_windows} instead of Lemma~\ref{lem:Cong Med q}, noting that with $\kappa<2/5$ and $\varrho \ge 1/5$ we obtain $$ \# {\mathcal N}_n(\vec{d}, q^2) \ll \#{\mathcal N}_n(\vec{d}) q^{2(-1+\kappa/4\varrho)} \ll \#{\mathcal N}_n(\vec{d}) q^{-1-\delta} $$ for some sufficiently small $\delta>0$, so again \begin{equation} \label{bound_s3}
S_3 = o\(\#{\mathcal N}_n(\vec{d})\). \end{equation}
To estimate $S_4$, we use Corollary~\ref{cor:Cong Aver q}, which by~\eqref{eq:zeta} applies with some sufficiently small $\varepsilon >0$, getting \begin{equation} \label{eq:S3} S_4 \ll \# {\mathcal N}_n(\mathbf{d}) V^{-\varepsilon/2} \log W = o\(\#{\mathcal N}_n(\vec{d})\). \end{equation}
Finally, we use the trivial bound $\# {\mathcal N}_n(\vec{d}, q^2) \le 2^n/q^2$ for $q > W$ and using $\zeta>\kappa$ we derive \begin{equation} \label{eq:S4} S_5 \ll 2^n W^{-1} \ll \#{\mathcal N}_n(\vec{d}) 2^{\kappa n - \zeta n} = o\(\#{\mathcal N}_n(\vec{d})\). \end{equation} Substituting~\eqref{eq:S1}, \eqref{eq:S2}, \eqref{bound_s3}, \eqref{eq:S3} and~\eqref{eq:S4} into~\eqref{eq:Si}, we now conclude the proof.
\end{proof}
\subsection{Average values of the Euler function}
\label{sec:Euler}
We now consider the average value $$ F_n(\vec{d}) = \sum_{s\in {\mathcal N}_n(\vec{d})} \frac{\varphi(s)}{s} $$ with the Euler function $\varphi(s)$, see~\cite[Section~16.3]{HardyWright}.
\begin{theorem} \label{thm:AverEuler} For any $\varepsilon > 0$, uniformly over integers $k <(1 - \varepsilon) n$ and $\vec{d} \in {\mathcal D}_{k,n}^*$, we have $$ F_n(\vec{d})= \(\frac{8}{\pi^2} +o(1)\) \#{\mathcal N}_n(\vec{d}). $$ \end{theorem}
\begin{proof} Using the the well-known formula $$
\frac{\varphi(s)}{s} = \sum_{q \mid s} \frac{\mu(q)}{q} $$ see~\cite[Equation~(16.3.1)]{HardyWright}, and changing the order of summation, we write $$ F_n(\vec{d}) = \sum_{q=1}^\infty \frac{\mu(q)}{q} \# {\mathcal N}_n(\vec{d}, q). $$
We now proceed very similarly to the proof of Theorem~\ref{thm:SF}.
Again we define $\kappa = k/n$ and we also use the function $\vartheta(\kappa, \varrho)$ that is given by~\eqref{eq:theta}.
Clearly, for $0 < \kappa < 1$ we have $$ \vartheta(\kappa, 1) = 1 - \sqrt{\kappa} > 0. $$ Thus, using Lemma~\ref{lem:theta}, we see that for any $\kappa< 1$ we can find $\xi$ to satisfy \begin{equation} \label{eq:xi11} 1> \xi > \kappa \end{equation} and \begin{equation} \label{eq:xi22} \vartheta(\kappa, \xi) > 0. \end{equation}
We set $$ Q = n^{1/10 \kappa} \qquad \mbox{and} \qquad W = 2^{\xi n}. $$ and write \begin{equation} \label{eq:Ti} F_n(\vec{d}) = T_1 + T_2 + T_3, \end{equation} where \begin{equation*} \begin{split} T_1 & =\sum_{q \le Q}\frac{\mu(q)}{q} \# {\mathcal N}_n(\vec{d}, q),\\ T_2 & =\sum_{Q< q \le W}\frac{\mu(q)}{q} \# {\mathcal N}_n(\vec{d}, q),\\ T_3 & =\sum_{q > W} \frac{\mu(q)}{q} \# {\mathcal N}_n(\vec{d}, q). \end{split} \end{equation*} We use Lemma~\ref{lem:Cong Small q} for $q \le Q$, and exactly as in the proof of of Theorem~\ref{thm:SF} we obtain the main term \begin{equation} \label{eq:T1} T_1 = \#{\mathcal N}_n(\vec{d}) \sum_{\substack{q \le Q\\ q~\text{odd}}} \frac{\mu(q)}{q^2} + O\( \#{\mathcal N}_n(\vec{d}) Q 2^{-\sqrt{n}}\) =\(\frac{8}{\pi^2} + o(1)\)\#{\mathcal N}_n(\vec{d}), \end{equation} see~\cite[Theorem~280]{HardyWright}.
To estimate $T_2$, we use Lemma~\ref{lem:Cong Med q} for $Q < q \le W$. First we note that by Lemma~\ref{lem:theta}, for $Q < q \le W$, we have $$ \vartheta(\kappa, \varrho) \ge \vartheta(\kappa, \xi), $$ where, in analogy to Lemma~\ref{lem:Cong Med q}, $\varrho$ is defined by $q = 2^{\varrho n}$. Hence in this range we have $$ \# {\mathcal N}_n(\vec{d}, q) \ll
\#{\mathcal N}_n(\vec{d}) q^{-\vartheta(\kappa, \varrho)} \ll \#{\mathcal N}_n(\vec{d}) q^{-\vartheta(\kappa, \xi)}. $$ Since by~\eqref{eq:xi22} we have $\vartheta(\kappa, \xi)> 0$, we now derive \begin{equation} \label{eq:T2} T_2 \ll \#{\mathcal N}_n(\vec{d}) Q^{-\vartheta(\kappa, \xi)} = o\(\#{\mathcal N}_n(\vec{d})\). \end{equation}
Finally, we use the trivial bound $\# {\mathcal N}_n(\vec{d}, q) \le 2^n/q$ for $q > W$ and using~\eqref{eq:xi11} derive \begin{equation} \label{eq:T3} T_3 \ll 2^n W^{-1} \ll \#{\mathcal N}_n(\vec{d}) 2^{\kappa n - \xi n} = o\(\#{\mathcal N}_n(\vec{d})\). \end{equation} Substituting~\eqref{eq:T1}, \eqref{eq:T2} and~\eqref{eq:T3} into~\eqref{eq:Ti}, we conclude the proof.
\end{proof}
\subsection{Non-residues with fixed digits} \label{sec:nonres} For a prime $p$, we use ${\mathcal N}_n^{+}(\vec{d}, p)$ and ${\mathcal N}_n^{-}(\vec{d}, p)$ to denote the sets of $s \in {\mathcal N}_n(\vec{d})$, that are quadratic residues and non-residues, respectively (we also use $ {\mathcal N}_n^{\pm}(\vec{d}, p)$ to denote either of these sets).
\begin{theorem} \label{thm:NonRes} For any $\varepsilon > 0$ there exists some $\delta > 0$ such that for $k < (1/2 - \varepsilon) n$, $\vec{d} \in {\mathcal D}_{k,n}$ and any prime $p$ with $2^{n}< p < 2^{n+1}$ we have $$ \# {\mathcal N}_n^{\pm}(\vec{d}, p) = \(\frac{1}{2} + O(p^{-\delta})\) \#{\mathcal N}_n(\vec{d}). $$ \end{theorem}
\begin{proof} We select arbitrary $s= \rf{\varepsilon n/2}$ free positions and denote by ${\mathcal B}$ the set of $2^s$ integers with all possible combinations of digits on these positions and zeros on all other positions. We also define by ${\mathcal A}$ the subset of $2^{n-k - s}$ elements of ${\mathcal N}_n(\vec{d})$ which also have zero digits on the positions that are allocated to ${\mathcal B}$. Clearly each element of ${\mathcal N}_n(\vec{d})$ has a unique representation as $a+b$ with $a \in {\mathcal A}$, $b \in {\mathcal B}$. The result is now instant from Corollary~\ref{cor:DoubleSums}. \end{proof}
\subsection{Primitive roots in Hilbert cubes} \label{sec:Hilb} We now present an improvement of~\cite[Theorem~1.3]{DES1}.
\begin{theorem} \label{thm:Hilb} We have $$ F(p) \le p^{3/19+o(1)}. $$ \end{theorem}
\begin{proof} Let ${{\mathcal H}}(a_0;a_1, \ldots, a_d)$ be a Hilbert cube, with $d$ distinct base elements $a_0, \ldots, a_d \in \mathbb{F}_p$. Suppose that $ {{\mathcal H}}(a_0;a_1, \ldots, a_d)$ does not contain primitive roots modulo $p$. We show that $d=O(p^{3/19+o(1)})$.
As in the proof of~\cite[Theorem~1.3]{DES1}, we fix some $\varepsilon > 0$ and assume that \begin{equation} \label{eq:d} d \ge p^{3/19 + \varepsilon}, \end{equation} and without loss of generality we may also assume that $d$ is even.
Let ${{\mathcal U}} = {{\mathcal H}}(a_0;a_1, \ldots, a_{d/2})$ and
${{\mathcal V}} = {\mathcal H}(0;a_{d/2+1}, \ldots, a_{d})$, both ${\mathcal U}$ and ${\mathcal V}$ understood as subsets of $\mathbb{F}_p$. It follows from Lemma~\ref{lem:hilbcube}, applied with $D = 7$ that ${\mathcal U}$ contains an arithmetic progression ${\mathcal A} \subseteq \mathbb{F}_p$ of length \begin{equation} \label{eq:L} L = \# {\mathcal A} \geq p^{7/38+o(1)}. \end{equation} Let $\Delta$ be the difference between consecutive terms of the progression. Let us consider the interval $$ {\mathcal I}=\{\Delta^{-1}a~:~a\in {\mathcal A}\} \subseteq \mathbb{F}_p $$ of $L$ consecutive residues modulo $p$.
On the other hand, it follows from Corollary~\ref{cor:subset} that $\# {\mathcal V} \gg d^2$. Now let $$ {\mathcal W} = \{\Delta^{-1}a~:~a\in {\mathcal V}\} \subseteq \mathbb{F}_p. $$ We now take an arbitrary element $w_1 \in {\mathcal W}$ and remove from
${\mathcal W}$ at most $O(L)$ elements $w$ with $|w-w_1| \le L$ and denote the remaining set as ${\mathcal W}_1$. We now choose an arbitrary element $w_2 \in {\mathcal W}_1$ and remove from
${\mathcal W}_1$ at most $O(L)$ elements $v$ with $|w-w_2| \le L$ and denote the remaining set as ${\mathcal W}_2$. Continuing, we obtain a set $\{w_1, \ldots, w_J\}$ of \begin{equation} \label{eq:J} J \gg \# {\mathcal W}/L = \# {\mathcal V}/L \gg d^2/L \end{equation} elements, which after renumbering satisfy the condition of Lemma~\ref{lem:MomentCharSums}.
By Lemma~\ref{lem:MomentCharSums} and \eqref{eq:J}, taking a sufficiently large $\nu$ after simple calculations we obtain that for any non-trivial multiplicative character $\chi$ of $\mathbb{F}_p$ we have $$
\sum_{j=1}^{J-1} \left|\sum_{i=1}^{L} \chi\(i+w_j\)\right|^{2\nu} \ll p^{1/2 + 1/2\nu + o(1)}L^{2\nu - 2} \ll JL^{2\nu} p^{-\eta}, $$ provided that $$
L^{1/2} d \gg p^{1/4+\varepsilon}. $$ Recalling \eqref{eq:d} and \eqref{eq:L}, we find that the latter condition is satisfied. Expressing, in a standard fashion, the counting function for primitive roots among the elements $i+w_j$, $i =1, \ldots, L$, $j = 1, \ldots, J$, via multiplicative characters, see, for example,~\cite[Lemma~2.4]{DES1}, we see that it is positive, provided $p$ is large enough. Since $\varepsilon$ is arbitrary, we obtain the desired result.
\end{proof}
\subsection{Elements with restricted digits in finite fields} \label{sec:rest dig}
Our next result improves~\cite[Theorem~2.1]{DMS} for any $n \ge 3$:
For ${\mathcal W}_{\mathfrak A}$,
given by~\eqref{eq:Set WA}, we use ${\mathcal W}_{\mathfrak A}^{+}$ and ${\mathcal W}_{\mathfrak A}^{-}$ to denote the sets of $w \in {\mathcal W}_{\mathfrak A}$, that are quadratic residues and non-residues, respectively (we also use ${\mathcal W}_{\mathfrak A}^{\pm}$ to denote either of these sets).
\begin{theorem} \label{thm:Nonres W} Let $n \ge 2$. For any $\varepsilon > 0$ there is some $\delta > 0$ such that for an arbitrary basis $\omega_1, \ldots, \omega_n$ of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$ and a collection of $n$ sets $$ {\mathfrak A} = \{{\mathcal A}_i\subseteq \mathbb{F}_p~:~i=1, \ldots, n\}, $$ satisfying \begin{equation} \label{eq:DES Cond1}
\prod_{1 \le i \le n}\# {\mathcal A}_i \ge p^{(1/2 + \varepsilon)n^2/(n-1)} \end{equation} and \begin{equation} \label{eq:DES Cond2} \min_{1 \le i \le n}\# {\mathcal A}_i \ge p^\varepsilon \end{equation} the following holds: for the set $$ {\mathcal W}_{\mathfrak A} = \left\{a_1 \omega_1+\ldots+a_n \omega_n~:~ a_i \in {\mathcal A}_i, \ i=1, \ldots, n\right\}, $$ we have $$ \# {\mathcal W}_{\mathfrak A}^{\pm} = \(\frac{1}{2} + O(p^{-\delta})\) \# {\mathcal W}_{\mathfrak A} $$ uniformly over $n$ and $p$. \end{theorem}
\begin{proof}Without loss of generality, we can assume that $\varepsilon \le 1/2$. We also set $$ n_0(\varepsilon) = \rf{4\varepsilon^{-1}}. $$
We first consider the case when \begin{equation} \label{eq:large n} n > n_0(\varepsilon). \end{equation}
Assuming that~\eqref{eq:large n} holds, we set $$ m = \rf{\frac{1 + \varepsilon}{1+2\varepsilon} n}. $$ By choosing ${\mathcal I}$ such that
the sets ${\mathcal A}_i$ with $i \in {\mathcal I}$ are the $m$ sets of largest cardinality we see that \begin{equation} \label{eq:large Ai}
\prod_{i\in{\mathcal I} }\# {\mathcal A}_i \ge
\(\prod_{1 \le i \le n}\# {\mathcal A}_i\)^{m/n} . \end{equation} We then define ${\mathcal J} = \{1, \ldots, n\} \setminus {\mathcal I}$ and \begin{equation*} \begin{split} {\mathcal A} & = \left\{\sum_{i\in {\mathcal I}} a_i \omega_i~:~ a_i \in {\mathcal A}_i, \ i\in {\mathcal I}\right\}, \\ {\mathcal B} & = \left\{\sum_{i \in {\mathcal J}} a_i \omega_i~:~ a_i \in {\mathcal A}_i, \ i\in {\mathcal J}\right\}. \end{split} \end{equation*} Thus, recalling~\eqref{eq:DES Cond1} and~\eqref{eq:large Ai}, we see that \begin{equation} \label{eq:Card A} \begin{split} \# {\mathcal A} \ge \(p^{(1/2 + \varepsilon)n^2/(n-1)}\)^{m/n} \ge \(p^{(1/2 + \varepsilon)n}\)^{m/n}\ge
p^{(1/2 + \varepsilon)m} \ge p^{(1/2 + \varepsilon/2)n} . \end{split} \end{equation} Furthermore for $n > n_0(\varepsilon)$ we have \begin{equation*} \begin{split} \#{\mathcal J} & = n-m \ge n - \frac{1 + \varepsilon}{1+2\varepsilon} n -1 = n - \frac{(1 + \varepsilon)n + 1+2\varepsilon}{1+2\varepsilon} \\ &\ge n - \frac{(1 + \varepsilon)n + 2}{1+2\varepsilon} = \frac{ \varepsilon n - 2}{1+2\varepsilon} \ge
\frac{ \varepsilon n - 2}{2} \ge \varepsilon n/4. \end{split} \end{equation*} Therefore, recalling~\eqref{eq:DES Cond2} again, we see that \begin{equation} \label{eq:Card B} \begin{split} \# {\mathcal B} \ge p^{\varepsilon^2 n/4}. \end{split} \end{equation} As $\{\omega_1, \ldots, \omega_n\}$ is a basis, every element $w = {\mathcal W}_{\mathfrak A}$ has a unique representation $w =a + b$ with $a \in {\mathcal A}$, $b \in {\mathcal B}$. Hence for any multiplicative character $\chi$ of $\mathbb{F}_r$ we have $$ \sum_{w\in {\mathcal W}_{\mathfrak A}} \chi(w) = \sum_{a\in {\mathcal A}}\sum_{b \in {\mathcal B}} \chi(a+b) . $$ Now using the bounds~\eqref{eq:Card A} and applying~\eqref{eq:Card B} and applying Corollary~\ref{cor:DoubleSums} with $\eta = \varepsilon^2/4$, we obtain \begin{equation} \label{eq:Bound W_A} \sum_{w\in {\mathcal W}_{\mathfrak A}} \chi(w) \ll \#{\mathcal A} \# {\mathcal B} p^{-\delta} = \#{\mathcal W}_{\mathfrak A} p^{-\delta} \end{equation} for any non-trivial multiplicative character $\chi$ of $\mathbb{F}_{p^n}$ where $\delta > 0$ depends only on $\varepsilon$. Thus, taking the quadratic character $\chi$, we obtain the desired result if the inequality~\eqref{eq:large n} holds.
Now, for small values of $n$, for which inequality~\eqref{eq:large n} fails, we simply take $m = n-1$ and choose ${\mathcal I}$ as before, to satisfy~\eqref{eq:large Ai}. Hence, instead of~\eqref{eq:Card A} we have $$ \# {\mathcal A} \ge p^{(1/2 + \varepsilon)n} $$ and also trivially, we have $$ \#{\mathcal B} \ge p^{ \varepsilon} \ge p^{\varepsilon n/n_0(\varepsilon)}. $$ Applying Corollary~\ref{cor:DoubleSums} with $\eta =\varepsilon/n_0(\varepsilon)$, we obtain the bound~\eqref{eq:Bound W_A} again, which concludes the proof. \end{proof}
\section{Comments}
We remark that motivated by a question of Erd{\H o}s, Mauduit and S{\'a}rk{\"o}zy~\cite[Problem~5]{EMS}, Banks and Shparlinski~\cite{BaSh1} have studied the average value of $\varphi(s)/s$ over integers with some digital restrictions, different from that of Theorem~\ref{thm:AverEuler}, see also~\cite{BaSh2}. As in~\cite{BaSh2}, one can also study the average value of $\sigma(s)/s$ for the sums of divisors function and obtain a full analogue of Theorem~\ref{thm:AverEuler} for this function. Furthermore, our argument can be used to give an asymptotic formula for the number of pairs $(s,r)$ with $r,s \in {\mathcal N}_n(\vec{d})$ such that $\gcd(s,r)=1$.
Clearly, the bound~\eqref{eq:Bound W_A} also allows to study the distribution of primitive roots in the set ${\mathcal W}_{\mathfrak A}$. Finally, we note that the case when the sets ${\mathcal A}_1, \ldots, {\mathcal A}_n$ are sets of consecutive residues corresponds to the settings of~\cite[Theorem~3.1]{DMS}. In this case, one can use the recent generalisations of the Burgess bounds that are due to Chang~\cite{Chang1,Chang2} and Konyagin~\cite{Kon2} (when $n$ is bounded) together with a classical bound of Davenport and Lewis~\cite{DaLe} (when $n \to \infty$) and improve the result of~\cite[Theorem~3.1]{DMS}.
\end{document} | arXiv |
\begin{document}
\title{A quantitative sharpening of Moriwaki's arithmetic Bogomolov inequality} \author{N. Naumann} \date{\ } \maketitle
A. Moriwaki proved the following arithmetic analogue of the Bogomolov unstability theorem. If a torsion-free hermitian coherent sheaf on an arithmetic surface has negative discriminant then it admits an arithmetically destabilising subsheaf. In the geometric situation it is known that such a subsheaf can be found subject to an additional numerical constraint and here we prove the arithmetic analogue. We then apply this result to slightly simplify a part of C. Soul\'e's proof of a vanishing theorem on arithmetic surfaces.
\section{Introduction and statement of result}
Let $K$ be a number field with ring of integers ${\mathcal O}_K$ and $X / \mathrm{Spec}\, ({\mathcal O}_K)$ an arithmetic surface, i.e. a regular, integral, purely two-dimensional scheme, proper and flat over $\mathrm{Spec}\, ({\mathcal O}_K)$ and with smooth and geometrically connected generic fibre. Attached to a hermitian coherent sheaf on $X$ are the usual characteristic classes with values in the arithmetic Chow-groups $\widehat{CH}^i (X)$ (cf. \cite{GS1}, 2.5), and in particular the discriminant of $\overline{E}$ \[ \Delta (\overline{E}) := (1 - r) \hat{c}_1 (\overline{E})^2 + 2r \hat{c}_2 (\overline{E}) \in \widehat{CH}^2 (X) \] where $r := \mathrm{rk} (E)$. The arithmetic degree map \[ \widehat{\deg}: \widehat{CH}^2(X)_{{\mathbb{R}}}\longrightarrow {\mathbb{R}} \] is an isomorphism \cite{GS2} and we will use the same symbol to to denote an element in $\widehat{CH}^2(X)_{{\mathbb{R}}}$ and its arithmetic degree in ${\mathbb{R}}$, see \cite{GS2}, 1.1 for the definition of arithmetic Chow-groups with real coefficients $\widehat{CH}^*(X)_{{\mathbb{R}}}$. Following \cite{Mo2} we define the positive cone of $X$ to be
\[ \hat{C}_{++}(X) := \{ x \in \widehat{CH}^1 (X)_{{\mathbb{R}}} \, | \, x^2 > 0 \; \mbox{and} \; \deg_K (x) > 0 \} \; . \]
Given a torsion-free hermitian coherent sheaf $\overline{E}$ of rank $r\ge 1$ on $X$ and a subsheaf $E'\subseteq E$ we endow $E'$ with the metric induced from $\overline{E}$ and consider the difference of slopes \[ \xi_{\overline{E}',\overline{E}} := \frac{\hat{c}_1 (\overline{E}')}{\mathrm{rk} (E')} - \frac{\hat{c}_1 (\overline{E})}{r} \in \widehat{CH}^1 (X)_{{\mathbb{R}}}. \]
Recall that a subsheaf $E'\subseteq E$ is {\em saturated} if the quotient $E/E'$ is torsion-free. Our main result is the following.
\begin{theorem} \label{theorem} Let $\overline{E}$ be a torsion-free hermitian coherent sheaf of rank $r\ge 2$ on the arithmetic surface $X$, satisfying \[ \Delta (\overline{E}) < 0 \; . \] Then there is a non-zero saturated subsheaf $\overline{E}' \subseteq \overline{E}$ such that $\xi_{\overline{E}' , \overline{E}} \in \hat{C}_{++} (X)$ and \begin{equation}\label{ineq} \xi^2_{\overline{E}',\overline{E}} \ge \frac{-\Delta}{r^2 (r-1)} \; . \end{equation}
\end{theorem}
\begin{remark} The existence of an $\overline{E}'\subseteq \overline{E}$ with $\xi_{\overline{E}' , \overline{E}} \in \hat{C}_{++} (X)$ is the main result of \cite{Mo2} and means that $\overline{E}'\subseteq \overline{E}$ is arithmetically destabilising with respect to any polarisation of $X$, c.f. {\em loc. cit.} for more details on this. The new contribution here is the inequality (\ref{ineq}) which is the exact arithmetic analogue of a known geometric result, c.f. for example \cite{HL}, Theorem 7.3.4. \end{remark} \begin{remark} A special case of Theorem \ref{theorem} appears in disguised form in the proof of \cite{So}, Theorem 2: Given a sufficiently positive hermitian line bundle $\overline{L}$ on the arithmetic surface $X$ and some non-torsion element $e \in \mathrm{H} ^1(X,L^{-1})\simeq \mathrm{Ext} ^1(L,{\mathcal O}_X)$, C. Soul\'e establishes a lower bound for
\[ ||e||^2:=\sup_{\sigma: K\hookrightarrow {\mathbb{C}}} \; ||\sigma(e)||^2_{L^2} \]
by considering the extension determined by $e$
\[ {\mathcal E}: 0 \longrightarrow \overline{{\mathcal O}_X} \longrightarrow \overline{E} \longrightarrow \overline{L} \longrightarrow 0 \]
and suitably metrised as to have $\hat{c}_1(\overline{E})=
\overline{L}$ and $2\hat{c}_2(\overline{E})=\sum_{\sigma} ||\sigma(e)||^2_{L^2}$, hence $\Delta(\overline{E})=-\overline{L}^2+2 \sum_{\sigma} ||\sigma(e)||^2_{L^2}$ (where we write $\overline{L}=\hat{c}_1(\overline{L})$ following the notation of {\em loc. cit.}).\\ If $E_{\overline{{\mathbb{Q}}}}$ is semi-stable the arithmetic Bogomolov inequality concludes the proof. Otherwise, the main point is to show the existence of of an arithmetic divisor $\overline{D}$ satisfying
\begin{eqnarray} \deg_K(\overline{D}) & \le & \deg_K(\overline{L})/2 \mbox{ and}\label{eins}\\
2(\overline{L}-\overline{D})\overline{D} & \leq & [K:{\mathbb{Q}}]\cdot ||e||^2, \label{zwei} \end{eqnarray}
c.f. $(28)$ and $(32)$ of {\em loc. cit.} where these inequalities are established by some direct argument. We wish to point out that the existence of some $\overline{D}$ satisfying (\ref{eins}) and (\ref{zwei}) is a special case of Theorem \ref{theorem}. In fact, let $\overline{E}'\subseteq \overline{E}$ be as in Theorem \ref{theorem} and define $\overline{D}:=\overline{L}-\hat{c}_1(\overline{E'})$. We then compute \[ \xi_{\overline{E}',\overline{E}}=\frac{\overline{L}}{2}-\overline{D} \] and $\xi_{\overline{E}',\overline{E}}\in \hat{C}_{++}(X)$ implies (\ref{eins}). Furthermore, the inequality (\ref{ineq}) in the present case reads
\[ \xi_{\overline{E}',\overline{E}}^2=\frac{\overline{L}^2}{4}+\overline{D}^2-\overline{L}\;\overline{D}\ge\frac{-\Delta}{4}=\frac{\overline{L}^2}{4}-\frac{1}{2}
\sum_{\sigma} ||\sigma(e)||^2_{L^2}\mbox{ , i.e.}\]
\[ 2(\overline{L}-\overline{D})\overline{D}\leq \sum_{\sigma} ||\sigma(e)||^2_{L^2},\]
hence the trivial estimate $[K:{\mathbb{Q}}]\cdot ||e||^2\geq
\sum_{\sigma} ||\sigma(e)||^2_{L^2}$ gives (\ref{zwei}). \end{remark}
I would like to thank K. K\"unnemann for useful conversations about a preliminary draft of the present note.
\section{Proof of Theorem \ref{theorem}}
We collect some lemmas first. We call a short exact sequence \[ {\mathcal E} : 0 \longrightarrow \overline{E}' \longrightarrow \overline{E} \longrightarrow \overline{E}'' \longrightarrow 0 \] of hermitian coherent sheaves on $X$ {\em isometric} if the metrics on $E'$ and $E''$ are induced from the one on $E$. This implies that $\hat{c}_1(\overline{E})=\hat{c}_1(\overline{E}')+\hat{c}_1(\overline{E}'')$ (i.e. $\tilde{c}_1({\mathcal E})=0$). We also have \[ \hat{c}_2 (\overline{E}) = \hat{c}_2 (\overline{E}' \oplus \overline{E}'') - a (\tilde{c}_2 ({\mathcal E})) \quad \mbox{in} \; \widehat{CH}^2 (X) \; , \]
where \[ a : \tilde{A}^{1,1} (X_{{\mathbb{R}}}) \longrightarrow \widehat{CH}^2 (X) \] is the usual map \cite{SABK}, chapter III.
\begin{lemma} \label{1}
If \[ {\mathcal E} : 0 \longrightarrow \overline{E}' \longrightarrow \overline{E} \longrightarrow \overline{E}'' \longrightarrow 0 \] is an isometric short exact sequence of hermitian coherent sheaves on $X$ with ranks $r' , r , r''\ge 1$ and discriminants $\Delta' , \Delta , \Delta''$, then \[ \frac{\Delta'}{r'} + \frac{\Delta''}{r''} - \frac{\Delta}{r} = \frac{rr'}{r''} \xi^2_{\overline{E}',\overline{E}} + 2a (\tilde{c}_2 ({\mathcal E})) \quad \mbox{in} \; \widehat{CH}^2 (X)_{{\mathbb{R}}} \; . \] \end{lemma}
\begin{proof} We omit the computation using the formulas for $\hat{c}_i(\overline{E})$ recalled above which shows that the left hand side of the stated equality equals
\[ \hat{c}_1(\overline{E})^2\left( \frac{r-1}{r}+\frac{1-r'}{r'}\right)+\hat{c}_1(\overline{E}'')^2\left( \frac{r-1}{r}+\frac{1-r''}{r''}\right)+ \] \[ + \hat{c}_1(\overline{E}')\hat{c}_1(\overline{E}'')\left( \frac{2(r-1)}{r}-2 \right) + 2a(\tilde{c}({\mathcal E})).\]
Similarly one writes $\xi_{\overline{E}',\overline{E}}^2$ as a rational linear combination of $\hat{c}_1(\overline{E})^2,\hat{c}_1(\overline{E}'')^2$ and $\hat{c}_1(\overline{E}')\hat{c}_1(\overline{E}'')$ and comparing the results, the stated formula drops out. \end{proof}
\begin{lemma} \label{2}
For ${\mathcal E}$ as in Lemma \ref{1} and $\overline{G}'' \subseteq \overline{E}''$ a saturated subsheaf of rank $s\ge 1$ carrying the induced metric, put \[ \overline{G} := \ker (E \longrightarrow E'' \longrightarrow E'' / G'') \subseteq \overline{E} \] with the induced metric. Then \[ \xi_{\overline{G},\overline{E}} = \frac{r' (r'' - s)}{(r' + s) r''} \xi_{\overline{E}',\overline{E}} + \frac{s}{r'+s} \xi_{\overline{G}'' , \overline{E}''} \quad \mbox{in} \; \widehat{CH}^1 (X)_{{\mathbb{R}}} \; . \] \end{lemma}
Observe that the coefficients in the last expression are non-negative rational numbers.
\begin{proof} We have a commutative diagram with exact rows and columns
\[ \xymatrix{ & & 0 & 0 & \\
& & \overline{E/G} \ar[u]\ar^{\simeq}[r] & \overline{E''/G''} \ar[u] \\ {\mathcal E}:0 \ar[r] & \overline{E}' \ar[r] & \overline{E} \ar[u]\ar[r] & \overline{E}'' \ar[r] \ar[u] & 0 \\ 0 \ar[r] & \overline{H} \ar^{\simeq}[u] \ar[r] & \overline{G} \ar[r]\ar[u] & \overline{G}''\ar[u]\ar[r] & 0 \\
& & 0 \ar[u] & 0. \ar[u] & } \]
Here, we have endowed $E/G,E''/G''$ and $H$ with the metrics induced from $\overline{E}$,$\overline{E}''$ and $\overline{G}$, hence all rows and columns are isometric by definition. A minor point to note is that with this choice of metrics the two indicated isomorphisms are isometric, indeed this only means that taking sub- (resp. quotient-)metrics is transitive. One has \[ \xi_{\overline{E}',\overline{E}}=\frac{r''\hat{c}_1(\overline{E}')-r'\hat{c}_1(\overline{E}'')}{r'r} \] and analogously for any isometric exact sequence in place of ${\mathcal E}$. Using this and the diagram one writes both sides of the stated equality as a ${\mathbb{Q}}$-linear combination of $\hat{c}_1(\overline{E}'),\hat{c}_1(\overline{G}'')$ and $\hat{c}_1(\overline{E''/G''})$ to obtain the same result, namely
\[ \frac{r''-s}{(r'+s)r}\hat{c}_1(\overline{E}')+ \frac{r''-s}{(r'+s)r}\hat{c}_1(\overline{G}'')-\frac{1}{r}\hat{c}_1(\overline{E''/G''}).\]
\end{proof}
Finally, we will need the following observation about the intersection theory on $X$ where, for $x \in \hat{C}_{++} (X)$, we write $|x| := (x^2)^{1/2} \in {\mathbb{R}}^+$.
\begin{lemma}\label{cone}
The subset $\hat{C}_{++}(X)\subseteq\widehat{CH}^1(X)_{{\mathbb{R}}}$ is an open cone, i.e. $x,y\in \hat{C}_{++}(X)$ and $\lambda\in{\mathbb{R}}^+$ implies that $x+y,\lambda x\in\hat{C}_{++}(X)$. For $x,y\in\hat{C}_{++}$ we have
$|x+y|\ge |x|+|y|$. \end{lemma}
\begin{proof} This is \cite{Mo2}, (1.1.2.2) except for the final assertion which is obvious if $x\in{\mathbb{R}} y$ and we can thus assume that $V:={\mathbb{R}} x+{\mathbb{R}} y\subseteq\widehat{CH}^1(X)_{{\mathbb{R}}}$ is two-dimensional. We claim that the restriction of the intersection-pairing makes $V$ a real quadratic space of type $(1,-1)$. As we have $x\in V$ and $x^2>0$ we only have to exhibit some $v\in V$ with $v^2<0$. To achieve this let $h\in\widehat{CH}^1(X)_{{\mathbb{R}}}$ be the first arithmetic Chern class of some sufficiently positive hermitian line bundle on $X$ such that the arithmetic Hodge index theorem holds for the Lefschetz operator defined by $h$, c.f. \cite{GS2}, Theorem 2.1, ii). Then $a:=xh$ (resp. $b:=yh$) are non-zero real numbers for otherwise we would have $x^2<0$ (resp. $y^2<0$). Thus $v:=\frac{x}{a}-\frac{y}{b}\in V$ satisfies $v\neq 0$ and $vh=0$ , hence $v^2<0$.\\ Fix a basis $e,f\in V$ with $e^2=1, f^2=-1$ and write \[ x=\alpha e + \beta f\mbox{ and}\] \[ y=\gamma e+\delta f.\]
To show that $|x+y|\ge |x|+|y|$ we can assume, changing both the signs of $x$ and $y$ if necessary, that $\alpha>0$. We then claim that $\gamma>0$. For otherwise there would be $\lambda_1,\lambda_2\in{\mathbb{R}}^+$ such that $v:=\lambda_1x+\lambda_2 y$ would have $e$- coordinate equal to zero, hence $v^2\leq 0$ contradicting the fact that either $-v$ or $v$ lies in $\hat{C}_{++}(X)$ (depending on whether or not we changed the signs of $x$ and $y$ above).\\
From $x^2=\alpha^2-\beta^2$, $y^2=\gamma^2-\delta^2>0$ we obtain $\alpha=|\alpha|\ge |\beta|$
and $\gamma=|\gamma|\ge|\delta|$ and then $\alpha\gamma\ge |\beta\delta| \ge \beta\delta$, i.e.
\begin{equation}\label{sternchen} xy=\alpha\gamma-\beta\delta\ge 0. \end{equation}
To conclude, we use the following chain of equivalent statements
\[ |x+y|\ge |x|+|y| \Leftrightarrow \]
\[ (x+y)^2-(|x|+|y|)^2\ge 0 \Leftrightarrow \]
\[ 2xy-2|x||y|\ge 0 \Leftrightarrow \]
\[ xy\ge |x||y| \stackrel{(\ref{sternchen})}{\Leftrightarrow} \]
\[ (xy)^2\ge |x|^2|y|^2 \Leftrightarrow \] \[ (\alpha\gamma-\beta\delta)^2\ge (\alpha^2-\beta^2)(\gamma^2-\delta^2) \Leftrightarrow \] \[ \alpha^2\gamma^2+\beta^2\delta^2-2\alpha\beta\gamma\delta\geq \alpha^2\gamma^2-\alpha^2\delta^2-\beta^2\gamma^2+\beta^2\delta^2 \Leftrightarrow \] \[ 2\alpha\beta\gamma\delta\leq \alpha^2\delta^2+\beta^2\gamma^2 \Leftrightarrow \] \[ 0\leq (\alpha\delta-\beta\gamma)^2. \]
\end{proof}
\begin{proofof} Theorem \ref{theorem}. We first remark that for a torsion-free hermitian coherent sheaf $\overline{F}$ of rank one on $X$ we always have $\Delta(\overline{F})\ge 0$. In fact, \[ F \simeq {\mathcal L} \otimes {\mathcal I}_Z \] for some line-bundle ${\mathcal L}$ and ${\mathcal I}_Z$ the ideal sheaf of some closed subscheme $Z \subseteq X$ of codimension $2$. This becomes an isometry for the trivial metric on ${\mathcal I}_Z$ and a suitable metric on ${\mathcal L}$ (since ${\mathcal I}_Z$ is trivial on the generic fibre of $X$). Then \[ \Delta (\overline{F}) = 2 \hat{c}_2 (\overline{{\mathcal L}} \otimes {\mathcal I}_Z) = 2 \hat{c}_2 ({\mathcal I}_Z) = 2 \, \mbox{length} (Z) \ge 0 \; . \] By the main result of \cite{Mo2}, there is $0 \neq \overline{E}' \subseteq \overline{E}$ saturated such that $\xi_{\overline{E}',\overline{E}} \in \hat{C}_{++} (X)$. We can assume that, as $E'$ varies through these subsheaves, the real numbers $\xi^2_{\overline{E}',\overline{E}}$ remain bounded for otherwise there is nothing to prove. So we can choose $0 \neq \overline{E}' \subseteq \overline{E}$ saturated with $\xi_{\overline{E}' , \overline{E}} \in \hat{C}_{++} (X)$ and $\xi^2_{\overline{E}',\overline{E}}$ maximal subject to these conditions. Put $E'' := E / E'$ and consider the isometric exact sequence \[ {\mathcal E} : 0 \longrightarrow \overline{E}' \longrightarrow \overline{E} \longrightarrow \overline{E}'' \longrightarrow 0 \] with discriminants $\Delta' , \Delta , \Delta''$ and ranks $r' , r, r''$. We claim that $\Delta'\ge 0$. This is clear in case $r=2$ from the remark made at the beginning of the proof. In case $r\ge 3$ we assume that $\Delta'<0$ and we let $\overline{G} \subseteq \overline{E}'$ be a saturated subsheaf with $\xi_{\overline{G},\overline{E}'} \in \hat{C}_{++}$. Then $\overline{G} \subseteq \overline{E}$ is saturated and using lemma \ref{cone} we get \[
|\xi_{\overline{G},\overline{E}}| = |\xi_{\overline{G},\overline{E}'} + \xi_{\overline{E}',\overline{E}}| \ge |\xi_{\overline{G},\overline{E}'}| + |\xi_{\overline{E}',\overline{E}}| > |\xi_{\overline{E}',\overline{E}}| \]
contradicting the maximality of $|\xi_{\overline{E}',\overline{E}}|$. So we have indeed $\Delta' \ge 0$. Assume now, contrary to our assertion, that \begin{equation}
\label{eq:1}
\frac{\Delta}{r} < -r (r-1) \xi^2_{\overline{E}',\overline{E}} \; . \end{equation} Then from Lemma \ref{1}, $\Delta' \ge 0$, (\ref{eq:1}) and $\tilde{c}_2 ({\mathcal E}) \le 0$ (\cite{Mo1}, 7.2) we get \begin{gather}
\frac{\Delta''}{r''} \le \frac{\Delta}{r} + \frac{rr'}{r''} \xi^2_{\overline{E}',\overline{E}} < \left( -r (r-1) + \frac{rr'}{r''} \right) \xi^2_{\overline{E}',\overline{E}} \notag \\ = - r^2 \frac{r''-1}{r''} \xi^2_{\overline{E}',\overline{E}} \le 0 \; , \notag \end{gather} hence $\Delta'' < 0$. By induction, there is $0 \neq \overline{G}'' \subseteq \overline{E}''$ saturated with $\xi_{\overline{G}'', \overline{E}''} \in \hat{C}_{++} (X)$ and \begin{equation}\label{nochnsternchen} \xi^2_{\overline{G}'',\overline{E}''} \ge \frac{-\Delta''}{r^{''2} (r'' - 1)} > \frac{r^2}{r^{''2}} \xi^2_{\overline{E}',\overline{E}} \; . \end{equation} Clearly $\overline{G} := \ker (E \to E'' / G'') \subseteq \overline{E}$ is saturated and from Lemma \ref{2}, the positivity of the coefficients appearing there and lemma \ref{cone} we get
\begin{eqnarray*}
|\xi_{\overline{G},\overline{E}}| & \ge & \frac{r' (r'' -s)}{(r' + s) r''} |\xi_{\overline{E}',\overline{E}}| + \frac{s}{r' + s} |\xi_{\overline{G}'',\overline{E}''}| \\
& \stackrel{(\ref{nochnsternchen})}{>} & \frac{r' (r'' - s)}{(r'+s) r''} |\xi_{\overline{E}' , \overline{E}}| + \frac{s}{r'+s} \frac{r}{r''} |\xi_{\overline{E}',\overline{E}}| \\
& = & \left( \frac{r' (r'' -s) + rs}{r'' (r' + s)} \right) |\xi_{\overline{E}',\overline{E}}| = |\xi_{\overline{E}',\overline{E}}| \; . \end{eqnarray*}
This again contradicts the maximality of $|\xi_{\overline{E}',\overline{E}}|$ and concludes the proof. \end{proofof}
\end{document} | arXiv |
August 2017 , Volume 45, Issue 8, pp 1852–1864 | Cite as
Design, Analysis and Testing of a Novel Mitral Valve for Transcatheter Implantation
Selim Bozkurt
Georgia L. Preston-Maher
Ryo Torii
Gaetano Burriesci
First Online: 03 April 2017
Mitral regurgitation is a common mitral valve dysfunction which may lead to heart failure. Because of the rapid aging of the population, conventional surgical repair and replacement of the pathological valve are often unsuitable for about half of symptomatic patients, who are judged high-risk. Transcatheter valve implantation could represent an effective solution. However, currently available aortic valve devices are inapt for the mitral position. This paper presents the design, development and hydrodynamic assessment of a novel bi-leaflet mitral valve suitable for transcatheter implantation. The device consists of two leaflets and a sealing component made from bovine pericardium, supported by a self-expanding wireframe made from superelastic NiTi alloy. A parametric design procedure based on numerical simulations was implemented to identify design parameters providing acceptable stress levels and maximum coaptation area for the leaflets. The wireframe was designed to host the leaflets and was optimised numerically to minimise the stresses for crimping in an 8 mm sheath for percutaneous delivery. Prototypes were built and their hydrodynamic performances were tested on a cardiac pulse duplicator, in compliance with the ISO5840-3:2013 standard. The numerical results and hydrodynamic tests show the feasibility of the device to be adopted as a transcatheter valve implant for treating mitral regurgitation.
Transcatheter mitral valve implantation (TMVI) Heart valve development Heart valve assessment Mitral valve Bioprosthetic bi-leaflet valve
Associate Editor Umberto Morbiducci oversaw the review of this article.
Selim Bozkurt and Georgia L. Preston-Maher share first authorship.
Mitral regurgitation is one of the major mitral valve pathologies leading to heart failure.27 It is a result of primary anatomical changes affecting the mitral valve leaflets, or left ventricular remodelling which may lead to dislocation of papillary muscles.15 Although mild and moderate mitral regurgitation may be tolerated and do not require surgical intervention, patients with severe symptomatic mitral regurgitation have a very low survival rate in absence of interventions40 which restore the coaptation of the mitral valve leaflets,11 or replace the mitral valve with a prosthetic device.30 While non-randomised reports suggest that repairing techniques have significantly lower mortality rates,54 randomised studies indicate no significant difference in the mortality rates3 between replacement and repair20 in ischemic related mitral regurgitation. Whenever practicable, surgical repair remains the best option for the treatment of degenerative mitral regurgitation.19,20 Nevertheless, in elderly patients surgical intervention is often associated with comorbidities such as diabetes, pulmonary disease, perioperative hemodialysis and low ejection fraction, which increase considerably the risk of operative mortality.5,49 As a result, only a small portion of patients suffering from functional mitral regurgitation and approximately half of those suffering from degenerative mitral regurgitation currently undergo surgery.7 Minimally invasive transcatheter implantation can reduce the risks in these patients and offer an alternative to surgical therapies for mitral valve diseases.34
Transcatheter techniques to treat mitral regurgitation can be classified as leaflet and chordae repair; indirect annuloplasty; left ventricular remodelling; and replacement.25 Leaflet and chordae repair techniques can be effective and durable in a wide variety of pathologies, even without annuloplasty in selected patients.21,36 Indirect annuloplasty releases devices which support remodelling of the annulus in the coronary sinus, improving leaflet coaptation. However this procedure is associated with adverse cardiovascular events, such as myocardial infarction and coronary sinus rupture,24,47 and data available on the short- and long-term outcome are still limited.32,37 Left ventricular remodelling is applied to reduce a dilated left ventricle diameter which may tether the mitral valve leaflets.22 Despite the initial attempts demonstrated benefits, this technique is not available commercially at the moment.
Although these transcatheter techniques can successfully reduce mitral regurgitation, a valve replacement would allow to restore the unidirectional blood flow in a wider patients' anatomical selection. Transcatheter mitral valve (TMV) replacements, which attempt to conjugate the lessons from surgical mitral valve interventions35,42 with the successful transcatheter aortic valve (TAV) experience, are still in developmental stages. A number of TMVs have been proposed, and are at different stages of evaluation.1,23,41 These are typically adapted from TAVs,41 and adopt the same three leaflets circular configuration. Possible issues that may arise with these devices include suboptimal placement in native mitral position, due to the irregular non-circular shape of the mitral annulus, and recurrence of paravalvular leakage.30 This is known to reduce the survival rates after TAV replacement, and is a more critical problem for mitral valve implants, where the implantation sizes and the peak transvalvular pressures are higher.25
In this paper, a novel mitral valve device suitable for transcatheter implantation, based on a bi-leaflet configuration with D-shaped orifice, is presented. In particular, the development of the proposed valve, in terms of design optimisation and in vivo hydrodynamic assessment is described.
Leaflet Design Optimisation and Manufacturing
Leaflets were designed to minimise structural and functional failure. Structural failure typically occurs due to excessive stresses, with the locations of structural failure in explanted bioprosthetic heart valves often associated with the peak regions of maximum principal stress.9 Design optimisation was performed using parametrically-varied CAD models by means of finite element analysis for both structural and functional criteria.
Leaflets were designed to lie, in their unstressed open configuration, on a ruled surface characterised by a D-shaped orifice cross section with a ratio between the antero-posterior and the inter-commissural diameters equal to 3:4 (Fig. 1). Similarly to healthy native mitral valve,58 leaflets were designed with a conical shape, reducing their cross section linearly form the inlet to the outlet. This solution was preferred to minimise the risk of ventricular outflow tract obstruction, by decreasing the tendency of the leaflets to diverge from their design configuration, especially when the valve is placed in annuli significantly smaller than the nominal valve dimension. Also, shorter free edges were observed to reduce the leaflets fluttering during diastole, which is typically associated with increased calcification, haemolysis, regurgitation and early fatigue failure.6 A scale factor (SF), defined as the ratio between the outlet (D V ) and inlet (D A ) intertrigonal dimensions of the device (Fig. 1a), was introduced to quantify the leaflets conicity in the free unloaded configuration. A set of five scale factors of 0.745, 0.798, 0.852, 0.906 and 0.960 were chosen for investigation, with the smallest corresponding to a maximum reduction of the D-shape cross sectional area from the base to the edge of the leaflets equal to 60%. A coaptation height parameter, CH, was defined, referring to the vertical distance from the arris between the aortic and mural leaflets to the middle of the leaflets free edge. This has the function to allow the adjustment of the leaflets edge and avoid excess of redundant material, which results in localised buckling, commonly associated with failure of pericardial leaflets.50 Five evenly spaced coaptation lengths were chosen for investigation, from 0 to 30% of the leaflets height. The combination of five scale factors and coaptation lengths resulted in twenty-five incrementally different bi-leaflet CAD models.
(a) Sketch of the leaflets design: CH represents the coaptation height, D V and D A are the dimensions used to define scale factor (SF) in the design. (b) Experimental data points describing the constitutive behavior of the used pericardium, and fitted curve with the adopted Ogden model.
The leaflets were designed in their assembled configuration as surfaces using 3D CAD software Rhinoceros 4.0 (Robert McNeel & Associates), using an inter-trigonal dimension equal to 26 mm. Numerical analyses of structural mechanics were performed using an explicit solver in LS-DYNA (Livermore Software Technology Corporation). The analysis of the twenty-five initial designs provided coaptation area and peak maximum principal stress data for hypertensive systolic loading conditions, i.e. when they are fully closed and a peak of transmitral pressure equal to 200 mmHg is applied.
Glutaraldehyde fixed bovine pericardium was selected as material for the leaflets, due to its long clinical use in bioprosthetic heart valves and favorable hemodynamic performance.26 Calf pericardial sacs were obtained from a local abattoir, and fixed in a 0.5% solution of glutaraldehyde for 48 h, after removing the fat and parietal pericardium by hand.26 Three sets of leaflets were obtained from visually homogeneous regions of the pericardial sac of thickness in the range of 400 μm ±10% (measured using a thickness gauge - Mitutoyo Corporation, Tokyo, Japan). One dumbbell-shaped sample of 4 mm width and 16 mm gauge length was extracted from the unused portion of each patch, using a die cutter.
Specimens were conditioned with uniaxial tensile cycles from 0 to 6 N with 20 mm/min rate until stabilisation, using a ZwickiLine testing machine (Zwick/Roell, Germany) equipped with a media container maintaining 40 °C, and used to determine the representative mechanical properties for the used material. The constitutive behaviour observed for the treated pericardium was modeled in the numerical analyses using a four parameters Ogden equation:
$$W = \frac{{\mu_{1} }}{{\alpha_{1} }}\left( {\lambda_{1}^{{\alpha_{1} }} + \lambda_{2}^{{\alpha_{1} }} + \lambda_{3}^{{\alpha_{1} }} - 3} \right) + \frac{{\mu_{2} }}{{\alpha_{2} }}\left( {\lambda_{1}^{{\alpha_{2} }} + \lambda_{2}^{{\alpha_{2} }} + \lambda_{3}^{{\alpha_{2} }} - 3} \right)$$
where the strain energy density W is expressed in terms of the principal stretches λ 1, λ 2 and λ 3, and the four material constants μ 1, μ 2, α 1 and α 2. The material constants best fitting the average stress–strain curve obtained from the experiments were: μ 1 = 7.6 × 10−6; μ 2 = 5.7 × 10−4; α 1 = α 2 = 26.26 (R 2 = 0.981). The experimental data points and fitted curve are reported in the graph in Fig. 1b.
The coaptation of the leaflets was modelled using a frictionless master-slave contact condition.9 The effect of the inertia of blood in reducing system oscillations was reproduced by using a damping coefficient of 0.9965, consistent with what identified in previous works based on similar simulations.9 Each leaflet was discretised with approximately 1820 quadrilateral 2D constant strain Belytschko-Lin-Tsay shell elements with 5 points of integration across the thickness. The leaflet thickness was set to 0.4 mm, approximating the value selected for the patches used for the valve manufacturing. To simulate leaflet closure, a uniformly distributed opening pressure of 4 mmHg was initially applied to the leaflets, starting from their unloaded position, and then reverted and ramped to a closing pressure of 115 mmHg. This corresponds to the typical mean transmitral systolic pressure difference obtained by testing the valve prototypes in the pulse duplicator, for a cardiac output of 5 L/min, a frequency of 70 beats per minute (with 65% of diastolic time) and a normotensive aortic pressure of 100 mmHg. A minimum safety factor of 3, based on the strength reported for glutaraldehyde fixed bovine pericardial tissue,4 was accepted for the predicted peaks of stress.
Frame Design and Optimisation
The TMV frame is designed to match and support the two leaflets along their constrained edge and provide their anchoring. Its structure is obtained from super elastic NiTi wires of 0.6 mm diameter.
The valve anchoring to the host anatomy is provided by the counteracting action from a set of proximal smoothly arched ribs, expanding into the atrium (portions 7 and 8 in Fig. 2a) and two petal-like structures protruding into the ventricle between the native mitral leaflets (portions 3 and 4 in Fig. 2a). The portion of the petals engaging with the anterior native leaflets (portions 4 in Fig. 2a) are designed to keep this in tension by expanding its anterioro-lateral and posterior-medial parts12 laterally, in the attempt to reduce its systolic motion without pushing it markedly in subaortic position and minimise the risk of left ventricular outflow tract obstruction.59
(a) Sketch of the valve wireframe; and (b) schematic representation of the implanted prosthetic valve.
The distal margin of the ventricular structures includes distal loops (portions 1 and 2 in Fig. 2) which act as torsion springs, reducing the levels of stress in the crimped frame and dampening the load experienced by the leaflets during the operating cycles. The loops are also used to host control tethers which allow the valve recollapse into a delivery sheath by adopting the same approach described in Rahmani et al. 45
3D solid models of the wireframe (Fig. 2) were developed using NX CAD (Siemens PLM Software) program. Each solid model was discretised with approximately 110,000 tetrahedral elements of maximum edge size equal to 0.2 mm. The wireframe was modeled as NiTi shape memory alloy by using an austenitic Young's modulus (E A) of 50 GPa, martensitic Young's modulus (E M) of 25 GPa, and 0.3 for the Poisson's ratio of both austenitic and martensitic (ν A , ν M) phases.56 The transformation stresses of the NiTi wire for the austenite start (σ as,s), austenite finish (σ as,f), martensite start (σ sa,s) and martensite finish (σ sa,f) were 380, 400, 250 and 220 MPa respectively.56 The sleeves were modeled as stainless steel by using a Young's modulus of 210 kN/mm2 and a Poisson's ratio of 0.3, and were connected to the wireframe by applying stress free projected glued contact to their surfaces. The relative motion between the TMV and catheter during crimping was simulated by fixing the displacement of the top of the loops.
The wireframe geometry was optimised to maintain the maximum von Mises stress below the martensitic yield stress, when crimped to 8 mm (24 French) diameter. Simulations were performed using the FEA software MSC.Marc/Mentat and an implicit solver utilizing single-step Houbolt time integration algorithm, by gradually reducing the diameter of a surround cylindrical contact surface. Critical regions subjected to the highest levels of stress during crimping were identified in the initial geometry and optimised iteratively, using the approach described in Burriesci et al. 10 For each portion indicated in Fig. 2, the length, curvature and angle values were updated in each simulation in order to obtain a parameter set minimising the crimping stress on the wireframe.
Valve Prototypes
Prototypes of the wireframe structure were manufactured by thermomechanical processing of nitinol wires, mechanically joined at specific locations by means of stainless steel crimping sleeves. The leaflets and the sealing cuff made from bovine pericardium were sutured to the inner portions of the frame extensions (portions 5 and 6 in Fig. 2) using polypropylene surgical sutures. The skirt, made from a polyester mesh (Surgical Mesh PETKM2004, Textile Development Associates, USA), was included to gently distribute the anchoring force over the annulus (between portions 5, 6 and 7 in Fig. 2). The nominal valve size of the prototypes, defined based on the inter-trigonal dimension of the designed leaflets, was equal to 26 mm. This is suitable for preclinical in vivo evaluation in large animal models.
Hydrodynamic Tests
The hydrodynamic performances of the three valve prototypes were assessed on a hydro-mechanical cardiovascular pulse duplicator system (ViVitro Superpump SP3891, ViVitro, BC) (Fig. 3). The flow through the heart valves is measured with two electromagnetic flow probes and two Carolina Medical flow meters (Carolina Medical Electronics, USA), and the pressures in the aorta, left ventricle and left atrium are acquired using Millar Mikro-Cath pressure transducers. The working fluid was buffers phosphate saline solution at 37 °C. Hydrodynamic assessment of the prototypes was performed at 70 bpm heart rate, 5 L/min mean cardiac output and 100 mmHg mean aortic pressure, in compliance with the ISO 5840-3:2013 standard. The pulse duplicator was operated to simulate systole/diastole ratio as 35/65 over a cardiac cycle and a bileaflet mechanical heart valve Sorin Bicarbon size 25 was used to represent the aortic valve. Silicone models of the mitral annulus and native leaflets were built, based on the geometry previously described in Lau et al.33 with inter-trigonal diameters ranging from 21 to 25 mm, and used to house the test valves. This dimensional range, at least one millimeter smaller than the nominal size of the test valve, was selected to allow some anchoring force and verify the valve securing and hydrodynamic performance over a large anatomical range.
Experimental set-up for the hydrodynamic assessment of the proposed device: (a) pulse duplicator; (b) picture of the valve prototype indicating the leaflets, the sealing cuff and the anchoring skirt (top); and picture of the device after positioning in the valve holder (bottom).
Hydrodynamic performances of the prototypes were assessed by calculating the effective orifice area (EOA), regurgitant fraction and mean transmitral diastolic pressure. The effective orifice area was estimated using the Gorlin Equation (Eq. 2), as described in the ISO 5840.
$${\text{EOA}} = \frac{{Q_{\text{mv,rms}} }}{{51.6\sqrt {\Delta p_{\text{mv}} /\rho } }}$$
where, Q mv,rms represents the root mean square of the flow rate through the mitral valve, Δp mv is the mean positive differential pressure across the mitral valve and ρ is the density of the circulating fluid. The regurgitant fraction is calculated as the ratio of the measured closing regurgitant volume (back flow during valve closure) plus the leakage volume (leaking flow after closure) and the forward flow volume during the ventricular filling.
Seventeen of the twenty five bi-leaflet designs simulated numerically were functionally patent, and all had an acceptable peak of maximum principal stress below 5 MPa.61 Due to the need to ensure adequate valve function for a wide range of possible expansion sizes and shapes, the design providing maximum coaptation area was selected (Fig. 4) and the wireframe was subsequently made to fit this.
Maximum principal stress distribution for the optimal transcatheter mitral valve leaflets in their critical loading mode when fully closed, peak value 3.51 N/mm2 at the arris between the leaflets.
The selected design, characterised by a coaptation area of 1.8 cm2, met the peak maximum principal stress design criteria, with an estimated peak value below 5 MPa (3.51 MPa), located at the arris between the leaflets. The resulting stress distributions for the optimal geometry of the crimped wireframe are shown in Fig. 5. The critical points of maximum stress during crimping occurred around the sleeves. The highest stress, as expected, occurred at the maximum collapse diameter of 8 mm, and was 835 N/mm2. This remains below the yield stress reported for martensite in superelastic Nitinol, at the operating range of temperature.46
Stress distributions for the optimal geometry of the transcatheter mitral valve wireframe, crimped to different diameter sizes.
The optimised wireframe geometry was closely replicated physically by thermomechanical processing of Nitinol wire, and mechanical crimping with stainless steel sleeves. Comparison between the free and crimped TMV wireframe geometries for the numerical model and prototype are given in Fig. 6.
The transcatheter mitral valve wireframe: (a) solid model; (b) numerical model crimped in a 8 mm diameter cylinder; (c) manufactured prototype; (d) prototype crimped in a 8 mm diameter tube
Elastic deformation of the wireframe in an 8 mm diameter tube shows that the portions functioning as springs (Fig. 2a: portions 3 and 4) and the portions holding the mitral valve leaflets (Fig. 2a: portions 5 and 6) do not intersect with each other, this leaves sufficient space for the leaflets and sealing cuff when crimped. Additionally, the geometry of the crimped wireframe was in good agreement with the numerical prediction.
Diagrams of the effective orifice area, regurgitant fraction and mean diastolic transmitral pressure difference for the prototypes in the different annulus sizes are represented in Fig. 7. The estimated EOA increased with the size of the host valve, with the mean for the three prototypes raising from 1.26 to 1.70 cm2 when moving from the 21 to the 25 mm annulus. All valves exceeded the effective orifice area required by the ISO 5840-3:2013 standard, for the different implantation sizes (larger than 1.05 cm2 and 1.25 cm2 for mitral annuluses of size 23 and 25 mm, respectively).
Hydrodynamic assessment results for the three tested prototypes (P1, P2, and P3; M represents the mean of the three tests) in six different annulus sizes: (a) effective orifice area; (b) regurgitation fraction; and (c) mean transmitral pressure difference during diastole. Minimum performance requirements for 23 and 25 mm, as per ISO 5840-3:2013, are indicated by the asterisk symbol, with the arrows pointing the allowed region.
Regurgitant fractions did not show a clear pattern with the implantation size, and ranged from 8.2 to 17.8%. However, all prototypes met the minimum performance requirements in the ISO5840-3:2013 standard (regurgitant flow fraction ≤20% for both 23 and 25 mm annuli—no specifications for smaller sizes). The mean diastolic transmitral pressure difference decreased in the larger annuluses and reached a maximum value of about 9 mmHg in the 21 mm annulus, reducing to 5 mmHg in the 25 mm annulus.
A sequence of snapshot images of one of the prototypes acquired during the forward mitral valve flow for 23 mm implantation size with 29 fps frame rate are shown in Fig. 8a. The valve leaflets fully opened at the beginning of the left ventricular filling. The anterior leaflet remained fully open during the forward mitral valve flow while the posterior leaflet was fluttering. Duration of the leaflet open phase was approximately 60% of the entire cardiac cycle.
Sequence of snapshot images of one of the tested prototypes during the forward mitral valve flow for 23 mm implantation (a–o). The anterior and posterior leaflets are on the left and right side, respectively. For the test in the sequence are also reported: (p) left ventricular, left atrial and aortic pressure signals (plv, pla and pao, respectively); (q) transmitral pressure difference signal (Δp mv); and (r) flow rate signal through the TMV (Q mv)
The peak (systolic) transmitral pressure difference was 125 mmHg, while the maximum diastolic opening pressure was about 45 mmHg. Regurgitant flow was observed over the ventricular systole, primarily due to paravalvular leakage between the mitral annulus and the device. The closing regurgitation (due to closure of the mitral valve leaflets) was higher in the larger annuluses. Anchoring was adequate for all tests, and no valve migration was observed for any of the test conditions. Typical pressure and flow rate diagrams through the valve, obtained for one of the three prototypes in an annulus of 23 mm over a cardiac cycle, are provided in Fig. 8.
Currently, no device specifically designed for TMV implantation has been approved for the European or American market. However, a number of solutions have been proposed, with many already at the stage of clinical trial (these include the CardiAQ 51,52 and Fortis,2,8 Edwards Lifescience; the Tendyne,39 Tendyne Holdings Inc., Roseville MN, USA; the Tiara,14 Neovasc, Richmond, Canada; the NaviGate, NaviGate Cardiac Structures Inc., Lake Forest, CA, USA; and the Intrepid, Medtronic, Dublin, Ireland).31 Despite the reduced number of patients involved in the trials and the large 30 days mortality rate, justified by the compassionate ground of the implants, this early experience has confirmed the potential benefit of the treatment and the ability of transcatheter solutions to successfully replace the mitral valve function.31 All devices under investigation are based on three occluding leaflet, replicating the configuration and function of semilunar valves. These are supported by self-expanding stents, obtained from laser-cut nitinol tubes, mechanically deformed and thermoset.41 The stents bulge or expand in a flange covered with a fabric material, designed to apply pressure on the atrial inflow portion, and used to minimise paravalvular leakage while counteracting the ventricular anchoring force providing the valve securing. From a technical point of view, a major distinction between the devices currently under investigation is represented by the method they use to generate the ventricular anchoring force, which can be based on ventricular tethers (e.g. Tendyne), native valve anchors (e.g. CardiAQ, Fortis, Tiara and NaviGate) or dual stent structures with barbs.38
The device presented in this paper introduces a number novel concepts, providing new and alternative features. Contrary to competing TMVs, the proposed solution is based on two asymmetric flexible leaflets, describing a D-shape cross section designed to better conform to the irregular anatomy of the valve annulus and minimise the disturbance to the sub-valvular apparati. This allows to maximise the geometrical orifice area of the prosthesis without interfering with the aortic valve anatomy and function. The leaflets are sutured onto a self-expanding frame, obtained from a nitinol wire, thermo-mechanically formed and mechanically crimped at five locations. This defines a set of arched ribs expanding into the atrium and two petal-like structures protruding into the ventricle between the native mitral leaflets, whose counteracting action generates the anchoring force, whilst limiting the systolic motion of the native anterior leaflet and the associated risk of left ventricular outflow tract obstruction. The wireframe configuration results in minimum metallic material, and relies on a skirt made from polymeric mesh (allowing integration from the host tissues), tensed between the atrial petals and the leaflets, to gently distribute the contact pressure over the annulus region. Paravalvular sealing is provided by a pericardial cuff extending around the entire framework of the valve, which inflates during systole as effect of the transvalvular closing pressure. The valve, designed in the presented version for transapical implantation, can be retrieved into the delivery system after complete expansion, using a similar mechanism to that described by the authors for a TAVI device.44
The structural numerical analyses, though inherently limited in their ability to represent the physics involved in heart valve leaflet closure, were adequate to predict the systolic function of the leaflets. In particular, this approximation does not take into account the interaction between the working fluid and the structural components, which determine the flow patterns and the pressure differences acting under real physiological conditions. Fluid structure interaction modelling would be more accurate for the simulation of the opening and closing leaflets dynamics. However, the peak of stress in the leaflets during the cardiac cycle is essentially led by the closing transvalvular pressure load,33 so that neglecting the local pressure variation and fluid shear stresses due to blood flow can still yield to sufficiently accurate results for the design evaluation stage.10
The valve wireframe optimisation was carried out until obtaining an optimal geometry which has lower stresses than NiTi yielding. Portions 5 and 6 in Fig. 2a were imposed by the leaflets geometry and kept unchanged for all wireframe models. The geometry of the wireframe is relatively complex, and includes a number of geometric parameters which needed to be optimised to obtain a suitable design. Each section was iteratively modified to minimise local stresses, resulting in a final geometry which fits adequately into the host mitral anatomy, maintaining acceptable levels of stress in the crimped configuration. The finite element analyses of a wireframe crimped to a diameter of 8 mm resulted in a maximum stress less than 900 MPa, which corresponds to a typical yield stress for Nitinol.46 The stress concentrations were predicted in the vicinity of the crimping sleeves, with local maxima around 600 MPa. Therefore, plastic deformation is not expected in the crimped wireframe, and this was confirmed by loading and unloading the physical prototype in an 8 mm diameter tube multiple times, without observable changes in shape. Besides, the presented version of the wireframe is designed to be ideally implantable from transapical route, which tolerates the use of larger sheath profiles (up to 33 French, 11 mm), resulting in further reduction of the stresses on the NiTi wireframe.60 Crimping of the TMV wireframe was simulated by gradually shrinking a cylindrical contact surface surrounding the prosthesis along its entire length. In the current application, the valve distal loops (Fig. 2a, portions 1 and 2) are engaged by a set of tethers, used to pull the valve into the catheter from the side at the outflow.45 Nevertheless, the resulting geometry of the crimped wireframe in the numerical simulations resulted visually accurate.
The valve design and prototypes were of a nominal size equal to 26 mm, corresponding to the largest inter-trigonal dimension of the prosthetic leaflets. This is suitable for patient's annuli with inter-trigonal diameters equal or lower than 25 mm. Though this range is smaller than the average size in adult humans, it is more suitable for preclinical in vivo evaluation in ovine models,43 which is expected to be one of the next developmental steps. The prototypes were tested in mock host annuli of inter-trigonal diameters ranging from 21 to 25 mm. As expected, the diastolic transmitral pressure difference raised nonlinearly as the dimensions of the host annulus reduced, increasing from about 5 mmHg for the 25 mm annulus, to about 9 mmHg for the 21 mm annulus. A high peak in the initial diastolic transmitral pressure drop is measured in the tests (up to 45 mmHg). This is often observed in tests performed on hydro-mechanical pulse duplicators,16,28,29,48,53,55 and could be due to the non-physiological ventricular compliance, which may determine steeper flow waves and higher pressure gradients associated with early passive filling during ventricular relaxation. The calculated EOA well reflected the variation in the area of the implantation annulus, varying proportionally. Regurgitant fraction did not show a clear pattern associated with the implantation size for the different prototypes, although the mean value reduced progressively from 21 to 24 mm, inverting the trend at 25 mm. The reduction with the size may be associated with the different length of the mock native leaflets, which were designed proportional to the annulus size and, therefore, provided different covering of the sealing cuff of the prosthetic valves. On the other hand, the increased regurgitant fraction in the 25 mm annulus may be justified by the presence of gaps between the device and the mitral annulus. Globally, the device met the hydrodynamic requirements requested for transcatheter mitral valves in the standard ISO5840-3:2013, for all implantation sizes. Direct comparison of the hydrodynamic performance with competing solutions is not possible, as these are not available in the market and no in vitro data quantifying their diastolic and systolic efficiency have been published. However, measured values of transmitral diastolic pressure drops are consistent with those reported for transcatheter mitral implantation of off-label TAVI devices in failed mitral valve bioprostheses or annuloplasty rings, and in severe calcific mitral stenosis.13,18 Regurgitant fractions were inferior to those previously measured on the same system for commercially available TAVI devices.45 This is very encouraging, in consideration of the fact that, for the mitral position, closure is associated with higher transvalvular pressure drop and longer durations with respect to the cardiac cycle.
In terms of anchoring, no migration was observed for any of the test configurations, covering host annuli with inter-trigonal diameters between 21 and 25 mm. However, it needs to be taken into account that the mock host valves did not model the physiological contraction, and cordae tendineae and papillary muscles were absent. Ex vivo isolated beating heart or pressurised animal heart platforms17,57 and acute in animal trials could provide more reliable insights on the fitting and performance of a transcatheter valve.44 These studies would also be essential to verify the efficacy of the anchoring mechanism to avoid left ventricular outflow tract obstruction by preventing the systolic motion of the native anterior leaflet.
A novel TMV was developed, consisting of two bovine pericardial leaflets designed to ensure proper functionality across a range of implantation configurations and a sealing cuff, supported by a wireframe, optimised to minimise stresses whilst crimped. The device exceeded the minimum performance requirement from the international standards, thereby proving its feasibility as a mitral valve substitute to treat mitral regurgitation. In vitro durability assessment of the valve by means of accelerated cyclic tests is now being conducted, with the aim of verifying that the solution guarantees a survival equal or superior to the requirement for flexible leaflets heart valves (200 × 106 cycles). The next steps in the development will include in vivo preclinical evaluation by means of in animal implants (possibly complemented by ex vivo studies), to validate the design principles and the efficacy of the device.
If these will confirm the predicted performance, the proposed device could provide a viable alternative to transcatheter repair techniques and, due to its geometric similarity to the human mitral valve anatomy, may result a more appropriate option compared to the other TMVs in development.
This work was supported by the British Heart Foundation (PG/13/78/30400). Authors wish also to acknowledge Dr Benyamin Rahmani and Dr Michael Mullen for their assistance and advices, and Lithotech Medical for their support in the frames manufacturing.
The authors do not have any conflict of interest to declare.
Supplementary material 1 (MOV 21234 kb)
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
1.UCL Mechanical Engineering, Cardiovascular Engineering LaboratoryUniversity College LondonLondonUK
2.Ri.MED Foundation, Bioengineering GroupPalermoItaly
Bozkurt, S., Preston-Maher, G.L., Torii, R. et al. Ann Biomed Eng (2017) 45: 1852. https://doi.org/10.1007/s10439-017-1828-2
Accepted 25 March 2017
First Online 03 April 2017
Publisher Name Springer US
Biomedical Engineering Society (BMES) | CommonCrawl |
Finally we come to the discussion of relations between variables measured in numerical scales. The most famous measure in this category is the Pearson's correlation coefficient, which population value is: \[\begin{equation} \rho_{x,y} = \frac{\sigma_{x,y}}{\sigma_x \sigma_y}, \tag{9.9} \end{equation}\] where \(\sigma_{x,y}\) is the covariance between variables \(x\) and \(y\) (see discussions in Sections 5.1 and 10.2), while \(\sigma_x\) and \(\sigma_y\) are standard deviations of these variables. Typically, we do not know the population values, so this coefficient can be estimated in sample via: \[\begin{equation} r_{x,y} = \frac{\mathrm{cov}(x,y)}{\sqrt{V(x)V(y)}}, \tag{9.10} \end{equation}\] where all the values from (9.9) are substituted by their in-sample estimates. This coefficient measures the strength of linear relation between variables and lies between -1 and 1, where the boundary values correspond to perfect linear relation and 0 implies that there is no linear relation between the variables. In some textbooks the authors claim that this coefficient relies on Normal distribution of variables, but nothing in the formula assumes that. It was originally derived based on the simple linear regression (see Section 10) and its rough idea is to get information about the angle of the straight line drawn on the scatterplot. It might be easier to explain this on an example:
plot(mtcarsData$disp,mtcarsData$mpg,
xlab="Displacement",ylab="Mileage")
abline(lm(mpg~disp,mtcarsData),col="red")
Figure 9.2: Scatterplot for dispalcement vs mileage variables in mtcars dataset
Figure 9.2 shows the scatterplot between the two variables and also has the straight line, going through the cloud of points. The closer the points are to the line, the stronger the linear relation between the two variables is. The line corresponds to the formula \(\hat{y}=a_0+a_1 x\), where \(x\) is the displacement and \(\hat{y}\) is the line value for the Mileage. The same relation can be presented if we swap the axes and draw the line \(\hat{x}=b_0+b_1 y\):
plot(mtcarsData$mpg,mtcarsData$disp,
xlab="Mileage",ylab="Displacement")
abline(lm(disp~mpg,mtcarsData),col="red")
Figure 9.3: Scatterplot for mileage vs dispalcement
The slopes for the two lines will in general differ, and will only coincide if the two variables have functional relations (all the point lie on the line). Based on this property, the correlation coefficient was originally constructed, as a geometric mean of the two parameters of slopes: \(r_{x,y}=\sqrt{a_1 b_1}\). We will come back to this specific formula later in Section 10. But this idea provides an explanation why the correlation coefficient measures the strength of linear relation. For the two variables of interest it will be:
cor(mtcarsData$mpg,mtcarsData$disp)
## [1] -0.8475514
Which shows strong negative linear relation between the displacement and mileage. This makes sense, because in general the cars with bigger engines will have bigger consumption and thus will make less miles per gallon of fuel. The more detailed information about the correlation is provided by the cor.test() function:
cor.test(mtcarsData$mpg,mtcarsData$disp)
## data: mtcarsData$mpg and mtcarsData$disp
## t = -8.7472, df = 30, p-value = 9.38e-10
## -0.9233594 -0.7081376
## cor
## -0.8475514
In addition to the value, we now have results of the hypothesis testing (where null hypothesis is \(\rho_{x,y}=0\)) and the confidence interval for the parameter. Given that the value of the parameter is close to its bound, we could conclude that the linear relation between the two variables is strong and statistically significant on 1% level.
Note that the value of correlation coefficient only depends on the distance of points from the straight line, it does not depend on the slope (excluding case, when slope is equal to zero and thus the coefficient is equal to zero as well). So the following two cases will have exactly the same correlation coefficients:
error <- rnorm(100,0,10)
x <- c(1:100)
y1 <- 10+0.5*x+0.5*error
y2 <- 2+1.5*x+1.5*error
# Produce the plots
par(mfcol=c(1,2))
plot(x,y1,ylim=c(0,200))
abline(lm(y1~x),col="red")
text(30,150,paste0("r=",round(cor(x,y1),5)))
Figure 9.4: Example of relations with exactly the same correlations, but different slopes.
There are other examples of cases, when correlation coefficient would be misleading or not provide the necessary information. One of the canonical examples is the Anscombe's quartet (Wikipedia, 2021), which shows very different types of relations, for which the Pearson's correlation coefficient would be exactly the same. An important lesson from this is to always do graphical analysis (see Section 5.2) of your data, when possible - this way misleading situations can be avoided.
Coming back to the scatterplot in Figure 9.2, it demonstrates some non-linearity in the relation between the two variables. So, it would make sense to have a different measure that could take it into account. This is where Spearman's correlation coefficient becomes useful. It is calculated using exactly the same formula (9.10), but applied to the data in ranks. By using ranks, we loose information about the natural zero and distances between values of the variable, but at the same time we linearise possible non-linear relations. So, Spearman's coefficient shows the strength of monotonic relation between the two variables:
cor.test(mtcarsData$mpg,mtcarsData$disp,
method="spearman")
## Warning in cor.test.default(mtcarsData$mpg, mtcarsData$disp, method =
## "spearman"): Cannot compute exact p-value with ties
## Spearman's rank correlation rho
## S = 10415, p-value = 6.37e-13
## alternative hypothesis: true rho is not equal to 0
## rho
We can notice that the value of the Spearman's coefficient in our case is higher than the value of the Pearson's correlation, which implies that there is indeed non-linear relation between variables. The two variables have a strong monotonic relation, which makes sense for the reasons discussed earlier. The non-linearity makes sense as well because the car with super powerful engines would still be able to do several miles on a gallon of fuel, no matter what. The relation will never be zero or even negative.
Note that while Spearman's correlation will tell you something about monotonic relations, it will fail to capture all other non-linear relations between variables. For example, in the following case the true relation is trigonometric:
y <- sin(x)
plot(x,y,type="l")
But neither Pearson's nor Spearman's coefficients will be able to capture it:
cor(x,y)
## [1] -0.04806497
cor(x,y,method="spearman")
In order to correctly diagnose such non-linear relation, either one or both variables need to be transformed to linearise the relation. In our case this implies measuring the relation between \(y\) and \(\sin(x)\) instead of \(y\) and \(x\):
cor(sin(x),y)
## [1] 1
• Wikipedia, 2021. Anscombe's quartet. https://en.wikipedia.org/wiki/Anscombe%27s_quartet (version: 2021-06-23) | CommonCrawl |
\begin{definition}[Definition:Disjunction/Truth Function]
The disjunction connective defines the truth function $f^\lor$ as follows:
{{begin-eqn}}
{{eqn | l = \map {f^\lor} {\F, \F}
| r = \F
}}
{{eqn | l = \map {f^\lor} {\F, \T}
| r = \T
}}
{{eqn | l = \map {f^\lor} {\T, \F}
| r = \T
}}
{{eqn | l = \map {f^\lor} {\T, \T}
| r = \T
}}
{{end-eqn}}
\end{definition} | ProofWiki |
Format: MarkdownItexEvery category -- indeed, every simplicial set -- admits a [[homotopy final functor]] into it out of a [[Reedy category]], namely its [[category of simplices]] (HTT 4.2.3.14). This makes me wonder: can every $(\infty,1)$-topos be presented as a localization of an $(\infty,1)$-topos of presheaves on a Reedy category?
Every category – indeed, every simplicial set – admits a homotopy final functor into it out of a Reedy category, namely its category of simplices (HTT 4.2.3.14). This makes me wonder: can every (∞,1)(\infty,1)-topos be presented as a localization of an (∞,1)(\infty,1)-topos of presheaves on a Reedy category?
Format: MarkdownItexAnd I guess with that point made, it makes sense to ask the question more generally about locally presentable $(\infty,1)$-categories. I'm thinking of something like this: suppose C is a small $(\infty,1)$-category and $(\Delta\downarrow C)$ its category of simplices; then we have a functor $t\colon (\Delta\downarrow C) \to C$ sending each simplex to the last object occurring in it. This induces a functor $t^* \colon sPre(C) \to sPre(\Delta\downarrow C)$, and every object in the image of this functor has the property that it sees as isomorphisms all the maps in $(\Delta\downarrow C)$ which fix the last object. Consider the localization of $sPre(\Delta\downarrow C)$ which forces all these maps to be invertible; it seems as though that has a decent chance to be equivalent to $sPre(C)$?
And I guess with that point made, it makes sense to ask the question more generally about locally presentable (∞,1)(\infty,1)-categories. I'm thinking of something like this: suppose C is a small (∞,1)(\infty,1)-category and (Δ↓C)(\Delta\downarrow C) its category of simplices; then we have a functor t:(Δ↓C)→Ct\colon (\Delta\downarrow C) \to C sending each simplex to the last object occurring in it. This induces a functor t *:sPre(C)→sPre(Δ↓C)t^* \colon sPre(C) \to sPre(\Delta\downarrow C), and every object in the image of this functor has the property that it sees as isomorphisms all the maps in (Δ↓C)(\Delta\downarrow C) which fix the last object. Consider the localization of sPre(Δ↓C)sPre(\Delta\downarrow C) which forces all these maps to be invertible; it seems as though that has a decent chance to be equivalent to sPre(C)sPre(C)?
Format: MarkdownItexAt least in the 1-categorical case, this is true. The functor $t^*$ has a left adjoint (left Kan extension), and by C3.3.8(i) in the Elephant, it is fully faithful; thus it exhibits $Pre(C)$ as a reflective subcategory of $Pre(\Delta\downarrow C)$. Does C3.3.8(i) have an $(\infty,1)$-categorical analogue?
At least in the 1-categorical case, this is true. The functor t *t^* has a left adjoint (left Kan extension), and by C3.3.8(i) in the Elephant, it is fully faithful; thus it exhibits Pre(C)Pre(C) as a reflective subcategory of Pre(Δ↓C)Pre(\Delta\downarrow C). Does C3.3.8(i) have an (∞,1)(\infty,1)-categorical analogue? | CommonCrawl |
What is the initial position of the ball
Types: Personalised Ball Markers, Personalised Golf Tee
ed instant of time when T = 0 seconds
Find the average velocity of the ball from t = 0 to t = 1.0s . Find the average speed of the ball between t = 1.0 s and t = 2.0s . Question: What is the initial position of the ball ? What is the initial velocity of the ball ? What is the acceleration of the ball? Find the average velocity of the ball from t = 0 to t = 1.0s . Find the average.
The initial position of the ball is 0 meter in 0 sec. While its final position is 1 5 meters in 1 5 sec
Ball B is half of the mass of ball A, so the velocity of ball B needs to be greater than the velocity of ball A in this example in order to make up for its smaller mass due to conservation of momentum A ball is thrown vertically upwards and returns to its initial position in 6.0 second
What is the initial position of the ball? 2. What is the position of the ball at 10 seconds? 3. At what time is the position of the ball equal to 5m.? 4. What is the total distance travelled by the ball? 1 See answer ishida0975 ishida0975 Answer: It is at the starting line, 0meters and 0 seconds
1.what is the initial position of . 1.what is the initial position of the ball? 0 meter. 2.what the final position of the ball? 15 meter. 3. what is the position of the ball at 10 secs? 10 meter. 4. At what time is position of the ball equal to 5 meters? 5 seconds. Explanation: HOPE IT HELPS: What is the initial position of the ball? 6. What is the final position? 7. What is the position of the ball at 10 seconds? 8. At what time is the position of the ball equal to 5 meters? Need ko po answer pls 1 See answer ClayCat ClayCat Answer: 1.)0 meters is the initial position. 2.)15 meters is the final position The position of a ball as a function of time is given by x=(5.5m/s)t+(−9m/s2)t2. a) what is the initial position of the ball? b)what is the initial velocity of the ball? c) what is the acceleration of the ball? d) find the average velocity of the ball from t=0 to t=1.0s. e) find the average speed of the ball between t=1.0s and t=2.0 s A bowling ball unintentionally falls out of anairliner's cargo bay as it flies along in a horizontaldirection. As detected by a person standing on the ground and viewing the plane as in the figure at right,which path would the bowling ball most closelyfollow after leaving the airplane,(B)(0
Answers: 3 on a question: Directions. answer the following questions. Write the answer on your answer sheet.1. What is the initial position of the ball?2. What is the position of the ball at 10 seconds?3. At what time is the position of the ball equal to 5 meters?4. What is the total distance travelled by the ball?obiect travelled The Position Of A Ball As A Function Of Time Is Given By X= (4.5m/s)t+ (−10m/s2)t2 For example, if you are asked to find the position of the ball at 10 seconds, all you need to do is to find the point along the diagonal line where the vertical line at the 10 second-mark intersects (Figure 4). Then find where the horizontal line from that point of intersection will cross the Y axis, which is the position axis Organized basketball is a game played by five players per team. Historically, these players have been assigned to positions defined by the role they play on the court, from a strategic point of view. Broadly speaking, the three main positions are guard, forward, and center, with the standard team featuring two guards, two forwards, and a center What is the position of the ball in T/3 seconds? A body moving with a constant acceleration travels the distances 3 m and 8 m respectively in 1 s and 2 s.Calculate the initial velocity of the body. Medium. View solution. Two balls are dropped to the ground from different heights
If we take the initial position y 0 to be zero, then the final position is y = −20.0 m. Now the initial vertical velocity is the vertical component of the initial velocity, found from v Oy = v 0 sin θ 0 = (25.0 m/s)(sin 35.0º) = 14.3 m/s. Substituting known values yield The position of a ball as a function of time is given by x=(5.3m/s)t+(−11m/s2)t2. A.What is the initial position of the ball? Express your answer to two significant figures and include appropriate units The initial position of the ball is {eq}x=0 m {/eq} or at the origin Solution: The position of the ball is given by {eq}x(t)=(4.4\frac{m}{s})t+(-8.. PRFFWC Presentation PRFFWC Answer The initial position of the ball is 0 meters 12. PRFFWC Presentation PRFFWC Answer The initial position of the ball after 10 seconds is 10m. 13. PRFFWC Presentation PRFFWC Answer 5 seconds is the time that the ball is equal to 5 meters. 14. PRFFWC Presentation PRFFWC Graphs 15
1. What is the potential energy of the spring in its compressed position? 256 J 2. To what maximum height above its initial (compressed) position does the ball reach? 6.53m 3. Earlier, just when the spring is returned to its equilibrium position, as the ball was moving upwards, how fast was the ball moving? undetermined? Homework Equations KE=1. A ball, with an initial position of x = 25.89 meters, undergoes a displacement of 32.2 meters. What is it's final position? - 1574147
Have a third person, the recorder, record the time in a data table. Repeat Step 4, stopping the times at the distances of 1.0 m, 1.5 m, 2.0 m, 2.5 m, and 3.0 m from the bottom of the ramp. Use your measurements of time and the displacement to make a position vs. time graph of the ball's motion A spring-loaded gun is aimed horizontally and is used to launch identical balls with different initial speeds. The gun is at a fixed position above the floor. The balls are fired one at a time. If the speed of the second ball fired is twice the speed of the first ball fired, how is the horizontal range (denoted R in the figure) affected
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Given an initial position and velocity of a receiver, find the velocity of a ball. I have a question involving kinematics and physics in a plane. The question is as follows: Quarterback Fred is going to throw a pass to tight end Doug. Doug is 20 m in front of Fred and running straight away at 6.0 m/s when Fred throws the 500 g football at a 40.
An easy way to check how the player imagines the contact on a tennis serve is to simulate the serve into a back fence. Slowly initiate your service motion from the beginning, and gently hit the fence and stop the racket there. Now check 4 key body positions/alignments: forearm/wrist angle. angle of your head
The initial velocity of the golf ball can be described by the vector . The vector-valued function that describes the velocity of the golf ball is: To find the vector-valued function that describes the ball's position, Jill can take the integral of the velocity function
Suppose that a tennis ball is dropped from the top of a skyscraper. Taking the ball's initial position zero {eq}\rm (y = 0) {/eq} and choosing {eq}\rm y {/eq}-axis positive in the downward.
A football player launches the ball from initial position x (0)=y (0)%=D0, with initial velocity v = v, cos 0 i + vo sin 0j. Neglecting drag, the ball's trajectory describes a parabola in the Oxy plane, where y is the vertical upward direction and x is the horizontal direction. After a distance d along the x-direction, the ball returns to the.
When the 0.500kg ball collides with the stationary ball, momentum is conserved. Meaning, initial momentum = final momentum. Momentum of an object is = mass(m) x velocity (v)
Initial and final position will be same if it is projected vertically upward Allaiza10 Allaiza10 02.01.2019 Science Secondary School answered What is the initial position of the ball?what is its final position? 1 See answer Allaiza10 is waiting for your help. Add your answer and earn points. krimusa7524 krimusa752
What is the initial position of the ball? - Answer
The position of the ball is given by the coordinates (x , y). The position of the ball depends on time t. The motion of the ball is defined by the motion functions: x(t) , y(t). Note that at time t = 0 , the ball is launched from the point (x , y) = (0 , yo) with the velocity vo. The initial velocity vector vo has magnitude vo and direction θo. The initial position of the ball will tell us how high the window is. From the y-axis on the graph we can see that the ball is \(\text{4}\) \(\text{m}\) from the ground. The window is therefore \(\text{4}\) \(\text{m}\) above the ground. Find the time taken to reach the maximum height. Q10. What is the initial position of the ball? What is its final position? Q11. What is the position of the ball at 10 seconds? Q12. At what time is the position of the ball equal to 5 meters? Using graphs Another way to describe the motion of the ball is by the use of motion graphs. Convert the diagram in Figure 2 to graph by following the. What is the initial position of the ball? What is its final position? Q11. What is the position of the ball at 10 seconds? Q12. At what time is the position of the ball equal to 5 meters? Using graphs Another way to describe the motion of the ball is by the use of motion graphs. Convert the diagram in Figure 2 to graph by following the guide below
What is the initial position of the ball ? What is Chegg
The following graph represents the position as a function of time of a moving object. Use this graph to answer questions 12 and 13. 12. What is the initial position of the object? A. 2 m B. 4 m C. 6 m D. 8 m E. 10 m 13. What is the velocity of the object
What do the softball position numbers mean? In fastpitch and slowpitch, they are used in the scorebook to tell positions apart. Each position on the field is designated with a 1-9 number. The position numbers go in order, starting with the pitcher as #1, however the shortstop is out of order and is labeled #6 instead of #5 - all other positions go naturally in order from pitcher to catcher.
What ball flight law is most responsible for the initial direction of a golf shot? In a properly struck chip shot, why should the ball position be right of the swing center with the club shaft leaning left? natural pendulum arm swing. According to most good putters, where should the ball position be at address when putting?.
The displacement is obviously zero, provided the ball actually does end up where it started. The distance traveled depends on the initial velocity. We can actually use the same equation you gave to calculate the distance as well. You can use calculus to find the peak height: $$\frac{d}{dt}(\frac{1}{2}at^2+v_0t) = 0$
The position of a ball as a function of time is given byx=(4.4m/s)t+(−8m/s2)t2What is the initial position of the ball? close. Start your trial now! First week only $4.99! arrow_forward. Question. The position of a ball as a function of time is given by x=(4.4m/s)t+(−8m/ s 2) t
This video screencast was created with Doceri on an iPad. Doceri is free in the iTunes app store. Learn more at http://www.doceri.co After impact the ball is raised from the initial equilibrium position of the pendulum to a height above that position. The kinetic energy of the system, ball and pendulum, at the instant following impact is 1 2 (M+m)V2. When the center of gravity reaches its highest position h 2, the potential energy of the syste means that its initial velocity is zero, v 0 = 0 • Then its present velocity = a •t, where a is the acceleration of gravity g ≈10 m/s2 or 32 ft/s2, for example: • What is the velocity of a ball 5 seconds after it is dropped from rest from the Sears Tower? Æv = 32 ft/s2 •5 s = 160 ft/s The position of a falling ball During downward motion, the signs of position, velocity, and acceleration are all positive. (d) Initial velocity of the ball, u = 2 9. 4 m/s Final velocity of the ball, v = 0 (At maximum height, the velocity of the ball becomes zero) Acceleration, a = g = 9. 8 m / s 2 From third equation of motion, height (s) can be calculated as: v 2 − u 2.
The position versus time graph of this object is. From the edge of a roof top you toss a green ball upwards with initial speed v0 and a blue ball downwards with the same initial speed. Air resistance is negligible. When they reach the ground below Initial position tells us the position of body at t = 0 and final position of the body tells position of body at time t. A 0.311 kg tennis racket moving 30.3 m/s east makes an elastic collision with a 0.0570 kg ball moving 19.2 m/s east. Find the velocity of the tennis r acket after the collision A ball is pushed with an initial velocity of 4.0 m/s. The ball rolls down a hill with a constant acceleration of 1.7 m/s2. The ball reaches the bottom of the hill in 7.0 s. What is the ball's velocity at the bottom of the hill? A. 15 m/s B. 13 m/s C. 14 m/s D. 16 m/s E. 19 m/ A cannon ball is fired with an initial velocity of 100.0 m/s at an angle of 45° above the horizontal. What maximum height will it reach and how far will it fly horizontally? The first step in the analysis of this motion is to resolve the initial velocity into its vertical and horizontal components Please refer to the attached figure describing the motion of the thrown ball. The position of the ball is at point (2,y) at that instant. The value of time t covered horizontally is equal to 2m / 3 m/s. This division indicates that the ball had tr..
Free Fall of a Ball Figure 3.27 shows the positions of a ball, at 1-s intervals, with an initial velocity of 4.9 m/s downward, that is thrown from the top of a 98-m-high building. (a) How much time elapses before the ball reaches the ground? (b) What is the velocity when it arrives at the ground Batter's Position in Batter's Box. Rules 5.04(b)(5), 6.03(a)(1): When the batter assumes a batting stance in the batter's box, he shall have both feet entirely within the batter's box; i.e., no part of either foot may extend beyond the outer edge of the lines defining the box when the batter assumes a position in the box. There is no penalty specified for violation other than the. Find the displacement of the ball during the time interval 0≤ t ≤ 3. b. Given that the initial position of the ball is s(0) = 9 feet, use the result from part a to determine its position at. You should see a ball and ground. 2. Input initial conditions #set up initial conditions ball.velocity=vector(0,5,0) ball.mass=0.25 ball.p=ball.velocity*ball.mass this plots t and the y-component of the position of the ball. 7. Printing data: To print the time and y position for each instance, simply add the following inside th
The initial position of the ball is at the height, {eq}y_0 {/eq} = 0 m. Acceleration due to Gravity near Surface of Earth: If air resistance is neglected and a body is dropped near the surface of. (Sometimes it is written as + ½ instead of - ½, but then you need to ensure g is a negative number).. Where: y() is the vertical position (height) and x() is the horizontal position x 0 and y 0 are the initial horizontal and vertical positions the projectile is launched from; v 0 is the initial velocity of the projectile; θ is the angle of the initial trajectory with the horizontal (i.e.
Remember my initial position s(0) equals 6. The ball was thrown from 6 feet above ground so when t equals zero, position is 6. So I'll have 6 equals -16(0)² plus 95 times zero plus d. that tells me these are just zero. It tells me that D is 6. And I have the position function s(t) equals -16t² plus 95t plus 6 The cannon is at height H=80 meters above the ground level, and the ball is fired with initial horizontal speed v. Assume acceleration due to gravity to be g=9.8 m/s{eq}^2 {/eq} The ball lands on the elevated green, yf = 5.3 m above the initial position near the hole, and stops Physics A ball rolls on a circular track of radius 0.65 m with a constant angular speed of 1.2 rad/s in the counterclockwise direction
What is initial position of the ball? - Answer
Click here��to get an answer to your question ️ on 67 t=0 A ball released from rest at time to hits the ground. It rebounds inelasically with a velocity of 5m/s and reaches the top at t=1.5S. What is net displacement of the ball from its initial position after 1.5s? (a) 1.25m (b) 3.75m (b) 5.00m (d) 6.25m t-1.55 An object of mass 'm' along a stright line with a velociy v collides with a.
The only way to return the ball to its initial position would be to follow a path that is associated with a line segment that connects the bottom left corner of the grid to a red dot and does not.
Example 3.14: Free Fall of a Ball. Figure \(\PageIndex{2}\) shows the positions of a ball, at 1-s intervals, with an initial velocity of 4.9 m/s downward, that is thrown from the top of a 98-m-high building. (a) How much time elapses before the ball reaches the ground? (b) What is the velocity when it arrives at the ground
through which the ball travels. The initial velocity can then be used to calculate where the ball . will land when the ball is shot at an angle. As with any other situation where motion has constant acceleration, the equations of . kinematics can be used to analyze an object in projectile motion. These equations can be see ANS: 10.7 m s 1 A ball is thrown directly downward, with an initial speed of 8.00 m s 1, from a height of 30.0 m. Calculate (a) the time taken for the ball to strike the ground, (b) the ball's speed when it reaches the ground. ANS: 1.79 s; 25.6 m s 1 2 The position of a ball as a function of time is given by x=(5.0 m/s)t+(-10 m/s2)t2.x=(5.0 m/s)t+(-10 m/s2)t2. (a) What are the initial position, initial velocity, and acceleration of the ball? (b) Plot x versus t for t=0t=0 to t=2.0 st=2.0 s . (c) Find the average velocity of the ball from t=0t=0 to t=1.0 st=1.0 s . (d) Find the average speed of the ball between t=1.0 st=1.0 s and t=2.0 st=2.0 s Here, I will shoot two balls straight up. In one case, I will measure the max height in order to find the launch speed (initial velocity). In the other cas.. We have given :- U(initial velocity) = 16.5 m/s (thrown upwards) A = g = 9.8 m/s^2 To find :- At what height the velocity of the ball would be half of its initial velocity. solution :- V^2 = U^2 + 2*A*S the final velocity would be zero so 0 = 16.5..
1. What is the initial position of the ball? What is its ..
Example. To make the arithmetic easy, let's use the approximation that g = 10 m/s 2 and throw a ball from the top of the Science Building and look at its velocity and position.. We throw the ball so it moves up with an initial vertical velocity of v yo = 20 m/s and so it moves horizontally with an initial horizontal velocity of v xo = 15 m/s
A football player launches the ball from initial position x(0)=y(0)=0, with initial velocity v = vo cos 0 i + vo sin 0j. Neglecting drag, the ball's trajectory describes a parabola in the Oxy plane, where y is the vertical upward direction and x is the horizontal direction
(c) Similarly, we must integrate to find the position function and use initial conditions to find the constant of integration. (d) Since the initial position is taken to be zero, we only have to evaluate the position function at [latex]t=0[/latex]. Solution. We take t = 0 to be the time when the boat starts to decelerate. Show Answe
When a ball is thrown upwards, what is the relationship between kinetic and potential energy at the initial position and maximum height reached by the ball? can neither be created or destroyed. It can only be converted from one form to another. Moreover the mechanical energy (sum of its potential and kinetic energy ) remains same
What is the initial position of the ball What is its final
When a ball is thrown vertically upward it starts its vertical motion with an initial velocity. How does an electroscope detect charge and tell the sign of a charge? What happens when a ball is thrown vertically upward? its velocity becomes zero at that height. ball thrown upward vertically, maximum height? As height rises, velocity falls which results in a reduction of KE and a corresponding.
A ball is kicked with an initial velocity of 13 m/s in the horizontal direction and 17 m/s in the vertical direction. (Assume the ball is kicked from the ground. For each answer, enter a number.) (a) At what speed (in m/s) does the ball hit the ground? m/s (b) For how long (in s) does the ball remain in the air
position of the ball, v. ix. is the initial velocity in the horizontal direction, t. is the elapsed time, v. fy. is the final velocity in the vertical direction, v. iy. is the initial velocity in the y-direction, a. is the acceleration in the y-direction, y. i. is the initial height of the ball, and . y
g apparatus to directly measure the time it takes for the ball to move a fixed distance. • Using the scales, find the mass of the metal ball,m
1. What is the initial position of the ball?2. What is the ..
The initial position is where the ball exits the launcher. Notice that the ball reaches a maximum height of 2 meters and stopsinstantaneously when it reaches that height. Repeat steps 1.5-1.6 five times and use the average time for calculations. 2. Motion in 2 dimensions In this situation, the initial position is and the final position is , which is the equilibrium position of the spring. What kind(s) of energy does the system spring-ball have at the initial position? ANSWER: Keep in mind that at the equilibrium position of the spring. The inital position defined at will have negativ
Consider the whole motion of the ball from time t = 0 to t = 3.6 to the vertical direction. When you consider the initial position it is point 1 as in your link and final position is 2 • A pool ball leaves a .60-meter high table with an initial horizontal velocity of 2.4 m/s. Predict the time required for the pool ball to fall to the ground and the horizontal distance between the table's edge and the ball's landing location The figure (Figure 1) shows the trajectory (i.e., the path) of a ball undergoing projectile motion over level ground. The time t0=0s corresponds to the moment just after the ball is launched from position x0=0m and y0=0m. Its launch velocity, also called the initial velocity, is v⃗ 0 A car starts at position x = 16 feet. After 8 seconds the car is 134 feet east of its initial position. What is the car's average velocity? Step One: Find the difference between the initial and final positions of the car. The car traveled from 16 feet to 134 feet (134 - 16 = 118). The car traveled east a total of 118 feet
1.what is the initial position of the ball?2.what the ..
If your ball position is too far forward in your stance (toward the foot closest to the target, left for a right-hander), the club will be released too early. Your shots will fly to low, because the clubface is closed at impact. If the ball is too far back in your stance, the club will not have enough time to release, causing high, pushed shots to the right due to an open clubface
The position of a particle moving along the x-axis varies with time according to . m. Find (a) the velocity and acceleration of the particle as functions of time, (b) the velocity and acceleration at t = 2.0 s, (c) the time at which the position is a maximum, (d) the time at which the velocity is zero, and (e) the maximum position
• The initial position of the ball is not level with the tabletop. You will need to measure this, but the apparatus has been designed well so that this value won't change when you change the launch angle. Caution Do not load the launcher while your head or body is in the line of fire. Do not allow anyone to get hit by the ball..
A forward ball position naturally makes my body fall into a power setup. (See No. 1 on the image.) That means my head is well behind the ball (No. 2) with the shaft leaning back and my right.
for the initial height of the ball , the initial speed of the ball , the distance to the net , and the launch angle (see Fig. 2). Fig. 2: Parameter Block settings Step 4 : Adjust the parameters Return to the main diagram ( > ) and, with a single click on the Parameters icon, enter the following parameters (Fig. 3) in the Inspector tab
swing down and around and just reach the vertically up position, with zero speed there. How much work is done on the ball by the gravitational force from the initial point to (a) the lowest point, (b) the highest point, and (c) the point on the right level with the initial point
A ball of mass 250 g is thrown with an initial velocity of 25 m/s at an angle of 30 ° 30° with the horizontal direction. Ignore air resistance. What is the momentum of the ball after 0.2 s? (Do this problem by finding the components of the momentum first, and then constructing the magnitude and direction of the momentum vector from the.
Figure shows the positions of a ball, at 1-s intervals, with an initial velocity of 4.9 m/s downward, that is thrown from the top of a 98-m-high building. (a) How much time elapses before the ball reaches the ground
initial ball position to where it lands.) Compare this to your predicted value. (Note: Be sure your gun is not cocked before you return it to the cart.) Lab Questions and Conclusions Write a description of how you used the equation for Conservation of Energy, Conservation of Momentum and Projectile Motion in this lab (next page)..
Free Fall of a Ball shows the positions of a ball, at 1-s intervals, with an initial velocity of 4.9 m/s downward, that is thrown from the top of a 98-m-high building. (a) How much time elapses before the ball reaches the ground? (b) What is the velocity when it arrives at the ground
A ball is rolling along a horizontal table (ignore friction) with a velocity of -36.3 m/s. At t= 0 s, the ball is at the initial position 4.3 m, when some unknown force causes a constant acceleration on the ball of a = + 3.7 m/s2. (Done on Blackboard, entered below for 3 pts. Explanation should be done on the Gradescope submission for 4 pts.
e the position and velocity functions for the coin. my work: based on the notes i thought the functions would be v (t) = -16t^2 and s (t)= -16t^2 + 1362. v (t) = u + at. in this case: v (t) = -32 * t
2. A ball is tossed from an upper-story window of a building. The ball is given an initial velocity of 8m/s at an angle of 200 below the horizontal, where it strikes the ground 3 seconds later. a. How far horizontally from the base of the building does the ball strike the ground? b. At what height was the ball thrown? c To find the displacement from the initial position where the ball reverses direction, we find the kinematics equation that contains and the given quantities. Examining our equations we see that we can use . Rearranging this equation to find yields . Notice that this value is bigger than the original 5.5 rad and is. Also, I can (to further overkill the point) create a plot of the trajectory (x position vs. y position) for a ball thrown at different angles. This is done with vpython. Here the angle is changed. Ball A, with a mass of 4 kg, is swung back to a point 0.8 m above its equilibrium position. Ball A is released from rest and swings down and hits ball B. After the collision, ball A rebounds to a height of 0.2 m above its equilibrium position, and ball B swings up to a height of 0.05 m. (a) How fast is ball A going just before the collision
Solved: The Position Of A Ball As A Function Of Time Is Gi
The center of the ball now moves another centimeter before the ball stops and v = 0. We can use v xf 2 - v xi 2 = 2a x (x f - x i) to find the average acceleration of the ball during this time interval.-(5 m/s) 2 = 2 a x 0.01 m. a x = -1205 m/s 2. Problem: A ball rolls up an incline, and then rolls back down to its initial position A ball thrown up vertically returns to the thrower after 6 seconds. Find (i) velocity (ii) Maximum height it reaches (iii) Position after 4 seconds - Get the answer to this question and access a vast question bank that is tailored for students travel from this initial position before it catches up with car B? A) 6.7 m vertically and more than 38 m horizontally B) 38 m horizontally and less than 6.7 m vertically C) more than 6.7 m vertically and less than 38 m horizontally D)less than 38 m horizontally and less than 6.7 m vertically 7.Without air resistance, a kicked ball would reach
1. Put on your safety glasses. 2. Measure the vertical distance from the bottom of the ball's launch position in the barrel (this position is marked on one side of the barrel) to the top of the strike plate. 3. Put the yellow plastic ball into the projectile launcher and cock it to the short range position. 4 initial position. We know that at t = 0, the velocity components are vx = 0 and vy = 50 m s and the coordinates are x = 0 and y = 0. From the acceleration a we do know something about the velocity. Since the acceleration is the time derivative of the velocity: a = dv dt, v = 1 1 1 3 3 2. Find the initial velocity, v0, of a ball rolling o↵the table in the figure below. The launch position is the origin of the coordinate system, positive directions as specified. (25 pts) 3. For a ball shot with an initial speed of 8.0 m/s at 0 = 30 ,findv0x and v0y. Always write the algebraic equation
The ball swings away in an arc and returns, barely missing the person's face. It helps people understand a harmonic oscillator and conservation of energy (and maybe scares them just a little, too. 9. Consider a ball that is thrown straight upward at the edge of a canyon with an initial velocity of 20 m/s. Three seconds later, where is it located? Take its initial position, at the edge of the canyon, to be the origin; that is, y i = 0. a) 30 m b) 15 m. c) - 10 m. d) - 30 m. Answers to the multiple-guess questions: 1 Reasoning Given the initial velocity, it is the acceleration due to gravity that determines how long the ball stays in the air. Thus, to find the time of flight we deal with the vertical part of the motion. Since the ball starts at and returns to ground level, the displacement in the y direction is zero
what is the position of the ball at 10 seconds? - Brainly
Projectile Motion - PhET Interactive Simulation
Suppose the ball in Figure 8.1 has an initial velocity v 0 and a mass m. If the spring constant is k, what is the maximum compression of the spring ? In the initial situation, the spring is in its relaxed position (U = 0). The total energy of the ball-spring system is given by. The maximum compression of the spring will occur when the ball is.
Just think about throwing a ball against a solid wall. The harder you throw the ball against the wall, the harder it bounces back. That is the reason it is easier to hit a home run on a fastball than on a curveball. Conservation of momentum also means that the bat can transfer some of its momentum to the ball
Section A Motion of a golf ball Suppose a golfer hits a ball with a velocity of 45 m s-1 at an angle of 20° to the horizontal. The projectile model can be used to answer some questions about what will happen to the ball later during its flight. Finding the position at a later time . Where will the ball be 2 seconds later? Horizontal motion.
3 Example 2: Find the velocity, acceleration, and speed of a particle given by the position function r(t) =2cost i +3sint j at t = 0.Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of t. Solution: We first calculate the velocity, speed, and acceleration formulas for an arbitrary value of t.In the process, we substitute and find each of.
12.3. The Calculus of Motion. A common use of vector-valued functions is to describe the motion of an object in the plane or in space. A position function →r(t) gives the position of an object at time t. This section explores how derivatives and integrals are used to study the motion described by such a function
Directions. answer the following questions. Write the ..
Calculate the magnitude of change of momentum occurred in the motion of the hockey ball by the force applied by the hockey stick. Solution. Given, mass of the ball (m) = 200g. Initial velocity of the ball (u) = 10 m/s. Final velocity of the ball (v) = - 5m/s. Initial momentum of the ball = mu = 200g × 10 ms-1 = 2000 g.m.s- A steel ball of mass m1 = 0:9kg and a cord of length of L = 2m of negligible mass make up a simple pendulum that can pivot without friction about the point O, as in the gure below. This pendulum is released from rest in a horizontal position, and when the ball is at its lowest point it strikes a block of mass m2 = 0:9kg sitting at rest on a. A lead ball of mass 0.25 kg is swung round on the end of a string so that the ball moves in a horizontal circle of radius 1.5 m. The ball travels at a constant speed of 8.6 m s-1. (a)€€€€ (i)€€€€€€Calculate the angle, in degrees, through which the string turns in 0.40 s
The ball, m 1, and bat, m 2, both have initial velocities before the collision (subscript b), with the ball's velocity being negative. After the collision (subscript a) both bat and ball have positive velocities
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The position of a ball as a function of time is given by x
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\begin{definition}[Definition:Time/Unit/Great Year of Plato]
The '''great year of Plato''' is a derived unit of time.
{{begin-eqn}}
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| r = 1
| c = '''great year of Plato'''
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{{eqn | r = 12 \, 960 \, 000
| c = days
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{{eqn | o = \approx
| r = 36 \, 000
| c = years (of $360$ days)
}}
{{end-eqn}}
{{NamedforDef|Plato|cat = Plato}}
\end{definition} | ProofWiki |
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In-plane behaviour of clay brick masonry wallettes retrofitted with single-sided fabric-reinforced cementitious matrix and deep mounted carbon fibre strips
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A. T. Vermeltfoort1
Bulletin of Earthquake Engineering volume 18, pages725–765(2020)Cite this article
The in-plane shear behaviour of a new seismic retrofit concept which combines two standalone retrofit measures for in-plane and out-of-plane strengthening of masonry walls was investigated. The in-plane reinforcement consists of a single-sided carbon fabric-reinforced cementitious matrix (FRCM) overlay, while the out-of-plane reinforcement consists of deep mounted carbon fibre reinforced polymer strips embedded in a viscous-elastic epoxy. An experimental program was undertaken in which clay brick masonry wallettes were subjected to the diagonal compression test to assess the effectiveness of the strengthening system on the in-plane behaviour. The obtained results showed that the single-sided carbon FRCM overlay increased the shear capacity with 80%, compared to the unstrengthened control specimens. Moreover, by testing two different FRCM overlay thicknesses it was found that a thicker matrix layer does not increase the shear capacity of wallettes. However, wallettes provided with a thicker FRCM overlay did show a higher level of ductility. Furthermore, the obtained experimental results showed that the presence of only the aforementioned out-of-plane reinforcement does not affect the in-plane strength of masonry wallettes loaded under shear, and even prevented the disintegration after reaching the failure load compared to the unstrengthened control specimens. Finally, an existing analytical model as well as the Eurocode 8 design provisions were compared to the found failure mechanisms and failure loads. The analytical model developed showed good correspondence with the experimental values for both the failure mechanism and failure load, with an experimental/model ratio \(\left( \varphi \right)\) of 0.98, while Eurocode 8 proved to lead to conservative values.
In Groningen, an area in the Northeast of the Netherlands, earthquakes occur because of gas production from the Groningen field. Decades of gas production led to the depletion of the pressure of hydrocarbon gas within the reservoir pore space, causing the reservoir to compact. In turn, this compaction increases the mechanical loads acting on pre-existing geological faults within and close to the reservoir. Some small fraction of these faults become unstable and are therefore prone to slip. Abrupt slip on such a fault results in an earthquake that radiates seismic energy (Bourne and Oates 2017). Although the magnitude of these induced earthquakes on Richter's scale is relatively low (< 3.5), they have a big impact on the buildings in the region due the soft surface soils in the area and the shallow depth (3 km beneath earth's surface) at which they occur (Van Thienen-Visser and Breunese 2015). As the majority of the buildings in Groningen are composed of cavity walls with single whyte load bearing walls of unstrengthened clay brick masonry, and are designed to only resist wind loads, it is essential to improve the earthquake resistance of the existing buildings in the area to prevent collapse, with likely casualties. A broad range of strengthening techniques for enhancing the capacity of Unstrengthened Masonry (URM) walls are available nowadays. Traditional strengthening methods such as reinforced concrete jacketing and steel frames, however, add considerable mass to the structure, are labor intensive, and generally alter the esthetics of a building (Triantafillou 1998). These disadvantages led to the idea of using Fibre Reinforced Polymers (FRP) composites for strengthening of masonry. Typically, these materials are made of Carbon (CFRP), Glass (GFRP), Basalt (BFRP) or Aramid (AFRP) fibres bonded together by an epoxy-resin. The main advantages of FRP include high strength, high stiffness, low weight and immunity to corrosion (Ianniruberto and Rinaldi 2001). Initially, FRP was used in the form of Externally Bonded (EB) sheets for both in-plane and out-of-plane strengthening of masonry. In this method firstly the surface of the substrate is prepared by removing contamination and weak surface layers, after which a FRP sheet is adhesively bonded to the substrate by means of an organic resin. This strengthening system has proven to be highly effective in enhancing both the shear capacity, the flexural capacity and the ductility of masonry walls. The main disadvantages of this method were however found to be vulnerability to environmental influences, vulnerability to fire, high cost of epoxies, lack of vapor permeability, inability to install the system on damp substrate and inability to install the system at low temperatures (Papanicolaou et al. 2008; Petersen 2009; Banijamali et al. 2015).
All these drawbacks can be mainly attributed to the organic resins used to bind the FRP to the substrate, and therefore a logical solution was the replacement of the organic binder by an inorganic binder (e.g. cement-based mortar). Moreover, continuous fibre sheets were replaced by textiles (FRP meshes) in order to achieve mechanical interlock between the textile and the cement-based mortar, since these inorganic binders lack the ability to penetrate and wet individual fibres (Papanicolaou et al. 2008). The resulting strengthening system of cement-based mortar matrix reinforced by continuous dry-fibre fabric is known under different appellations, one being fabric-reinforced cementitious matrix (FRCM) (Nanni 2012). Arboleda et al. (2014) studied the durability characteristics of the carbon FRCM composite system. Environmental stresses such as frost and chemical attack were addressed with exposure environments such as freeze/thaw cycles, high temperature water vapor and immersion in seawater. The authors concluded that no significant loss of residual tensile strength and bond strength were observed under the aforementioned conditions.
Previous experimental studies on FRCM reinforced masonry have highlighted a significant improvement for in-plane shear capacity. Mantegazza et al. (2006) performed diagonal compression tests on 11 single whyte clay brick masonry wallettes, both unstrengthened and strengthened. Based on the tests the author stated that the masonry portion involved in the load resisting mechanism is larger in FRCM strengthened specimens than that involved in unstrengthened specimens. In contrary to the unstrengthened specimens, multiple cracks were visible on these specimens. The specimens provided with a double layer of FRCM showed debonding failure. The authors found that the strengthening system modified the failure mechanism and increased both the in-plane strength and stiffness of a wallette. The increase in strength was found to be greatest for the double-sided FRCM specimens. Babaeidarabad, De Caso and Nanni (2013) carried out an experimental campaign aimed at assessing the effectiveness of carbon FRCM for the strengthening of clay brick masonry and evaluating the validity of an existing analytical model. Test results showed that the increase in ultimate in-plane strength is proportional to the amount of FRCM and ranged between 2.4 and 4.7 times that of the unstrengthened specimens. Moreover, the authors reported that substrate toe-crushing failure occurred for wallettes with a calibrated reinforcement ratio higher than 4%, and therefore increments of FRCM beyond this value are ineffective according to the researchers. Additionally, test results revealed that the FRCM strengthening method also effectively increases both the stiffness and ductility of wallettes. The increase in ductility was found to be higher for the specimens provided with 1-ply than for 4-ply FRCM to both sides. From these test results the authors inferred that the ductility of 4-ply strengthened masonry wallettes were limited by toe-crushing failure prior to FRCM failure. Thus, the failure modes of FRCM strengthened panels is directly influenced by the strengthening scheme. Ismail (2012) investigated the in-plane behaviour of double wythe clay masonry wallettes strengthened with different types of FRCM systems. The shear strength of single-sided retrofitted wallettes ranged from 113 to 148% compared with the strength of the unstrengthened wallettes, whereas the shear strength of test wallettes with a double-sided FRCM retrofit ranged from 446 to 481%. The author attributed the lower increase in shear strength for the single-sided retrofitted specimens to the unrestrained boundary conditions of the diagonal compression tests, as these specimens showed out-of-plane bending behaviour. Therefore, the researcher proclaimed that these values must be regarded as conservative. The author reported that in reality walls have more restrained boundary conditions and super imposed vertical loads and therefore larger shear strength increments can be achieved.
For the out-of-plane strengthening of masonry walls the FRCM system requires application to both faces of a wall. In case of strengthening of load bearing leafs of cavity walls, this would require the removal of the façade of a building for the installation of the FRCM layer on the cavity side of the wall, as well as temporary rehousing of the occupants in order to install the FRCM layer from within the building. This would therefore be a very costly operation. The cost effective retrofitting can be enhanced by implementing the near surface mounted (NSM) out-of-plane reinforcement technique, where FRP strips or rods are inserted into grooves cut in the surface of a wall. The NSM technique proved to be a feasible strengthening method. Compared to the EB technique this method leads to a significantly higher axial strain at debonding and a reduced construction time (Seracino et al. 2007; Petersen et al. 2009). However, since the FRP strips or rods are placed right underneath the walls' surface, double-sided application is required for strengthening for both out-of-plane loading directions, and therefore leading to the same drawbacks as mentioned for the FRCM system. The deep mounted technique was accordingly developed where deeper grooves are cut in the masonry, after which FRP strips are installed in the center of the wall. The FRP strips therefore offer additional out-of-plane flexural strength to the wall for both out-of-plane loading directions whilst only installing the reinforcement from one side of the wall, leading to cost-effective retrofitting (Türkmen et al. 2016; Türkmen et al. 2017). This system uses a viscous-elastic epoxy instead of a conventional stiff epoxy for the installation of the FRP strips, since previous out-of-plane bending tests on strengthened masonry panels by Türkmen et al. (2016) showed that by using a viscous-elastic epoxy (Young's modulus < 50 N/mm2) instead a conventional stiff adhesive (Young's modulus ~ 10,000 N/mm2), a significant increase in terms of ductility and maximum withstandable load is reached and critical crack development is prevented. Similar findings were previously reported by Kwiecień (2012) and Derkowski et al. (2013). Additionally, the improved stress distributions over the bonded length due to the application of a flexible adhesive (Türkmen et al. 2018) made the deep mounting of the CFRP strips possible. The use of a conventional stiff adhesive for deep mounting results in premature splitting failure in the masonry as observed by Dizhur et al. (2014) during their direct pull-test experimental campaign.
A schematic overview of the proposed combined reinforcement concept is shown in Fig. 1. Previously static-cyclic in-plane shear tests were performed on full-scaled masonry specimens strengthened with this combined reinforcement system (Türkmen et al. 2018). Within the full-scale wall experimental program rocking and sliding failure were observed, but no shear failure was observed in the surfaces for any of the specimens. It was therefore considered essential to perform additional tests to determine the in-plane shear capacity of walls strengthened with this combined system. The diagonal compression test was selected for this purpose. While the influence of FRCM reinforcement on the in-plane behaviour of masonry wallettes has been a popular subject of research for the past years, the influence of the aforementioned combination of retrofit measures on the in-plane shear capacity of masonry has not been investigated before. Moreover, this experimental program aims to investigate the possible degrading effect of the proposed out-of-plane strengthening system on the in-plane shear strength of masonry panels. Finally, the experimental results will be compared to the outcomes of existing analytical models and design codes, to check the validity of these models for this combined retrofit system.
Reinforcement concept for a house with cavity walls, where the FRCM reinforcement and deep mounted CFRP strips are installed from either the inside (temporary rehousing occupants) or the outside (removal of the façade) on the load bearing walls to secure cost-effective retrofitting
Experimental program
Materials and characterization
The clay bricks used in this research had dimensions of 205(± 4) × 95(± 2) × 50(± 2) mm3\(\left( {l_{b} \times w_{b} \times h_{b} } \right).\) Several mechanical characteristics of the used clay bricks were determined with an experimental program conform the corresponding standards. The bricks had a mean compressive strength of 31.7 N/mm2 (n = 12; COV = 7.4%), where n and COV are the amount of tested specimens and the Coefficient of Variation respectively. The compressive strength of the clay bricks were determined following the EN 772-1 (2015) standard using gypsum capping and half bricks. The mean splitting tensile strength (determined conform ASTM C1006-07 2007) and flexural tensile strength (obtained following ASTM C67-03 2003) of the bricks were found to be 3.34 N/mm2 (n = 12; COV = 8.7%) and 5.89 N/mm2 (n = 9; COV = 7.4%). The mean flexural tensile strength of the mortar specimens was found to be 3.6 N/mm2 (n = 8, COV = 16.5%), and the mean compressive strength of the mortar was 10.6 N/mm2 (n = 16; COV = 20.7%). Both the flexural tensile strength and the compressive strength of the mortar specimens were determined according to EN 1015-11 (2007a). Compression tests were performed on three masonry specimens, consisting of 6 brick high masonry prisms, under displacement control with a loading speed of 0.20 mm/min. The average compressive strength \(\left( {f_{m} } \right)\) of the specimens was 14.8 N/mm2 (COV = 6.1%). The Young's modulus was determined as a secant modulus at 35% of the compressive strength in accordance with EN 1052-1 (1998). The average modulus of elasticity of the masonry prims was found to be 3100 N/mm2 (COV = 2.5%).
In order to determine the mechanical properties of the masonry under shear in accordance with EN 1052-3 (2007b), a total of 9 triplet shear tests was performed at three different normal stress levels: 0.2, 0.6 and 1.0 N/mm2. For each specimen the relation between the applied normal stress and the shear strength has been established. The ratio between the compressive stress and the shear strength of the masonry was obtained using a linear regression. The parameters for the Coulomb's friction criterion follow from Eq. (1):
$$f_{v} = f_{v,0} + \mu_{ma} \sigma_{n}$$
with \(f_{v,0}\), \(\mu_{ma}\) and \(\sigma_{n}\) being the initial shear strength, friction coefficient and axial load respectively. The residual shear strength \(\left( {f_{{v,0,{\text{res}}}} } \right)\) and residual coefficient of friction \(\left( {\mu_{ma,res} } \right)\) were determined by applying the same linear regression analysis when a plateau was reached in the post-peak phase. The mechanical properties of the materials used for building the specimens for this study are summarized in Table 1.
Table 1 Mechanical properties of the masonry materials used for building the specimens
The obtained values regarding the mechanical properties of the masonry under shear were compared with the results of the study carried out by Jafari et al. (2017) on the material properties characterization of Dutch URM, and the values proposed in the Dutch Practical Guideline for the seismic assessment of local buildings in Groningen, NPR 9998 (2018). From the comparison it was concluded that the shear properties of masonry used in this study showed an acceptable agreement with shear properties as obtained by Jafari et al. (2017) and proposed in NPR 9998 (2018).
The reinforced mortar used for the mortar matrix was a polymer modified mortar based on organic binders, polymer fibres and selected aggregates, with a maximum grain size of 1.8 mm. The polymer fibres are shown in Fig. 2a. The additional reactive components, which were mixed into the reinforcement mortar, bonded with the amorphous silica on the carbon FRP mesh. This ensured an improved adhesion between the mesh and the cementitious matrix. For the preparation of the reinforced mortar, a plastic bonding agent was used in order to improve the adhesion of the cementitious matrix to the clay brick substrate. This was done by mixing 110 grams of the plastic bonding agent per 10 kg of prepared mortar. The reinforced mortar was prepared following the manufacturer's instructions by adding 2.6L of water to a bag of 25 kg dry mortar. Both the flexural tensile strength and the compressive strength of the reinforced mortar specimens were determined according to EN 1015-11 (2007a). The average flexural tensile strength of the reinforced mortar specimens was found to be 7.58 N/mm2 (n = 9; COV = 11.7%). The mean compressive strength of the reinforced mortar was 62.6 N/mm2 (n = 12; COV = 1.6%). The weight density was 2138 kg/m3 (n = 6; COV = 1.7%).
Photo of the materials used for reinforcement: a close-up of the dry reinforced mortar, showing the polymer fibres b carbon FRP mesh with the amorphous silica intended for improved adhesion; c CFRP strip
The bidirectional carbon FRP mesh, with a fibre weight density of 1.79 g/cm3 and about 3 mm width per thread, had a square aperture dimension of approximately 50 × 50 mm2. The theoretical cross section of the C-fibre for design was 44 mm2. The Young's modulus, tensile strength and elongation at rupture of the mesh as provided by the supplier are > 240 kN/mm2, > 4300 N/mm2 and 1.75% respectively for the carbon FRP mesh. The prefabricated (pultruded) CFRP strips were 20 mm in width and 1.4 mm in thickness and have a fibre volume content of > 68%. The Young's modulus, tensile strength and elongation at rupture of the CFRP strip were found to be 215 kN/mm2, 2876 N/mm2 and 1.59% respectively.
The material properties for the two-component viscous-elastic adhesive were obtained following ISO 527-1 (2012) using three specimens at a loading rate of 10 mm/min. The Young's modulus was determined as the secant modulus between 0.5% and 5% of the tensile strength, and was found to be 16.0 N/mm2 (COV = 1.7%). The tensile strength and elongation at rupture were determined as 4.3 N/mm2 (COV = 0.9%) and 72.1% (COV = 3.5%)
Tensile tests were performed using a clevis-type gripping mechanism following the American guideline AC434.13 (2013) with a metal tab contact length of 150 mm as recommended by Donnini and Corinaldesi (2017). The uncracked FRCM slab had a tensile strength \(\left( {\sigma_{m} } \right)\) of 4.31 N/mm2 (n = 9; COV = 14.9%), a corresponding strain \(\left({\epsilon_{m}} \right)\) of 0.016% (n = 8; COV = 13.5%) and a Young's modulus of 27,680 N/mm2 (n = 8; COV = 6.9%). The stresses for the uncracked case were calculated with respect to the cross section area of the reinforced mortar. For the cracked FRCM slab, the stresses were calculated with respect to the cross section area of the CFRP mesh. The ultimate stress in the mesh, \(\sigma_{ult}\), was found to be of 1628 N/mm2 (n = 9; COV = 10.2%), with a corresponding ultimate strain \(\left( {\varepsilon_{ult} } \right)\) of 1.91% (n = 6; COV = 14.9%). The Young's modulus of the cracked specimen, calculated as the slope of the segment of the stress–strain diagram between 0.90 \(\sigma_{ult}\) and 0.60 \(\sigma_{ult}\) (following AC434.13 2013), was 70,920 N/mm2 (n = 6; COV = 15.3%). The main failure mode observed was slippage of the CFRP mesh within the mortar matrix. The mechanical properties of the materials used for reinforcing the specimens for this study are summarized in Table 2.
Table 2 Mechanical properties of the materials used for reinforcing the specimens
Building the test specimens
The specimens for the diagonal compression tests were built in the testing laboratory of QuakeShield in Grijpskerk, the Netherlands. A total of 13 half brick clay masonry wallettes \(\left( {n_{test} } \right)\) were built by an experienced mason. All specimens had a square geometry of about 700 × 700 mm2\(\left( {h_{w} \times l_{w} } \right)\), a nominal thickness of 95 mm \(\left( {t_{w} } \right)\) and a nominal mortar thickness of 12 mm. The panels had reduced dimensions compared to the prescriptions of the ASTM E 519-02 (2010) standard (1200 × 1200 mm2) due to the geometrical limitations of the test setup. The masonry specimens were constructed against a vertical sideboard to ensure minimum horizontal deviation. Because of this construction method, the mortar layer thickness of the sideboard side seemed thicker due to the mortar flowing out in the gap between the masonry specimen and the sideboard. The masonry specimens were left to cure for at least 28 days in the unheated laboratory (8–18 °C) before retrofitting.
Reinforcing the masonry specimens
Figure 3 presents photographs of the installment of the reinforcement system in a practical application. The reinforcement process however is uniform. A schematic overview of the different specimens in this study is provided in Fig. 4. Details and geometrical properties of the specimens are provided in Fig. 5 and Table 3. Four of the 13 specimens were left untreated (URM). After the walls were sufficiently cured the retrofitting process of the other specimens started by milling vertical grooves of 65 mm deep \(\left( {d_{f} } \right)\) and 10 mm wide \(\left( {b_{f} } \right)\) at the center of the wallettes (Fig. 3a). The dust in the groove was removed with compressed air. The CFRP strips with a cross-section of 20 × 1.4 mm2\(\left( {b_{p} \times t_{p} } \right)\) were cleaned with acetone after cutting the strips into the specified length. A layer of primer was then applied to the groove (Fig. 3b) to obtain an improved bond of the applied adhesive to the masonry. After partially filling the groove with the flexible adhesive (Fig. 3c), the CFRP strip was inserted into the groove using a positioning fork (Fig. 3d). Excess adhesive till a depth of 30 mm \(\left( {d_{ff} } \right)\) in the grooves was removed by using a scraper. After the placement of the strips, the adhesive was left to cure for one day.
Photos showing the different stages of the reinforcement process, taken at a retrofitted building by QuakeShield: milling the grooves (a), cleaning the grooves with acetone (b), injecting flexible adhesive in the groove (c), pushing the CFRP strip into position with a positioning fork (d), if only CFRP strips need to be installed, filling the remaining part of the groove with mortar (e), if an FRCM layer needs to be added, installing the first layer of reinforced mortar (f), pressing the CFRP mesh into the mortar (g), application of the second layer of mortar matrix (h)
Schematic overview of the URM, STRIP, COMB10 and COMB20 specimens
Detail of the reinforced specimens
Table 3 Geometrical properties of the reinforced specimens
The remaining unfilled parts of the grooves were with filled mortar (same mortar used for the FRCM layer). This was done with the purpose of partially restoring the compressive and shear capacity in the groove in order to prevent possible vertical shear failure. The masonry surface was wetted prior to the mortar application to prevent shrinkage. On three specimens no FRCM layer was installed (STRIP). A photo of a STRIP specimen is provided in Fig. 6. For the remaining specimens (COMB10 and COMB20), a thin layer of mortar was subsequently applied to the masonry surface by hand. The CFRP mesh was then applied on the mortar matrix surface and was pressed into the matrix. After placing the CFRP mesh in the mortar a new thin layer of mortar was applied to embed the CFRP mesh, resulting in a nominal FRCM layer thickness \(\left( {t_{FRCM} } \right)\) of 10 mm and 20 mm for the COMB10 and COMB20 specimens respectively. Due to the added FRCM layer, the mass of the COMB10 and COMB20 specimens increased with approximately 10.5 kg (21.4 kg/m2) and 21.0 kg (42.8 kg/m2) respectively. To ensure the compression load being applied only on the masonry, the FRCM thickness was reduced close to panel boundaries. The specimens were left to cure for an additional 28 days. Figure 7 shows a photo of a COMB specimen.
Photo of a STRIP specimen with a CFRP strip (marked) embedded in a flexible adhesive
Photo of a COMB specimen with a CFRP strip (marked) embedded in a flexible adhesive and FRCM reinforcement layer with CFRP mesh (marked)
When only the CFRP mesh, CFRP strip or the FRCM layer is considered, the specimens had a reinforcement ratio of \(\rho_{r,mesh} = 0.046\%\), \(\rho_{r,CFRP\,strip} = 0.042\%\) and \(\rho_{r,FRCM} = 10.5\%\) (per 10 mm layer thickness) based on the cross sectional areas. It should be noted that in practice, the reinforcement ratio of the CFRP strip is variable, as the CFRP strips can be positioned closer or further apart from each other depending on the design lateral load.
Test setup and procedure
To investigate the behaviour of the retrofit system under in-plane loading, the diagonal compression test was chosen. The diagonal compression test, as described in ASTM E 519-02 (2010), is regarded as a simple procedure to determine the shear strength of masonry elements. The principle of the test is depicted in Fig. 8a. The diagonal compression test was introduced to simulate a pure shear stress state, in accordance with the situation depicted in Fig. 8b. Under these conditions the Mohr's circle of the stress states are reduced (Fig. 8c), leading to the corresponding value of average shear stress following Eq. (2):
$$\tau = \frac{P}{{\sqrt 2 A_{n} }}$$
where \(P\) and \(A_{n}\) are respectively the compressive force applied to the specimen and the cross sectional area (parallel to the bed joint) of the specimen. The principal tensile stress (σI) is hence equal to the shear stress. Using Eq. (2) and the ultimate force \(P_{max}\) leads to the shear strength as provided in Eq. (3), where \(f_{v}\) and \(P_{max}\) are respectively the shear strength and the compressive failure load of the specimen:
Illustrations showing: principle of test (a), pure shear stress state (b) and Mohr circle (c)
$$f_{v} = \frac{{P_{max} }}{{\sqrt 2 A_{n} }}$$
The diagonal compression tests were performed at the Structures Laboratory of Eindhoven University of Technology. The tests were performed on a Schenk-Trebel servo hydraulic compression machine with a maximum capacity of 2.5 MN. The test setup consisted of a data acquisition system and a monitoring system consisting of four Linear Variable Displacement Transducers (LVDT's) with a measuring range of + 2 to − 2 mm and an accuracy of ± 1/500 mm. A vertically orientated LVDT in the middle of both sides of the specimens measured the vertical deformation, while two horizontally positioned LVDT's (one on each side of the specimen) monitored the horizontal deformations. A schematic overview and photo of the setup are provided in Figs. 9 and 10 respectively. A steel v-shaped loading shoe at the top and bottom side of the specimens was used to apply the compressive load to the specimens. The steel shoe consisted of two 20 mm thick steel plates with two 50 mm thick steel blocks in between (attached with M16 bolts). The steel blocks were perpendicular to each other and had a length of 100 mm, as illustrated in Fig. 11. The steel shoes were provided with 10 mm thick softboard to prevent local stress concentrations near the supports. Figure 12 shows a photo of the loading shoe. It should be noted that due to the reduced dimensions of the test specimens with respect to the ASTM E 519-02 (2010) standard, the confining effect produced by the v-shaped steel shoes could become more prominent and, consequently, result in a greater loading capacity of the tested specimens.
Illustration of the test setup
Photo of the test setup
Illustration of the loading shoe
Photo of the loading shoe
Each test was performed under displacement control by using the displacement measurement system of the testing machine. A displacement rate of 0.08 mm/min was used until a compressive force of 12 kN was reached (corresponding to the force needed to close the spacing of the ball hinge of the compression machine) after which the displacement rate was lowered to 0.04 mm/min for the remainder of the test. The tests were stopped when the compressive force dropped to zero or when significant damage occurred. During the tests the cracks were marked on the specimens and photographs were taken of the crack propagation.
Test results and discussion
The test results are summarized in Table 4. The failure load \(P_{max}\) and shear strength will be discussed first. Failure modes, shear strains \(\left( \gamma \right)\), shear strengths \(\left( {f_{v} } \right)\), shear moduli \(\left( G \right)\) and pseudo-ductility factors \(\left( \mu \right)\) will be covered in the following sections.
Table 4 Overview of the diagonal compression test results
The URM specimens had an average shear strength of 0.75 N/mm2, while the average shear strength of the masonry specimens reinforced with solely a DM CFRP strip was 0.77 N/mm2. From these results it can be concluded that despite the deep grooves, the shear strength of a masonry element is not affected by the out-of-plane reinforcement system. The average shear strength of COMB10 specimens was 1.24 N/mm2, which is 1.7 times the unstrengthened specimens' shear strength. For the COMB20 the shear strength was 1.36 N/mm2, resulting in a shear strength amplification factor of 1.8 compared to the URM specimens.
Table 4 also presents the spread of the of the strength values. A relatively high scatter in results was obtained for the unstrengthened specimens and the specimens reinforced with only a DM CFRP strip, compared to the FRCM reinforced specimens. This is however expectable considering the general behaviour of the URM, and the brittle failure that occurred during these tests. For the COMB20 specimens a considerably lower scatter in results was found. Due to the significant lower FRCM layer thickness of specimen COMB10-1 compared to COMB10-2 and COMB10-3, the shear strength was also considerably lower. This indicates that the matrix mortar layer thickness has an influence on the strength. However, no strong correlation was found between the FRCM layer thickness and the failure load \(\left( {R_{linear}^{2} = 0.4} \right)\).
Depending on physical and mechanical properties of a wall, four possible failure modes have been identified for URM (Li et al. 2005; Silva et al. 2008; Petersen 2009; Babaeidarabad et al. 2013; Babaeidarabad et al. 2014) and described by Li et al. (2005) and (Babaeidarabad et al. 2014):
Shear sliding\(\left( {V_{ss} } \right)\) failure takes place along a single bed joint caused by bond failure between clay brick and mortar.(Figure 13a).
Possible failure modes: Shear sliding (a), shear friction (b), diagonal tension (c) and crushing (d)
Shear friction\(\left( {V_{sf} } \right)\) failure is controlled by the loss of bond between the mortar and masonry units in the stepped-stair format. (Figure 13b).
Diagonal tension\(\left( {V_{dt} } \right)\) failure occurs when the principal tension stress produced by the combination of shear and compressive forces reaches the tensile strength of the wall. (Figure 13c).
Crushing\(\left( {V_{c} } \right)\) when the maximum stress on the edges of block exceeds the compressive strength of the masonry, compression failure can occur. (Figure 13d).
During the diagonal compression tests several types of failure mechanisms were observed. The crack patterns of the tested specimens are illustrated in Fig. 14. Photos of some specimens after testing are provided in Fig. 15.
Crack patterns and propagation of the specimens (as-built side for COMB10 and COMB20)
Photos after testing of URM-2 (a), STRIP-3 (b), COMB20-1 as-built side (c) and COMB20-1 reinforced side (d)
The failure behaviour of the unstrengthened specimens was brittle. Failure of these specimens was sudden and no considerable crack development was observed prior to failure. All URM specimens except URM-4, failed by the formation of one large crack parallel to the loading direction. The crack occurred sudden and immediately propagated over the height of the specimen, leading to brittle failure. The crack mainly followed a stair-stepped pattern, where cracking predominantly occurred at the interface between the units and the mortar (i.e. shear friction failure). Unlike the other control specimens, specimen URM-4 failed by shear sliding at the bed joint located at the second layer from the bottom of the specimen.
Specimens provided with only a DM CFRP strip showed mainly the same failure behaviour as the unstrengthened control specimens. Specimen STRIP-1 failed by shear sliding while the other two specimens, STRIP-2 and STRIP-3, showed stair-stepped diagonal cracking (shear friction failure). The STRIP specimens did not disintegrate like the unstrengthened specimens after reaching the failure load. This is attributed to the CFRP strip, holding the specimen together after failure. Specimens provided with both a DM CFRP strip and a single-sided FRCM overlay (COMB10 and COMB20) showed completely different failure behaviour. Contrary to the control specimens, these specimens behaved more ductile. When the failure load was reached a large diagonal tension crack was formed within these specimens on the as-built side, covering the complete vertical diagonal of the panels. Unlike the control specimens the strengthened specimens still possessed a considerable amount of capacity after reaching the failure load. During the course of the tests multiple cracks developed on the as-built surface of these specimens. Eventually hairline cracks were observed at the strengthened side (typical cracking displayed in Fig. 15d). Specimen COMB20-1 showed some additional masonry crushing near the bottom support at the final test stage.
Where the COMB10-1 and COMB10-2 specimens had two diagonal cracks parallel to the vertical diagonal on the as-built side, specimen COMB10-3 (with a nominal FRCM layer thickness of only 6.8 mm) showed two diagonal cracks at the bottom half and one diagonal crack at the top half of the specimen. In contrary to the COMB10 specimens, the COMB20 specimens showed three to four diagonal cracks over a wider area. A possible explanation for this discrepancy in crack pattern may lie in the difference in thickness of the upper mortar layer of the FRCM overlay. Grande et al. (2018) conducted a parametric analysis on the interaction between the CFRP reinforcement and the mortar matrix at the level of the interface under shear bond test conditions. The researchers served that an increase of the thickness of the upper mortar layer (within certain boundaries) and thereby an increase in axial stiffness, led to an increase of the force sustained by the reinforcement. It was found that the maximum force of a coupon with an upper mortar layer of a certain thickness was 1.4 times higher than the maximum force in absence of an upper mortar layer. This effect was reported to be strictly correlated to the increase of the length of the transfer zone (effective bond length) due to the increase of the axial stiffness of the upper mortar layer (Grande et al. 2018). The thicker upper mortar layer leading to an increased utilization of the carbon FRP mesh is in line with the experimental observations of this study of more cracks occurring at a wider area with an increased FRCM layer thickness.
Next to the mentioned failure modes, out-of-plane bending deformation on all the COMB10 and COMB20 specimens were observed towards the end of the conducted experiments. This observation is shown in Fig. 16 for the COMB20-3 specimen, where the dashed lines represent the specimen at initial condition and the solid lines illustrate the specimen at the end of the experiment. The out-of-plane bending was confirmed by the difference in cracks between the reinforced side and the as-built side of the specimens. Small cracks on the strengthened side and large cracks on the as-built side are associated with out-of-plane bending deformation (Prota et al. 2006; Parisi et al. 2019) for one side strengthened specimens subjected to diagonal compression tests. The cracks on the as-built side, as shown in Fig. 17 for COMB10-2, closed partly as the load was removed.
The out-of-plane deformation of the COMB20-3 specimen at the end of the test (solid line) with respect to the initial condition (dashed line)
Photos showing the opening of cracks towards the end of the test of the COMB10-2 specimen on the as-built side
Shear stress–strain diagram
The vertical shortening and horizontal elongation were computed from the mean displacement readings on both sides divided by the gauge length (g), using Eqs. (4) and (5) respectively.
$$\epsilon_{v} = \frac{{\Delta V_{1} + \Delta V_{2}}}{{2g_{v}}}$$
$$\epsilon_{h} = \frac{{\Delta H_{1} + \Delta H_{2}}}{{2g_{h}}}$$
The shear stress versus strain diagrams following from the experiments are shown in Fig. 18. Specimen URM-4 is not presented in Fig. 18 due to measurement errors. Additionally, measurements after a 20% drop in the post-peak phase are also not shown. The averaged shear stress versus strain diagram for the different configurations is shown in Fig. 19.
Shear stress versus strain diagrams
Averaged shear stress–strain diagram of the URM, STRIP, COMB10 and COMB20 specimens
Comparing the URM and STRIP specimens, no significant differences are noticeable. Both specimen types showed linear behaviour up to the point of sudden failure. In contradiction to the absence of residual strength for the URM specimen, the STRIP specimens had a mean residual strength of 11.2 kN, as shown in Fig. 20. The residual strength was determined as the mean value between the point with the first positive slope after the peak (marked with "o" in Fig. 20) and the end of the diagonal compression experiment (marked with "x" in Fig. 20).
Force-time diagram of the STRIP specimens, showing the residual strength
For specimens COMB10 and COMB20, both the strains of the FRCM-side and the as-built side of the specimen are presented separately with an additional subscript "r" and "u" respectively (for example: εh,u is the axial strain in the horizontal direction of the as-built side of the specimen). The strain is defined as the mean of the strains measured on both sides of the specimens following Eq. (6):
$$\epsilon = \frac{{\epsilon_{u} + \epsilon_{r}}}{2}$$
For the specimens reinforced with a FRCM layer, the strains along the as-built side were significantly different from the opposite side where FRCM was installed. On the FRCM-side, lower deformation values were measured in both the horizontal and vertical direction. This was in line with the expectations considering the significant difference in modulus of elasticity between the mortar matrix and the masonry. For the COMB specimens it was noticed that the mean horizontal strain was higher than the mean vertical strain during the post-peak phase. This was primarily caused by the diagonal tension cracks on the as-built side of the specimens. Noticeable was the difference in strain on the as-built side of the COMB-specimens. Despite the higher mean initial stiffness (caused by the FRCM thickness), the mean strains in particular the horizontal direction was significantly higher for the COMB20 specimens when compared with the COMB10 specimens. Looking at the crack patterns that were presented in Fig. 14, the difference in horizontal strain can be explained with the amount of cracks that were formed, more cracks leading to higher deformation values.
Shear modulus
The shear strain is defined in Eq. (7):
$$\gamma = \epsilon_{v} + \epsilon_{h}$$
The slope of the elastic portion of the τ–γ diagram is denoted as the shear modulus of rigidity, (Ge,) according to ASTM E 519-02 (2010), following Eq. (8):
$$G_{e} = \frac{{\tau_{e} }}{{\gamma_{e} }}$$
where \(\tau_{e} = 0.7\tau_{max}\) was assumed to be the cracking shear strength and γe was the corresponding cracking shear strain identified on the experimental τ–γ diagram. The cracking shear strain and shear modulus are provided in columns 10 and 11 respectively in Table 4. Comparing the URM and STRIP specimens, considering only the cases where the predominant failure mechanism was shear friction, it can be observed that the STRIP specimens result in a 25.3% lower shear modulus. This indicates that the deep grooves resulted in a reduction in shear modulus. For the specimens where both shear sliding and shear friction occurred (URM-4 and STRIP-1), the shear modulus was found to be approximately the same, but higher than the mean value of the corresponding specimen group. This error was likely caused by the failure plane concentrating outside the horizontal and/or vertical diagonals, where the deformation measurements were made. No significant difference in shear moduli was found between the COMB10 and COMB20 specimens, indicating that the thickness of the FRCM had limited influence on the shear modulus. A possible explanation could be the formation of shrinkage cracks during the curing stage of the FRCM layer, and that therefore the enhancement in stiffness and strength is primarily based on the presence of the CFRP mesh. Compared with the URM specimens, the application of a single sided FRCM layer resulted in an increase of approximately 40% of the shear modulus. It was noticeable that the mean shear modulus of COMB20 specimens was more scattered (COV 15.6%) compared to the COMB10 specimens (COV 3.68%). The shear stress-shear strain diagrams are presented in Fig. 21. Specimen COMB10-1 is not included due to the faulty attachment of two LVDT's, leading to missing data near the failure load. For the remaining specimens, the LVDT sensors malfunctioned due to the crack development in the post-peak phase. Because of this, the shear strain relation of specimens COMB10-2, COMB10-3 and COMB20-1 have been linear extrapolated to obtain the ultimate shear strain \(\gamma_{u}\) (associated with a maximum 20% strength drop on the post-peak softening branch).
Shear stress versus shear strain diagrams
Pseudo-ductility
The wallette's pseudo-ductility μ (see Column 12 in Table 4), is calculated using Eq. (9), where \(\gamma_{u} = \gamma_{e}\) for specimens without post peak strength.
$$\mu = \frac{{\gamma_{u} }}{{\gamma_{e} }}$$
In general, a higher pseudo-ductility ratio leads to an increased ability of strengthened masonry walls to redistribute stresses, a higher global deformation capacity and an improved energy dissipation (Babaeidarabad et al. 2014). The pseudo-ductility factors obtained were in the range 8.5–9.3 and 15.4–15.6 for the COMB10 and COMB20 specimens respectively. The out-of-plane deformations could result in optimistic factors for the pseudo-ductility.
Evaluation unstrengthened masonry
The in-plane shear capacity of unstrengthened walls were determined using two approaches: the analytical model developed by Li et al. (2005) and the design provisions according to Eurocode 8-3 (2005a, b).
Analytical model
For the walls subjected to a diagonal compressive force, all of the clamping force on the wallette is provided by the vertical component of the diagonal compression force (Li et al. 2005), as shown in Fig. 22. The relationship between the v-component \(\left( {P_{v} } \right)\) and the u-component \(\left( {P_{u} } \right)\) of force \(P\) is provided by Eq. (10), with \(\theta\) being the angle between the bed joint direction (u-axis) and the main diagonal of the wallette (y-axis).
Forces acting on wallette during a diagonal compression test
$$P_{v} =\upsigma_{n} A_{n} = P_{u} tan\theta$$
As the angle between the horizontal and the main diagonal of the wallette was kept constant at 45° during the experimental campaign, Eq. (10) can be reduced to Eq. (11):
$$P_{v} =\upsigma_{n} A_{n} = P_{u} = \sqrt 2 P$$
An unstrengthened masonry wall fails when the value of the applied shear force reaches the minimum shear capacity, Vm, computed in accordance with Eq. (12):
$$V_{m} = { \hbox{min} }\left( {V_{ss};\, V_{sf};\,V_{dt} ;\,V_{c} } \right)$$
Shear sliding\(\left( {V_{ss} } \right)\) Recognizing that shear strength results from the combination of bond strength and friction resistance between mortar joint and blocks (Li et al. 2005), the shear strength is typically modelled with the Mohr–Coulomb relationship provided in Eq. (13).
where \(f_{v,0}\) is the shear bond strength and \(\mu_{ma}\) is the average coefficient of friction. The cohesive strength obtained with the triplet experiments was 0.38 N/mm2. An average value for cohesive strength τ0 of 3% of the masonry gross area compressive strength \(\left( {f'_{m} } \right)\) is suggested in various researches (Li et al. 2005; Silva et al. 2008; Petersen 2009; Babaeidarabad et al. 2014), resulting in 0.44 N/mm2 for the masonry used in this study. For the coefficient of friction μ0, a typical range from 0.3 to 1.2 is assumed (Li et al. 2005), with an average of 0.75. This corresponds well with the value obtained via the triplet experiments.
The shear capacity due to shear sliding failure is derived from Eq. (14):
$$V_{ss} = \left( {f_{v,0} + \mu_{ma} \sigma_{n} } \right)A_{n}$$
Substituting equation Eq. (11) into Eq. (14), the horizontal force to resist shear sliding failure along a bed joint can be rewritten as:
$$V_{ss} = \frac{{f_{v,0} }}{{1 - \mu_{ma} }}A_{n}$$
Shear friction\(\left( {V_{sf} } \right)\) Crisafulli et al. (1995) revised the theory of Mann and Muller (1982) and presented a more realistic distribution of normal and shear stresses acting on a block (Li et al. 2005). The reduced shear strength \(f_{sf}\) is determined using a modification of Eq. (13), resulting in:
$$f_{sf} = f_{v,0}^{*} + \mu_{ma}^{*} \sigma_{n}$$
where \(f_{v,0}^{*}\) and \(\mu_{ma}^{*}\) are the reduced shear bond strength and the reduced coefficient of friction respectively, which are the confirmed determinative factors for the friction failure instead of the actual coefficients \(f_{v0}\) and \(\mu_{ma}\) (Mann and Muller (1982). The shear capacity due to shear sliding failure is derived from Eq. (17):
$$V_{ss} = \left( {f_{v,0}^{*} + \mu_{ma}^{*} \sigma_{n} } \right)A_{n}$$
Substituting Eq. (11) into Eq. (17), the horizontal force to resist shear friction failure can be rewritten as:
$$V_{ss} = \frac{{f_{v,0}^{*} }}{{1 - \mu_{ma}^{*} }}A_{n}$$
Diagonal tension\(\left( {V_{dt} } \right)\) The required force to induce diagonal tensile crack of the brick is determined using Eq. (19). The tensile strength of the clay brick masonry \(\left( {f_{tb}^{'} } \right)\) is determined by Silva et al. (2008) using Eq. (20):
$$V_{dt} = \frac{{f_{tb}^{{\prime }} }}{2.3}\sqrt {1 + \frac{{\sigma_{n} }}{{f_{tb}^{{\prime }} }}} A_{n}$$
$$f_{tb}^{{\prime }} = \frac{2}{3}\sqrt {f_{m}^{{\prime }} }$$
Substituting Eq. (11) into Eq. (20), the expression of \(V_{dt}\) for the discussed condition and present failure mode can be rewritten as:
$$V_{dt} = 1.44f_{tb}^{{\prime }} A_{n}$$
Crushing\(\left( {V_{c} } \right)\) The shear strength to initiate crushing is evaluated as
$$V_{C} = \left( {f_{m} - \sigma_{n} } \right)\frac{{2l_{b} }}{{3h_{b} }}A_{c}$$
with \(A_{c}\) being interface loading area between the steel shoe and the wallette, parallel to the bed joint. Substituting Eq. (11) into Eq. (22), the horizontal force to initiate crushing can be obtained:
$$V_{C} = \frac{{2l_{b} }}{{3h_{b} + 2l_{b} }}f_{m} A_{c}$$
Equations (14), (17), (19), and (22) completely represent the failure envelope for the shear strength of masonry. Using the relevant parameters provided in Table 5, the failure envelope for the unstrengthened masonry used in this experimental research was determined as shown in Fig. 23. The failure envelope presented here is a function of the compressive stress applied to the wallette, ranging from zero to the compressive strength of the masonry.
Table 5 Masonry properties for the analytical model
Failure envelope of the unstrengthened masonry determined with the analytical model
Design provisions Eurocode 8
According to Eurocode 8-3 (2005b), the shear force capacity of an unstrengthened masonry wall controlled by shear under an axial load \(N\) is determined with Eq. (24):
$$V_{sf,EC} = f_{v} \cdot D^{{\prime }} \cdot t_{w}$$
where \(D^{{\prime }}\) is the depth of the compressed area of the wall and \(f_{v}\) is the masonry shear strength accounting for the presence of vertical load. In this study, the depth of the compressed area in the bed joints is assumed to be equal to the length of the specimen. The masonry shear strength is determined according to Eq. (25):
$$f_{v} = f_{v,0} + 0.4\frac{N}{{A_{n} }} \le 0.065 \cdot f_{b}$$
where \(f_{v,0}\) is the initial shear strength in the absence of vertical load and \(f_{b}\) the normalized mean compressive strength of the masonry unit, obtained from either in situ tests or additional sources of information, and divided by the confidence factor (= 1 for KL3). In primary seismic walls, both these material strengths are further divided by the partial factor \(\left( {\gamma_{M} } \right)\) for masonry in accordance with Eurocode 8-1 (2005a). Characteristic initial shear strength of masonry \(\left( {f_{v,0} } \right)\) is provided as 0.3 N/mm2 for clay masonry with M10-M20 mortar strength class in Eurocode 6 (2006).
Substituting Eq. (11) in Eq. (24), the horizontal force to resist shear friction failure following Eurocode 8-3 (2005b) can be rewritten as:
$$V_{sf,EC} = \frac{{f_{v,0} }}{0.6 }A_{n}$$
The upper limit \(0.065f_{m}\) takes care of the possibility that failure in shear tension will occur in the compression area subjected to a combination of a significant normal compressive stress and a shear stress. When failure due to shear tension will occur, cracks will run through the units. The shear force capacity for this failure mechanism is provided in Eq. (27):
$$V_{dt,EC} = 0.065 \cdot f_{m} \cdot A_{n}$$
In contrast with the ASCE/SEI 41-13 (2014), Eurocode 8 does not differentiate the rocking mechanism from the toe-crushing mechanism. In Eurocode 8, the shear force capacity of an unstrengthened masonry wall as controlled by flexure under an axial load \(N\) is obtained via Eq. (28):
$$V_{fl,EC} = \frac{{l_{w} }}{{h_{w} }} \cdot \frac{N}{2} \cdot \left( {1 - 1.15\frac{N}{{l_{w} \cdot t_{w} \cdot f_{m} }}} \right)$$
where \(f_{mas}\) is the compressive strength of the masonry divided by the confidence factor (= 1 for KL3). Regarding the normal stress distribution, the Eurocode 8-3 (2005a, b) refers to a stress block distribution by adopting a reduction coefficient of the compressive strength (0.87 = 1/1.15). The mechanical scheme to obtain Eq. (28) is provided in Fig. 24a.
Mechanical scheme of Eurocode 8 to determine the shear force capacity during flexural failure (a) and the modification for this study (b)
Since during the diagonal compression experiments the axial load was not introduced at the center of the wallette, as illustrated in Fig. 24b, a modification of Eq. (28) also has been considered. First, the depth of the compressed area was determined using Eq. (29):
$$D^{{\prime }} = \frac{N}{{0.85 \cdot t_{w} \cdot f_{m} }}$$
The moment with respect to the bottom right corner in Fig. 24b equals:
$$\left( {l_{w} - D^{{\prime }} } \right) \cdot N = \left( {h_{w} - 2\delta } \right) \cdot V$$
with \(\delta\) being the distance between the corner of the specimen and the location where the concentrated force V is assumed to be introduced. Substituting Eq. (29) into Eq. (30), the modified shear force capacity for flexural failure becomes:
$$V_{fl,EC'} = \frac{N}{{\left( {h_{w} - 2\delta } \right)}}\left( {l_{w} - 1.15\frac{N}{{t_{w} \cdot f_{m} }}} \right)$$
With the compressive depth during the experiments being limited to 100 mm due to the dimension of the steel shoe, combining Eq. (29) with Eq. (11) results in the shear strength needed to initiate crushing:
$$V_{c} = 0.85 \cdot f_{m} \cdot 100 \cdot t_{w}$$
Equations (24), (27) and (28) completely represent the failure envelope for the shear strength of masonry following the Eurocode 8-3 (2005a, b). Similar to Eq. (12), a wall fails when the value of the applied shear force reaches the minimum shear capacity:
$$V_{m,EC} = { \hbox{min} }\left( {V_{ss,EC} ;V_{dt,EC} ;V_{fl,EC} } \right)$$
The failure envelope including the modified shear force rocking/toe crushing is obtained with Eq. (34):
$$V_{m,EC'} = { \hbox{min} }\left( {V_{ss,EC} ;V_{dt,EC} ;V_{fl,EC'} } \right)$$
Using the relevant parameters provided in Table 6, the failure envelope for the unstrengthened masonry used in this experimental research was determined as shown in Fig. 25. The failure envelope of the modified shear for rocking/toe crushing is also shown. It should be noted that for comparison reasons the partial factor for masonry is not taken into account.
Table 6 Masonry properties for the (modified) Eurocode 8-3 (2005a, b) approach
Failure envelope of the unstrengthened masonry determined with the standard Eurocode 8 design provisions (Vm) and a modified version (Vm′)
Comparing Figs. 23 and 25, it can be observed that despite leaving out the partial factors for masonry, the Eurocode 8-3 (2005a, b) approach to determine the failure envelope of unstrengthened masonry is conservative when compared to the failure envelope obtained from the analytical model. This is primarily caused by the different approaches to determine the shear strength at diagonal tension failure, where the difference builds up to 292% with respect to the Eurocode 8-3 (2005a, b) approach.
Comparison experimental results
The strengths obtained with the analytical model and Eurocode 8-3 (2005a, b) are compared with the mean experimental shear strength of the URM specimens. The mean experimental shear strength is determined by using Eq. (10). The results are summarized in Table 7.
Table 7 Experimental and analytical results of the URM and STRIP specimens
The analytical model showed good correspondence with the experimental values for both the failure mechanism and the failure load, with an experimental/model ratio \(\left( \varphi \right)\) of 0.98. Despite leaving out the partial factor for masonry, the Eurocode 8-3 (2005a, b) approach resulted in lower values \(\left( {\varphi = 1.43} \right)\). Including the partial factor for masonry, the Eurocode approach results in even more conservative values \(\left( {\varphi = 2.14} \right)\).
Limitation presented models
It should be noted that the presented models assume a uniform axial stress and shear stress through the cross section of the specimen (u-axis in Fig. 22). This assumption does not reflect the reality as there will be a non-uniform distribution of stresses. The experiment for an unstrengthened specimen has been recreated with a Finite Element (FE) model. The masonry panel was modelled as a homogenous material with the dimensions 700 (length) × 700 (height) × 95 (thickness) mm3. Linear elastic material behaviour was assumed, with a young's modulus of 3200 N/mm2 and Poisson's ratio of 0.27. Linear hexahedral finite elements with reduced integration (Abaqus C3D8R) were used to generate the mesh of the masonry panel, with a global seed size of 20 mm. With this model, displayed in Fig. 26, the interface between the specimen and the bottom loading shoe, was fixed for displacement in the y-direction (marked with orange triangles in Fig. 26). The load was introduced as a uniform stress of 4.73 N/mm2 (marked with purple arrows in Fig. 26), covering the entire interface between the specimen and the upper loading shoe, resulting in a total force of 67 kN (mean experimental failure load of URM) in the negative y direction.
FE model geometry of the URM specimen
The distribution of the axial stresses over the vertical diagonal of the wall (y-axis in Fig. 26) and horizontal diagonal (x-axis in Fig. 26) of the wallette is shown in Fig. 27a. Here a considerable variation in the axial compressive stress can be observed. The maximum compressive stress at (x,y) = (0,0) is 122% higher than the mean value for compressive stress over the horizontal diagonal (y = 0). Rotating the x–y coordinate system to the u-v coordinate system, as presented in Fig. 22, the shear stress and axial stress as used by the analytical model can be obtained. The distribution of the compressive stress and shear stress over cross section of the wallette at mid-height (v = 0.5hw) is shown in Fig. 27b. In contrast to the compressive stress which is roughly constant over the discussed cross-section (COV = 14.1%), the shear stresses show strong variation (COV = 46.2%). The highest shear stress occurs at the center of the specimen (u = 350 mm) and differs 49.4% from the mean shear stress value. The analytical model as proposed by Li et al.(2005) does not take this variation in shear stress into account. Whereas the Eurocode 8 also does not take this non-uniform stress distribution into account. The NZSEE, NTC and ASCE introduce a corrective factor (named b, β or (1 + αv)) to account for the shear stress distribution at the center of the panel and relate the peak value to the mean one (Cattari et al. 2015).
Results of the linear-elastic FE simulation on the distribution of: axial stresses σI and σII over the vertical and horizontal diagonal respectively (a) and compressive stress σn and shear stress τ over cross section of wallette at mid-height (b)
Evaluation FRCM reinforced masonry
In order for the FRCM reinforcement system to be applied on a large scale for the in-plane strengthening of masonry walls, simple practitioner oriented design models are essential. However, due to the novelty of this technique and the wide variety of FRCM materials on the market, design provisions are generally not provided by international building codes (Ceroni and Salzano 2018). Previous theoretical studies have led to various analytical formulations for determining the shear strength of FRCM reinforced masonry (Babaeidarabad et al. 2014; Gattesco and Boem 2015; Cascardi et al. 2016; Triantafillou 2016). Cascardi et al. (2016) presented an advanced analytical model based on artificial neural network (ANN). For the construction of the model a number of 75 samples were selected from previous diagonal compression tests found in scientific literature, varying in both material and geometry. By comparing the proposed model the authors showed that the proposed model is competitive with the consolidated analytical formulations. Because of the reduced specimen dimensions and the non-standard single-sided reinforcement configuration used in the current study, the model proposed by Cascardi et al. (2016) was not considered.
At the end of 2013 the first design guide for FRCM reinforcement (ACI 549-13 2013) was published. This guideline provides structural engineers with easily applicable design models for determining the shear resistance of FRCM reinforced masonry walls. Past studies have shown that the ACI design models show reasonable agreement with experimental data and can be considered as conservative (Babaeidarabad et al. 2014; Almeida et al. 2015). The European building codes (Eurocode), which in general differs significantly from the American design philosophy, do not provide any design models for the shear strength of FRCM reinforced masonry. Kolyvas et al. (2012) and Triantafillou (2016), however provided similar practitioner oriented design models in Eurocode framework.
In this research both the approach according to ACI 549-13 (2013) and Triantafillou (2016) to determine the FRCM contribution on the shear strength of masonry were considered. Concerning the nominal shear strength \(V_{RM}\) of FRCM reinforced masonry, both approaches pose that this is the result of the summation of the shear strength of the masonry and of the FRCM-overlay, in accordance with Eq. (35). It should be noted that the FRCM contribution is considered only after masonry cracking (Li et al. 2005; Silva et al. 2008; Petersen 2009; ACI 549-13 2013; Babaeidarabad et al. 2014).
$$V_{RM} = V_{m} + V_{t}$$
The design shear strength of FRCM reinforced masonry according to Triantafillou (2016) is determined using Eqs. (36–38), as provided in Table 8. Triantafillou defines the maximum design stress \(\left( {f_{td} } \right)\) allowed to the CFRP net as the lowest value between the design characteristic strength of the mesh \(\left( {f_{tk} } \right)\) divided by the material factor \(\left( {\gamma_{t} } \right)\), and the stress corresponding to the design tensile strain \(\varepsilon_{fv}\) where debonding is assumed to be initiated, as shown in Eq. (36). The contribution of the FRCM \(\left( {V_{Rd,t} } \right)\) is determined using Eq. (37), where \(A_{f}\) is the area of mesh per unit width (mm2/mm) and n is the number of mesh layers. It should be noted that a reduction factor of 0.9 is present. The design shear strength of FRCM reinforced masonry, including a partial factor for shear \(\left( {\gamma_{Rd} } \right)\) of 1.2 and the design shear strength of the unstrengthened masonry \(\left( {V_{Rd,m} } \right)\), is limited by a maximum value \(\left( {V_{Rd,max,c} } \right).\) This limitation corresponds to compression failure of the struts in the truss, as shown in Eq. (38)
Table 8 Approaches to determine the in-plane shear capacity of FRCM retrofitted masonry walls
The design shear resistance of masonry walls strengthened with FRCM according to ACI 549-13 (2013) is obtained by using Eqs. (39–41). The ACI 549-13 (2013) directly uses the design tensile strain \(\varepsilon_{fv}\) to determine the maximum design stress, as shown in Eq. (39). The contribution of the FRCM is determined using Eq. (40), where in contrast to Triantafillou (2016) no reduction factor is used. The design shear strength of FRCM reinforced masonry \(\left( {V_{Rd,RM} } \right)\), including a strength reduction factor for shear \(\left( {\Phi _{v} } \right)\) of 0.75, is limited to 50% of the un-strengthened wall's shear capacity to limit the total force transferred to the substrate of the masonry per unit width (Babaeidarabad et al. 2014), as shown in Eq. (41).
With the parameters presented in Table 9, the shear strength of the masonry reinforced with a FRCM-overlay can be determined. The results are provided in Table 10. The contribution of the FRCM for the shear capacity \(V_{Rd,t}\) was determined as 20.0 kN and 29.6 kN using the approach proposed by Triantafillou (2016) and ACI 549-13 (2013) respectively. The mean shear contribution of the FRCM reinforcement, as estimated by the experiments and analytical model, was 38.7 kN \(\left( {V_{FRCM} } \right)\). The approach proposed by Triantafillou (2016), with an experimental/model ratio \(\left( \varphi \right)\) of 1.9, resulted in more conservative results when compared to the ACI 549-13 (2013) \(\left( {\varphi = 1.3} \right)\).
Table 9 Values used in the design codes to obtain the shear capacity of FRCM reinforced masonry
Table 10 Experimental and analytical results of the URM and STRIP specimens
When taking the partial factor for shear \(\left( {\gamma_{Rd} } \right)\), the strength reduction factor for shear \(\left( {\Phi _{v} } \right)\), and limiting the shear capacity of the strengthened wall to 50% of the un-strengthened wall shear capacity into account, the shear resistance \(\left( {V_{{Rd,RM^{{\prime }} }} } \right)\) of masonry walls strengthened with FRCM as determined with the two approaches is approximately the same (using the masonry shear strength as determined with the analytical model). This is mainly due to the design shear strength being limited to 50% of the un-strengthened wall shear capacity according to ACI 549-13 (2013). The experimental/design value ratio's \(\left( \rho \right)\) for the approaches following Triantafillou (2016) and ACI 549-13 (2013) were 2.14 and 2.25 respectively. When for the masonry contribution the design value as obtained using Eurocode 8 is used, the experimental/design value ratios \(\left( \rho \right)\) reduce to 3.5 and 4.9 for the approaches following Triantafillou (2016) and ACI 549-13 (2013) respectively. It can be observed that the presented design provisions, both for the masonry part and the FRCM contribution, are conservative.
An experimental program was undertaken to assess the effectiveness of a combined retrofit method to improve the in-plane behaviour of clay brick URM walls. The diagonal compression test was used for the evaluation of the in-plane shear behavior of these retrofitted wallettes. From the experiments the following conclusions can be drawn:
The out-of-plane reinforcement, which consisted of deep mounted CFRP strips embedded with a flexible adhesive in a deep groove (partly filled with mortar), did not affect the strength of masonry elements loaded under in-plane shear. It was however found that the deep grooves resulted in a 25.3% lower shear modulus compared to the unstrengthened control specimens. Moreover the experiments showed that in contrast to the unstrengthened specimens, the specimens with solely the out-of-plane reinforcement did not disintegrate after reaching the failure load. This can be attributed to the DM CFRP strip holding the specimens together.
The single-sided carbon FRCM overlay increased the shear capacity with 1.7 and 1.8 times that of the unstrengthened control specimens with a 10 mm and 20 mm FRCM layer thickness respectively. The application of a single sided FRCM layer resulted in an increase of approximately 40% of the shear modulus compared to the unstrengthened control specimens.
No strong correlation was found between the thickness of the mortar matrix of the FRCM layer and the failure load. Additionally, FRCM layer thickness was found to have limited influence on the shear modulus. A possible explanation could be the formation of shrinkage cracks during the curing stage of the FRCM layer, and that therefore the enhancement in stiffness and strength is primarily based on the presence of the CFRP mesh.
The FRCM layer thickness did have an influence on the number of diagonal cracks that were observed on the as-built side of the combined DM CFRP and FRCM reinforced specimens. With a 20 mm FRCM layer thickness, one to two additional diagonal tensile cracks occurred over a wider area when compared to the specimens provided with a 10 mm FRCM layer. A possible explanation for this discrepancy in crack pattern is the difference in thickness of the upper mortar layer of the FRCM overlay. A thicker upper mortar layer leads to an improved utilization of the carbon FRP mesh.
Stiffness differences between the as-built side and the FRCM strengthened side led out-of-plane bending during the final stages of the diagonal compression experiments. With more restrained boundary conditions and superimposed vertical loads, as is the case in practice, larger shear strength increments could be achieved (Ismail 2012).
The pseudo-ductility factors obtained were in the range 8.5–9.3 and 15.4–15.6 for the reinforced specimens with a 10 mm and 20 mm FRCM layer thickness respectively. Comparison of these values with the pseudo-ductility of URM showed that a one sided FRCM overlay leads to a significant increase in ductility.
For the evaluation of unstrengthened masonry, the analytical model developed by Li et al. (2005) showed good correspondence with the experimental values for both the failure mechanism and the failure load, with an experimental/model ratio \(\left( \varphi \right)\) of 0.98. Despite leaving out the partial factor for masonry, the Eurocode 8-3 (2005a, b) approach resulted in a lower ratio \(\left( {\varphi = 1.43} \right)\). Including the partial factor for masonry, the Eurocode approach results in even more conservative values \(\left( {\varphi = 2.14} \right)\). An important limitation of both approaches is that the non-uniform shear stress distribution at the center of the panel is not taken into account.
For the FRCM contribution on the in-plane shear capacity, the approach proposed by Triantafillou (2016) (experimental/model ratio \(\left( \varphi \right)\) of 1.94) resulted in more conservative results when compared to the ACI 549-13 (2013) \(\left( {\varphi = 1.31} \right)\).
The obtained design values for the shear strength of FRCM reinforced masonry were conservative, especially when for the masonry contribution the design value as obtained using Eurocode 8-3 (2005a, b) was used (experimental/design value ratio \(\left( \rho \right)\) range 3.47–4.89).
As for the recommendations, firstly, even though the emerging line of experimental results seems quite consistent, the results could be confirmed by a larger experimental campaign as the number of specimens tested in this study was limited. Secondly, as stated in the introduction, no shear damage was observed within the applied load range on the previously conducted static-cyclic in-plane shear tests on full-scaled masonry specimens strengthened with the combined reinforcement system (Türkmen et al. 2018). With these cantilever shear walls there are regions of: (a) nearly pure tension stress; (b) nearly pure compression stress; (c) combined tension and shear stresses; and (d) combined compression and shear stresses. The diagonal compression test as conducted in the current study is applicable only to the latter case, the compression-shear region. Thus, the mechanical characteristics and effectiveness of the repair techniques will not be fully revealed by diagonal compression testing alone. Just like the additional experimental program for the combined compression and shear stresses covered in the current study, testing other normal-shear stress combinations are recommended to fully understand the response of reinforced masonry shear walls.
\(A_{c}\) :
Interface loading area between the steel shoe and the wallette (\({\text{mm}}^{2}\))
\(A_{f}\) :
Area of mesh per unit width (\({\text{mm}}^{2} /{\text{mm}}\))
\(A_{n}\) :
Cross sectional area of the specimen, parallel to the bed joint (\({\text{mm}}^{2}\))
\(b_{f} , b_{p}\) :
Width groove; width CFRP strip (\({\text{mm}}\))
\(d_{f} , d_{ff}\) :
Depth groove; Depth mortar (\({\text{mm}}\))
\(D'\) :
Depth of the compressed area (\({\text{mm}}\))
\(E,E_{f}\) :
Modulus of elasticity, tensile modulus of elasticity of the CFRP mesh (\({\text{N}}/{\text{mm}}^{2}\))
\(f_{c}\) :
Compressive strength (\({\text{N}}/{\text{mm}}^{2}\))
\(f_{m} , f_{b} , f_{mas}\) :
Compressive strength masonry; mean: normalized mean; mean/confidence factor (\({\text{N}}/{\text{mm}}^{2}\))
\(f_{sf}\) :
Reduced shear strength (\({\text{N}}/{\text{mm}}^{2}\))
\(f_{st}\) :
Splitting tensile strength of the clay brick (\({\text{N}}/{\text{mm}}^{2}\))
\(f_{t}\) :
Tensile strength (\({\text{N}}/{\text{mm}}^{2}\))
\(f_{td} ,f_{tk}\) :
Design stress allowed to the CFRP mesh: maximum; characteristic (\({\text{N}}/{\text{mm}}^{2}\))
\(f_{tb}^{\prime }\) :
Tensile strength masonry (\({\text{N}}/{\text{mm}}^{2}\))
\(f_{v} , f_{v,0} , f_{v,0}^{*}\) :
Shear strength; Initial shear strength; reduced initial shear strength (\({\text{N}}/{\text{mm}}^{2}\))
\({\text{g}}\) :
Gauge length (\({\text{mm}}\))
\(G_{e}\) :
Shear modulus of rigidity (\({\text{kN}}/{\text{mm}}^{2}\))
\(h_{b} , h_{w}\) :
Height brick; height wallette (\({\text{mm}}\))
\(l_{b} , l_{w}\) :
Length brick; length wallette (\({\text{mm}}\))
\(n\) :
Number of mesh layers (–)
\(n_{test}\) :
Number of test specimens (–)
Axial load (\({\text{kN}}\))
\(P\) :
Compressive force applied to the specimen (\({\text{kN}}\))
\(P_{max}\) :
Maximum value of the compressive force applied to the specimen (\({\text{kN}}\))
\(t_{FRCM}\) :
Thickness FRCM layer (\({\text{mm}}\))
\(t_{w} , t_{p}\) :
Thickness specimen (as-built); Thickness CFRP strip (\({\text{mm}}\))
\(V_{c}\) :
Shear force capacity for crushing failure (\({\text{kN}}\))
\(V_{COMB}\) :
Mean shear strength of the COMB specimens as obtained with the experiments (\({\text{kN}}\))
\(V_{dt} , V_{dt,EC}\) :
Shear force capacity for diagonal tension failure: analytical; EC8 (\({\text{kN}}\))
\(V_{fl,EC} ,V_{fl,EC'}\) :
Shear force capacity for flexural/toe crushing failure (EC8): original; modified (\({\text{kN}}\))
\(V_{FRCM}\) :
Estimation of the mean shear contribution of the FRCM reinforcement (\({\text{kN}}\))
\(V_{m}\) :
Shear strength for unstrengthened masonry (\({\text{kN}}\))
\(V_{m,EC} , V_{m,EC'}\) :
Minimum shear capacity, following EC8: original; modified (\({\text{kN}}\))
\(V_{Rd,m} , V_{Rd,m,EC}\) :
Design shear resistance of unstrengthened masonry: analytical model; EC8 (\({\text{kN}}\))
\(V_{Rd,RM} , V_{Rd,RM'}\) :
Design shear resistance of FRCM reinforced masonry: no reduction factors; with reduction factors (\({\text{kN}}\))
\(V_{Rd,max,c}\) :
Maximum design shear resistance corresponding to compression failure (\({\text{kN}}\))
\(V_{Rd,t}\) :
Design shear resistance contribution of the FRCM overlay (\({\text{kN}}\))
\(V_{RM}\) :
Nominal shear strength of FRCM reinforced masonry (\({\text{kN}}\))
\(V_{sf} , V_{sf,EC}\) :
Shear force capacity for shear friction failure: analytical; EC8 (\({\text{kN}}\))
\(V_{ss}\) :
Shear force capacity for shear sliding failure (\({\text{kN}}\))
\(V_{t}\) :
Nominal shear strength contribution of FRCM overlay (\({\text{kN}}\))
\(w_{b}\) :
Width brick (\({\text{mm}}\))
\(\gamma_{e} , \gamma_{max} , \gamma_{u}\) :
Shear strain: elastic; at peak stress; ultimate (‰)
\(\gamma_{M} , \gamma_{t} ,\gamma_{Rd}\) :
Partial factor; material factor; partial factor for shear (–)
\(\delta\) :
Distance between edge of specimen and concentrated force V (\({\text{mm}}\))
\(\Delta H, \Delta V\) :
Horizontal elongation; Vertical shortening (\({\text{mm}}\))
\(\varepsilon_{h} , \varepsilon_{u} , \varepsilon_{r} , \varepsilon_{v}\) :
Horizontal strain, strain at as-built side; strain at reinforced side; vertical strain (‰)
\(\varepsilon_{m} , \varepsilon_{ult}\) :
Cracking strain of the reinforced mortar; ultimate strain of the CFRP mesh
\(\varepsilon_{fv}\) :
Design value of the tensile strain of the CFRP mesh (ACI 549-13) (‰)
\(\theta\) :
Angle between the bed joint direction and the main diagonal of the wallette
\(\mu\) :
Pseudo-ductility ratio (–)
\(\mu_{ma} , \mu_{ma}^{*}\) :
Friction coefficient; reduced friction coefficient (–)
\(\rho ,\rho_{r}\) :
Density; reinforcement ratio (–)
\(\sigma_{m} , \sigma_{n} , \sigma_{ult}\) :
Cracking stress of the reinforced mortar; axial stress; ultimate stress of the mesh (\({\text{N}}/{\text{mm}}^{2}\))
\(\tau , \tau_{e} , \tau_{u}\) :
Shear stress: average; at crack initiation; at ultimate shear strain (\({\text{N}}/{\text{mm}}^{2}\))
\(\varphi\) :
Experimental/model ratio (–)
\(\varPhi_{v}\) :
Strength reduction factor for shear (–)
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The authors wish to gratefully acknowledge the support by QuakeShield, a joint venture between Royal Oosterhof Holman and SealteQ Group. Special thanks to our beloved and retiring colleague A.T. Vermeltfoort, who provided insight and expertise that greatly assisted the research. Wishing you a long and joyous retirement.
Department of the Built Environment, Section Structural Design, Eindhoven, University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands
Ö. S. Türkmen
, S. N. M. Wijte
& A. T. Vermeltfoort
QuakeShield (Joint Venture Royal Oosterhof Holman and SealteQ Group), P.O. Box 6, 9843 ZG, Grijpskerk, The Netherlands
B. T. De Vries
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Correspondence to Ö. S. Türkmen.
Türkmen, Ö.S., De Vries, B.T., Wijte, S.N.M. et al. In-plane behaviour of clay brick masonry wallettes retrofitted with single-sided fabric-reinforced cementitious matrix and deep mounted carbon fibre strips. Bull Earthquake Eng 18, 725–765 (2020) doi:10.1007/s10518-019-00596-2
Received: 30 October 2018
Issue Date: January 2020
DOI: https://doi.org/10.1007/s10518-019-00596-2
FRCM | CommonCrawl |
\begin{document}
\title{The optimal control of storage for arbitrage
and buffering, with energy applications}
\begin{abstract}
We study the optimal control of storage which is used for both
arbitrage and buffering against unexpected events, with particular
applications to the control of energy systems in a stochastic and
typically time-heterogeneous environment. Our philosophy is that of
viewing the problem as being formally one of stochastic dynamic
programming, but of using coupling arguments to provide good
estimates of the costs of failing to provide necessary levels of
buffering. The problem of control then reduces to that of the
solution, dynamically in time, of a deterministic optimisation
problem which must be periodically re-solved. We show that the
optimal control then proceeds locally in time, in the sense that the
optimal decision at each time~$t$ depends only on a knowledge of the
future costs and stochastic evolution of the system for a time
horizon which typically extends only a little way beyond~$t$. The
approach is thus both computationally tractable and suitable for the
management of systems over indefinitely extended periods of time.
We develop also the associated strong Lagrangian theory (which may
be used to assist in the optimal dimensioning of storage), and we
provide characterisations of optimal control policies.
We give examples based on Great Britain electricity price data. \end{abstract}
\section{Introduction} \label{sec:introduction}
The control of complex stochastic systems, for example modern power networks which must cope with many sources of uncertainty in both generation and demand, requires real-time optimisation of decision problems which are often computationally intractable---notably so in a time-heterogeneous environment. This clearly also poses difficulties for the design of such systems. As in the case of the well studied areas of communication and manufacturing networks, our belief is that what is required is the careful specification of the stochastic models governing the behaviour of such systems, coupled with the analytical derivation of accurate approximation techniques.
In the present paper we use an economic framework to consider the optimal control of a single storage facility. The problem is made interesting because, at least in power networks, storage may be simultaneously used for many different purposes, with potentially conflicting objective functions. However, if storage is to be economically viable, it must be capable of meeting these competing objectives. We concentrate on energy storage in a time-heterogeneous environment, and consider two of the main uses of such storage systems: (a) price arbitrage, i.e.\ the buying and selling of energy over time (whether to earn revenue for the store owner or for the benefit of the consumer), and (b) the provision of buffering services, so as to react rapidly to sudden and unexpected changes, for example the loss of a generator or transmission line, or a sudden surge in demand. Our general approach is likely to be applicable to other uses of storage, and also to the optimal control of other facilities used for the provision of multiple services.
There is considerable literature on the control of storage for each of the above two purposes considered on its own. In the case of the use of storage for arbitrage, and with linear cost functions for buying and selling at each instant in time, the problem of optimal control is the classical \emph{warehouse problem} (see \cite{Cahn, Bell, Drey} and also \cite{Sec2010} for a more recent example). Cruise et al \cite{CFGZ} consider the optimal control of storage---in both a deterministic and a stochastic setting---in the case where the store is a price maker (i.e.\ the size of the store is sufficiently large that its activities influence prices in the market in which it operates) and is subject to both capacity and rate constraints; they develop the associated Lagrangian theory, and further show that the optimal control at any point in time usually depends only on the cost functions associated with a short future time horizon. Recent alternative approaches for studying the value and use of storage for arbitrage can be found in the papers~\cite{KHT,PADS,SDJ,VHMS,WAM}---see also the text~\cite{WW}, and the further references given in~\cite{CFGZ}. For an assessment of the potential value of energy storage in the UK electricity system see~\cite{TMTSBP}.
There have been numerous studies into the use of storage for buffering against both the increased variability and the increased uncertainty in electrical power systems, due to higher penetration of renewable penetration---the former due to the natural variability of such resources as wind power, and the latter due to the inherent uncertainty of forecasting. These studies have considered many different more detailed objectives; these range from the sizing and control of storage facilities co-located with the renewable generation so as to provide a smoother supply and so offset the need for network reinforcement \cite{CL, DS, KHH}, to studies on storage embedded within transmission networks so as to increase wind power utilisation and so reduce overall generation costs \cite{HD,RJM, TO}. In addition there have been a number of studies into the more general use of storage for buffering, for example, so as to provide fast frequency response to power networks \cite{MMA, OCO, TMTSBP}, or to provide quality of service as part of a microgrid \cite{BLLBP,HMN}.
In general the problem of using a store for buffering is necessarily stochastic. The natural mathematical approach is via stochastic dynamic programming. This, however, is liable to be computationally intractable, especially in the case of long time horizons and the likely time heterogeneity of the stochastic processes involved. Therefore much of the literature considers necessarily somewhat heuristic but nevertheless plausible control policies---again often adapted to meeting a wide variety of objectives. For example, for storage embedded in a distribution network, two control policies are considered in \cite{BI1}; the first policy aims to feed into a store only when necessary to keep local voltage levels within a predefined range and to empty the store again as soon as possible thereafter; the second policy aims to maintain a constant level of load in the network. For larger stores operating within transmission networks, the buffering policies studied have included that of a fixed target level policy \cite{BGK}, a dynamic target level policy \cite{GTL}, and a two stage process with day ahead generation scheduling and a online procedure to adapt load levels \cite{AA}.
Control policies have been studied via a range analytic and simulation based methods. Examples of an analytic approach can be found in \cite{HPSPB}, where partial differential equations are utilised to model the behaviour and control of a store, and in \cite{BI1, BI2}, where spectral analysis of wind and load data is used with models which also incorporate turbine behaviour. Simulation-based studies include \cite{BGK, GTL}, which use a bootstrap approach based on real wind forecast error data, and \cite{AA}, which uses Monte Carlo simulation of the network state.
In the present paper we study the optimal control of a store which is used both for arbitrage and for buffering against unpredictable events. As previously indicated we use an economic framework, so that the store sees costs (positive or negative) associated with buying and selling, and with the provision of buffering services. The store seeks to operate in such a way as to minimise over time the sum of these costs. We believe such an economic framework to be natural when the store operates as part of some larger and perhaps very complex system, provided the price signals under which the store operates are correctly chosen. The store may be sufficiently large as to have market impact, leading to nonlinear cost functions for buying and selling, may be subject to rate (as well as capacity) constraints, and, as will typically be the case, may suffer from round-trip inefficiencies. We formulate a stochastic model which is realistic in many circumstances and characterise some of the properties of an optimal control, relating the results to the existing experimental literature. We develop the associated strong Lagrangian theory and, by making a modest approximation---the validity of which may be tested in practical applications---show how to construct a computationally tractable optimal control. These latter results form a nontrivial extension of those of the ``arbitrage-only'' case studied in~\cite{CFGZ}, and require significant new developments of the necessary optimization theory; as in~\cite{CFGZ}, the optimal control at any time usually depends on a relatively short time horizon (though one which is typically somewhat longer than in the earlier case), so that the algorithm is suitable for the optimal control of the store over an indefinite period of time.
The optimal control is given by the solution, at the start of the control period, of a deterministic optimisation problem which can be regarded as that of minimising the costs associated with the store buying and selling added to those of notionally ``insuring'' for each future instant in time against effects of the random fluctuations resulting from the provision of buffering services. The cost of such ``insurance'' depends on the absolute level of the store at that time. Thus this deterministic problem is that of choosing the vector of successive levels of the store so as to minimise a cost function $\sum_t[C_t(x_t)+A_t(s_t)]$, subject to rate and capacity constraints, where $C_t(x_t)$ is the cost of incrementing the level of the store (positively or negatively) at time~$t$ by $x_t$, and the ``penalty'' function~$A_t$ is such that $A_t(s_t)$ is the expected cost of any failure to provide the required buffering services at the time~$t$ when the level of the store is then $s_t$. We define this optimisation problem~$\mathbf{P}$ more carefully in Sections~\ref{sec:problem} and \ref{sec:simpl-optim-probl}. In the stochastic environment in which the store operates, the solution of this deterministic problem determines the future control of the store until such time as its buffering services are actually required, following which the level of the store is perturbed and the optimisation problem must be re-solved starting at the new level. The continuation of this process provides what is in principle the exactly optimal stochastic control of the store on a potentially indefinite time scale.
In Section~\ref{sec:problem} we formulate the relevant stochastic model and discuss its applicability. This enables us, in Section~\ref{sec:char-optim-solut} to provide some characteristic properties of optimal solutions, which we relate to empirical work in the existing literature. In Sections~\ref{sec:determ-funct-a_t-2} and \ref{sec:simpl-optim-probl} we develop the approach to an optimal control outlined above. Section~\ref{sec:solution} considers the deterministic optimisation problem associated with the stochastic control problem and derives the associated strong Lagrangian theory, while in Section~\ref{sec:algorithm} we develop an efficient algorithm. Section~\ref{sec:example} gives examples.
\section{Problem formulation} \label{sec:problem}
Consider the management of a store over a finite time interval $[0,T]$ where the time horizon $T$ is integer, and where $[0,T]$ is divided into a succession of periods $t=1,\dots,T$ of integer length. At the start of each time period~$t$ the store makes a decision as to how much to buy or sell during that time period; however, the level of the store at the end of that time period may be different from that planned if, during the course of the period, the store is called upon to provide buffering services to deal with some unexpected problem or \emph{shock}. Such a shock might be the need to supply additional energy during the time period~$t$ due to some unexpected failure---for example that of a generator---or might simply be the difference between forecast and actual renewable generation or demand. We suppose that the capacity of the store during the time period~$t$ is $E_t$ units of energy. (Usually $E_t$ will be constant over time, but need not be, and there are some advantages---see in particular Section~\ref{sec:determ-funct-a_t-2}---in allowing the time dependence.) Similarly we suppose that the total energy which may be input or output during the time period~$t$ is subject to rate (i.e.\ power) constraints $P_{It}$ and $P_{Ot}$ respectively. This slotted-time model corresponds, for example, to real world energy markets where energy is typically traded at half-hourly or hourly intervals, with the actual delivery of that energy occurring in the intervening continuous time periods. Detailed descriptions of the operation of the UK market can be found in \cite{NAO,ELEXON}.
For each~$t$ let $X_t=\{x:-P_{Ot}\le x\le P_{It}\}$. Both buying and selling prices associated with any time period~$t$ may be represented by a convex function~$C_t$ defined on $X_t$ which is such that, for positive $x$, $C_t(x)$ is the price of buying $x$ units of energy for delivery during the time period~$t$, while, for negative $x$, $C_t(x)$ is the negative of the price for selling $-x$ units of energy during that time period. Thus, in either case, $C_t(x)$ is the cost of a planned change of $x$ to the level of the store during the time period~$t$, in the absence of any buffering services being required during the course of that time period. The convexity assumption corresponds, for each time~$t$, to an increasing cost to the store of buying each additional unit, a decreasing revenue obtained for selling each additional unit, and every unit buying price being at least as great as every unit selling price. When, as is usually the case, the store is not perfectly \emph{efficient} in the sense that only a fraction $\eta\le1$ of the energy input in available for output, then this may be captured in the cost function by reducing selling prices by the factor~$\eta$; under the additional assumption that the cost functions~$C_t$ are increasing it is easily verified that this adjustment preserves the above convexity of the functions~$C_t$. We thus assume that the cost functions are so adjusted so as to capture any such round-trip inefficiency.
\begin{remark}
\label{rmk:1}
A further form of possible inefficiency of a store is
\emph{leakage}, whereby a fraction of the contents of the store is
lost in each unit of time. We do not explicitly model this here.
However, only routine modifications are required to do so, and are
entirely analogous to those described in~\cite{CFGZ}. \end{remark} \begin{remark}
\label{rmk:2}
Note also that, in the above model, it is possible to absorb the
rate constraints into the cost functions---by setting the costs
associated with $x\notinX_t$ to be prohibitively high---and to
preserve the convexity of these functions. However, in general we
prefer to avoid this approach here. \end{remark}
Suppose that at the end of the time period $t-1$, or equivalently at the start of the time period~$t$, the level of the store is $s_{t-1}$ (where we take $s_0$ to be the initial level of the store). We assume that one may then choose a \emph{planned} adjustment (contract to buy or sell) $x_t\inX_t$---and such that additionally $s_{t-1}+x_t\in[0,E_t]$---to the level of the store during the time period~$t$, the cost of this adjustment being $C_t(x_t)$. Subsequent to this, during the course of the time period~$t$, the the store may subject to some \emph{shock} or random disturbance, corresponding perhaps to the need to provide unexpected buffering services, which may both disturb the final level of the store at the end of that time period---and perhaps also at the end of subsequent time periods---and have further associated costs, the latter being typically those of the store not being able to provide the required services.
For each $t$, and for each possible level~$s_{t-1}$ of the store at the end of the time period~$t-1$, define $V_{t-1}(s_{t-1})$ to be the expected future cost of subsequently managing the store under an optimal strategy (i.e.\ one under which this expected cost is minimised), under the assumption that either no shocks have occurred by the end of the time period~$t-1$ or that, given the level $s_{t-1}$, such past shocks as have occurred by that time do not influence the optimal future management of the store or its associated costs. Under these conditions, and for a planned adjustment~$x_t$ to the level of the store during the time period~$t$ (at an immediate cost~$C_t(x_t)$ as indicated above), in the absence of any shock during the time period~$t$, the expected cost of optimally managing the store thereafter is then $V_t(s_{t-1}+x_t)$. We assume that the expected \emph{additional} cost to the store, both immediate and future, of dealing optimally with any shock which may occur during the time period~$t$ is a function~$A_t(s_{t-1}+x_t)$ of the planned level~$s_{t-1}+x_t$ of the store for the end of the time period~$t$. We then have that \begin{equation}
\label{eq:1}
V_{t-1}(s_{t-1}) = \min_{\substack{x_t\in X_t\\s_{t-1}+x_t\in\cap[0,E_t]}}
\left[
C_t(x_t) + A_t(s_{t-1}+x_t) + V_t(s_{t-1}+x_t)
\right], \end{equation} and that the optimal planned increment to the level of the store for the time period~$t$ (given that an optimal policy is to be followed thereafter) is given by $\hat{x}_t(s_{t-1})$ where this is defined to be the value of $x_t\inX_t$ which achieves the minimisation in the recursion~\eqref{eq:1}.
We also define the terminal condition \begin{equation}
\label{eq:2}
V_T(s_T) = 0 \end{equation} for all possible levels $s_T$ of the store at the end of the time period~$T$.
Note that $A_t(s_{t-1}+x_t)$ (which may be alternatively be interpreted as the ``insurance'' cost associated with the planed level of the store for the time period~$t$ as described in the Introduction) may be understood via a coupling argument, in which the possibly disturbed and subsequently optimally controlled process of store levels---following any shock in the time period~$t$---is coupled to the process which is undisturbed in that time period and subsequently optimally controlled; $A_t(s_{t-1}+x_t)$ is then the expected difference in the costs of operating the two processes until such time (if ever) as they subsequently merge. As we discuss further below, this interpretation proves useful in finding workable approximations to the functions~$A_t$.
\begin{remark}
We make the assumption above that each function~$A_t$, representing
the extra cost of dealing with a shock occurring during the time
period~$t$, may be represented as a function of the planned level
$s_{t-1}+x_t$ of the store for the end of that time period and,
given this, does not further depend on the level~$s_{t-1}$ of the
store at the beginning of that time period. The accuracy of this
assumption will vary according to the precise characteristics of the
store, the way in which it interacts with its external environment
in the event of shocks, and the various cost functions which form
part of the model. The assumption is likely to be at its most
accurate when rate constraints do not play a major role in the
management of the store, as the store may adjust to its target
levels quickly. Elsewhere, when the level of the store does not
change too much during a single time period, the assumption may
still be regarded as a reasonable approximation. Its
relaxation---for example by allowing $A_t$ to be a more general
function of $s_{t-1}$ and $x_t$---simply complicates without
essentially changing the analysis below. \end{remark}
Our aim is to determine the optimal control of the store over the time interval $[0,T]$. Such a control will necessarily be stochastic. In principle some form of stochastic dynamic programming approach is required. However, particularly within a time heterogeneous environment (in which there may be no form of regularity in either the functions~$C_t$ or in the shock processes), such an approach would be unlikely to be efficient, and might well prove computationally intractable, on account of (a) the need, in such an approach, to completely determine each of the functions~$V_t$ defined above, and (b) the need to solve the problem over the entire time interval $[t,T]$ in order to determine the optimal control at any time~$t$.
Our method of proceeding is therefore as follows. We assume that the functions $A_t$ are known, at least to within reasonable approximations. (We argue in Section~\ref{sec:determ-funct-a_t-2} that in many cases the functions~$A_t$ may be determined either exactly or to within a very good approximation; this follows from the coupling characterisation of these functions introduced above.) Given the initial level~$s_0$ of the store we may then use the argument leading to the recursion~\eqref{eq:1} and \eqref{eq:2} to determine very efficiently a control which remains optimal up to the end of the first time period in which a shock actually occurs. Following such a shock (and, if necessary, once its knock-on effects have cleared from the system---again see the discussion of Section~\ref{sec:determ-funct-a_t-2}), the current state (level) of the store is reexamined and the optimal future control strategy recalculated. Iteration of this process leads to an efficient (stochastic) dynamic control for the entire time interval $[0,T]$. We also show below that typically the optimal decision at (the start of) any time~$t$ depends only on the functions $C_{t'}$ and $A_{t'}$ for values of time~$t'$ extending only a little beyond the time~$t$. The approach outlined above is therefore generally also suitable for the ongoing optimal management of the store over an indefinite period of time.
\section{Characterisation of optimal solutions} \label{sec:char-optim-solut}
In this section we establish some properties of the functions $\hat{x}_t(\cdot)$ defined in the previous section and determining the optimal control of the store.
One case which will be of particular interest is that where the store is a price taker (i.e.\ the store is not so large as to impact itself on market prices), so that, for each~$t$, the cost function~$C_t$ is given by \begin{equation}
\label{eq:3}
C_t(x) =
\begin{cases}
c^{(b)}_t x, & \quad\text{if $x\ge0$}\\
c^{(s)}_t x, & \quad\text{if $x<0$}.
\end{cases} \end{equation} and where the unit ``buying'' price~$c^{(b)}_t$ and the unit ``selling'' price~$c^{(s)}_t$ are such that $c^{(s)}_t\le c^{(b)}_t$. (That, at any time~$t$, the reward obtained in the market resulting from decreasing the level of the store by a single unit may be less than the cost of increasing the level of the store by a single unit may primarily reflect the fact that the store may be less than perfectly efficient---see the discussion of Section~\ref{sec:problem}.)
Proposition~\ref{prop:simple} below is a very simple result which shows that in the case where buying and selling prices are equal (typically corresponding to a perfectly efficient store), and provided rate constraints are nonbinding, the optimal policy is a ``target'' one. By this we mean that for each time period~$t$ there exists a target level~$\hat{s}_t$: given that the level of the store at the end of the immediately preceding time period is $s_{t-1}$ and that shocks prior to that time have no further ongoing effects on the management of the store, the optimal planned level~$s_{t-1}+x_t$ of the store to be achieved during the time period~$t$ is set to some value~$\hat{s}_t$, independently of $s_{t-1}$.
\begin{proposition}\label{prop:simple}
Suppose that, for each $t$, we have $c^{(b)}_t=c^{(s)}_t=c_t$ say;
define
\begin{equation}
\label{eq:4}
\hat{s}_t = \argmin_{s\in[0,E_t]}[c_t s + A_t(s) + V_t(s)].
\end{equation}
Then, for each~$t$ and for each $s_{t-1}$, we have
$\hat{x}_t(s_{t-1})=\hat{s}_t-s_{t-1}$ provided only that this
quantity belongs to the set $X_t$. \end{proposition}
\begin{proof}
The recursion~\eqref{eq:1} here becomes, for each $t$,
\begin{equation}
\label{eq:5}
V_{t-1}(s_{t-1}) = \min_{\substack{x_t\in X_t\\s_{t-1}+x_t\in\cap[0,E_t]}}
\left[
c_tx_t + A_t(s_{t-1}+x_t) + V_t(s_{t-1}+x_t)
\right],
\end{equation}
and the above minimisation is achieved by $x_t$ such that
$s_{t-1}+x_t=\hat{s}_t$, provided only that $x_t\inX_t$. \end{proof}
In order to deal with the possibility of rate constraint violation, with the more general price-taker case where $c^{(s)}_t<c^{(b)}_t$, and with the quite general case where the cost functions~$C_t$ are merely required to be convex, we require the additional assumption of convexity of the functions~$A_t$. This latter condition, while not automatic, is reasonably natural in many applications---see the examples of Section~\ref{sec:example}.
\begin{proposition}\label{prop:aconvex}
Suppose that, in addition to convexity of the functions~$C_t$, each
of the functions~$A_t$ is convex. Then, for each~$t$:
\begin{compactenum}[(i)]
\item the function~$V_{t-1}$ is convex;
\item $\hat{x}_t(s_{t-1})$ is a decreasing function of $s_{t-1}$;
\item $s_{t-1}+\hat{x}_t(s_{t-1})$ is an increasing function of
$s_{t-1}$.
\end{compactenum} \end{proposition}
\begin{proof}
It is helpful to define, for each $t=1,\dots,T$, the
function~$U_{t-1}$ of each possible level~$s_{t-1}$ of the store at
the end of the time period~$t-1$, and each possible planned
increment~$x_t$ to the level of the store for the time period~$t$ by
\begin{equation}
\label{eq:6}
U_{t-1}(s_{t-1},\,x_t) = C_t(x_t) + A_t(s_{t-1}+x_t) + V_t(s_{t-1}+x_t).
\end{equation}
The recursion~\eqref{eq:1} now becomes
\begin{equation}
\label{eq:7}
V_{t-1}(s_{t-1})
= \min_{\substack{x_t\in X_t\\s_{t-1}+x_t\in\cap[0,E_t]}} U_{t-1}(s_{t-1},\,x_t).
\end{equation}
To show (i) we use backwards induction in time. The function $V_T$
is convex. Suppose that, for any given $t\le T$, the function~$V_t$
is convex; we show that the function~$V_{t-1}$ is convex. For each
of given values $s^{(i)}_{t-1}$, $i=1,\dots,n$ of $s_{t-1}$, let
$x^{(i)}_t$ be the value of $x_t$ which achieves the minimisation in
\eqref{eq:7}, and for any convex combination
$\bar s_{t-1}=\sum_{i=1}^n\kappa_is^{(i)}_{t-1}$, where each
$\kappa_i\ge0$ and where $\sum_{i=1}^n\kappa_i=1$, define also
$\bar{x}_t=\sum_{i=1}^n\kappa_ix^{(i)}_t$. Note that
$\bar{x}_t\inX_t$ and that $\bar{s}_{t-1}+\bar{x}_t\in[0,E_t]$. Then,
from~\eqref{eq:7},
\begin{align*}
V_{t-1}(\bar s_{t-1})
& \le U_{t-1}(\bar s_{t-1},\,\bar x_t)\\
& \le \sum_{i=1}^n\kappa_i U_{t-1}(s^{(i)}_{t-1},\,x^{(i)}_t)\\
& = \sum_{i=1}^n\kappa_iV_{t-1}(s^{(i)}_{t-1}),
\end{align*}
where the second line in the above display follows from the
definition~\eqref{eq:6} of the function~$U_{t-1}$ and the convexity
of the functions~$C_t$, $A_t$ and $V_t$ (the latter by the inductive
hypothesis). Thus $V_{t-1}$ is convex as required.
To show (ii) and (iii), given values
$s^{(1)}_{t-1}\le s^{(2)}_{t-1}$ of $s_{t-1}$, again let
$x^{(1)}_t$, $x^{(2)}_t$ be the respective values of $x_t$ which
achieves the minimisation in \eqref{eq:7}. Since for the function
$U_{t-1}(s^{(1)}_{t-1},\,\cdot)$ is minimised in $X_t\cap E_t$ at
$x^{(1)}_t$, it follows straightforwardly, from the
definition~\eqref{eq:6} of the function~$U_{t-1}$ and the convexity
of the function~$C_t$ and that of the function $A_t+V_t$, that,
since $s^{(1)}_{t-1}\le s^{(2)}_{t-1}$, the minimisation of the
function $U_{t-1}(s^{(2)}_{t-1},\,\cdot)$ is achieved (or, in the
case of nonuniqueness, may be achieved) at $x^{(2)}_t\le x^{(1)}_t$.
Thus the result~(ii) follows. Similarly, it is again
straightforward from the convexity of the function $C_t$ and that of
the function $A_t+V_t$ and since $s^{(1)}_{t-1}\le s^{(2)}_{t-1}$,
that $x^{(2)}_t$ is (or, in the case of nonuniqueness, may be taken
to be) such that
$s^{(2)}_{t-1}+x^{(2)}_t\ge s^{(1)}_{t-1}+x^{(1)}_t$. The
result~(iii) thus similarly follows.
\end{proof}
\begin{remark}
\label{rmk:3}
Note that the rate constraints $x_t\inX_t$ for all $t$ cause no
difficulties for the above proof---a result which may alternatively
be seen by absorbing these constraints into the cost functions $C_t$
as described in Remark~\ref{rmk:2}. \end{remark}
We now return to the price-taker case, in which the cost functions are as defined by~\eqref{eq:3}, and which corresponds to a store which is not sufficiently large as to have market impact. Here we may prove a strengthened version of Proposition~\ref{prop:aconvex}. For each $t$, given that the function~$A_t$ is convex, define \begin{equation}
\label{eq:8}
s^{(b)}_t = \argmin_{s\in[0,E_t]}[c^{(b)}_t s + A_t(s) + V_t(s)] \end{equation} and similarly define \begin{equation} \label{eq:9}
s^{(s)}_t = \argmin_{s\in[0,E_t]}[c^{(s)}_t s + A_t(s) + V_t(s)]. \end{equation} Note that the above convexity assumption and the condition that, for each~$t$, we have $c^{(s)}_t\le c^{(b)}_t$ imply that $s^{(b)}_t\le s^{(s)}_t$.
We now have the following result. \begin{proposition}
\label{prop:interval}
Suppose that the cost functions~$C_t$ are as given by \eqref{eq:3}
and that the functions~$A_t$ are convex. Then the optimal policy is
given by: for each~$t$ and given $s_{t-1}$, take
\begin{equation}
\label{eq:10}
x_t =
\begin{cases}
\min(s^{(b)}_t-s_{t-1},\,P_{It}) & \quad\text{if $s_{t-1}<s^{(b)}_t$,}\\
0 & \quad\text{if $s^{(b)}_t\le s_{t-1}\le s^{(s)}_t$,}\\
\max(s^{(s)}_t-s_{t-1},\,-P_{Ot}) & \quad\text{if $s_{t-1}>s^{(s)}_t$.}
\end{cases}
\end{equation} \end{proposition}
\begin{proof}
For each $t$, it follows from the convexity of the functions~$C_t$,
$A_t$ and $V_t$ (the latter by the first part of
Proposition~\ref{prop:aconvex}) that, for $s_{t-1}<s^{(b)}_t$ the
function $C_t(x_t)+A_t(s_{t-1}+x_t)+V_t(s_{t-1}+x_t)$ is minimised
by $x_t=s^{(b)}_t-s_{t-1}$, for $s^{(b)}_t\le s_{t-1}\le s^{(s)}_t$
it is minimised by $x_t=0$, while for $s_{t-1}>s^{(s)}_t$, it is
minimised by $x_t=s^{(s)}_t-s_{t-1}$. The required result now
follows from the recursion~\eqref{eq:1}. \end{proof}
Thus in general in the price-taker case there exists, for each time period~$t$, a ``target interval'' $[s^{(b)}_t,s^{(s)}_t]$ such that, if the level of the store at the end of the previous time period is $s_{t-1}$ (and again given that the shocks prior to this time have no ongoing effects on the optimal management of the store), the optimal policy is to chose $x_t$ so that $s_{t-1}+x_t$ is the nearest point (in absolute distance) to $s_{t-1}$ lying within, or as close as possible to, the above interval. In the case where $c^{(b)}_t=c^{(s)}_t=c_t$, the above interval shrinks to the single point~$\hat{s}_t$ defined by~\eqref{eq:4}.
These results shed some light on earlier, more applied, papers of Bejan et al~\cite{BGK} and Gast et al~ \cite{GTL}, in which the uncertainties in the operation of a energy store result from errors in wind power forecasts. The model considered in those papers is close to that of the present paper, as we now describe. The costs of operating the store result (a) from round-trip inefficiency, which in the formulation of the present paper would be captured by the cost functions $C_t$ as defined by~\eqref{eq:3} with $C_t$ the same for all~$t$, and (b) from buffering events, i.e.\ from failures to meet demand through insufficient energy available to be supplied from the store when it is needed, and from energy losses through store overflows. In the formulation of the present paper these costs would be captured by the functions~$A_t$. In contrast to the present paper decisions affecting the level of the store (the amount of conventional generation to schedule for a particular time) are made $n$ time steps---rather than a single time step---in advance. The shocks to the system result from the differences between the available wind power as forecasted $n$ steps ahead of real time (when conventional generation is scheduled) and the wind power actually obtained. Although the model of the above papers is therefore not exactly the same as that of the present paper, the underlying arguments leading to Propositions~\ref{prop:simple}--\ref{prop:interval} continue to apply, at least to a good approximation. In particular sample path arguments suggest that the reduction of round-trip efficiency slows the rate at which the store-level trajectories---started from different initial levels but with the same stochastic description of future shock processes---converge over subsequent time.
Bejan et al~\cite{BGK} consider only the case where the round-trip efficiency is $1$. They study the efficiency of policies---analogous to those suggested by Proposition~\ref{prop:simple}---whereby, for each~$t$, the generation scheduled for time~$t$ at the earlier time $t-n$ is such as would, given perfect forecasting, achieve a given target level $\hat{s}_t$ of the store at time~$t$; this target level is independent of the level $s_{t-n}$ of the store at the time $t-n$ and of earlier scheduling decisions. However, Bejan et al~\cite{BGK} further take $\hat{s}_t$ to be independent of $t$, something which may not be optimal given the likely nonstationarity of the process of forecast errors.
Gast et al~\cite{GTL} subsequently study the same time series of available wind power, but allowed for round-trip efficiencies which are less than~$1$. They find (as might be expected here) that simple ``target'' policies such as that described above do not work well under these circumstances, and compare the behaviour of a variety of time-homogeneous policies.
\section{Determination of the functions~$A_t$} \label{sec:determ-funct-a_t-2}
We described in Section~\ref{sec:problem} how, given a knowledge of the functions~$A_t$, the optimal control of the store could be determined. In Sections~\ref{sec:simpl-optim-probl}--\ref{sec:algorithm} we develop such an approach, which is based on strong Lagrangian theory and which is very much more efficient, in senses explained there, than the application of standard dynamic programming or nonlinear optimisation techniques. In this section we consider conditions under which the functions~$A_t$ may be thus known, either exactly or to good approximations.
Suppose that, as in Section~\ref{sec:problem}, at the end of the time period~$t-1$ the level of the store is $s_{t-1}$ and that, given $s_{t-1}$, any shocks prior to that time have no further effect on the optimal management of the store. Suppose further that an increase of $x_t$ (positive or negative) is planned for the time period~$t$ (at a cost of $C_t(x_t)$). Recall that $A_t(s_{t-1}+x_t)$ is then defined to be the expected additional cost to the store of dealing optimally with any shock which may occur during the time period~$t$, and may be conveniently characterised in terms of the coupling defined in that Section~\ref{sec:problem}. Now define also $\bar{A}_t(s_{t-1}+x_t)$ to be the expected additional cost to the store of dealing with any shock which may occur during the time period~$t$ and \emph{immediately} returning the level of the store to its planned level $s_{t-1}+x_t$ at the end of the time period~$t$. As in the case of the function~$A_t$, we assume that each function $\bar{A}_t$ depends on $s_{t-1}$ and $x_t$ through their sum $s_{t-1}+x_t$---the extent to which this approximation is reasonable being as discussed for the functions~$A_t$. Given the costs of dealing with any shocks, and the known costs of making any immediate subsequent adjustments to the level of the store, the functions $\bar{A}_t$ are readily determinable, and in particular do not depend on how the store is controlled outside the time period~$t$.
Note that, in the case of linear cost functions (i.e.\ $C_t(x)=c_tx$ for all~$t$) and when shocks do not have effects which persist beyond the end of the time period in which they occur, the argument of Proposition~\ref{prop:simple} implies immediately that $A_t=\bar{A}_t$ for all~$t$: the linearity of $C_t$ implies that, at the end of the time period~$t-1$ and when the level of the store is then $s_{t-1}$, if $s_{t-1}+x_t$ is the optimal planned level of the store for the end of the time period~$t$, then it remains the optimal level of the store for the end of that time period following any shock which occurs during it.
More generally the functions $\bar{A}_t$ provide reasonable approximations to the functions~$A_t$ to the extent to which it is reasonable, following any shock with which the store is required to deal, to return immediately the level of the store to that which would have obtained in the absence of the shock. In particular, when shocks are relatively rare but are potentially expensive (as might be the case when the store is required to pay the costs of failing to have sufficient energy to deal with an emergency), then the major contribution to both the functions~$A_t$ and $\bar{A}_t$ will be this cost, regardless of precisely how the level of the store is adjusted in the immediate aftermath of the shock.
If necessary, better approximations to the functions $A_t$ may be obtained by allowing longer periods of time in which to optimally couple the trajectory of the store level, following a shock, to that which would have obtained in its absence. In applications one would wish to experiment a little here.
In applications there is also a need, when the costs of a shock arise from a failure to have insufficient energy in the store to deal with it, to identify what these costs are. There are various possible candidates. Two simple such---natural in the context of risk metrics for power systems, where they correspond respectively to \emph{loss of
load} and \emph{energy unserved} (see, for example, \cite{Billinton})---are: \begin{compactenum}[(i)] \item for each $t>0$, the cost of a shock occurring during the time
period~$t$ is simply some constant $a_t>0$ if there is insufficient
energy within the store to meet it, and is otherwise~$0$. \item for each $t>0$, the cost of a shock occurring during the time
period~$t$ is proportional to the shortfall in the energy necessary
to meet that shock. \end{compactenum} Given the planned level $s_{t-1}+x_t$ of the store to be achieved during any time period~$t$, the total additional cost of dealing with any shock occurring during that time period (as defined for example in terms of the coupling introduced in Section~\ref{sec:problem}) is a random variable which is a function of the size of the shock. The distribution of this random variable, and so also its expectation $A_t(s_{t-1}+x_t)$ may need to be determined by observation.
Note finally that the effects of shocks may persist over several time periods (as, for example, when the store is required to provide ongoing support for the sudden loss of major piece of equipment such as a generator), so that each of the functions~$A_t$---which will in general be decreasing---need not be flat for values of its argument in excess of the output rate constraint~$P_{Ot}$. In particular a reasonable way of dealing with a shock whose effects do persist over several time periods may simply be to reserve notionally sufficient energy in the store to deal with it; then, following such a shock, the level of the store will temporally become the excess over that reserve and the capacity of the store will correspondingly be temporally reduced. This causes no problems for the present theory, and is a reason for allowing a possible time dependence (which may be dynamic) for the capacity of the store.
We consider some plausible functional forms of the functions~$A_t$ in Section~\ref{sec:example}.
\section{The optimal control problem} \label{sec:simpl-optim-probl}
We now assume that the functions $A_t$ defined in Section~\ref{sec:problem} are known, at least to a sufficiently good approximation---see the discussion of the previous section.
Define (the random variable) $\hat{s}=(\hat{s}_0,\dots,\hat{s}_T)$ (with $\hat{s}_0=s^*_0$) to be the levels of the store at the end of the successive time periods $t=0,\dots,T$ under the (stochastic) optimal control as defined in Section~\ref{sec:problem}.
Recall also from Section~\ref{sec:problem} that, for each~$t$ and each level $s_{t-1}$ of the store at the end of the time period~$t-1$, the quantity~$\hat{x}_t(s_{t-1})$ is the value of $x_t\inX_t$ which achieves the minimisation in the recursion~\eqref{eq:1}.
For any vector $s=(s_0,\dots,s_T)$ and for each $t=1,\dots,T$, define \begin{equation}
\label{eq:13}
x_t(s) = s_t - s_{t-1}. \end{equation} Define also the following (deterministic) optimisation problem: \begin{compactitem} \item[$\mathbf{P}$:]
choose $s=(s_0,\dots,s_T)$ with $s_0=s^*_0$ so as to minimise
\begin{equation}
\label{eq:14}
\sum_{t=1}^T [C_t(x_t(s)) + A_t(s_t)]
\end{equation}
subject to the capacity constraints
\begin{gather}
\label{eq:15}
0 \le s_t\le E_t,
\quad 1 \le t \le T,
\end{gather}
and the rate constraints
\begin{equation}
\label{eq:16}
x_t(s) \in X_t,
\qquad 1 \le t \le T.
\end{equation} \end{compactitem} Let $s^*=(s^*_0,\dots,s^*_T)$ denote the solution to the above problem~$\mathbf{P}$. It follows from direct iteration of the recursion~\eqref{eq:1}, using also~\eqref{eq:2}, that $x_1(s^*)$ achieves the minimisation in~\eqref{eq:1} for $t=1$ and when $s_0=s^*_0$, i.e.\ that $\hat{x}_1(s^*_0)=\hat{x}_1(\hat{s}_0)=x_1(s^*)$. Thus, from~\eqref{eq:13}, provided no shock occurs during the time period~$1$ so that $\hat{s}_1=\hat{s}_0+\hat{x}_1(\hat{s}_0)$, we have also that $\hat{s}_1=s^*_1$. More generally, let the random variable~$T'$ index the first time period during which a shock does occur. Then repeated application of the above argument gives immediately the following result.
\begin{proposition}
For all $t<T'$, we have $\hat{s}_t=s^*_t$. \end{proposition}
The solution to the problem $\mathbf{P}$ therefore defines the optimal control of the store up to the end of the time period~$T'$ defined above. At that time, and the end of each subsequent time period during which there occurs a shock, it is of course necessary to reformulate the problem~$\mathbf{P}$, starting at the end of the time period~$T'$ (or as soon any shock occurring during that time period has been fully dealt with), instead of at time~$0$, and replacing the initial level $s^*_0=\hat{s}_0$ by the perturbed level $\hat{s}_{T'}$ of the store at that time. Thus the stochastic optimal control problem may be solved dynamically by the solution of the problem~$\mathbf{P}$ at time~$0$, and the further solution of (a reformulated version) of this problem at the end of each subsequent time period in which a shock occurs. The solution of the problem, which we now consider, is very much simpler than that of the corresponding stochastic dynamic programming approach.
\section{Lagrangian theory and characterisation of solution} \label{sec:solution}
We showed in the previous section that, to the extent that the functions~$A_t$ are known, an optimal control for the store may be developed via the solution of the optimisation problem~$\mathbf{P}$ defined there. In Section~\ref{sec:determ-funct-a_t-2} we discussed how to make what are in many cases good and readily determinable approximations for the functions~$A_t$.
We again assume convexity of the functions~$A_t$ (see Section~\ref{sec:char-optim-solut}), in addition to that of the functions~$C_t$. We develop the strong Lagrangian theory~\cite{Boyd,Whi} associated with the problem~$\mathbf{P}$. This leads to both an efficient algorithm for its solution, and to the identification of the Lagrange multipliers necessary for the proper dimensioning of the store. In particular Theorem~\ref{thm:exist} establishes the existence of a pair of vectors $(s^*,\,\lambda^*)$ such that $s^*$ solves the problem~$\mathbf{P}$ and $\lambda^*$ is a function of the associated Lagrange multipliers corresponding to the capacity constraints (see below); the theorem further gives conditions necessarily satisfied by the pair $(s^*,\,\lambda^*)$.
We now introduce the more general problem~$\mathbf{P}(a,\,b)$ in which $s_0$ is kept fixed at the value $s^*_0$ of interest above, but in which $s_1,\dots,s_T$ are allowed to vary between quite general upper and lower bounds: \begin{compactitem} \item[$\mathbf{P}(a,\,b)$:]
choose $s=(s_0,\dots,s_T)$, with $s_0=s^*_0$ so as to minimise
\begin{equation}
\label{eq:17}
\sum_{t=1}^T [C_t(x_t(s)) + A_t(s_t)]
\end{equation}
subject to the capacity constraints
\begin{equation}
\label{eq:18}
a_t \le s_t\le b_t,
\quad 1 \le t \le T,
\end{equation}
and the rate constraints
\begin{equation}
\label{eq:19}
x_t(s) \in X_t,
\qquad 1 \le t \le T.
\end{equation} \end{compactitem} Here $a=(a_1,\dots,a_T)$ and $b=(b_1,\dots,b_T)$ are such that $a_t\le b_t$ for all $t$. Let also $a^*$ and $b^*$ be the values of $a$ and $b$ corresponding to our particular problem~$\mathbf{P}$ of interest, i.e.\ \begin{equation}
\label{eq:20}
a^*_t = 0, \quad b^*_t = E_t, \quad 1\le t\le T. \end{equation}
Note that the convexity of the functions $C_t$ and $A_t$ guarantees their continuity, and, since for each $a$, $b$ as above the space of allowed values of $s$ is compact, a solution $s^*(a,\,b)$ to the problem~$\mathbf{P}(a,\,b)$ always exists. Let $V(a,\,b)$ be the corresponding minimised value of the objective function, i.e.\ \begin{displaymath}
V(a,\,b)=\sum_{t=1}^T[C_t(x_t(s^*(a,\,b)))+A_t(s^*(a,\,b))]. \end{displaymath} Observe also that the function $V(a,\,b)$ is itself convex in $a$ and $b$. To see this, consider any convex combination $(\bar a,\bar b)=(\kappa a_1+(1-\kappa)a_2,\kappa b_1+(1-\kappa)b_2)$ of any two values $(a_1,b_1)$, $(a_2,b_2)$ of the pair $(a,b)$, where $0\le\kappa\le1$; since the constraints~\eqref{eq:18} and \eqref{eq:19} are linear, it follows that the vector $\bar s=\kappa s^*(a_1,b_1)+(1-\kappa)s^*(a_2,b_2)$ is feasible for the problem~$\mathbf{P}(\bar a,\,\bar b)$; hence, from the convexity of the functions~$C_t$ and $A_t$, \begin{align*}
V(\bar a,\bar b)
& \le \sum_{t=1}^T[C_t(x_t(\bar s)) + A_t(\bar s_t)]\\
& = \sum_{t=1}^T[C_t(\kappa x_t(s^*(a_1,b_1)) + (1-\kappa)x_t(s^*(a_2,b_2)))
+ A_t(\kappa s^*_t(a_1,b_1) + (1-\kappa)s^*_t(a_2,b_2))]\\
& \le \kappa\sum_{t=1}^T[C_t(x_t(s^*(a_1,b_1)))+A_t(s^*_t(a_1,b_1))]
+ (1-\kappa)\sum_{t=1}^T[C_t(x_t(s^*(a_2,b_2)))+A_t(s^*_t(a_2,b_2))]\\
& = \kappa V(a_1,b_1) + (1-\kappa)V(a_2,b_2). \end{align*}
We now have the following result, which encapsulates the relevant strong Lagrangian theory.
\begin{theorem}
\label{thm:exist}
Let $s^*$ denote the solution to the
problem~$\mathbf{P}$. Then there exists a vector
$\lambda^*=(\lambda^*_1,\dots,\lambda^*_T)$ such that
\begin{compactenum}[(i)]
\item \label{con1} for all vectors~$s$ such that $s_0=s^*_0$ and
$x_t(s)\inX_t$ for all $t$ ($s$ is not otherwise constrained),
\begin{equation}
\label{eq:21}
\sum_{t=1}^T \left[C_t(x_t(s)) + A_t(s_t) - \lambda^*_t s_t \right]
\ge
\sum_{t=1}^T \left[C_t(x_t(s^*)) + A_t(s^*_t)- \lambda^*_t s^*_t \right].
\end{equation}
\item \label{con2} the pair $(s^*,\lambda^*)$ satisfies the
complementary slackness conditions, for $1\le t\le T$,
\begin{equation}
\label{eq:22}
\begin{cases}
\lambda^*_t = 0 & \quad\text{if $0 < s^*_t < E_t$,}\\
\lambda^*_t \ge 0 & \quad\text{if $s^*_t = 0$,}\\
\lambda^*_t \le 0 & \quad\text{if $s^*_t = E_t$.}
\end{cases}
\end{equation}
\end{compactenum}
Conversely, suppose that there exists a pair of vectors
$(s^*,\,\lambda^*)$, with $s_0=s^*_0$, satisfying the
conditions~\eqref{con1} and \eqref{con2} and such that $s^*$ is
additionally feasible for the problem~$\mathbf{P}$. Then $s^*$
solves the problem~$\mathbf{P}$. \end{theorem}
\begin{proof}
Consider the general problem~$\mathbf{P}(a,\,b)$ defined above.
Introduce slack (or surplus) variables $z=(z_1,\dots,z_t)$ and
$w=(w_1,\dots,w_t)$ and rewrite $\mathbf{P}(a,\,b)$ as:
\begin{compactitem}
\item[$\mathbf{P}(a,\,b)$:] minimise
\begin{math}
\sum_{t=1}^T [C_t(x_t(s)) + A_t(s_t)]
\end{math}
over all $s=(s_0,\dots,s_T)$ with $s_0=s^*_0$, over all $z\ge0$,
over all $w\ge0$, and subject to the further constraints
\begin{align}
s_t - z_t & = a_t, \qquad 1 \le t \le T, \label{eq:23}\\
s_t + w_t & = b_t, \qquad 1 \le t \le T, \label{eq:24}
\end{align}
and also $x_t(s)\inX_t$ for $1 \le t \le T$.
\end{compactitem}
Since the function $V(a,\,b)$ is also convex in $a$ and $b$, it
follows from the supporting hyperplane theorem (see \cite{Boyd} or
\cite{Whi}), that there exist Lagrange multipliers
$\alpha^*=(\alpha^*_1,\dots,\alpha^*_T)$ and
$\beta^*=(\beta^*_1,\dots,\beta^*_T)$ such that
\begin{equation}
\label{eq:25}
V(a,\,b) \ge V(a^*,\,b^*) + \sum_{t=1}^T\alpha^*_t(a_t-a^*_t) +
\sum_{t=1}^T\beta^*_t(b_t-b^*_t)
\qquad\text{for all $a$, $b$}
\end{equation}
Thus also, for all $s$ with $s_0=s^*_0$ and such that $x_t(s)\inX_t$
for $1 \le t \le T$, for all $z\ge0$, and for all $w\ge0$, by
defining $a$ and $b$ via \eqref{eq:23} and \eqref{eq:24}, we have
\begin{multline}
\label{eq:26}
\sum_{t=1}^T \left[C_t(x_t(s)) + A_t(s_t) - \alpha^*_t(s_t-z_t)
-\beta^*_t(s_t+w_t)\right]\\
\ge
\sum_{t=1}^T \left[C_t(x_t(s^*)) + A_t(s^*_t) - \alpha^*_ta^*_t
-\beta^*_tb^*_t\right].
\end{multline}
Since the components of $z$ and $w$ may take arbitrary positive
values, we obtain at once the following complementary slackness
conditions for the vectors of Lagrange multipliers $\alpha^*$ and
$\beta^*$:
\begin{align}
\alpha^*_t \ge 0, \qquad &
\text{$\alpha^*_t=0$ whenever $s^*_t>a^*_t$},
\qquad 1\le t \le T,
\label{eq:27}\\
\beta^*_t \le 0, \qquad &
\text{$\beta^*_t=0$ whenever $s^*_t<b^*_t$},
\qquad 1\le t \le T.
\label{eq:28}
\end{align}
Thus, from \eqref{eq:26}--\eqref{eq:28}, by taking $z_t=w_t=0$ for
all $t$ on the left side of~\eqref{eq:26}, it follows that, for all
$s$ with $s_0=s^*_0$ and $x_t(s)\inX_t$ for $1 \le t \le T$,
\begin{equation}
\label{eq:29}
\sum_{t=1}^T
\left[C_t(x_t(s)) + A_t(s_t) - (\alpha^*_t+\beta^*_t)s_t \right]
\ge
\sum_{t=1}^T
\left[C_t(x_t(s^*)) +A_t(s^*_t) - (\alpha^*_t+\beta^*_t)s^*_t \right].
\end{equation}
The condition~\eqref{con1} of the theorem now follows on defining
\begin{equation}
\label{eq:30}
\lambda^*_t=\alpha^*_t+\beta^*_t,
\qquad 1 \le t \le T.
\end{equation}
while the condition~\eqref{con2} follows from \eqref{eq:30} on using
also the complementary slackness conditions \eqref{eq:27} and
\eqref{eq:28}.
To prove the converse result, suppose that a pair
$(s^*,\,\lambda^*)$ (with $s_0=s^*_0$) satisfies the
conditions~\eqref{con1} and \eqref{con2} and that $s^*$ is feasible
for the problem~$\mathbf{P}$. From the condition~\eqref{con2}, we may
define (unique) vectors $\alpha^*=(\alpha^*_1,\dots,\alpha^*_T)$ and
$\beta^*=(\beta^*_1,\dots,\beta^*_T)$ such that the
conditions~\eqref{eq:27}, \eqref{eq:28} and \eqref{eq:30} hold. The
condition~\eqref{con1} of the theorem now translates to the
requirement that, for all vectors~$s$ such that $s_0=s^*_0$ and
$x_t(s)\inX_t$ for all $t$, the relation~\eqref{eq:29} holds.
Finally, it follows from this and from the conditions~\eqref{eq:27}
and \eqref{eq:28} that, for any vector $s$ which is feasible for the
problem~$\mathbf{P}$---and so in particular satisfies $0\le s_t\le E_t$ for
all $t$,
\begin{equation}
\label{eq:31}
\sum_{t=1}^T \left[C_t(x_t(s)) + A_t(s_t) \right]
\ge
\sum_{t=1}^T \left[C_t(x_t(s^*)) +A_t(s^*_t) \right],
\end{equation}
so that $s^*$ solves the problem~$\mathbf{P}$ as required. \end{proof}
\begin{remark}
Note that the second part of Theorem~\ref{thm:exist}, i.e.\ the
converse result, does not require the convexity assumptions on the
functions~$C_t$ and $A_t$. \end{remark}
The above Lagrangian theory---which we require for the determination of the optimal control as described in Section~\ref{sec:algorithm}---further enables a determination of the sensitivity of the value of the store with respect to variation of its capacity constraints. For the given problem~$\mathbf{P}$, the cost of optimally operating the store (the negative of its value) is given by $V(a^*,\,b^*)$, where we recall that $a^*$ and $b^*$ are as given by \eqref{eq:20}. For any~$t$, the derivative of this optimised cost with respect to $E_t$, assuming this derivative to exist, is given by the Lagrange multiplier~$\beta^*_t$ defined in the above proof (the differentiability assumption ensuring that $\beta_t$ is here uniquely defined). Note further that when $s^*_t<E_t$ then (from~\eqref{eq:28}) the Lagrange multiplier~$\beta^*_t$ is equal to zero, and when $s^*_t=E_t$ then (from \eqref{eq:27} and \eqref{eq:30}) we have $\beta^*_t=\lambda^*_t$.
A further determination of the sensitivity of the value of the store with respect to variation of its rate constraint may be developed along the lines of Theorem~5 of Cruise et al~\cite{CFGZ}, but we do not pursue this here.
\section{Determination of $(s^*,\lambda^*)$} \label{sec:algorithm}
The structure of the objective function causes some difficulties for the solution of the problem~$\mathbf{P}$. As previously observed, a dynamic programming approach might seem natural but, even for this deterministic problem, typically remains too computationally complex---on account of both the likely time-heterogeneity of the functions $A_t$ and $C_t$, and of the need, even for small $t$, to consider the problem over the entire time interval~$[0,\,T]$.
We continue to assume convexity of the functions $C_t$ and $A_t$. Under the further assumption of differentiability of the functions $A_t$, we give an efficient algorithm for the construction of a pair $(s^*,\lambda^*)$ satisfying the conditions of Theorem~\ref{thm:exist}---so that, in particular, $s^*$ solves the problem~$\mathbf{P}$. This algorithm is further sequential and local in time, in the sense that the determination of the solution to any given time~$t'\le T$ typically requires only the consideration of the problem, i.e.\ a knowledge of the functions $C_t$ and $A_t$, for those times~$t$ extending to some time horizon which is typically only a short distance beyond~$t'$.
We have already shown in Section~\ref{sec:simpl-optim-probl} that the ability to dynamically solve the deterministic problem~$\mathbf{P}$, or updates of this problem, at the times of successive shocks enables an efficient (stochastically) optimal control of the store.
We give conditions necessarily satisfied by the pair $(s^*,\lambda^*)$. Under the further assumption of strict convexity of the functions~$C_t$, we show how these conditions may be used to determine $(s^*,\lambda^*)$ uniquely. We then indicate how the strict/ convexity assumption may be relaxed.
\begin{proposition}
\label{prop:existnu}
Suppose that the functions~$A_t$ are differentiable, and that the
pair~$(s^*,\lambda^*)$ is such that $s^*$ is feasible for the
problem~$\mathbf{P}$, while $(s^*,\lambda^*)$ satisfies the
condition~(\ref{con2}) of Theorem~\ref{thm:exist}. For each~$t$
define
\begin{equation}
\label{eq:32}
\nu^*_t = \sum_{u=t}^T[\lambda^*_u - A'_u(s^*_u)].
\end{equation}
Then the condition that $(s^*,\lambda^*)$ satisfies the
condition~(\ref{con1}) of Theorem~\ref{thm:exist} is equivalent to the
condition that
\begin{equation}
\label{eq:33}
\text{$x_t(s^*)$ minimises $C_t(x)-\nu^*_tx$ in $x\inX_t$},
\qquad 1 \le t \le T.
\end{equation} \end{proposition}
\begin{proof}
Assume that the pair $(s^*,\lambda^*)$ is as given. Suppose first
that additionally $(s^*,\lambda^*)$ satisfies the
condition~(\ref{con1}) of Theorem~\ref{thm:exist}. The
condition~\eqref{eq:33} is then straightforward when the
functions~$C_t$ are additionally differentiable: for each~$t$ the
partial derivative of the left side of \eqref{eq:21} with respect to
$x_t(s)$ (with $x_u(s)$ being kept constant for $u\neq t$) is
necessarily zero at $s=s^*$, so that~\eqref{eq:33} follows from the
assumed convexity of the functions~$C_t$. For the general case,
note that it follows from the condition~\eqref{con1} of
Theorem~\ref{thm:exist} (by considering $s$ such that $s_0=s^*_0$ ,
$x_t(s)=x_t(s^*)+h$, $x_u(s)=x_u(s^*)$ for $u\ne t$), that, for
all~$t$ and for all real~$h$,
\begin{equation}
\label{eq:34}
C_t(x_t(s^*)+h) + \sum_{u=t}^T[A_u(s^*_u+h) - \lambda^*_uh]
\end{equation}
is minimised at $h=0$, and so, for all (small)~$h$,
\begin{equation}
\label{eq:35}
C_t(x_t(s^*)+h) - \nu_th \ge C_t(x_t(s^*)) + o(h),
\quad\text{as $h\to0$}.
\end{equation}
Thus~\eqref{eq:33} again follows from the assumed convexity of the
functions~$C_t$.
To prove the converse result, suppose now that $(s^*,\lambda^*)$
satisfies the condition~\eqref{eq:33}. This condition, together
with the convexity and differentiability of the functions~$A_t$,
then implies that, for all~$t$, the expression \eqref{eq:34} is
minimised at $h=0$.
It is now straightforward that the hyperplane in $\mathbb{R}^T$ whose vector
of slopes is $\lambda^*$ supports the function
$\sum_{t=1}^T\left[C_t(x_t(s))+A_t(s_t)\right]$ at the point
$(s^*,\,\sum_{t=1}^T[C_t(x_t(s^*))+A_t(s^*_t)])$, so that finally
the condition~\eqref{con1} of Theorem~\ref{thm:exist} holds as
required. \end{proof}
It now follows from Theorem~\ref{thm:exist} and Proposition~\ref{prop:existnu} that if the pair~$(s^*,\lambda^*)$ is such that $s^*$ is feasible for the problem~$\mathbf{P}$, and that $(s^*,\lambda^*)$ satisfies the both condition~\eqref{eq:33} and the condition~(\ref{con2}) of Theorem~\ref{thm:exist}, then $s^*$ further solves the problem~$\mathbf{P}$.
We now show how to construct such a pair $(s^*,\lambda^*)$. We assume, for the moment, strict convexity of the functions~$C_t$; we subsequently indicate how to relax this assumption. It follows from the assumed strict convexity that, for each $t$ and for each $\nu_t$, there is a unique $x\inX_t$, which we denote by $x^*_t(\nu_t)$, which minimises $C_t(x)-\nu_tx$ in $X_t$. Further $x^*_t(\nu_t)$ is continuous and increasing in $\nu_t$---strictly so for $\nu_t$ such that $x^*_t(\nu_t)$ lies in the interior of $X_t$. In particular, from~\eqref{eq:13}, the condition~\eqref{eq:33} may now be rewritten as \begin{equation}
\label{eq:36}
s^*_t = s^*_{t-1} + x^*_t(\nu^*_t),
\qquad 1 \le t \le T. \end{equation} It further follows from \eqref{eq:32} that \begin{equation}
\label{eq:37}
\nu^*_{t+1} = \nu^*_t + A'_t(s^*_t) - \lambda^*_t.
\qquad 1 \le t \le T-1. \end{equation}
Thus, were the vector~$\lambda^*$ known, together with the value of the constant~$\nu^*_1$, the pair $(s^*,\,\nu^*)$ could be constructed sequentially via \eqref{eq:36} and \eqref{eq:37}. We observe that, while $\lambda^*$ is not known, it does satisfy the conditions~\eqref{eq:22} and in particular the requirement that $\lambda^*_t=0$ for all $t$ such that $0<s^*_t<E_t$. We now follow a procedure which is a generalisation of one described by Cruise et al~\cite{CFGZ}, and which involves an essentially one-dimensional search so as to identify the constant~$\nu^*_1$. This search, which may be thought of as being carried out at time zero and which is not computationally intensive (see the further remarks at the end of this section), then needs to be repeated at each of a number of subsequent times as described below. We show how to define inductively a sequence of times $0=T_0<T_1<\dots<T_k=T$ such that $s^*(T_i)=0$ or $s^*(T_i)=E_{T_i}$ for $1\le i\le k$ and such that $\lambda^*_t=0$ for all values of $t$ not in the above sequence.
The time $T_1$ is chosen as follows. Consider \emph{trial} values $\nu_1$ of $\nu^*_1$. For each such $\nu_1$, define a pair of vectors $\nu=(\nu_1,\dots,\nu_T)$ and $s=(s_1,\dots,s_T)$ by \begin{align}
s_t & = s_{t-1} + x^*_t(\nu_t),
\qquad 1 \le t \le T, \label{eq:38}\\
\nu_{t+1} & = \nu_t + A'_t(s_t),
\qquad 1 \le t \le T-1. \label{eq:39} \end{align}
Define $M$ and $M'$ to be the sets of values of $\nu_1$ for which the vector $s$ defined via \eqref{eq:38} and \eqref{eq:39} violates one of the capacity constraints~\eqref{eq:15} and first does so respectively below or above---in either case at a time which we denote by $\overline T_1(\nu_1)$. Since, for each $t$, $x^*_t(\nu_t)$ is increasing in $\nu_t$ and $A'_t(s_t)$ is increasing in $s_t$ (by the convexity of $A_t$), it follows that if $\nu_1\in M$ then $\nu'_1\in M$ for all $\nu_1'<\nu_1$ and that if $\nu_1\in M'$ then $\nu_1'\in M'$ for all $\nu_1'>\nu_1$; further the sets $M$ and $M'$ are disjoint, and (since the solution set for the problem~$\mathbf{P}$ is nonempty) neither the set $M$ nor the set $M'$ can be the entire real line. Let $\bar\nu_1=\sup M$. (In the extreme case where $M$ is empty we may set $\bar\nu_1=-\infty$). We now consider the behaviour of the corresponding vector~$s$ defined via \eqref{eq:38} and \eqref{eq:39} where we take $\nu_1=\bar\nu_1$; for this vector~$s$ there are three possibilities: \begin{compactenum}[(a)] \item the quantity $\bar\nu_1$ belongs neither to the set $M$ nor to
the set $M'$, i.e.\ the vector $s$ generated as above is feasible
for the problem~$\mathbf{P}$; in this case we take $T_1=T$ and $s^*=s$ with
$\nu^*_1=\bar\nu_1$ and $\lambda^*_t=0$ for $1\le t\le T-1$ (so that
the remaining values of $\nu^*$ are given by \eqref{eq:37}); \item the quantity $\bar\nu_1$ belongs to the set $M$; in this case
there exists at least one $t<\overline T_1(\bar\nu_1)$ such that
$s_t=E_t$ (were this not so then, by the continuity of each
$x^*_t(\nu_t)$ in $\nu_t$, the value of $\nu_1$ could be increased
above $\bar\nu_1$ while remaining within the set $M$); define $T_1$
to be any such $t$, say the largest, and take $s^*_t=s_t$ for
$1\le t\le T_1$ with $\nu^*_1=\bar\nu_1$ and $\lambda^*_t=0$ for
$1\le t\le T_1-1$; \item the quantity $\bar\nu_1$ belongs to the set $M'$; in this case,
similarly to the case (b), there exists at least one
$t<\overline T_1(\bar\nu_1)$ such that $s_t=0$; define $T_1$ to be
any such $t$, again say the largest, and again take $s^*_t=s_t$ for
$1\le t\le T_1$ with $\nu^*_1=\bar\nu_1$ and $\lambda^*_t=0$ for
$1\le t\le T_1-1$. \end{compactenum}
In each of the cases~(b) and (c) above, we now repeat the above procedure, starting at the time~$T_1$ instead of the time~$0$, and considering trial values of $\nu^*_{T_1+1}$, thereby identifying $\nu^*_{T_1+1}$, the time~$T_2$ and the values of $s^*_t$ for $T_1+1\le t\le T_2$, and taking $\lambda^*_t=0$ for $T_1+1\le t\le T_2-1$. The quantity~$\lambda^*_{T_1}$ is now defined via \eqref{eq:37}. Further consideration of the sets $M$ and $M'$ defined above in relation to the identification of $\nu^*_1=\bar\nu_1$ shows easily that in the case $\bar\nu_1\in M$---so that $s^*_{T_1}=E_{T_1}$---the quantity $\nu^*_{T_1+1}=\bar\nu_{T_1+1}$ is necessarily such that $\lambda^*_{T_1}\ge0$ (since in this case, by the above construction, the quantity $\nu^*_{T_1+1}$ has a value which is necessarily at least as great as would have been the case had $\lambda^*_{T_1}$ been equal to $0$), whereas in the case $\bar\nu_1\in M'$---so that $s^*_{T_1}=0$---the quantity $\nu^*_{T_1+1}=\bar\nu_{T_1+1}$ is necessarily such that $\lambda^*_{T_1}\le0$.
For $T_2\neq T$ we continue in this manner until the entire sequence $0=T_0<T_1<\dots<T_k=T$ is identified. We thus obtain vectors~$s^*$, $\lambda^*$ and $\nu^*$ such that $s^*$ is feasible for the problem~$\mathbf{P}$, while $(s^*,\lambda^*)$ satisfies the condition~\eqref{eq:33} and the condition~(\ref{con2}) of Theorem~\ref{thm:exist} and so solves the problem~$\mathbf{P}$ as required.
In the case where, for at least some $t$, the cost function $C_t$ is convex, but not necessarily strictly so, some extra care is required. Here, for such $t$, the function $\nu\rightarrow x^*_t(\nu)$ is not in general uniquely defined; further, for any given choice, this function is not in general continuous. However, the above construction of $(s^*,\lambda^*)$ continues to hold provided that, where necessary, we choose the right value of $x^*_t(\nu)$. The latter may always be identified by considering, for example, a sequence of strictly convex functions $C^{(\epsilon)}_t$ converging to $C_t$ and identifying $x^*_t(\nu)$ as the limit of its corresponding values within this sequence.
Note that the above construction proceeds locally in time, in the sense that, at each successive time~$T_i$, the determination of the subsequent time~$T_{i+1}$ and of the values of $s^*_t$ and $\nu^*_t$ for $T_i+1\le t\le T_{i+1}$ only requires consideration of the functions~$C_t$ and $A_t$ up to some time $\overline T_{i+1}$ (necessarily beyond $T_{i+1}$) the identification of which does not depend on the functions $C_t$ and $A_t$ at any subsequent times. More precisely we have $\overline T_1=\overline T_1(\bar\nu_1)$, where $\overline T_1(\bar\nu_1)$ is as identified above, and the remaining $\overline T_i$, $2\le i\le k$, are similarly identified. In particular we have that, for each time~$t$ and given $s^*_{t-1}$, the optimal choice of store level~$s^*_t$ depends only on the functions~$C_{t'}$ and $A_{t'}$ for $t\le t'\le\overline T(t)$ where we define $\overline T(t)=\overline T_{i+1}$ for $i$ such that $T_i+1\le t\le T_{i+1}$. The function $\overline T(t)$ is piecewise constant in~$t$, and so the \emph{time horizon} or \emph{look-ahead
time} $\overline T(t)-t$ required for the optimal decision at each time~$t$ has the ``sawtooth'' shape which we illustrate in our examples of Section~\ref{sec:example}.
Note further that a lengthening of the total time $T$ over which the optimization is to be performed does not in general change the values of the times $T_i$, but rather simply creates more of them. In particular the solution to the problem~$\mathbf{P}$ involves computation which grows essentially linearly in $T$, and the algorithm is suitable for the management of a store with an infinite time horizon.
The typical length of the intervals between the successive times $T_i$ depends on the shape of the functions $C_t$ and $A_t$ and in particular on the rate at which they fluctuate in time. Thus, for example, the long-run management of a store for which the functions~$C_t$ show strong daily fluctuations typically involves decision making on a running time horizon of the order of a day or so.
Finally note that, as already indicated, in the implementation of the above construction, some form of one-dimensional search is usually required to determine each of the successive $\bar\nu_{T_i+1}$: each trial value of this quantity provides either an upper or lower bound to the true value, so that, for example, a simple binary search is sufficient. Given also the ``locality'' property referred to above, the numerical effort involved in the implementation of the above algorithm is usually very slight.
\section{Examples} \label{sec:example}
We give some examples, in which we solve (exactly) the optimal control problem $\mathbf{P}$ formally defined in Section~\ref{sec:simpl-optim-probl}. We investigate how the optimal solution depends on the cost functions~$C_t$ defined there which reflecting buying and selling costs and hence the opportunity to make money from price arbitrage, and on the functions~$A_t$ which reflect the costs of providing buffering services.
The cost functions~$C_t$ are derived from half-hourly electricity prices in the Great Britain spot market over the entire year 2011, adjusted for a modest degree of market impact, as described in detail below. Thus we work in half-hour time units, with the time horizon~$T$ corresponding to the number of half-hour periods in the entire year. These spot market prices show a strong daily cyclical behaviour (corresponding to daily demand variation), being low at night and high during the day. This price variation can be seen in Figure~\ref{fig:price} which shows half-hourly GB spot prices (in pounds per megawatt-hour) throughout the month of March 2011. There is a similar patter of variation throughout the rest of the year.
\begin{figure}
\caption{GB half-hourly spots prices (\pounds/MWh) for March 2011.}
\label{fig:price}
\end{figure}
Without loss of generality, we choose energy units such that the rate (power) constraints are given by $P_{It}=P_{Ot}=1$ unit of energy per half-hour period. For illustration, we take the capacity of the store to be given by $E=10$ units of energy; thus the store can completely fill or empty over a 5-hour period, which is the case, for example, for the large Dinorwig pumped storage facility in Snowdonia \cite{Din}.
We choose cost functions $C_t$ of the form \begin{equation}
\label{eq:40}
C_t(x) =
\begin{cases}
c_t x (1+\delta x), & \quad\text{if $x\ge0$}\\
\eta c_t x (1+\delta x), & \quad\text{if $x<0$},
\end{cases} \end{equation} where the $c_t$ are proportional to the half-hourly electricity spot prices referred to above, where $\eta$ is an adjustment to selling prices representing in particular round-trip efficiency as described in Section~\ref{sec:problem}, and where the factor $\delta>0$ is chosen so as to represent a degree of market impact (higher unit prices as the store buys more and lower unit prices as the store sells more). For our numerical examples we take $\eta=0.85$ which is a typical round-trip efficiency for a pumped-storage facility such as Dinorwig. We choose $\delta=0.05$; since the rate constraints for the store are $P_{It}=P_{Ot}=1$ this corresponds to a maximum market impact of~$5$\%. While this is modest, our results are qualitatively little affected as $\delta$ is varied over a wide range of values less than one, covering therefore the range of possible market impact likely to be seen for storage in practice.
Finally we need to choose the functions~$A_t$ reflecting the costs of providing buffering services. Our aim here is to give an understanding of how the optimal control of the store varies according to the relative economic importance of cost arbitrage and buffering, i.e.\ according to the relative size of the functions~$C_t$ and $A_t$. We choose functions~$A_t$ which are constant over time~$t$ and of the form $A_t(s)=ae^{-\kappa s}$ and $A_t(s)=b/s$ for a small selection of the parameters $a$, $\kappa$ and $b$. The extent to which a store might provide buffering services in applications is extremely varied, and so the likely balance between arbitrage and buffering cannot be specified in advance. Rather we choose just sufficient values of the above parameters to show the effect of varying this balance. For a possible justification of the chosen forms of the functions~$A_t$ (including why it should not necessarily be truncated to $0$ for values of $s$ greater than the rate constraint of $1$), see Section~\ref{sec:determ-funct-a_t-2}; in particular the form $A_t(s)=ae^{-\kappa s}$ is plausible in the case of light-tailed shocks, while the form $A_t(s)=b/s$ shows the effect of a slow rate of decay in~$s$.
In each of our examples, we determine the optimal control of the store over the entire year, with both the initial level~$S^*_0$ and the final level~$S^*_T$ given by $S^*_0=S^*_T=0$. In each of the corresponding figures, the upper panel shows the optimally controlled level of the store throughout the month of March. The lower panel shows, for each time~$t$ in the same month, the time horizon (or look-ahead time) $\overline T(t)-t$, defined in Section~\ref{sec:algorithm}, i.e.\ the length of time beyond the time~$t$ for which knowledge of the cost functions is required in order to make the optimal decision at time~$t$.
Figure~\ref{fig:exp} shows the optimal control of the store when the functions~$A_t$ are given by $A_t(s)=ae^{-\kappa s}$. The uppermost panels correspond to $a=0$, so that the store incurs no penalty for failing to provide buffering services and optimises its control solely on the basis of arbitrage between energy prices at different times. The daily cycle of prices is sufficiently pronounced that here the store fills and empties---or nearly so---on a daily basis, notwithstanding the facts that the round-trip efficiency of $0.85$ is considerably less than $1$ and that the minimum time for the store to fill or empty is $5$ hours. It will be seen also that the time horizon, or look-ahead time, required for the determination of optimal decisions is in general of the order of one or two days.
\begin{figure}
\caption{Store level and time horizon throughout March 2011 for the
example with $A_t(s)=ae^{-\kappa s}$. The top panels correspond
to $a=0$, the central panels to $a=1$, $\kappa=1$, and the bottom
panels to $a=10$, $\kappa=1$.}
\label{fig:exp}
\end{figure}
The central panels of Figure~\ref{fig:exp} correspond to $\kappa=1$ and $a=1$. The choice of $a$ in particular is such that the store is just sufficiently incentivised by the need to reduce buffering costs that it rarely empties completely (though it does so very occasionally). Otherwise the behaviour of the store is very similar to that in the case $a=0$. Note also that in this case the time horizons or look-ahead times are in general somewhat longer; an intuitive explanation (backed by a careful examination of the figure) is that, starting from a time when the store is full, the determination of by how much the store should avoid emptying completely requires taking account of the cost functions for a longer period of future time than is the case where the store does empty completely.
Finally the bottom two panels of Figure~\ref{fig:exp} correspond to $\kappa=1$ and $a=10$. Here the costs of failing to provide buffering services are much higher, and so the optimised level of the store rarely falls below 25\% of its capacity. Curiously the look-ahead times are in general less than in the case $a=1$---presumably since the store level is more often reaching the capacity constraint.
Variation of the exponential parameter~$\kappa$ does not result in dramatically different behaviour, so we do not pursue this here.
Figure~\ref{fig:inv} shows the optimal control of the store when the functions~$A_t$ are given, for each~$t$, by $A_t(s)=b/s$. The upper panels correspond to $b=0$, so that we again have $A_t(s)=0$ for all $s$ and the control is as observed previously. The lower panels correspond to the case $b=1$, and, as might be expected, the behaviour here is somewhat intermediate between that for the two nonzero exponentially decaying exponential functions.
\begin{figure}
\caption{Store level and time horizon throughout March 2011 for the
example with $A_t(s)=b/s$. The upper panels correspond to
$A_t(s)= 0$ and the lower panels to $A(t)= 1/s$.}
\label{fig:inv}
\end{figure}
\end{document} | arXiv |
On the lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation
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June 2018, 7(2): 247-273. doi: 10.3934/eect.2018012
Robust Stackelberg controllability for linear and semilinear heat equations
Víctor Hernández-Santamaría 1, and Luz de Teresa 2,
Departamento de Control Automático, CINVESTAV-IPN, Apartado Postal 14-740, 0700, México, D.F., México
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510 D.F., México
The first author was supported by CONACyT (Mexico) and both authors were supported by project IN102116 of DGAPA, UNAM. (Mexico).
Received May 2017 Revised December 2017 Published May 2018
In this paper, we present a Stackelberg strategy to control a semilinear parabolic equation. We use the concept of hierarchic control to combine the concepts of controllability with robustness. We have a control named the leader which is responsible for a controllability to trajectories objective. Additionally, we have a control named the follower, that solves a robust control problem. That means we solve for the optimal control in the presence of the worst disturbance case. In this way, the follower control is insensitive to a broad class of external disturbances.
Keywords: Hierarchic control, controllability to trajectories, robust control, Carleman inequalities.
Mathematics Subject Classification: Primary: 49J20, 93B05; Secondary: 49K35.
Citation: Víctor Hernández-Santamaría, Luz de Teresa. Robust Stackelberg controllability for linear and semilinear heat equations. Evolution Equations & Control Theory, 2018, 7 (2) : 247-273. doi: 10.3934/eect.2018012
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T. Tachim Medjo, Louis Tcheugoue Tebou. Robust control problems in fluid flows. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 437-463. doi: 10.3934/dcds.2005.12.437
Jian-Xin Guo, Xing-Long Qu. Robust control in green production management. Journal of Industrial & Management Optimization, 2022, 18 (2) : 1115-1132. doi: 10.3934/jimo.2021011
M. Arisawa, P.-L. Lions. Continuity of admissible trajectories for state constraints control problems. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 297-305. doi: 10.3934/dcds.1996.2.297
T. Tachim Medjo. On the Newton method in robust control of fluid flow. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1201-1222. doi: 10.3934/dcds.2003.9.1201
T. Tachim Medjo. Robust control problems for primitive equations of the ocean. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 769-788. doi: 10.3934/dcdsb.2011.15.769
Magdi S. Mahmoud, Omar Al-Buraiki. Robust control design of autonomous bicycle kinematics. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 181-191. doi: 10.3934/naco.2014.4.181
Xavier Litrico, Vincent Fromion, Gérard Scorletti. Robust feedforward boundary control of hyperbolic conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 717-731. doi: 10.3934/nhm.2007.2.717
Ariela Briani, Hasnaa Zidani. Characterization of the value function of final state constrained control problems with BV trajectories. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1567-1587. doi: 10.3934/cpaa.2011.10.1567
Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305
Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations & Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039
Klaus-Jochen Engel, Marjeta Kramar FijavŽ. Exact and positive controllability of boundary control systems. Networks & Heterogeneous Media, 2017, 12 (2) : 319-337. doi: 10.3934/nhm.2017014
Mehdi Badra. Global Carleman inequalities for Stokes and penalized Stokes equations. Mathematical Control & Related Fields, 2011, 1 (2) : 149-175. doi: 10.3934/mcrf.2011.1.149
Maria Grazia Naso. Controllability to trajectories for semilinear thermoelastic plates. Conference Publications, 2005, 2005 (Special) : 672-681. doi: 10.3934/proc.2005.2005.672
El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations & Control Theory, 2013, 2 (3) : 441-459. doi: 10.3934/eect.2013.2.441
M. S. Mahmoud, P. Shi, Y. Shi. $H_\infty$ and robust control of interconnected systems with Markovian jump parameters. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 365-384. doi: 10.3934/dcdsb.2005.5.365
Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control & Related Fields, 2021, 11 (1) : 95-117. doi: 10.3934/mcrf.2020029
T. Tachim Medjo. Robust control of a Cahn-Hilliard-Navier-Stokes model. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2075-2101. doi: 10.3934/cpaa.2016028
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Based on the idea and the provided source code of Andrej Karpathy (arxiv-sanity)
Galaxy And Mass Assembly (GAMA): The environmental dependence of the galaxy main sequence (1802.08456)
L. Wang, P. Norberg, S. Brough, M. J. I. Brown, E. da Cunha, L. J. Davies, S. P. Driver, B. W. Holwerda, A. M. Hopkins, M. A. Lara-Lopez, J. Liske, J. Loveday, M. W. Grootes, C. C. Popescu, A. H. Wright
Feb. 23, 2018 astro-ph.GA
Aims. We aim to investigate if the environment (characterised by the host dark matter halo mass) plays any role in shaping the galaxy star formation main sequence (MS). Methods. The Galaxy and Mass Assembly project (GAMA) combines a spectroscopic survey with photometric information in 21 bands from the far-ultraviolet (FUV) to the far-infrared (FIR). Stellar masses and dust-corrected star-formation rates (SFR) are derived from spectral energy distribution (SED) modelling using MAGPHYS. We use the GAMA galaxy group catalogue to examine the variation of the fraction of star-forming galaxies (SFG) and properties of the MS with respect to the environment. Results. We examine the environmental dependence for stellar mass selected samples without preselecting star-forming galaxies and study any dependence on the host halo mass separately for centrals and satellites out to z ~ 0.3. We find the SFR distribution at fixed stellar mass can be described by the combination of two Gaussians (referred to as the star-forming Gaussian and the quiescent Gaussian). Using the observed bimodality to define SFG, we investigate how the fraction of SFG F(SFG) and properties of the MS change with environment. For centrals, the position of the MS is similar to the field but with a larger scatter. No significant dependence on halo mass is observed. For satellites, the position of the MS is almost always lower (by ~0.2 dex) compared to the field and the width is almost always larger. F(SFG) is similar between centrals (in different halo mass bins) and field galaxies. However, for satellites F(SFG) decreases with increasing halo mass and this dependence is stronger towards lower redshift.
GitHub 0
Galaxy And Mass Assembly (GAMA): The mechanisms for quiescent galaxy formation at $z<1$ (1707.07989)
K. Rowlands, V. Wild, N. Bourne, M. Bremer, S. Brough, S. P. Driver, A. M. Hopkins, M. S. Owers, S. Phillipps, K. Pimbblet, A. E. Sansom, L. Wang, M. Alpaslan, J. Bland-Hawthorn, M. Colless, B. W. Holwerda, E. N. Taylor
Jan. 15, 2018 astro-ph.GA
One key problem in astrophysics is understanding how and why galaxies switch off their star formation, building the quiescent population that we observe in the local Universe. From the GAMA and VIPERS surveys, we use spectroscopic indices to select quiescent and candidate transition galaxies. We identify potentially rapidly transitioning post-starburst galaxies, and slower transitioning green-valley galaxies. Over the last 8 Gyrs the quiescent population has grown more slowly in number density at high masses (M$_*>10^{11}$M$_\odot$) than at intermediate masses (M$_*>10^{10.6}$M$_\odot$). There is evolution in both the post-starburst and green valley stellar mass functions, consistent with higher mass galaxies quenching at earlier cosmic times. At intermediate masses (M$_*>10^{10.6}$M$_\odot$) we find a green valley transition timescale of 2.6 Gyr. Alternatively, at $z\sim0.7$ the entire growth rate could be explained by fast-quenching post-starburst galaxies, with a visibility timescale of 0.5 Gyr. At lower redshift, the number density of post-starbursts is so low that an unphysically short visibility window would be required for them to contribute significantly to the quiescent population growth. The importance of the fast-quenching route may rapidly diminish at $z<1$. However, at high masses (M$_*>10^{11}$M$_\odot$), there is tension between the large number of candidate transition galaxies compared to the slow growth of the quiescent population. This could be resolved if not all high mass post-starburst and green-valley galaxies are transitioning from star-forming to quiescent, for example if they rejuvenate out of the quiescent population following the accretion of gas and triggering of star formation, or if they fail to completely quench their star formation.
Galaxy And Mass Assembly (GAMA): Blue spheroids within 87 Mpc (1712.03644)
Smriti Mahajan, Michael J. Drinkwater, S. Driver, A. M. Hopkins, Alister W. Graham, S. Brough, Michael J.I. Brown, B.W. Holwerda, Matt S. Owers, Kevin A. Pimbblet
Dec. 11, 2017 astro-ph.GA
In this paper we test if nearby blue spheroid (BSphs) galaxies may become the progenitors of star-forming spiral galaxies or passively-evolving elliptical galaxies. Our sample comprises 428 galaxies of various morphologies in the redshift range 0.002<z<0.02 (8-87 Mpc) with panchromatic data from the Galaxy and Mass Assembly survey. We find that BSph galaxies are structurally (mean effective surface brightness, effective radius) very similar to their passively-evolving red counterparts. However, their star-formation and other properties such as colour, age and metallicity are more like star-forming spirals than spheroids (ellipticals and lenticulars). We show that BSph galaxies are statistically distinguishable from other spheroids as well as spirals in the multi-dimensional space mapped by luminosity-weighted age, metallicity, dust mass and specific star formation rate. We use HI data to reveal that some of the BSphs are (further) developing their disks, hence their blue colours. They may eventually become spiral galaxies --- if sufficient gas accretion occurs --- or more likely fade into low-mass red galaxies.
Galaxy And Mass Assembly (GAMA): The environments of high- and low- excitation radio galaxies (1705.04502)
J. H. Y. Ching, A. S. G. Robotham, M. Colless, A. M. Hopkins, J. Liske, O. Steele University of St Andrews, International Centre for Radio Astronomy Research, Liverpool John Moores University, University of Louisville, University of the Western Cape, University of Sussex, SRON Netherlands Institute for Space Research,
May 12, 2017 astro-ph.GA
We study the environments of low- and high- excitation radio galaxies (LERGs and HERGs respectively) in the redshift range $0.01 < z < 0.4$, using a sample of 399 radio galaxies and environmental measurements from the Galaxy And Mass Assembly (GAMA) survey. In our analysis we use the fifth nearest neighbour density ($\Sigma_{5}$) and the GAMA galaxy groups catalogue (G3Cv6) and construct control samples of galaxies matched in {\update stellar mass and colour} to the radio-detected sample. We find that LERGs and HERGs exist in different environments and that this difference is dependent on radio luminosity. High-luminosity LERGs ($L_{\rm NVSS} \gtrsim 10^{24}$ W Hz$^{-1}$) lie in much denser environments than a matched radio-quiet control sample (about three times as dense, as measured by $\Sigma_{5}$), and are more likely to be members of galaxy groups ($82^{+5}_{-7}$ percent of LERGs are in GAMA groups, compared to $58^{+3}_{-3}$ percent of the control sample). In contrast, the environments of the HERGs and lower luminosity LERGs are indistinguishable from that of a matched control sample. Our results imply that high-luminosity LERGs lie in more massive haloes than non-radio galaxies of similar stellar mass and colour, in agreement with earlier studies (Wake et al. 2008; Donoso et al. 2010). When we control for the preference of LERGs to be found in groups, both high- and low- luminosity LERGs are found in higher-mass haloes ($\sim 0.2$ dex; at least 97 percent significant) than the non-radio control sample.
Galaxy And Mass Assembly (GAMA): The galaxy stellar mass function to $z=0.1$ from the r-band selected equatorial regions (1705.04074)
A. H. Wright, A. S. G. Robotham, S. P. Driver, M. Alpaslan, S. K. Andrews, I. K. Baldry, J. Bland-Hawthorn S. Brough, M. J. I. Brown, M. Colless, E. da Cunha, L. J. M. Davies, Alister W. Graham, B. W. Holwerda, A. M. Hopkins, P. R. Kafle, L. S. Kelvin, J. Loveday, S. J. Maddox, M. J. Meyer, A. J. Moffett, P. Norberg, S. Phillipps, K. Rowlands, E. N. Taylor, L. Wang, S. M. Wilkins
We derive the low redshift galaxy stellar mass function (GSMF), inclusive of dust corrections, for the equatorial Galaxy And Mass Assembly (GAMA) dataset covering 180 deg$^2$. We construct the mass function using a density-corrected maximum volume method, using masses corrected for the impact of optically thick and thin dust. We explore the galactic bivariate brightness plane ($M_\star-\mu$), demonstrating that surface brightness effects do not systematically bias our mass function measurement above 10$^{7.5}$ M$_{\odot}$. The galaxy distribution in the $M-\mu$-plane appears well bounded, indicating that no substantial population of massive but diffuse or highly compact galaxies are systematically missed due to the GAMA selection criteria. The GSMF is {fit with} a double Schechter function, with $\mathcal M^\star=10^{10.78\pm0.01\pm0.20}M_\odot$, $\phi^\star_1=(2.93\pm0.40)\times10^{-3}h_{70}^3$Mpc$^{-3}$, $\alpha_1=-0.62\pm0.03\pm0.15$, $\phi^\star_2=(0.63\pm0.10)\times10^{-3}h_{70}^3$Mpc$^{-3}$, and $\alpha_2=-1.50\pm0.01\pm0.15$. We find the equivalent faint end slope as previously estimated using the GAMA-I sample, although we find a higher value of $\mathcal M^\star$. Using the full GAMA-II sample, we are able to fit the mass function to masses as low as $10^{7.5}$ $M_\odot$, and assess limits to $10^{6.5}$ $M_\odot$. Combining GAMA-II with data from G10-COSMOS we are able to comment qualitatively on the shape of the GSMF down to masses as low as $10^{6}$ $M_\odot$. Beyond the well known upturn seen in the GSMF at $10^{9.5}$ the distribution appears to maintain a single power-law slope from $10^9$ to $10^{6.5}$. We calculate the stellar mass density parameter given our best-estimate GSMF, finding $\Omega_\star= 1.66^{+0.24}_{-0.23}\pm0.97 h^{-1}_{70} \times 10^{-3}$, inclusive of random and systematic uncertainties.
The SAMI Galaxy Survey: The cluster redshift survey, target selection and cluster properties (1703.00997)
M. S. Owers, J. T. Allen, I. Baldry, J. J. Bryant, G. N. Cecil, L. Cortese, S. M. Croom, S. P. Driver, L. M. R. Fogarty, A. W. Green, E. Helmich, J. T. A. de Jong, K. Kuijken, S. Mahajan, J. McFarland, M. B. Pracy, A. G. S. Robotham, G. Sikkema, S. Sweet, E. N. Taylor, G. Verdoes Kleijn, A. E. Bauer, J. Bland-Hawthorn, S. Brough, M. Colless, W. J. Couch, R. L Davies, M. J. Drinkwater, M. Goodwin, A. M. Hopkins, I. S. Konstantopoulos, C. Foster, J. S. Lawrence, N. P. F Lorente, A. M. Medling, N. Metcalfe, S. N. Richards, J. van de Sande, N. Scott, T. Shanks, R. Sharp, A. D. Thomas, C. Tonini
March 3, 2017 astro-ph.GA
We describe the selection of galaxies targeted in eight low redshift clusters (APMCC0917, A168, A4038, EDCC442, A3880, A2399, A119 and A85; $0.029 < z < 0.058$) as part of the Sydney-AAO Multi-Object integral field Spectrograph Galaxy Survey (SAMI-GS). We have conducted a redshift survey of these clusters using the AAOmega multi-object spectrograph on the 3.9m Anglo-Australian Telescope. The redshift survey is used to determine cluster membership and to characterise the dynamical properties of the clusters. In combination with existing data, the survey resulted in 21,257 reliable redshift measurements and 2899 confirmed cluster member galaxies. Our redshift catalogue has a high spectroscopic completeness ($\sim 94\%$) for $r_{\rm petro} \leq 19.4$ and clustercentric distances $R< 2\rm{R}_{200}$. We use the confirmed cluster member positions and redshifts to determine cluster velocity dispersion, $\rm{R}_{200}$, virial and caustic masses, as well as cluster structure. The clusters have virial masses $14.25 \leq {\rm log }({\rm M}_{200}/\rm{M}_{\odot}) \leq 15.19$. The cluster sample exhibits a range of dynamical states, from relatively relaxed-appearing systems, to clusters with strong indications of merger-related substructure. Aperture- and PSF-matched photometry are derived from SDSS and VST/ATLAS imaging and used to estimate stellar masses. These estimates, in combination with the redshifts, are used to define the input target catalogue for the cluster portion of the SAMI-GS. The primary SAMI-GS cluster targets have $R< \rm{R}_{200}$, velocities $|v_{\rm pec}| < 3.5\sigma_{200}$ and stellar masses $9.5 \leq {\rm log(M}^*_{approx}/\rm{M}_{\odot}) \leq 12$. Finally, we give an update on the SAMI-GS progress for the cluster regions.
Galaxy and Mass Assembly (GAMA): Probing the merger histories of massive galaxies via stellar populations (1703.00465)
I. Ferreras, A. M. Hopkins, M. L. P. Gunawardhana, A. E. Sansom, M. S. Owers, S. Driver, L. Davies, A. Robotham, E. N. Taylor, I. Konstantopoulos, S. Brough, P. Norberg, S. Croom, J. Loveday, L. Wang, M. Bremer
The merging history of galaxies can be traced with studies of dynamically close pairs. These consist of a massive primary galaxy and a less massive secondary (or satellite) galaxy. The study of the stellar populations of secondary (lower mass) galaxies in close pairs provides a way to understand galaxy growth by mergers. Here we focus on systems involving at least one massive galaxy - with stellar mass above $10^{11}M_\odot$ in the highly complete GAMA survey. Our working sample comprises 2,692 satellite galaxy spectra (0.1<z<0.3). These spectra are combined into high S/N stacks, and binned according to both an "internal" parameter, the stellar mass of the satellite galaxy (i.e. the secondary), and an "external" parameter, selecting either the mass of the primary in the pair, or the mass of the corresponding dark matter halo. We find significant variations in the age of the populations with respect to environment. At fixed mass, satellites around the most massive galaxies are older and possibly more metal rich, with age differences ~1-2Gyr within the subset of lower mass satellites ($\sim 10^{10}M_\odot$). These variations are similar when stacking with respect to the halo mass of the group where the pair is embedded. The population trends in the lower-mass satellites are consistent with the old stellar ages found in the outer regions of massive galaxies.
Galaxy and Mass Assembly (GAMA): Formation and Growth of Elliptical Galaxies in the Group Environment (1702.07641)
Simon Deeley, Michael J. Drinkwater, Daniel Cunnama, Joss Bland-Hawthorn, Sarah Brough, Michelle Cluver, Matthew Colless, Luke J. M. Davies, Simon P. Driver, Caroline Foster, Meiert W. Grootes, A. M. Hopkins, Prajwal R. Kafle, Maritza A. Lara-Lopez, Jochen Liske, Smriti Mahajan, Steven Phillipps, Chris Power, Aaron Robotham
There are many proposed mechanisms driving the morphological transformation of disk galaxies to elliptical galaxies. In this paper, we determine if the observed transformation in low mass groups can be explained by the merger histories of galaxies. We measured the group mass-morphology relation for groups from the Galaxy and Mass Assembly group catalogue with masses from 10$^{11}$ - 10$^{15}$ M$_{\odot}$. Contrary to previous studies, the fraction of elliptical galaxies in our more complete group sample increases significantly with group mass across the full range of group mass. The elliptical fraction increases at a rate of 0.163$\pm$0.012 per dex of group mass for groups more massive than 10$^{12.5}$ M$_{\odot}$. If we allow for uncertainties in the observed group masses, our results are consistent with a continuous increase in elliptical fraction from group masses as low as 10$^{11}$M$_{\odot}$. We tested if this observed relation is consistent with merger activity using a GADGET-2 dark matter simulation of the galaxy groups. We specified that a simulated galaxy would be transformed to an elliptical morphology either if it experienced a major merger or if its cumulative mass gained from minor mergers exceeded 30 per cent of its final mass. We then calculated a group mass-morphology relation for the simulations. The position and slope of the simulated relation were consistent with the observational relation, with a gradient of 0.184$\pm$0.010 per dex of group mass. These results demonstrate a strong correlation between the frequency of merger events and disk-to-elliptical galaxy transformation in galaxy group environments.
Galaxy And Mass Assembly: The 1.4GHz SFR indicator, SFR-M* relation and predictions for ASKAP-GAMA (1701.06242)
L. J. M. Davies, M. T. Huynh, A. M. Hopkins, N. Seymour, S. P. Driver, A. G. R. Robotham, I. K. Baldry, J. Bland-Hawthorn, N. Bourne, M. N. Bremer, M. J. I. Brown, S. Brough, M. Cluver, M. W. Grootes, M. Jarvis, J. Loveday, A. Moffet, M. Owers, S. Phillipps, E. Sadler, L. Wang, S. Wilkins, A. Wright
We present a robust calibration of the 1.4GHz radio continuum star formation rate (SFR) using a combination of the Galaxy And Mass Assembly (GAMA) survey and the Faint Images of the Radio Sky at Twenty-cm (FIRST) survey. We identify individually detected 1.4GHz GAMA-FIRST sources and use a late-type, non-AGN, volume-limited sample from GAMA to produce stellar mass-selected samples. The latter are then combined to produce FIRST-stacked images. This extends the robust parametrisation of the 1.4GHz-SFR relation to faint luminosities. For both the individually detected galaxies and our stacked samples, we compare 1.4GHz luminosity to SFRs derived from GAMA to determine a new 1.4GHz luminosity-to-SFR relation with well constrained slope and normalisation. For the first time, we produce the radio SFR-M* relation over 2 decades in stellar mass, and find that our new calibration is robust, and produces a SFR-M* relation which is consistent with all other GAMA SFR methods. Finally, using our new 1.4GHz luminosity-to-SFR calibration we make predictions for the number of star-forming GAMA sources which are likely to be detected in the upcoming ASKAP surveys, EMU and DINGO.
Galaxy and Mass Assembly (GAMA): Exploring the WISE Cosmic Web in G12 (1607.01190)
T.H. Jarrett, M.E. Cluver, C. Magoulas, M. Bilicki, M. Alpaslan, J. Bland-Hawthorn, S. Brough, M.J.I. Brown, S. Croom, S. Driver, B. W. Holwerda, A. M. Hopkins, J. Loveday, P. Norberg, J.A. Peacock, C.C. Popescu, E.M. Sadler, E.N. Taylor, R.J. Tuffs, L. Wang
Jan. 2, 2017 astro-ph.CO, astro-ph.GA
We present an analysis of the mid-infrared WISE sources seen within the equatorial GAMA G12 field, located in the North Galactic Cap. Our motivation is to study and characterize the behavior of WISE source populations in anticipation of the deep multi-wavelength surveys that will define the next decade, with the principal science goal of mapping the 3D large scale structures and determining the global physical attributes of the host galaxies. In combination with cosmological redshifts, we identify galaxies from their WISE W1 3.4um resolved emission, and by performing a star-galaxy separation using apparent magnitude, colors and statistical modeling of star-counts. The resultant galaxy catalog has ~590,000 sources in 60 deg^2, reaching a W1 5-sigma depth of 31 uJy. At the faint end, where redshifts are not available, we employ a luminosity function analysis to show that approximately 27% of all WISE extragalactic sources to a limit of 17.5 mag (31 uJy) are at high redshift, z > 1. The spatial distribution is investigated using two-point correlation functions and a 3D source density characterization at 5 Mpc and 20 Mpc scales. For angular distributions, we find brighter and more massive sources are strongly clustered relative to fainter and lower mass source; likewise, based on WISE colors, spheroidal galaxies have the strongest clustering, while late-type disk galaxies have the lowest clustering amplitudes. Along the radial direction, the strongest clustering is in the largest redshift shell, while the weakest is in the nearest redshift shell, consistent with the stellar mass and morphological type dependency results. In three dimensions, we find a number of distinct groupings, often bridged by filaments and super-structures. Using special visualization tools, we map these structures, exploring how clustering may play a role with stellar mass and galaxy type.
The Large Area Radio Galaxy Evolution Spectroscopic Survey (LARGESS): Survey design, data catalogue and GAMA/WiggleZ spectroscopy (1609.05578)
John H. Y. Ching, Elaine M. Sadler, Scott M. Croom, Helen M. Johnston, Michael B. Pracy, Warrick J. Couch, A. M. Hopkins, Russell J. Jurek, K. A. Pimbblet
Sept. 19, 2016 astro-ph.GA
We present the Large Area Radio Galaxy Evolution Spectroscopic Survey (LARGESS), a spectroscopic catalogue of radio sources designed to include the full range of radio AGN populations out to redshift z = 0.8. The catalogue covers roughly 800 square degrees of sky, and provides optical identifications for 19,179 radio sources from the 1.4 GHz Faint Images of the Radio Sky at Twenty-cm (FIRST) survey down to an optical magnitude limit of i_mod < 20.5 in Sloan Digital Sky Survey (SDSS) images. Both galaxies and point-like objects are included, and no colour cuts are applied. In collaboration with the WiggleZ and Galaxy And Mass Assembly (GAMA) spectroscopic survey teams, we have obtained new spectra for over 5,000 objects in the LARGESS sample. Combining these new spectra with data from earlier surveys provides spectroscopic data for 12,329 radio sources in the survey area, of which 10,856 have reliable redshifts. 85% of the LARGESS spectroscopic sample are radio AGN (median redshift z = 0.44), and 15% are nearby star-forming galaxies (median z = 0.08). Low-excitation radio galaxies (LERGs) comprise the majority (83%) of LARGESS radio AGN at z < 0.8, with 12% being high-excitation radio galaxies (HERGs) and 5% radio-loud QSOs. Unlike the more homogeneous LERG and QSO sub-populations, HERGs are a heterogeneous class of objects with relatively blue optical colours and a wide dispersion in mid-infrared colours. This is consistent with a picture in which most HERGs are hosted by galaxies with recent or ongoing star formation as well as a classical accretion disk.
The SAMI Galaxy Survey: Spatially resolving the environmental quenching of star formation in GAMA galaxies (1609.02635)
A. L. Schaefer, S. M. Croom, J. T. Allen, S. Brough, A. M. Medling, I.-T. Ho, N. Scott, S. N. Richards, M. B. Pracy, M. L. P. Gunawardhana, P. Norberg, M. Alpaslan, A. E. Bauer, K. Bekki, J. Bland-Hawthorn, J. V. Bloom, J. J. Bryant, W. J. Couch, S. P. Driver, L. M. R. Fogarty, C. Foster, G. Goldstein, A. W. Green, A. M. Hopkins, I. S. Konstantopoulos, J. S. Lawrence, A. R. López-Sánchez, N. P. F. Lorente, M. S. Owers, R. Sharp, S. M. Sweet, E. N. Taylor, J. van de Sande, C. J. Walcher, O. I. Wong
Sept. 9, 2016 astro-ph.GA
We use data from the Sydney-AAO Multi-Object Integral Field Spectrograph (SAMI) Galaxy Survey and the Galaxy And Mass Assembly (GAMA) survey to investigate the spatially-resolved signatures of the environmental quenching of star formation in galaxies. Using dust-corrected measurements of the distribution of H$\alpha$ emission we measure the radial profiles of star formation in a sample of 201 star-forming galaxies covering three orders of magnitude in stellar mass (M$_{*}$; $10^{8.1}$-$10^{10.95}\, $M$_{\odot}$) and in $5^{th}$ nearest neighbour local environment density ($\Sigma_{5}$; $10^{-1.3}$-$10^{2.1}\,$Mpc$^{-2}$). We show that star formation rate gradients in galaxies are steeper in dense ($\log_{10}(\Sigma_{5}/$Mpc$^{2})>0.5$) environments by $0.58\pm 0.29\, dex\, $r$_{e}^{-1}$ in galaxies with stellar masses in the range $10^{10}<$M$_{*}/$M$_{\odot}<10^{11}$ and that this steepening is accompanied by a reduction in the integrated star formation rate. However, for any given stellar mass or environment density the star-formation morphology of galaxies shows large scatter. We also measure the degree to which the star formation is centrally concentrated using the unitless scale-radius ratio ($r_{50,H\alpha}/r_{50,cont}$), which compares the extent of ongoing star formation to previous star formation. With this metric we find that the fraction of galaxies with centrally concentrated star formation increases with environment density, from $\sim 5\pm 4\%$ in low-density environments ($\log_{10}(\Sigma_{5}/$Mpc$^{2})<0.0$) to $30\pm 15\%$ in the highest density environments ($\log_{10}(\Sigma_{5}/$Mpc$^{2})>1.0$). These lines of evidence strongly suggest that with increasing local environment density the star formation in galaxies is suppressed, and that this starts in their outskirts such that quenching occurs in an outside-in fashion in dense environments and is not instantaneous.
Galaxy And Mass Assembly (GAMA): The absence of stellar mass segregation in galaxy groups and consistent predictions from GALFORM and EAGLE simulations (1609.01800)
P. R. Kafle, A. S. G. Robotham, C. del P. Lagos, L. J. Davies, A. J. Moffett, S. P. Driver, S. K. Andrews, I. K. Baldry, J. Bland-Hawthorn, S. Brough, L. Cortese, M. J. Drinkwater, R. Finnegan, A. M. Hopkins, J. Loveday
Sept. 7, 2016 astro-ph.CO, astro-ph.GA
We investigate the contentious issue of the presence, or lack thereof, of satellites mass segregation in galaxy groups using the Galaxy And Mass Assembly (GAMA) survey, the GALFORM semi-analytic and the EAGLE cosmological hydrodynamical simulation catalogues of galaxy groups. We select groups with halo mass $12 \leqslant \log(M_{\text{halo}}/h^{-1}M_\odot) <14.5$ and redshift $z \leqslant 0.32$ and probe the radial distribution of stellar mass out to twice the group virial radius. All the samples are carefully constructed to be complete in stellar mass at each redshift range and efforts are made to regularise the analysis for all the data. Our study shows negligible mass segregation in galaxy group environments with absolute gradients of $\lesssim0.08$ dex and also shows a lack of any redshift evolution. Moreover, we find that our results at least for the GAMA data are robust to different halo mass and group centre estimates. Furthermore, the EAGLE data allows us to probe much fainter luminosities ($r$-band magnitude of 22) as well as investigate the three-dimensional spatial distribution with intrinsic halo properties, beyond what the current observational data can offer. In both cases we find that the fainter EAGLE data show a very mild spatial mass segregation at $z \leqslant 0.22$, which is again not apparent at higher redshift. Interestingly, our results are in contrast to some earlier findings using the Sloan Digital Sky Survey. We investigate the source of the disagreement and suggest that subtle differences between the group finding algorithms could be the root cause.
The XXL survey XV: Evidence for dry merger driven BCG growth in XXL-100-GC X-ray clusters (1608.01223)
S. Lavoie, J. P. Willis, J. Democles, D. Eckert, F. Gastaldello, G. P. Smith, C. Lidman, C. Adami, F. Pacaud, M. Pierre, N. Clerc, P. Giles, M. Lieu, L. Chiappetti, B. Altieri, F. Ardila, I. Baldry, A. Bongiorno, S. Desai, A. Elyiv, L. Faccioli, B. Gardner, B. Garilli, M. W. Groote, L. Guennou, L. Guzzo, A. M. Hopkins, J. Liske, S. McGee, O. Melnyk, M. S. Owers, B. Poggianti, T. J. Ponman, M. Scodeggio, L. Spitler, R. J. Tuffs
Aug. 3, 2016 astro-ph.GA
The growth of brightest cluster galaxies is closely related to the properties of their host cluster. We present evidence for dry mergers as the dominant source of BCG mass growth at $z\lesssim1$ in the XXL 100 brightest cluster sample. We use the global red sequence, H$\alpha$ emission and mean star formation history to show that BCGs in the sample possess star formation levels comparable to field ellipticals of similar stellar mass and redshift. XXL 100 brightest clusters are less massive on average than those in other X-ray selected samples such as LoCuSS or HIFLUGCS. Few clusters in the sample display high central gas concentration, rendering inefficient the growth of BCGs via star formation resulting from the accretion of cool gas. Using measures of the relaxation state of their host clusters, we show that BCGs grow as relaxation proceeds. We find that the BCG stellar mass corresponds to a relatively constant fraction 1\%\ of the total cluster mass in relaxed systems. We also show that, following a cluster scale merger event, the BCG stellar mass lags behind the expected value from the M$_{cluster}$ - M$_{BCG}$ relation but subsequently accretes stellar mass via dry mergers as the BCG and cluster evolve towards a relaxed state.
GAMA/H-ATLAS: Common star-formation rate indicators and their dependence on galaxy physical parameters (1607.02971)
L. Wang, P. Norberg, M. L. P. Gunawardhana, S. Heinis, I. K. Baldry, J. Bland-Hawthorn, N. Bourne, S. Brough, M. J. I. Brown, M. E. Cluver, A. Cooray, E. da Cunha, S. P. Driver, L. Dunne, S. Dye, S. Eales, M. W. Grootes, B. W. Holwerda, A. M. Hopkins, E. Ibar, R. Ivison, C. Lacey, M. A. Lara-Lopez, J. Loveday, S. J. Maddox, M. J. Micha lowski, I. Oteo, M. S. Owers, C. C. Popescu, D. J. B. Smith, E. N. Taylor, R. J. Tuffs, P. van der Werf
July 25, 2016 astro-ph.GA
We compare common star-formation rate (SFR) indicators in the local Universe in the GAMA equatorial fields (around 160 sq. deg.), using ultraviolet (UV) photometry from GALEX, far-infrared (FIR) and sub-millimetre (sub-mm) photometry from H-ATLAS, and Halpha spectroscopy from the GAMA survey. With a high-quality sample of 745 galaxies (median redshift 0.08), we consider three SFR tracers: UV luminosity corrected for dust attenuation using the UV spectral slope beta (SFRUV,corr), Halpha line luminosity corrected for dust using the Balmer decrement (BD) (SFRHalpha,corr), and the combination of UV and IR emission (SFRUV+IR). We demonstrate that SFRUV,corr can be reconciled with the other two tracers after applying attenuation corrections by calibrating IRX (i.e. the IR to UV luminosity ratio) and attenuation in the Halpha (derived from BD) against beta. However, beta on its own is very unlikely to be a reliable attenuation indicator. We find that attenuation correction factors depend on parameters such as stellar mass, z and dust temperature (Tdust), but not on Halpha equivalent width (EW) or Sersic index. Due to the large scatter in the IRX vs beta correlation, when compared to SFRUV+IR, the beta-corrected SFRUV,corr exhibits systematic deviations as a function of IRX, BD and Tdust.
H-ATLAS/GAMA: The nature and characteristics of optically red galaxies detected at submillimetre wavelengths (1511.08018)
A. Dariush, S. Dib, S. Hony, D. J. B. Smith, S. Zhukovska, L. Dunne, S. Eales, E. Andrae, M. Baes, I. Baldry, A. Bauer, J. Bland-Hawthorn, S. Brough, N. Bourne, A. Cava, D. Clements, M. Cluver, A. Cooray, G. De Zotti, S. Driver, M. W. Grootes, A. M. Hopkins, R. Hopwood, S. Kaviraj, L. Kelvin, M. A. Lara-Lopez, J. Liske, J. Loveday, S. Maddox, B. Madore, M. J. Michalowski, C. Pearson, C. Popescu, A. Robotham, K. Rowlands, M. Seibert, F. Shabani, M. W. L. Smith, E.N. Taylor, R. Tuffs, E. Valiante, J.S. Virdee
Nov. 25, 2015 astro-ph.GA
We combine Herschel/SPIRE sub-millimeter (submm) observations with existing multi-wavelength data to investigate the characteristics of low redshift, optically red galaxies detected in submm bands. We select a sample of galaxies in the redshift range 0.01$\leq$z$\leq$0.2, having >5$\sigma$ detections in the SPIRE 250 micron submm waveband. Sources are then divided into two sub-samples of $red$ and $blue$ galaxies, based on their UV-optical colours. Galaxies in the $red$ sample account for $\approx$4.2 per cent of the total number of sources with stellar masses M$_{*}\gtrsim$10$^{10}$ Solar-mass. Following visual classification of the $red$ galaxies, we find that $\gtrsim$30 per cent of them are early-type galaxies and $\gtrsim$40 per cent are spirals. The colour of the $red$-spiral galaxies could be the result of their highly inclined orientation and/or a strong contribution of the old stellar population. It is found that irrespective of their morphological types, $red$ and $blue$ sources occupy environments with more or less similar densities (i.e., the $\Sigma_5$ parameter). From the analysis of the spectral energy distributions (SEDs) of galaxies in our samples based on MAGPHYS, we find that galaxies in the $red$ sample (of any morphological type) have dust masses similar to those in the $blue$ sample (i.e. normal spiral/star-forming systems). However, in comparison to the $red$-spirals and in particular $blue$ systems, $red$-ellipticals have lower mean dust-to-stellar mass ratios. Besides galaxies in the $red$-elliptical sample have much lower mean star-formation/specific-star-formation rates in contrast to their counterparts in the $blue$ sample. Our results support a scenario where dust in early-type systems is likely to be of an external origin.
Galaxy And Mass Assembly (GAMA): growing up in a bad neighbourhood - how do low-mass galaxies become passive? (1511.02245)
L. J. M. Davies, A. S. G. Robotham, S. P. Driver, M. Alpaslan, I. K. Baldry, J. Bland-Hawthorn, S. Brough, M. J. I. Brown, M. E. Cluver, B. W. Holwerda, A. M. Hopkins, M. A. Lara-Lopez, S. Mahajan, A. J. Moffett, M. S. Owers, S. Phillipps
Nov. 6, 2015 astro-ph.GA
Both theoretical predictions and observations of the very nearby Universe suggest that low-mass galaxies (log$_{10}$[M$_{*}$/M$_{\odot}$]<9.5) are likely to remain star-forming unless they are affected by their local environment. To test this premise, we compare and contrast the local environment of both passive and star-forming galaxies as a function of stellar mass, using the Galaxy and Mass Assembly survey. We find that passive fractions are higher in both interacting pair and group galaxies than the field at all stellar masses, and that this effect is most apparent in the lowest mass galaxies. We also find that essentially all passive log$_{10}$[M$_{*}$/M$_{\odot}$]<8.5 galaxies are found in pair/group environments, suggesting that local interactions with a more massive neighbour cause them to cease forming new stars. We find that the effects of immediate environment (local galaxy-galaxy interactions) in forming passive systems increases with decreasing stellar mass, and highlight that this is potentially due to increasing interaction timescales giving sufficient time for the galaxy to become passive via starvation. We then present a simplistic model to test this premise, and show that given our speculative assumptions, it is consistent with our observed results.
The ASKAP/EMU Source Finding Data Challenge (1509.03931)
A. M. Hopkins, R. P. Norris, C. Ferrari, C. A. Hales, D. Herranz, M. Massardi, R. Paladino (21, 18), M. Pestalozzi, H. J. A. Rottgering, J. Swinbank CSIRO Astronomy & Space Science, Instituto de Fisica de Cantabria, Jodrell Bank Centre for Astrophysics, Laboratoire Lagrange Universite Cote d'Azur, Laboratoire AIM CEA/DSM-CNRS-Universite Paris Diderot, National Radio Astronomical Observatory, School of Physics The University of Sydney, ARC Centre of Excellence for All-Sky Astrophysics, University of Groningen Kapteyn Astronomical Institute, International Centre for Radio Astronomy Research UWA, European Space Agency ESAC Planck Science Office, National Centre for Radio Astrophysics Tata Institute of Fundamental Research, Department of Physics, Astronomy University of Bologna, Hamburger Sternwarte Universitat Hamburg, Leiden Observatory Leiden University, Department of Astrophysical Sciences Princeton University, Department of Astronomy University of Cape Town, Department of Physics University of the Western Cape, Department of Physics The George Washington University)
Sept. 14, 2015 astro-ph.GA, astro-ph.IM
The Evolutionary Map of the Universe (EMU) is a proposed radio continuum survey of the Southern Hemisphere up to declination +30 deg., with the Australian Square Kilometre Array Pathfinder (ASKAP). EMU will use an automated source identification and measurement approach that is demonstrably optimal, to maximise the reliability, utility and robustness of the resulting radio source catalogues. As part of the process of achieving this aim, a "Data Challenge" has been conducted, providing international teams the opportunity to test a variety of source finders on a set of simulated images. The aim is to quantify the accuracy of existing automated source finding and measurement approaches, and to identify potential limitations. The Challenge attracted nine independent teams, who tested eleven different source finding tools. In addition, the Challenge initiators also tested the current ASKAPsoft source-finding tool to establish how it could benefit from incorporating successful features of the other tools. Here we present the results of the Data Challenge, identifying the successes and limitations for this broad variety of the current generation of radio source finding tools. As expected, most finders demonstrate completeness levels close to 100% at 10sigma dropping to levels around 10% by 5sigma. The reliability is typically close to 100% at 10sigma, with performance to lower sensitivities varying greatly between finders. All finders demonstrate the usual trade-off between completeness and reliability, whereby maintaining a high completeness at low signal-to-noise comes at the expense of reduced reliability, and vice-versa. We conclude with a series of recommendations for improving the performance of the ASKAPsoft source-finding tool.
Galaxy and Mass Assembly (GAMA): Projected Galaxy Clustering (1509.02159)
D. J. Farrow, N. Metcalfe, A. M. Hopkins, David P. Palamara ICC, Durham, Liverpool John Moores University, Hamburger Sternwarte, University of Western Australia, ICRAR,
We measure the projected 2-point correlation function of galaxies in the 180 deg$^2$ equatorial regions of the GAMA II survey, for four different redshift slices between z = 0.0 and z=0.5. To do this we further develop the Cole (2011) method of producing suitable random catalogues for the calculation of correlation functions. We find that more r-band luminous, more massive and redder galaxies are more clustered. We also find that red galaxies have stronger clustering on scales less than ~3 $h^{-1}$ Mpc. We compare to two different versions of the GALFORM galaxy formation model, Lacey et al (in prep.) and Gonzalez-Perez et al. (2014), and find that the models reproduce the trend of stronger clustering for more massive galaxies. However, the models under predict the clustering of blue galaxies, can incorrectly predict the correlation function on small scales and under predict the clustering in our sample of galaxies with ~3$L_r$ . We suggest possible avenues to explore to improve these cluster- ing predictions. The measurements presented in this paper can be used to test other galaxy formation models, and we make the measurements available online to facilitate this.
The ATLAS 5.5 GHz Survey of the Extended Chandra Deep Field South: The Second Data Release (1509.01344)
M. T. Huynh, M. E. Bell, A. M. Hopkins, R. P. Norris, N. Seymour
We present a new image of the 5.5 GHz radio emission from the extended Chandra Deep Field South. Deep radio observations at 5.5 GHz were obtained in 2010 and presented in the first data release. A further 76 hours of integration has since been obtained, nearly doubling the integration time. This paper presents a new analysis of all the data. The new image reaches 8.6 microJy rms, an improvement of about 40% in sensitivity. We present a new catalogue of 5.5 GHz sources, identifying 212 source components, roughly 50% more than were detected in the first data release. Source counts derived from this sample are consistent with those reported in the literature for S_{5.5GHz} > 0.1 mJy but significantly lower than published values in the lowest flux density bins (S_{5.5GHz} < 0.1 mJy), where we have more detected sources and improved statistical reliability. The 5.5 GHz radio sources were matched to 1.4 GHz sources in the literature and we find a mean spectral index of -0.35 +- 0.10 for S_{5.5GHz} > 0.5 mJy, consistent with the flattening of the spectral index observed in 5 GHz sub-mJy samples. The median spectral index of the whole sample is \alpha_{med} = -0.58, indicating that these observations may be starting to probe the star forming population. However, even at the faintest levels (0.05 < S_{5.5GHz} < 0.1 mJy), 39% of the 5.5 GHz sources have flat or inverted radio spectra. Four flux density measurements from our data, across the full 4.5 to 6.5 GHz bandwidth, are combined with those from literature and we find 10% of sources (S_{5.5GHz} >~ 0.1 mJy) show significant curvature in their radio spectral energy distribution spanning 1.4 to 9 GHz.
Galaxy And Mass Assembly (GAMA): The Bright Void Galaxy Population in the Optical and Mid-IR (1508.06186)
S. J. Penny, M. J. I. Brown, K. A. Pimbblet, M. E. Cluver, D. J. Croton, M. S. Owers, R. Lange, M. Alpaslan, I. Baldry, J. Bland-Hawthorn, S. Brough, S. P. Driver, B. W. Holwerda, A. M. Hopkins, T. H. Jarrett, D. Heath Jones, L. S. Kelvin, M. A. Lara-Lopez, J. Liske, A. R. Lopez-Sanchez, J. Loveday, M. Meyer, P. Norberg, A. S. G. Robotham, M. Rodrigues
Aug. 25, 2015 astro-ph.GA
We examine the properties of galaxies in the Galaxies and Mass Assembly (GAMA) survey located in voids with radii $>10~h^{-1}$ Mpc. Utilising the GAMA equatorial survey, 592 void galaxies are identified out to z~0.1 brighter than $M_{r} = -18.4$, our magnitude completeness limit. Using the $W_{\rm{H\alpha}}$ vs. [NII]/H$\alpha$ (WHAN) line strength diagnostic diagram, we classify their spectra as star forming, AGN, or dominated by old stellar populations. For objects more massive than $5\times10^{9}$ M$_{\odot}$, we identify a sample of 26 void galaxies with old stellar populations classed as passive and retired galaxies in the WHAN diagnostic diagram, else they lack any emission lines in their spectra. When matched to WISE mid-IR photometry, these passive and retired galaxies exhibit a range of mid-IR colour, with a number of void galaxies exhibiting [4.6]-[12] colours inconsistent with completely quenched stellar populations, with a similar spread in colour seen for a randomly drawn non-void comparison sample. We hypothesise that a number of these galaxies host obscured star formation, else they are star forming outside of their central regions targeted for single fibre spectroscopy. When matched to a randomly drawn sample of non-void galaxies, the void and non-void galaxies exhibit similar properties in terms of optical and mid-IR colour, morphology, and star formation activity, suggesting comparable mass assembly and quenching histories. A trend in mid-IR [4.6]-[12] colour is seen, such that both void and non-void galaxies with quenched/passive colours <1.5 typically have masses higher than $10^{10}$ M$_{\odot}$, where internally driven processes play an increasingly important role in galaxy evolution.
The SAMI Galaxy Survey: instrument specification and target selection (1407.7335)
J. J. Bryant, M. S. Owers, A. S. G. Robotham, S. M. Croom, S. P. Driver, M. J. Drinkwater, N. P. F. Lorente, L. Cortese, N. Scott, M. Colless, A. Schaefer, E. N. Taylor, I. S. Konstantopoulos, J. T. Allen, I. Baldry, L. Barnes, A. E. Bauer, J. Bland-Hawthorn, J. V. Bloom, A. M. Brooks, S. Brough, G. Cecil, W. Couch, D. Croton, R. Davies, S. Ellis, L. M. R. Fogarty, C. Foster, K. Glazebrook, M. Goodwin, A. Green, M. L. Gunawardhana, E. Hampton, I. -T. Ho, A. M. Hopkins, L. Kewley, J. S. Lawrence, S. G. Leon-Saval, S. Leslie, G. Lewis, J. Liske, A. R. Lopez-Sanchez, S. Mahajan, A. M. Medling, N. Metcalfe, M. Meyer, J. Mould, D. Obreschkow, S. O'Toole, M. Pracy, S. N. Richards, T. Shanks, R. Sharp, S. M. Sweet, A. D. Thomas, C. Tonini, C. J. Walcher
The SAMI Galaxy Survey will observe 3400 galaxies with the Sydney-AAO Multi-object Integral-field spectrograph (SAMI) on the Anglo-Australian Telescope (AAT) in a 3-year survey which began in 2013. We present the throughput of the SAMI system, the science basis and specifications for the target selection, the survey observation plan and the combined properties of the selected galaxies. The survey includes four volume limited galaxy samples based on cuts in a proxy for stellar mass, along with low-stellar mass dwarf galaxies all selected from the Galaxy And Mass Assembly (GAMA) survey. The GAMA regions were selected because of the vast array of ancillary data available, including ultraviolet through to radio bands. These fields are on the celestial equator at 9, 12, and 14.5 hours, and cover a total of 144 square degrees (in GAMA-I). Higher density environments are also included with the addition of eight clusters. The clusters have spectroscopy from 2dFGRS and SDSS and photometry in regions covered by the Sloan Digital Sky Survey (SDSS) and/or VLT Survey Telescope/ATLAS. The aim is to cover a broad range in stellar mass and environment, and therefore the primary survey targets cover redshifts 0.004 < z < 0.095, magnitudes r$_{pet}$ < 19.4, stellar masses $10^{7} - 10^{12}$ M$_{sol}$, and environments from isolated field galaxies through groups to clusters of $10^{15}$ M$_{sol}$.
Galaxy And Mass Assembly (GAMA): Bivariate functions of H$\alpha$ star forming galaxies (1411.2557)
M. L. P. Gunawardhana, A. M. Hopkins, E. N. Taylor, J. Bland-Hawthorn, P. Norberg, I. K. Baldry, J. Loveday, M. S. Owers, S. M. Wilkins, M. Colless, M. J. I. Brown, S. P. Driver, M. Alpaslan, S. Brough, M. Cluver, S. Croom, L. Kelvin, M. A. Lara-López, J. Liske, A. R. López-Sánchez, A. S. G. Robotham
Nov. 10, 2014 astro-ph.CO, astro-ph.GA
We present bivariate luminosity and stellar mass functions of H$\alpha$ star forming galaxies drawn from the Galaxy And Mass Assembly (GAMA) survey. While optically deep spectroscopic observations of GAMA over a wide sky area enable the detection of a large number of $0.001<{SFR}_{H\alpha}$ (M$_{\odot}$ yr$^{-1}$)$<100$ galaxies, the requirement for an H$\alpha$ detection in targets selected from an $r$-band magnitude limited survey leads to an incompleteness due to missing optically faint star forming galaxies. Using $z<0.1$ bivariate distributions as a reference we model the higher-$z$ distributions, thereby approximating a correction for the missing optically faint star forming galaxies to the local SFR and stellar mass densities. Furthermore, we obtain the $r$-band LFs and stellar mass functions of H$\alpha$ star forming galaxies from the bivariate LFs. As our sample is selected on the basis of detected H$\alpha$ emission, a direct tracer of on-going star formation, this sample represents a true star forming galaxy sample, and is drawn from both photometrically classified blue and red sub-populations, though mostly from the blue population. On average 20-30% of red galaxies at all stellar masses are star forming, implying that these galaxies may be dusty star forming systems.
The SAMI Galaxy Survey: Cubism and covariance, putting round pegs into square holes (1407.5237)
R. Sharp, J. T. Allen, L. M. R. Fogarty, S. M. Croom, L. Cortese, A. W. Green, J. Nielsen, S. N. Richards, N. Scott, E. N. Taylor, L. A. Barnes, A. E. Bauer, M. Birchall, J. Bland-Hawthorn, J. V. Bloom, S. Brough, J. J. Bryant, G. N. Cecil, M. Colless, W. J. Couch, M. J. Drinkwater, S. Driver, C. Foster, M. Goodwin, M. L. P. Gunawardhana, I.-T. Ho, E. J. Hampton, A. M. Hopkins, H. Jones, I. S. Konstantopoulos, J. S. Lawrence, S. K. Leslie, G. F. Lewis, J. Liske, A.R. Lopez-Sanchez, N. P. F. Lorente, R. McElroy, A. M. Medling, S. Mahajan, J. Mould, Q. Parker, M. B. Pracy, D. Obreschkow, M. S. Owers, A. L. Schaefer, S. M Sweet, A. Thomas, C. Tonini, C. J. Walcher
Oct. 30, 2014 astro-ph.GA, astro-ph.IM
We present a methodology for the regularisation and combination of sparse sampled and irregularly gridded observations from fibre-optic multi-object integral-field spectroscopy. The approach minimises interpolation and retains image resolution on combining sub-pixel dithered data. We discuss the methodology in the context of the Sydney-AAO Multi-object Integral-field spectrograph (SAMI) Galaxy Survey underway at the Anglo-Australian Telescope. The SAMI instrument uses 13 fibre bundles to perform high-multiplex integral-field spectroscopy across a one degree diameter field of view. The SAMI Galaxy Survey is targeting 3000 galaxies drawn from the full range of galaxy environments. We demonstrate the subcritical sampling of the seeing and incomplete fill factor for the integral-field bundles results in only a 10% degradation in the final image resolution recovered. We also implement a new methodology for tracking covariance between elements of the resulting datacubes which retains 90% of the covariance information while incurring only a modest increase in the survey data volume.
The SAMI Galaxy Survey: Early Data Release (1407.6068)
J. T. Allen, S. M. Croom, I. S. Konstantopoulos, J. J. Bryant, R. Sharp, G. N. Cecil, L. M. R. Fogarty, C. Foster, A. W. Green, I.-T. Ho, M. S. Owers, A. L. Schaefer, N. Scott, A. E. Bauer, I. Baldry, L. A. Barnes, J. Bland-Hawthorn, J. V. Bloom, S. Brough, M. Colless, L. Cortese, W. J. Couch, M. J. Drinkwater, S. P. Driver, M. Goodwin, M. L. P. Gunawardhana, E. J. Hampton, A. M. Hopkins, L. J. Kewley, J. S. Lawrence, S. G. Leon-Saval, J. Liske, Á. R. López-Sánchez, N. P. F. Lorente, R. McElroy, A. M. Medling, J. Mould, P. Norberg, Q. A. Parker, C. Power, M. B. Pracy, S. N. Richards, A. S. G. Robotham, S. M. Sweet, E. N. Taylor, A. D. Thomas, C. Tonini, C. J. Walcher
We present the Early Data Release of the Sydney-AAO Multi-object Integral field spectrograph (SAMI) Galaxy Survey. The SAMI Galaxy Survey is an ongoing integral field spectroscopic survey of ~3400 low-redshift (z<0.12) galaxies, covering galaxies in the field and in groups within the Galaxy And Mass Assembly (GAMA) survey regions, and a sample of galaxies in clusters. In the Early Data Release, we publicly release the fully calibrated datacubes for a representative selection of 107 galaxies drawn from the GAMA regions, along with information about these galaxies from the GAMA catalogues. All datacubes for the Early Data Release galaxies can be downloaded individually or as a set from the SAMI Galaxy Survey website. In this paper we also assess the quality of the pipeline used to reduce the SAMI data, giving metrics that quantify its performance at all stages in processing the raw data into calibrated datacubes. The pipeline gives excellent results throughout, with typical sky subtraction residuals in the continuum of 0.9-1.2 per cent, a relative flux calibration uncertainty of 4.1 per cent (systematic) plus 4.3 per cent (statistical), and atmospheric dispersion removed with an accuracy of 0."09, less than a fifth of a spaxel. | CommonCrawl |
John Venn
John Venn, FRS,[2][3] FSA[4] (4 August 1834 – 4 April 1923) was an English mathematician, logician and philosopher noted for introducing Venn diagrams, which are used in logic, set theory, probability, statistics, and computer science. In 1866, Venn published The Logic of Chance, a groundbreaking book which espoused the frequency theory of probability, arguing that probability should be determined by how often something is forecast to occur as opposed to "educated" assumptions. Venn then further developed George Boole's theories in the 1881 work Symbolic Logic, where he highlighted what would become known as Venn diagrams.
John Venn
FRS FSA
Born(1834-08-04)4 August 1834
Kingston upon Hull, Yorkshire, England
Died4 April 1923(1923-04-04) (aged 88)
Cambridge, England
Alma materGonville and Caius College, Cambridge
Known for
• Frequentist probability
• Reference class problem
• Venn diagram
AwardsFellow of the Royal Society (1883)
Scientific career
Fields
• Mathematics
• Logic[1]
• Philosophy
InstitutionsGonville and Caius College, Cambridge
Signature
Early life
John Venn was born on 4 August 1834 in Kingston upon Hull, Yorkshire,[5] to Martha Sykes and Rev. Henry Venn, who was the rector of the parish of Drypool. His mother died when he was three years old.[6] Venn was descended from a long line of church evangelicals, including his grandfather John Venn.[7] Venn was brought up in a very strict atmosphere at home. His father Henry had played a significant part in the Evangelical movement and he was also the secretary of the Society for Missions to Africa and the East, establishing eight bishoprics overseas. His grandfather was pastor to William Wilberforce of the abolitionist movement, in Clapham.
He began his education in London joining Sir Roger Cholmeley's School,[8] now known as Highgate School, with his brother Henry in September 1846. He moved on to Islington Proprietary School.[4][5]
University life and career
In October 1853, he went to Gonville and Caius College, Cambridge. He found the Mathematical Tripos unsuited to his mathematical style, complaining that the handful of private tutors he worked with "always had the Tripos prominently in view". In contrast, Venn wished to investigate interesting ideas beyond the syllabus. Nonetheless, he was Sixth Wrangler upon sitting the exams in January 1857.[9]
Venn experienced, in his words, a "reaction and disgust" to the Tripos which led him to sell his books on mathematics and state that he would never return to the subject.[9] Following his family vocation, he was ordained as an Anglican priest in 1859, serving first at the church in Cheshunt, Hertfordshire, and later in Mortlake, Surrey.[10]
In 1862, he returned to Cambridge as a lecturer in moral science, studying and teaching political economy, philosophy, probability theory and logic.[5][9] He reacquainted himself with logic and became a leading scholar in the field through his textbooks The Logic of Chance (1866), Symbolic Logic (1881) and The Principles of Empirical or Inductive Logic (1889). His academic writing was influenced by his teaching: he saw Venn diagrams, which he called "Eulerian Circles" and introduced in 1880, as a pedagogical tool. Venn was known for teaching students across multiple Cambridge colleges, which was rare at the time.[9]
In 1883, he resigned from the clergy, having concluded that Anglicanism was incompatible with his philosophical beliefs.[5]
In 1903 he was elected President of the College, a post he held until his death.[5]
I began at once somewhat more steady work on the subjects and books which I should have to lecture on. I now first hit upon the diagrammatical device of representing propositions by inclusive and exclusive circles. Of course the device was not new then, but it was so obviously representative of the way in which any one, who approached the subject from the mathematical side, would attempt to visualise propositions, that it was forced upon me almost at once.
— John Venn[11]
He built rare machines. A certain machine was meant to bowl cricket balls. The machine was so fascinating that when Australian cricketers were visiting Cambridge, the machines were used to entertain their arrival. The bowling machine that Venn built actually bowled out the top ranked player of the team four times consecutively.[12]
In 1883, Venn was elected a Fellow of the Royal Society,[13] and in 1884, he was awarded a Sc.D. by Cambridge.[14]
He died on 4 April 1923.[5]
Civic and personal life
In 1868, Venn married Susanna Carnegie Edmonstone with whom he had one son, John Archibald Venn. His son entered the mathematics field as well.[3]
Newspaper archives show that Venn was a very active member of local civic society in Cambridge, and a committee member of the Cambridge Charitable Organisations Society, later elected vice-chairman in December 1884.[15]
Venn was president of the Cambridge Antiquarian Society in 1908–1909.[16] He is also listed as a vice president of the Cambridge Provident Medical Institution.[17]
Venn was a prominent supporter of votes for women. He co-signed with his wife Susanna, a letter to the Cambridge Independent Press published 16 October 1908, encouraging women to put themselves forward as candidates for the up-and-coming Cambridge town council elections.[18] The letter was co-sponsored by Lady Maud Darwin, wife of Sir George Darwin, and Florence Ada Keynes.
The newspaper archives reveal that Venn was also a passionate gardener, regularly taking part in local competitions organised by groups such as the Cambridgeshire Horticultural Society, winning prizes for his roses in July 1885[19] and for his white carrots later that September.[20]
Memorials
• In 2017 The Drypool Bridge in Hull was decorated with intersecting circles, in honour of Venn[21] and an unofficial 'blue plaque' is installed near the same location on Clarence Street.
• Venn is commemorated at the University of Hull by the Venn Building.[22]
• A stained glass window in the dining hall of Gonville and Caius College, Cambridge, commemorates Venn's work.
• In commemoration of the 180th anniversary of Venn's birth, on 4 August 2014, Google replaced its normal logo on global search pages with an interactive and animated Google doodle that incorporated the use of a Venn diagram.[23][24]
• Venn Street in Clapham, London, which was the home of his grandfather, shows a Venn diagram on the street sign.[25]
Publications
Venn compiled Alumni Cantabrigienses, a biographical register of former members of the University of Cambridge.[26] His other works include:
• Venn, John (January 1876). "Consistency and Real Inference". Mind. 1 (1).
• Venn, John (1881). Symbolic Logic. London: Macmillan and Company. ISBN 978-1-4212-6044-0.
• Venn, John (1880). "On the Employment of Geometrical Diagrams for the Sensible Representation of Logical Propositions". Proceedings of the Cambridge Philosophical Society. 4: 47–59.
• Venn, John (1866). The Logic of Chance: An Essay on the Foundations and Province of the Theory of Probability, with Especial Reference to Its Application to Moral and Social Science (First ed.). London and Cambridge: Macmillan.. Two further editions were published.[27][28]
• Venn, John (1901). Caius College. London: F. E. Robinson & Co.
• Caius, John (1904). Venn, John (ed.). The Annals of Gonville and Caius College. Printed for the Cambridge Antiquarian Society, sold by Deighton, Bell & Co.
• Venn, John (1904). Annals of a Clerical Family: Being Some Account of the Family and Descendants of William Venn, Vicar of Otterton, Devon, 1600–1621. Cambridge: Cambridge University Press. ISBN 978-1-108-04492-9.
• Venn, John (1870). On Some of the Characteristics of Belief. London and Cambridge: Macmillan and Co.
References
1. Venn, John (July 1880). "I. On the Diagrammatic and Mechanical Representation of Propositions and Reasonings" (PDF). The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 5. 10 (59): 1–18. doi:10.1080/14786448008626877. Archived (PDF) from the original on 16 May 2017. Google Books
2. Anonymous (1926). "Obituary Notices of Fellows Deceased: Rudolph Messel, Frederick Thomas Trouton, John Venn, John Young Buchanan, Oliver Heaviside, Andrew Gray". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 110 (756): i–v. doi:10.1098/rspa.1926.0036.
3. Pickles, John D. "Venn, John Archibald". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/40972. (Subscription or UK public library membership required.)
4. Gibbins, John R. (2004) "Venn, John (1834–1923)", Oxford Dictionary of National Biography, Oxford University Press. doi:10.1093/ref:odnb/36639
5. Duignan, Brian (22 May 2014). "John Venn (English logician and philosopher)". Encyclopædia Britannica. Retrieved 3 August 2014.
6. Anonymous (20 January 2012). "John Venn – Mathematician Biography, Facts and Pictures". Famous-mathematicians.com. Retrieved 3 August 2014.
7. Anonymous (October 2003). "Venn biography". School of Mathematics and Statistics University of St Andrews, Scotland. Retrieved 3 August 2014.
8. Highgate School Roll 1833–1912, Unwin Brothers Ltd 1913
9. Verburgt, Lukas M. (April 2023). "The Venn Behind the Diagram". Mathematics Today. Vol. 59, no. 2. Institute of Mathematics and its Applications. pp. 53–55.
10. Soylent Communications (2014). "John Venn". Retrieved 3 August 2014.
11. Edwards, Anthony William Fairbank (2004). Cogwheels of the Mind: The Story of Venn Diagrams. Baltimore, Maryland, USA: Johns Hopkins University Press. p. 3. ISBN 978-0-8018-7434-5.
12. "John Venn". Famous Inventors. Archived from the original on 2 October 2014. Retrieved 2 August 2018.
13. "Portrait of John Venn". Royal Society Picture Library. Royal Society. Retrieved 2 August 2018.
14. Edwards, A. W. F. (2009). "Statistical Methods for Evolutionary Trees". Genetics. 183 (1): 5–12. doi:10.1534/genetics.109.107847. PMC 2746166. PMID 19797062.
15. "Cambridge Independent Press". Cambridge Independent Press. 6 December 1884. Retrieved 13 April 2017.
16. "Cambridge Independent Press". Cambridge Independent Press. 29 October 1909. Retrieved 13 April 2017.
17. "Cambridge Independent Press". Cambridge Independent Press. 13 February 1886. Retrieved 13 April 2017.
18. "Cambridge Independent Press". Cambridge Independent Press. 16 October 1908. Retrieved 13 April 2017.
19. "Cambridge Independent Press". Cambridge Independent Press. 11 July 1885. Retrieved 13 April 2017.
20. "Cambridge Independent Press". Cambridge Independent Press. 19 September 1885. Retrieved 13 April 2017.
21. Young, Angus (5 June 2017). "John Venn inspired £325k makeover of Hull's Drypool Bridge is now complete". Hull Daily Mail. Retrieved 12 November 2017.
22. "John Venn". Carnegie Heritage Centre. Retrieved 2 August 2018.
23. Antonimuthu, Rajamanickam (2014). "John Venn Google Doodle". YouTube. Archived from the original on 12 December 2021.
24. "4 August: Remembering John Venn on Birthday". Observer Voice. 11 August 2023. Retrieved 11 August 2023.
25. "Rev and Dr Venn". London Remembers. Retrieved 2 August 2018.
26. Venn, John (1922). Alumni Cantabrigienses: A Biographical List of All Known Students, Graduates and Holders of Office at the University of Cambridge, from the Earliest Times to 1900. Cambridge: Cambridge University Press.
27. Venn, John (1876). The Logic of Chance: An Essay on the Foundations and Province of the Theory of Probability, with Especial Reference to Its Logical Bearings and Its Application to Moral and Social Science (Second ed.). Macmillan.
28. Venn, John (1888). The logic of chance: an essay on the foundations and province of the theory of probability, with especial reference to its logical bearings and its application to moral and social science, and to statistics (Third ed.). Macmillan.
External links
Wikimedia Commons has media related to John Venn.
Wikiquote has quotations related to John Venn.
• A Cambridge Alumni Database
• The Venn archives Archived 21 October 2017 at the Wayback Machine clarify the confusing timeline of the various Venns.
• Obituary of John Venn (New York Times)
• Portrait of Venn by Charles Brock, and a link to a site about Venn
• Another (clearer) view of the Venn stained glass window
• John Venn at Find a Grave
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| Wikipedia |
\begin{document}
\title{Classification of cubic homogeneous polynomial maps with Jacobian matrices of rank two}
\author{Michiel de Bondt\footnote{[email protected]}} \affil{\small Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, The Netherlands} \author{Xiaosong Sun\footnote{Corresponding author, E-mail: [email protected]}} \affil{\small School of Mathematics, Jilin University, Changchun 130012, China}
\maketitle
\begin{abstract} Let $K$ be any field with $\textup{char}K\neq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ with $\textup{rk} JH\leq 2$. In particular, we show that, for such an $H$, if $F=x+H$ is a Keller map then $F$ is invertible, and furthermore $F$ is tame if the dimension $n\neq 4$. \end{abstract}
\section{Introduction}
Let $K$ be an arbitrary field and $K[x]:=K[x_1,x_2,\ldots,x_n]$ the polynomial ring in $n$ variables. For a polynomial map $F=(F_1,F_2,\ldots,F_m)\in K[x]^m$, we denote by ${\mathcal{J}} F:=(\frac{\partial F_i}{\partial x_j})_{m\times n}$ the Jacobian matrix of $F$ and $\deg F:=\max_i \deg F_i$ the degree of $F$. A polynomial map $H\in K[x]^m$ is called homogeneous of degree $d$ if each $H_i$ is zero or homogeneous of degree $d$.
A polynomial map $F\in K[x]^n$ is called a Keller map if $\det {\mathcal{J}} F\in K^*$. The Jacobian conjecture asserts that any Keller map is invertible if $\textup{char} K=0$; see \cite{essen2000} or \cite{bass1982}. It is still open for any dimension $n\geq 2$.
Following \cite{shestakov2}, we call a polynomial automorphism elementary if it is of the form $(x_1,\ldots,x_{i-1},cx_i+a,x_{i+1},\ldots,x_n)$, where $c\in K^*$ and $a\in K[x]$ contains no $x_i$. Furthermore, we call a polynomial automorphism tame if it is a finite composition of elementary ones. The definitions of elementary and tame may be different in other sources, but (as long as $K$ is a generalized Euclidean ring) the definitions of tame are equivalent. The Tame Generators Problem asks if every polynomial automorphism is tame. It has an affirmative answer in dimension 2 for arbitrary characteristic (see \cite{jung, kulk}) and a negative answer in dimension 3 for the case of $\textup{char} K=0$ (see \cite{shestakov2}), and is still open for any $n\geq 4$.
A polynomial map $F=x+H\in K[x]^n$ is called triangular if $H_n\in K$ and $H_i\in K[x_{i+1},\ldots,x_n],$ $1\leq i \leq n-1$. A polynomial map $F$ is called linearly triangularizable if it is linearly conjugate to a triangular map, i.e., there exists an invertible linear map $T\in \operatorname{GL}_n(K)$ such that $T^{-1}F(Tx)$ is triangular. A linearly triangularizable map is tame.
Some special polynomial maps have been investigated in the literature. For example, when $\textup{char}K=0$, a Keller map $F=x+H\in K[x]^n$ is shown to be linearly triangularizable in the cases: (1) $n=3$ and $H$ is homogeneous of arbitrary degree $d$ (de Bondt and van den Essen \cite{bondt05}); (2) $n=4$ and $H$ is quadratic homogeneous (Meisters and Olech \cite{mei91}); (3) $n=9$ and $F$ is a quadratic homogeneous quasi-translation (Sun \cite{sun10}); (4) $n$ arbitrary and $H$ is quadratic with $\operatorname{rk} {\mathcal{J}} H\leq 2$ (De Bondt and Yan \cite{bondt-yan}), and to be tame in the case (5) $n=5$ and $H$ is quadratic homogeneous (de Bondt \cite{bondt09} and Sun \cite{sun} independently), and to be invertible in the case (6) $n=4$ and $H$ is cubic homogeneous (Hubbers \cite{hub94}). For the case of arbitrary characteristic, de Bondt \cite{bondt17} described the Jacobian matrix ${\mathcal{J}} H$ of rank two for any quadratic polynomial map $H$ and showed that if ${\mathcal{J}} H$ is nilpotent then ${\mathcal{J}} H$ is similar to a triangular one.
In this paper, we investigate cubic homogeneous polynomial maps $H$ with $\textup{rk} {\mathcal{J}} H\leq 2$ for any dimension $n$ when $\textup{char}K\neq 2, 3$. In Section 2, we classify all such maps (Theorem \ref{rkle2}). And in Section 3, we show that for such an $H$, if $F=x+H$ is a Keller map, then it is invertible and furthermore it is tame if the dimension $n\neq 4$ (Theorem \ref{uporkle2}).
\section{Cubic homogeneous maps $H$ with $\textup{rk} JH\leq 2$}
For a polynomial map $H\in K[x]^m$, we write $\operatorname{trdeg}_K K(H)$ for the transcendence degree of $K(H)$ over $K$. It is well-known that $\operatorname{rk} {\mathcal{J}} H = \operatorname{trdeg}_K K(H)$~ if $K(H) \subseteq K(x)$ is separable, in particular if $\textup{char}K=0$; see \cite[Proposition 1.2.9]{essen2000}. And for arbitrary characteristic, one has $\operatorname{rk} {\mathcal{J}} H \leq \operatorname{trdeg}_K K(H)$; see \cite{bondt15} or \cite{pss}.
It was shown in \cite{bondt17} that when $\textup{char}K\neq 2$, for any quadratic polynomial map $H$ with $\operatorname{rk} {\mathcal{J}} H\leq 2$, one has $\operatorname{rk} {\mathcal{J}} H = \operatorname{trdeg}_K K(H)$. We will show that when $\textup{char}K\neq 2,3$, for any cubic homogeneous polynomial map $H$ with $\operatorname{rk} {\mathcal{J}} H\leq 2$, one has $\operatorname{rk} {\mathcal{J}} H = \operatorname{trdeg}_K K(H)$. The notation $a|_{x=c}$ below means to substitute $x$ by $c$ in $a$.
\begin{theorem} \label{detdep} Let $s \le n$. Take $$ \tilde{x} := (x_1,x_2,\ldots,x_s) ~~ \mbox{and} ~~ L := K(x_{s+1},x_{s+2},\ldots,x_n)\mbox{.} $$ To prove that for (homogeneous) polynomial maps $H \in K[x]^m$ of degree $d$, \begin{equation} \label{eq1} \operatorname{rk} {\mathcal{J}} H = r ~~\mbox{implies } \operatorname{trdeg}_K K(H) = r,~ \mbox{~for every } r < s\mbox{,} \end{equation} it suffices to show that for (homogeneous) polynomial maps $\tilde{H} \in L[\tilde{x}]^s$ of degree $d$, \begin{equation} \label{eq2}\textup{trdeg}_LL(\widetilde{H})=s ~~ \mbox{implies}~~ \operatorname{rk} {\mathcal{J}}_{\tilde{x}} \tilde{H}=s\mbox{.} \end{equation} \end{theorem}
\begin{proof} Suppose that $H \in K[x]^m$ is (homogeneous) of degree $d$, such that \eqref{eq1} does not hold. Then there exists an $r < s$ such that $\operatorname{rk} {\mathcal{J}} H = r < \operatorname{trdeg}_K K(H)$. We need to show that \eqref{eq2} does not hold.
Let $s'= \operatorname{trdeg}_K K(H)$. Assume without loss of generality that $H_1, H_2, \ldots, H_{s'}$ are algebraically independent over $K$, and that the components of $$ H' := \big(H_1, H_2, \ldots, H_{s'},
x_{s'+1}^d, x_{s'+2}^d,\ldots,x_s^d\big) $$ are algebraically independent over $K$ if $s'< s$. Then $$ \operatorname{rk} {\mathcal{J}} H' \le r + (s - s') < s = \operatorname{trdeg}_K K(H')\mbox{.} $$
For the case of $s'\geq s$, just take $H'=(H_1,H_2,\ldots,H_s)$, and we have also $ \operatorname{rk} {\mathcal{J}} H' \le r < s. $
Notice that \eqref{eq1} is also unsatisfied for $H'$. So, replacing $H$ by $H'$, we may assume that $H\in K[x]^s$ with $\operatorname{rk} {\mathcal{J}} H=r<\operatorname{trdeg}_K K(H) = s$.
One may observe that $H_1(x_1,x_1x_2,x_1x_3,\ldots,x_1x_n)$ is algebraically independent over $K$ of $x_2,x_3,\allowbreak\ldots,x_n$. On account of the Steinitz Mac Lane exchange lemma, we may assume without loss of generality that the components of $$ \big(H(x_1,x_1x_2,x_1x_3,\ldots,x_1x_n),x_{s+1},x_{s+2},\ldots,x_n\big) $$ are algebraically independent over $K$. Then the components of $H(x_1,x_1x_2,x_1x_3,$ $\ldots,x_1x_n)$ are algebraically independent over $L:= K(x_{s+1},x_{s+2},\ldots,x_n)$, and so are the components of $$ \tilde{H} := H(x_1,x_2,\ldots,x_s, x_1x_{s+1},x_1x_{s+2},\ldots,x_1x_n)\in L[\widetilde{x}]^s\mbox{,} $$ where $\widetilde{x}=(x_1,x_2,\ldots,x_s)$. That is, $\textup{trdeg}_LL(\widetilde{H})=s$.
Let $G := (x_1,x_2,\ldots,x_s, x_1x_{s+1},x_1x_{s+2},\ldots,x_1x_n)$. Then it follows from the chain rule that $$
{\mathcal{J}}_{\tilde{x}} \tilde{H} = ({\mathcal{J}} H)|_{x = G} \cdot {\mathcal{J}}_{\tilde{x}} G\mbox{,} $$
so $\operatorname{rk} {\mathcal{J}}_{\tilde{x}} \tilde{H}\leq \operatorname{rk} ({\mathcal{J}} H)|_{x = G}\leq \operatorname{rk}{\mathcal{J}} H<s$. Therefore \eqref{eq2} does not hold for $\widetilde{H}$, which completes the proof. \end{proof}
\begin{lemma} \label{rkform}
Let $H \in K[x]^m$ be a polynomial map of degree $d$ and $r := \operatorname{rk} {\mathcal{J}} H$. Denote by $|K|$ the cardinality of $K$. \begin{enumerate}[\upshape (i)]
\item If $|K|>(d-1)r$ and ${\mathcal{J}} H \cdot x = 0$, then there exist $S \in \operatorname{GL}_m(K)$ and $T \in \operatorname{GL}_n(K)$, such that for $\tilde{H} := SH(Tx)$, $$
\tilde{H}|_{x=e_{r+1}} = \left(\begin{array}{cc} I_r & 0 \\ 0 & 0 \end{array} \right)\mbox{.} $$
\item If $|K|>(d-1)r + 1$ and ${\mathcal{J}} H \cdot x \ne 0$, then there exist $S \in \operatorname{GL}_m(K)$ and $T \in \operatorname{GL}_n(K)$, such that for $\tilde{H} := SH(Tx)$, $$
\tilde{H}|_{x=e_1} = \left( \begin{array}{cc} I_r & 0 \\ 0 & 0 \end{array} \right)\mbox{.} $$
\end{enumerate}
Moreover, $|K|$ may be one less (i.e. at least $(d-1)r$ and $(d-1)r + 1$ respectively) if every nonzero component of $H$ is homogeneous. \end{lemma}
\begin{proof} (i) Assume without loss of generality that $$ a_0:= \det {\mathcal{J}}_{x_1,x_2,\ldots,x_r} (H_1,H_2,\ldots,H_r) \ne 0\mbox{.} $$
Suppose that $|K|>(d-1)r$. It follows by \cite[Lemma 5.1 (i)]{bondt13} that there exists a vector $w \in K^n$ such that $a_0(w) \ne 0$. So $\operatorname{rk} \big({\mathcal{J}} H\big)\big|_{x=w} = r$. There exist $n-r$ independent vectors $v_{r+1}, v_{r+2}, \ldots, \allowbreak v_n \in K^n$, such that $
\big({\mathcal{J}} H\big)\big|_{x=w} \cdot v_i = 0 $ for $i = r+1, r+2, \allowbreak \ldots, n$. And we may take $v_{r+1} = w$ since $$
\big({\mathcal{J}} H\big)\big|_{x=w} \cdot w = \big({\mathcal{J}} H \cdot x\big)\big|_{x=w} = 0\mbox{.} $$ Take $T =(v_1,v_2,\cdots,v_n)\in \operatorname{GL}_n(K)$. From the chain rule, we deduce that $$
\big({\mathcal{J}} (H(Tx))\big)\big|_{x=e_{r+1}} \cdot e_i
= ({\mathcal{J}} H)|_{x=Te_{r+1}} \cdot T e_i = ({\mathcal{J}} H)|_{x=w} \cdot v_i \quad (1\leq i\leq n)\mbox{.} $$ In particular,
$\operatorname{rk} {\mathcal{J}} (H(Tx))\big|_{x=e_{r+1}} = r$ and the last
$n - r$ columns of $\big({\mathcal{J}} (H(Tx))\big)\big|_{x=e_{r+1}}$ are zero. There exists $S \in \operatorname{GL}_m(K)$ such that $$
\big({\mathcal{J}} (SH(Tx))\big)\big|_{x=e_{r+1}} =
S \cdot \big({\mathcal{J}} (H(Tx))\big)\big|_{x=e_{r+1}} = \left( \begin{array}{cc} I_r & 0\\ 0 & 0 \end{array} \right)\mbox{.} $$
(2) Suppose that $|K|>(d-1)r + 1$. Since ${\mathcal{J}} H \cdot x \ne 0$, we may assume that $$ \operatorname{rk} \Big( {\mathcal{J}} H \cdot x, {\mathcal{J}}_{x_2,x_3,\ldots,x_r} H \Big) = r\mbox{,} $$ and that $$ a_1 := \det \Big( {\mathcal{J}} (H_1,H_2,\ldots,H_r) \cdot x, {\mathcal{J}}_{x_2,x_3,\ldots,x_r} (H_1,H_2,\ldots,H_r) \Big) \ne 0\mbox{.} $$
It follows by \cite[Lemma 5.1 (i)]{bondt13} that there exists $w \in K^n$ such that $a_1(w) \ne 0$. One may observe that $\operatorname{rk} \big({\mathcal{J}} H\big)\big|_{x=w} = r$
and thus there exist independent vectors $v_{r+1}, v_{r+2}, \ldots, v_n \in K^n$, such that $\big({\mathcal{J}} H\big)\big|_{x=w} \cdot v_i = 0$ for $i = r+1, \allowbreak r+2, \ldots, n$. Since
$\big({\mathcal{J}} H \cdot x\big)\big|_{x=w}$ is the first column of a full column rank matrix, we have $$
\big({\mathcal{J}} H\big)\big|_{x=w} \cdot w = \big({\mathcal{J}} H \cdot x\big)\big|_{x=w} \ne 0\mbox{.} $$ So $v_1 := w$ is independent of $v_{r+1}, \allowbreak v_{r+2}, \ldots, v_n$.
Take $T =(v_1,v_2,\cdots,v_n)\in \operatorname{GL}_n(K)$. Then $$
\big({\mathcal{J}} (H(Tx))\big)\big|_{x=e_1} \cdot e_i
= ({\mathcal{J}} H)|_{x=Te_1} \cdot T e_i = ({\mathcal{J}} H)|_{x=w} \cdot v_i \quad (1\leq i\leq n). $$ The rest of the proof of (ii) is similar to that of (i).
The last claim follows from \cite[Lemma 5.1 (ii)]{bondt13}, as an improvement to \cite[Lemma 5.1 (i)]{bondt13}. \end{proof}
\begin{proposition} \label{propA} Assume that $\textup{char}K\notin \{1,2,\ldots,d\}$. Then for any cubic homogeneous polynomial map $H\in K[x]^m$ of degree $d$ with $\operatorname{rk} {\mathcal{J}} H\leq 1$, the components of $H$ are linearly dependent over $K$ in pairs, and one has $\operatorname{rk} {\mathcal{J}} H = \operatorname{trdeg}_K K(H)$. \end{proposition}
\begin{proof} The case $\operatorname{rk} {\mathcal{J}} H=0$ is obvious, so let $\operatorname{rk} {\mathcal{J}} H=1$. On account of Lemma \ref{rkform}, we may assume that
${\mathcal{J}} H|_{x=e_1}=E_{11}$. Let $j \ge 2$. Since $\deg_{x_1} H_j < d$, we infer that either $H_j = 0$, or $\deg_{x_1} \parder{}{x_1} H_j < \deg_{x_1} \parder{}{x_i} H_j$ for some $i \ge 2$, where $\deg_{x_1} 0 = -\infty$. The latter is impossible due to $\operatorname{rk} {\mathcal{J}} H=1$, so $H_j = 0$. This holds for all $j \ge 2$, which yields the desired results. \end{proof}
\begin{lemma}\label{lemB} Let $H=(h,x_1^2x_2,x_2^2x_3)$ or $(h,x_1^2x_3,x_2^2x_3)\in K[x_1,x_2,x_3]^3$, where $h$ is cubic homogeneous, and assume that $\operatorname{char} K\neq 2,3$. Then $\operatorname{rk} {\mathcal{J}} H=\textup{trdeg}_KK(H)$. \end{lemma}
\begin{proof} It suffices to consider the case of $\operatorname{rk} {\mathcal{J}} H=2$. Define a derivation $D$ on $A=K[x_1,x_2,x_3]$ as follows: for any $f\in A$, $$D(f)=\frac{x_1x_2x_3}{H_2H_3}\det {\mathcal{J}} H\mbox{.}$$ In the case $H=(h,x_1^2x_2,x_2^2x_3)$, an easy calculation shows that $D=x_1\partial_{x_1}-2x_2\partial_{x_2}+4x_3\partial_{x_3}$. Then for any term $u=x_1^{d_1}x_2^{d_2}x_3^{d_3}\in A$, $D(u)=(d_1-2d_2+4d_3)u$. And thus $\ker D:=\{g\in A \mid D(g)=0\}$, the kernel of $D$, is linearly spanned by all terms $u$ with $d_1-2d_2+4d_3=0$. So the only cubic terms in $\ker D$ are $x_1^2x_2$ and $x_2^2x_3$. Since $\operatorname{rk} {\mathcal{J}} H=2$, we have $\det{\mathcal{J}} H=0$ and thus $h\in \ker D$, which implies that $h$ is a linear combinations of $x_1^2x_2$ and $x_2^2x_3$. Thus $\textup{trdeg}_KK(H)=2$.
In the case $H=(h,x_1^2x_3,x_2^2x_3)$, one may verify that $x_1^2 x_3$, $x_1 x_2 x_3$ and $x_2^2 x_3$ are the only cubic terms in $\ker D$. The conclusion follows similarly. \end{proof}
\begin{theorem} \label{eqthm} Assume that $\textup{char}K\neq 2,3$. Then for any cubic homogeneous polynomial map $H\in K[x]^m$ with $\operatorname{rk} {\mathcal{J}} H\leq 2$, one has $\operatorname{rk} {\mathcal{J}} H = \operatorname{trdeg}_K K(H)$. \end{theorem}
\begin{proof} Due to Theorem \ref{detdep}, and replacing $L$ there by $K$, we may assume that $H\in K[x_1,x_2,x_3]^3,$ and it suffices to show that $$\textup{trdeg}_KK(H)=3 ~ ~\mbox{implies}~ \operatorname{rk} {\mathcal{J}} H=3\mbox{,}$$ or equivalently, \begin{equation} \label{detdep3} \det {\mathcal{J}} H = 0 ~\mbox{implies} ~ \textup{trdeg}_KK(H)<3\mbox{.} \end{equation} So assume that $\det {\mathcal{J}} H=0$. Since we may replace $K$ by an extension field to make it large enough, it follows by Lemma \ref{rkform} that we may assume that $
\big({\mathcal{J}} H\big)\big|_{x=e_1} =E_{11}+E_{22}. $ Then ${\mathcal{J}} H$ is of the form $$\left(
\begin{array}{ccc}
x_1^2+\ast & \ast & \ast \\
\ast & x_1^2+\ast & \ast \\
\ast & \ast & \frac{\partial H_3}{\partial x_3} \\
\end{array}
\right)\mbox{,} $$ where the $x_1$-degree of each element $\ast$ is less than 2. Observing the terms with $x_1$-degree $\geq 5$ in $\det {\mathcal{J}} H$, we have that $\frac{\partial H_3}{\partial x_3}\in K[x_2,x_3]$. Notice that $H_2$ and $H_3$ are of the form: \begin{align*} H_2&=x_1^2x_2+b_{10}x_1x_3^2+b_{11}x_1x_2x_3+b_{12}x_1x_2^2+b_0(x_2,x_3)\mbox{;}\\ H_3&=c_{12}x_1x_2^2+c_{00}x_3^3+c_{01}x_2x_3^2+c_{02}x_2^2x_3+c_{03}x_2^3\mbox{.} \end{align*} We shall show that $x_2^2 \mid H_3$, i.e., $c_{00} = c_{01} = 0$.
Noticing that the part of $x_1$-degree 4 of $\det {\mathcal{J}} H$ is $\big(\frac{\partial H_3}{\partial x_3}-\frac{\partial H_2}{\partial x_1\partial x_3}\frac{\partial H_3}{\partial x_1\partial x_2}\big)x_1^4$, we see that $\frac{\partial H_3}{\partial x_3}-\frac{\partial H_2}{\partial x_1\partial x_3}\frac{\partial H_3}{\partial x_1\partial x_2}=0$. Consequently, $$ (3 c_{00}x_3^2 + 2 c_{01}x_2 x_3 + c_{02}x_2^2) = (2 b_{10} x_3 + b_{11} x_2) (2 c_{12} x_2) $$ so \begin{align*} c_{00} &= 0 & c_{01} &= 2 b_{10} c_{12} & c_{02} &= 2 b_{11} c_{12} \end{align*} One may observe that the coefficient of $x_1^3x_3^3$ in $\det {\mathcal{J}} H$ is $2c_{01}b_{10}=0$, which we can combine with $c_{01} = 2 b_{10} c_{12}$ to obtain $c_{01} = 0$. Therefore, $$H_3=(c_{12}x_1+c_{03}x_2+c_{02}x_3)x_2^2\mbox{.}$$ Moreover, if $c_{12}=0$ then $c_{02} = 2 b_{11} c_{12} = 0$ and thus $H_3=c_{03} x_2^3$.
We distinguish two cases. \begin{itemize}
\item\emph{Case 1:} $c_{12} \ne 0$ and $c_{12}x_1+c_{03}x_2+c_{02}x_3 \nmid H_i$ for some $i$.
Then $H_3$ is the product of two linear forms, of which two are distinct. Hence we can compose $H$ with invertible linear maps on both sides, to obtain a map $H'$ for which $H'_2 = x_1^2 x_2$, and $x_2\nmid H'_1$.
Notice that $H'_1(1,0,t) \ne 0$. As $K$ has at least $5$ elements, it follows from [3, Lemma 5.1 (i)] that there exists a $\lambda \in K$, such that $H'_1(1,0,\lambda) \ne 0$. Hence the coefficient of $x_1^3$ in $H'_1(x_1,x_2,x_3+\lambda x_1)$ is nonzero. Furthermore, $H'_2(x_1,x_2,x_3+\lambda x_1) = x_1^2 x_2$.
Replacing $H'$ by $H'(x_1,x_2,x_3+\lambda x_1)$, we may assume that $H'_2=x_1^2x_2$ and that $H'_1$ contains $x_1^3$ as a term. We may even assume that the coefficient of $x_1^3$ in $H'_1$
equals $1$. Then ${\mathcal{J}} H'|_{x=e_1}$ is of the form $$ \left( \begin{array}{ccc} 1 & \ast & a \\ 0 & 1 & 0 \\ \ast & \ast & \ast \\ \end{array} \right)\mbox{,} $$ and has rank 2. Furthermore, $v_3=(-a,0,1)^t$ belongs to its null space. We may apply the proof of Lemma \ref{rkform} on $H'$ by taking $T=(e_1,e_2,v_3)$ and taking an appropriate $S\in \operatorname{GL}_3(K)$ such that $\widetilde{H}:=SH'(Tx)$ satisfies
${\mathcal{J}} \widetilde{H}|_{x=e_1}=S{\mathcal{J}} H'|_{x=Te_1}T=E_{11}+E_{22}$. Notice that $Tx$ is of the form $(L_1,x_2,L_3)$, and observing
the form of ${\mathcal{J}} H'|_{x=e_1}$ one may also choose $Sx$ to be of the form $(\ast, x_2,\ast)$. Then $\widetilde{H}_2=L_1^2x_2$.
So we can compose $\widetilde{H}$ with an invertible linear map on the right, to obtain a map $\widetilde{H}'$ for which $\widetilde{H}'_2 = x_1^2 x_2$ and $\widetilde{H}'_3 = x_2^2 L'$ for some linear form $L'$.
Suppose first that $L'$ is a linear combination of $x_1$ and $x_2$. If $\widetilde{H}'_1 \in K[x_1,x_2]$, then we are done. Otherwise, we have $\det {\mathcal{J}}_{x_1,x_2} (\widetilde{H}'_2,\widetilde{H}'_3)=0$, and then by Proposition \ref{propA}, $\textup{trdeg}_K K(H'_2,H'_3)<2$.
Suppose next that $L'$ is not a linear combination of $x_1$ and $x_2$. Then we may assume that $\widetilde{H}'_3 = x_2^2 x_3$. By Lemma \ref{lemB} (i), $\textup{trdeg}_K K(\widetilde{H}')<3$.
\item\emph{Case 2:} $c_{12} = 0$ or $c_{12}x_1+c_{03}x_2+c_{02}x_3 \mid H_i$ for all $i$.
Since $x_2^2 \mid H_3$, we can compose $H$ with invertible linear maps on both sides, to obtain a map $H'$ for which $H'_1 \in \{x_1^3, x_1^2 x_2\}$. After a possible interchange of $H'_2$ and $H'_3$, the first two rows of ${\mathcal{J}} H'$ are independent. Now we may apply the proof of Lemma \ref{rkform} to $H'$, more precisely, there exist $S,T\in \operatorname{GL}_3(K)$ such that $\widetilde{H} := SH'(Tx)$ satisfies
${\mathcal{J}} \widetilde{H}|_{x=e_1}=E_{11}+E_{22}$. If we choose $w$ such that first two rows of $({\mathcal{J}} H')_{x=w}$ are independent, then we can take $S$ such that $Sx=(f_1x_1+f_2x_2,g_1x_1+g_2x_2,\ast)$. By repeating the discussion for $\widetilde{H}$ as for $H$ above, we may assume that $\widetilde{H}_3=Lx_2^2$ for some linear form $L$.
Let $Tx=(L_1,L_2,L_3)$. Notice that $H'_1(Tx) \in \{L_1^3,L_1^2L_2\}$ and that $H'_1(Tx)$ is a linear combination of $\widetilde{H}_1$ and $\widetilde{H}_2$. Hence we can compose $\widetilde{H}$ with a linear map on the left, to obtain a map $\widetilde{H}'$ for which $\widetilde{H}'_2 \in \{L_1^3,L_1^2L_2\}$ and $\widetilde{H}'_3 = Lx_2^2$.
Suppose first that $\widetilde{H}'_2 = L_1^2L_2$. Then $c_{12} \ne 0$, so $c_{12}x_1+c_{03}x_2+c_{02}x_3 \mid H_i$ for all $i$. From this, we infer that $L_2 \mid \widetilde{H}_i$ and $L_2 \mid \widetilde{H}'_i$ for all $i$. As $x_2 \nmid \widetilde{H}_1$, we deduce that $L$ and $L_2$ are dependent linear forms, which are independent of $x_2$. If $L$ and $L_2$ are linear combinations of $L_1$ and $x_2$, then we can reduce to Proposition \ref{propA}, and otherwise we can reduce to Lemma \ref{lemB} (ii).
Suppose next that $\widetilde{H}'_2 = L_1^3$. If $L$, $L_1$ and $x_2$ are linearly dependent over $K$, then we can reduce to Proposition \ref{propA}. Otherwise, $\tilde{H}$ is as $H$ in the previous case. \qedhere
\end{itemize} \end{proof}
\begin{remark} Inspired by Lemma \ref{lemB}, we investigated maps $H$ of which the components are terms, and searched for $H$ with algebraically independent components for which $\det {\mathcal{J}} H = 0$. One can infer that $H$ is as such, if and only if the matrix with entries $\deg_{x_i} H_j$ has determinant zero over $K$, but not over $\mathbb{Z}$.
We found the following non-homogeneous $H$ as above over fields of characteristic $5$: $$ (x_1^3 x_2, x_1 x_2^2), \quad (x_1^2 x_2, x_1 x_3^2, x_2 x_3) $$ with the following homogenizations respectively: $$ (x_1^3 x_2, x_1 x_2^2 x_3, x_3^4), \quad (x_1^2 x_2, x_1 x_3^2, x_2 x_3 x_4, x_4^3) $$ Besides these homogenizations, we found the following homogeneous $H$ over fields of characteristic $5$: $$ (x_1^2 x_3^2, x_1 x_2^3, x_2 x_3^3), \quad (x_4 x_1^2, x_1 x_2^2, x_2 x_3^2, x_3 x_4^2) $$ We conclude with a homogeneous $H$ over fields of characteristic $7$, and a homogeneous $H$ over any characteristic $p \in \{1,2,\ldots,d\}$ respectively: $$ (x_3 x_1^3, x_1 x_2^3, x_2 x_3^3), \quad (x_1^d, x_1^{d-p} x_2^p) $$ These examples show that the conditions in Proposition \ref{propA} and Theorem \ref{eqthm} cannot be relaxed. \end{remark}
\begin{theorem} \label{rkle2} Suppose that $\textup{char} K\neq 2,3$ and let $H \in K[x]^m$ be cubic homogeneous. Let $r := \operatorname{rk} {\mathcal{J}} H$ and suppose that $r \le 2$. Then there exist $S \in \operatorname{GL}_m(K)$ and $T \in \operatorname{GL}_n(K)$, such that for $\tilde{H} := SH(Tx)$, one of the following statements holds: \begin{enumerate}[\upshape (1)]
\item $\tilde{H}_{r+1} = \tilde{H}_{r+2} = \cdots = \tilde{H}_m = 0$;
\item $r=2$ and $\tilde{H} \in K[x_1,x_2]^m$;
\item $r=2$ and $K \tilde{H}_1 + K \tilde{H}_2 + \cdots + K \tilde{H}_m = K x_3 x_1^2 \oplus K x_3 x_1 x_2 \oplus K x_3 x_2^2$. \end{enumerate} Furthermore, we may take $S = T^{-1}$ if $m = n$. \end{theorem}
\begin{proof} By Theorem \ref{eqthm}, $\operatorname{trdeg}_KK(H)=\operatorname{rk} {\mathcal{J}} H=r\leq 2$. Since $H$ is homogeneous, we have $\operatorname{trdeg}_K K(tH) = r$ as well, where $t$ is a new variable.
Suppose first that $r \le 1$. It follows by \cite[Theorem 2.7]{bondt15} that we may take $\tilde{H}$ as in (1).
Suppose next that $r = 2$. By \cite[Theorem 2.7]{bondt15}, $H$ is of the form $g \cdot h(p,q)$, such that $g, h$ and $(p,q)$ are homogeneous and $\deg g + \deg h \cdot \allowbreak \deg (p,q) = 3$.
If $\deg h \le 1$, then every triple of components of $h$ is linearly dependent over $K$, and thus we may take $\tilde{H}$ as in (1). If $\deg h = 3$, then $\deg (p,q) = 1$ and $\deg g = 0$, whence we may take $\tilde{H}$ as in (2).
So assume that $\deg h = 2$. Then $\deg(p,q) = 1$ and $\deg g = 1$. If $g$ is a linear combination of $p$ and $q$, then we may take $\tilde{H}$ as in (2). If $g$ is not a linear combination of $p$ and $q$, then we may take $\tilde{H}$ as in (3) or (1).
Finally, if $m = n$ and $\tilde{H}=SH(Tx)$ is as in (1), then $SH(S^{-1}x)=\tilde{H}(T^{-1}S^{-1}x)$ is still as in (1). So we may take $S=T^{-1}$. If $m = n$ and $\tilde{H}=SH(Tx)$ is as in (2) or (3), then $T^{-1}H(Tx)=T^{-1}S^{-1}\tilde{H}$ is still as in (2) or (3), whence we may also take $S=T^{-1}$. \end{proof}
\section{Cubic homogeneous Keller maps $x+H$ with $\textup{rk} JH\leq 2$}
For two matrices $M, N\in \operatorname{Mat}_n(K[x])$, we say that $M$ is similar over $K$ to $N$, if there exists $T \in \operatorname{GL}_n(K)$ such that $N = T^{-1}MT$.
\begin{theorem} \label{trdeg1} Let $F = x + H\in K[x]^n$ be a Keller map with $\operatorname{trdeg}_K K(H) = 1$. Then ${\mathcal{J}} H$ is similar over $K$ to a triangular matrix, and the following statements are equivalent: \begin{enumerate}[\upshape (1)]
\item $\det {\mathcal{J}} F = 1$;
\item ${\mathcal{J}} H$ is nilpotent;
\item $({\mathcal{J}} H) \cdot ({\mathcal{J}} H)|_{x=y} = 0$, where $y=(y_1,y_2,\ldots,y_n)$ are $n$ new variables.
\end{enumerate} \end{theorem}
\begin{proof} Since $\operatorname{trdeg}_K K(H) = 1$, by \cite[Corollary 3.2]{bondt15} there exists a polynomial $p \in K[x]$ such that $H_i \in K[p]$ for each $i$. Say that $H_i = h_i(p)$, where $h_i \in K[t]$ for each $i$. Write $h_i' = \parder{}{t} h_i$, then \begin{equation} \label{JHhp} {\mathcal{J}} H = h'(p) \cdot {\mathcal{J}} p\mbox{.} \end{equation} Assume without loss of generality that $$ h_1' = h_2' = \cdots = h_s' = 0\mbox{,} $$ and that $$ 0 \le \deg h_{s+1}' < \deg h_{s+2}' < \cdots < \deg h_n'\mbox{.} $$ For $s < i < n$, $$ \deg h_i'(p) = \deg h_i' \cdot \deg p \le (\deg h_{i+1}' - 1) \cdot \deg p = \deg h_{i+1}'(p) - \deg p. $$ Since the degrees of the entries of ${\mathcal{J}} p$ are less than $\deg p$, we deduce from \eqref{JHhp} that the nonzero entries on the diagonal of ${\mathcal{J}} H$ have different degrees in increasing order. Furthermore, the nonzero entries beyond the $(s+1)$th entry on the diagonal of ${\mathcal{J}} H$ have positive degrees.
By \eqref{JHhp}, $\operatorname{rk} (- {\mathcal{J}} H) \le 1$, and thus $n-1$ eigenvalues of $-{\mathcal{J}} H$ are zero. It follows that the trailing degree of the characteristic polynomial of $-{\mathcal{J}} H$ is at least $n-1$. More precisely, $$ \det (t I_n+{\mathcal{J}} H) = t^n - \operatorname{tr} (-{\mathcal{J}} H) \cdot t^{n-1}\mbox{,} $$ and thus $$
\det {\mathcal{J}} F = \big(t^n - \operatorname{tr} (-{\mathcal{J}} H) \cdot t^{n-1}\big)\big|_{t=1} = 1 + \operatorname{tr} {\mathcal{J}} H\mbox{.} $$ Observe that the diagonal of ${\mathcal{J}} H$ is totally zero, except maybe the $(s+1)$th entry, which is a constant.
So $\parder{}{x_i} p = 0$ for all $i > s+1$, and ${\mathcal{J}} H$ is lower triangular. If the $(s+1)$th entry on the diagonal of ${\mathcal{J}} H$ is nonzero, then (1), (2) and (3) do not hold. If the $(s+1)$th entry on the diagonal of ${\mathcal{J}} H$ is zero, then $\parder{}{x_i} p = 0$ for all $i > s$, whence (1), (2) and (3) hold. \end{proof}
Let $H \in K[x]^n$ be homogeneous of degree $d\geq 2$. Then $x + H$ is a Keller map if and only if ${\mathcal{J}} H$ is nilpotent; see for example \cite[Lemma 6.2.11]{essen2000}. So we first investigate nilpotent matrices over $K[x]$.
\begin{lemma} \label{2x2} Let $N \in \operatorname{Mat}_2(K[x])$ such that $N$ is nilpotent. Then there exist $a,b,c \in K[x]$ such that $$ N = c \left( \begin{array}{cc} ab & -b^2 \\ a^2 & -ab \end{array}\right)\mbox{.} $$ Furthermore, $N$ is similar over $K$ to a triangular matrix if and only if $a$ and $b$ are linearly dependent over $K$. \end{lemma}
\begin{proof} Since $\det N = 0$, we may write $N$ in the form $$ N = c \cdot \binom{b}{a} \cdot \big(\,a ~~ {-\tilde{b}}\,\big)\mbox{,} $$ where $a,b \in K[x]$ and $\tilde{b},c \in K(x)$. Since $\operatorname{tr} N = 0$, we have $\tilde{b} = b$. If we choose $a$ and $b$ to be relatively prime, then $c \in K[x]$ as well.
Furthermore, $a$ and $b$ are linearly dependent over $K$ if and only if the rows of $N$ are linearly dependent over $K$, if and only if $N$ is similar over $K$ to a triangular matrix. \end{proof}
\begin{lemma} \label{2x2nilp} Let $H \in K[x]^2$ be cubic homogeneous, such that ${\mathcal{J}}_{x_1,x_2} H$ is nilpotent. Then there exists $T \in \operatorname{GL}_2(K)$ such that for $\tilde{H} := T^{-1} H\big(T(x_1,x_2),x_3,\allowbreak x_4,\allowbreak \ldots,x_n\big)$, one of the following statements holds: \begin{enumerate}[\upshape (1)]
\item ${\mathcal{J}}_{x_1,x_2} \tilde{H}$ is a triangular matrix;
\item there are independent linear forms $a, b \in K[x]$, such that $$ {\mathcal{J}}_{x_1,x_2} \tilde{H} = \left( \begin{array}{cc} ab & -b^2 \\ a^2 & -ab \end{array} \right) ~~ \mbox{ and } ~~ {\mathcal{J}}_{x_1,x_2} \left( \begin{array}{c} a \\ b \end{array} \right) = \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right)\mbox{;} $$
\item $\textup{char} K=3$ and there are independent linear forms $a, b \in K[x]$, such that $$ {\mathcal{J}}_{x_1,x_2} \tilde{H} = \left( \begin{array}{cc} ab & -b^2 \\ a^2 & -ab \end{array} \right) ~~\mbox{and} ~~ {\mathcal{J}}_{x_1,x_2} \left( \begin{array}{c} a \\ b \end{array} \right) = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)\mbox{.} $$ \end{enumerate} \end{lemma}
\begin{proof} Suppose that (1) does not hold. By Lemma \ref{2x2}, there are $a,b,c \in K[x]$, such that $$ {\mathcal{J}}_{x_1,x_2} H = c \left( \begin{array}{cc} ab & - b^2 \\ a^2 & -ab \end{array} \right) $$ where $a$ and $b$ are linearly independent over $K$. As $H$ is cubic homogeneous, the entries of ${\mathcal{J}}_{x_1,x_2} H$ are quadratic homogeneous, so $c \in K$ and $a$ and $b$ are independent linear forms.
If we take $$ T = \left( \begin{array}{cc} c & 0 \\ 0 & 1 \end{array} \right) \mbox{,} \quad \mbox{then} \quad {\mathcal{J}}_{x_1,x_2} \tilde{H} = \left( \begin{array}{cc} \tilde{a}\tilde{b} & -\tilde{b}^2 \\ \tilde{a}^2 & -\tilde{a}\tilde{b} \end{array} \right) \mbox{,} $$
where $\tilde{a} = c \cdot a|_{x_1=cx_1}$ and $\tilde{b} =c^{-1}\cdot b|_{x_1=cx_1}$.
We claim that the coefficient $k_2$ of $x_2$ in $\tilde{b}$ is $0$. Suppose conversely that $k_2\neq 0$. Then the coefficient of $x_2^3$ in $$ 3 \tilde{H}_1 = {\mathcal{J}}_{x_1,x_2} \tilde{H}_1 \cdot \binom{x_1}{x_2}=\tilde{b}(x_1\tilde{a}-x_2\tilde{b}) $$ is nonzero. In particular, $\operatorname{char} K\neq 3$. One may verify that $$ {\mathcal{J}}_{x_1,x_2} (\tilde{H}_1 + \tfrac13 k_2^{-1}\tilde{b}^3) = (\tilde{c}\tilde{b},~0)\mbox{,} $$ where $\tilde{c} := \tilde{a} + k_2^{-1}\tilde{b} (\parder{}{x_1} \tilde{b})$. As a consequence, $\parder{}{x_2} (\tilde{c}\tilde{b}) = \parder{}{x_1} 0 = 0$. Furthermore, $\tilde{c}$ and $\tilde{b}$ are independent, just like $\tilde{a}$ and $\tilde{b}$. By $\parder{}{x_2} (\tilde{c}\tilde{b}) = 0$, we have $\tilde{c}\tilde{b} \in K[x_1,x_3,x_4,\ldots,x_n]$ if $\textup{char} K\neq 2$. Since $\tilde{c}$ and $\tilde{b}$ are independent, we deduce that if $\textup{char} K= 2$ then $\tilde{c}\tilde{b} \in K[x_1,x_3,x_4,\ldots,x_n]$ as well. Since the coefficient $\lambda$ of $x_2$ in $\tilde{b}$ is nonzero, we have $\tilde{c} = 0$, a contradiction.
So the coefficient of $x_2$ in $\tilde{b}$ is $0$. Similarly, the coefficient of $x_1$ in $\tilde{a}$ is $0$. Consequently, $$ {\mathcal{J}}_{x_1,x_2} \left( \begin{array}{c} \tilde{a} \\ \tilde{b} \end{array} \right) = \left( \begin{array}{cc} 0 & \lambda \\ \mu & 0 \end{array} \right) \mbox{,}$$ where $\lambda, \mu \in K$. Therefore $$ {\mathcal{J}}_{x_1,x_2} \tilde{H} = \left(\begin{array}{cc} (\lambda x_2+\cdots)(\mu x_1+\cdots) & -(\mu x_1+\cdots)^2 \\ (\lambda x_2+\cdots)^2 & - (\lambda x_2+\cdots)(\mu x_1+\cdots) \end{array} \right)\mbox{.} $$ So the coefficient of $x_1^2 x_2$ in $2\tilde{H}_1$ is equal to both $\lambda\mu$ and $-2\mu^2$. Similarly, the coefficient of $x_1 x_2^2$ in $2\tilde{H}_2$ is equal to both $\lambda\mu$ and $-2\lambda^2$. It follows that either $\lambda = \mu = 0$ or $0\neq \lambda = -2\mu = 4\lambda$. In the former case, $\widetilde{H}$ satisfies (2). In the latter case, $\textup{char}K=3$ and $\lambda = \mu$. Replacing $\tilde{H}$ by $\lambda \tilde{H}\big(\lambda^{-1}(x_1,x_2),x_3,x_4,\ldots,x_n\big)$, we have that $\widetilde{H}$ satisfies (3). \end{proof}
\begin{theorem} \label{uporkle2} Suppose that $\operatorname{char} K\neq 2, 3$. Let $H \in K[x]^n$ be cubic homogeneous such that $x+H$ is a Keller map, i.e., ${\mathcal{J}} H$ is nilpotent. \begin{enumerate}[\upshape (i)]
\item If $\operatorname{rk} {\mathcal{J}} H = 1$, then there exists $T \in \operatorname{GL}_n(K)$ such that for $\tilde{H} := T^{-1} H(Tx)$, \begin{align*} \tilde{H}_1 &\in K[x_2,x_3,x_4,\ldots,x_n]\mbox{,} \\ \tilde{H}_2 &= \tilde{H}_3 = \tilde{H}_4 = \cdots = \tilde{H}_n = 0\mbox{.} \end{align*}
\item If $\operatorname{rk} {\mathcal{J}} H = 2$, then either $H$ is linearly triangularizable or there exists $T \in \operatorname{GL}_n(K)$ such that for $\tilde{H} := T^{-1} H(Tx)$, \begin{align*} \tilde{H}_1 &- (x_1x_3x_4-x_2x_4^2) \in K[x_3,x_4,\ldots,x_n]\mbox{,} \\ \tilde{H}_2 &- (x_1x_3^2-x_2x_3x_4) \in K[x_3,x_4,\ldots,x_n]\mbox{,} \\ \tilde{H}_3 &= \tilde{H}_4 = \cdots = \tilde{H}_n = 0\mbox{.} \end{align*} \end{enumerate} Furthermore, $x+tH$ is invertible over $K[t]$ if $\operatorname{rk} {\mathcal{J}} H\leq 2$, where $t$ is a new variable. Moreover, $x+tH$ is even tame over $K[t]$ if either $\operatorname{rk} {\mathcal{J}} H=1$ or $\operatorname{rk} {\mathcal{J}} H=2$ and $n\neq 4$. In particular, $x + \lambda H$ is invertible and tame under the above condition respectively for every $\lambda \in K$. \end{theorem}
\begin{proof} We may take $\tilde{H}$ as in (1), (2) or (3) of Theorem \ref{rkle2}. If $\operatorname{rk} {\mathcal{J}} H = 1$, then $\tilde{H}$ is as in (1) of Theorem \ref{rkle2}, i.e., $\tilde{H}_{i}= 0, 2\leq i\leq n$, whence (i) holds because $\operatorname{tr} {\mathcal{J}} \tilde{H} = 0$. So assume that $\operatorname{rk} {\mathcal{J}} H = 2$. Notice that ${\mathcal{J}} H$ is nilpotent.
If $\tilde{H}$ is as in (1) or (2) of Theorem \ref{rkle2}, i.e., $\tilde{H}_{i}= 0, 3\leq i\leq n$ or $\tilde{H} \in K[x_1,x_2]^n$, then ${\mathcal{J}}_{x_1,x_2}(\tilde{H}_1,\tilde{H}_2)$ is nilpotent.
If $\tilde{H}$ is as in (3) of Theorem \ref{rkle2}, i.e., $K \tilde{H}_1 + K \tilde{H}_2 + \cdots + K \tilde{H}_n = K x_3 x_1^2 \oplus K x_3 x_1 x_2 \oplus K x_3 x_2^2$, then $\tilde{H}_3 = 0$, because $x_3^{-1} \tilde{H}_3$ is the constant part with respect to $x_3$ of $\operatorname{tr} {\mathcal{J}} \tilde{H} = 0$. So ${\mathcal{J}}_{x_1,x_2}(\tilde{H}_1,\tilde{H}_2)$ is nilpotent in any case.
One may observe that, in all the cases (1), (2) and (3) of Theorem \ref{rkle2}, if ${\mathcal{J}}_{x_1,x_2}(\tilde{H}_1,\tilde{H}_2)$ is similar over $K$ to a triangular matrix, then ${\mathcal{J}} \tilde{H}$ is similar over $K$ to a triangular matrix, and so is ${\mathcal{J}} H$, and thus $H$ is linearly triangularizable.
Now suppose ${\mathcal{J}}_{x_1,x_2}(\tilde{H}_1,\tilde{H}_2)$ is not similar over $K$ to a triangular matrix. Noticing that $\textup{char}K\neq 2,3$, ${\mathcal{J}}_{x_1,x_2}(\tilde{H}_1,\tilde{H}_2)$ must be as in (2) of Lemma \ref{2x2nilp}, i.e., $$ {\mathcal{J}}_{x_1,x_2} \tilde{H} = \left( \begin{array}{cc} ab & -b^2 \\ a^2 & -ab \end{array} \right) ~~ \mbox{ and } ~~ {\mathcal{J}}_{x_1,x_2} \left( \begin{array}{c} a \\ b \end{array} \right) = \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right)\mbox{,} $$ where $a,b$ are linearly independent linear forms.
If $\tilde{H}_1 \in K[x_1,x_2,x_3]$, then $a,b\in k[x_3]$, a contradiction. So $\tilde{H}$ is not as in (2) or (3) of Theorem \ref{rkle2}, and thus is as in (1) of Theorem \ref{rkle2}, i.e., $\tilde{H}_3 = \tilde{H}_4 = \cdots = \tilde{H}_n = 0$. Consequently, by linear coordinate transformation, we may take $\tilde{H}$ such that $a = x_3$ and $b = x_4$. So (ii) holds.
For the last claim, when $\operatorname{rk} {\mathcal{J}} H=1$, $\widetilde{H}$ is of the form in (i), whence $x+t\widetilde{H}$ is elementary and thus tame. When $\operatorname{rk} {\mathcal{J}} H=2$, $\widetilde{H}$ is of the form in (ii), and it suffices to show the following automorphism $$ F=\big(x_1 + tx_4(x_3x_1-x_4x_2),x_2 + tx_3(x_3x_1-x_4x_2),x_3,x_4,x_5\big) $$ is tame over $K[t]$.
For that purpose, let $w=t(x_3x_1-x_4x_2)$ and let $D:=x_4\partial_{x_1}+x_3\partial_{x_2}$ be a derivation of $K[t][x_1,x_2,x_3,x_4]$. Observe that $D$ is triangular and $w\in \ker D$, and that $F=(\textup{exp} (wD), x_5).$ Therefore $F$ is tame over $K[t]$ due to the following Lemma \ref{smithlemma}. \end{proof}
Recall that a derivation $D$ of $K[x]$ is called locally nilpotent if for every $f\in K[x]$ there exists an $m$ such that $D^{m}(f)=0$. For such a derivation, $\textup{exp} D:=\sum_{i=0}^\infty \frac{1}{i!}D^i$ is a polynomial automorphism of $K[x]$. A derivation $D$ of $K[x]$ is called triangular if $D(x_i)\in K[x_{i+1},\ldots,x_n]$ for $i=1,2,\ldots,n-1$ and $D(x_n)\in K.$ A triangular derivation is locally nilpotent.
\begin{lemma}\label{smithlemma} Let $D$ be a triangular derivation of $K[t][x]$ and $w\in \ker D$ i.e. $D(w)=0$. Then $(\textup{exp}(wD),x_{n+1})$ is tame over $K[t]$. \end{lemma}
\begin{proof} From \cite[Corollary]{smith}, it follows that there exists a $k$ such that $(\exp(wD),\allowbreak x_{n+1},x_{n+2}, \ldots,x_{n+k})$ is tame over $K(t)$. Inspecting the proof of \cite[Corollary]{smith} yields that $(\exp(wD),x_{n+1})$ is tame over $K[t]$. \end{proof}
\paragraph{Acknowledgments} The first author has been supported by the Netherlands Organisation of Scientific research (NWO). The second author has been partially supported by the NSF of China (grant no. 11771176 and 11601146).
\end{document} | arXiv |
You are here: Home / IPA Recipients for January 2020
IPA Recipients for January 2020
Josefina T. Dizon
Institute for Governance and Rural Development
College of Public Affairs and Development
UP Los Baños
Water Governance Framework in Sta. Cruz River Watershed, Laguna, Philippines, Journal of Environmental Science and Management, 22(1): 54-66, 2019
One of the farmers drying palay on the road
Deforestation due to illegal cutting, charcoal making, quarrying, mining, and destructive farming practices resulted to insufficient water supply for irrigation in the Sta. Cruz River Watershed. Since the National Irrigation Administration has transferred to the Irrigators' Association (IA) the ownership of the irrigation system, the model proposes that the IA plays a central role in the management of the irrigation system. Thus, its leadership, capability, participation and its existing water policies are important inputs into the governance framework. Leadership is important because it is an administrative function of the IA to influence its membership towards achieving their goal. Capacity building is the process of changing attitudes and behaviors and imparting knowledge and skills. Participation, meanwhile, which is the process whereby people find ways to meet collective needs and overcome common problems to attain efficiency, effectiveness, self-reliance, coverage and sustainability, is important in addressing the hindering factors to attain sustainability of irrigation water. In the system, however, a number of factors impinge upon the IA and these include the economics (irrigation efficiency and economic efficiency), social (water allocation), political (accountability and transparency) and administrative systems (regulations, water resources management and distribution, and decision-making). Thus, the model espouses that to achieve good governance which is imperative to achieve water security, and finally rice security, these different factors should be considered. The model below summarizes the proposed governance framework.
Significance:
Through irrigation, agriculture withdraws water accounting for 70% of water withdrawals and uses 90% of the water consumption in the world (FAO 2011) and 95% in developing countries (Podimata and Yannopoulos 2015). Since the country's major water user is the agriculture sector, there is a need to develop a framework for water governance since it involves many institutions and it is quite complex. The Global Water Partnership defines water governance as "the range of political, social, economic and administrative systems that are in place to develop and manage water resources, and the delivery of water services at different levels of society". However, Birongo and Le (2005) and Rogers and Hall (2003) stressed that there is no single model of effective water governance. The challenges confronting water governance requires a holistic approach of determining the different factors affecting its achievement. This research identified the hindering factors in the sustainability of irrigation water and determined the interaction between the different actors to develop a water governance framework using the case of the Sta. Cruz River Watershed. This research addresses the concerns of water scholars and policy makers regarding the need to develop a water governance model.
Link to the article: https://ovcre.uplb.edu.ph/journals-uplb/index.php/JESAM/article/view/86
Impact factor:(2018/2019) 0.266
Hazel Joyce M. Ramirez
Rural High School
Co-creating Scripts in Computer-supported Collaborative Learning and its Effects on Students' Logical Thinking in Earth Science, Interactive Learning Environments, https://doi.org/10.1080/10494820.2019.1702063, 2019
Computer-supported collaborative learning (CSCL) is a technology-driven inquiry-based approach that encourages social interaction and shared knowledge construction in completing computer-aided tasks. Although there were researches carried out on CSCL, no research to date has extensively examined how CSCL enhanced with scripts containing student-generated questions that can facilitate the development of logical thinking, an essential skill to effectively comprehend science concepts. In this light, this research examined the effects of co-creating scripts in CSCL on students' logical thinking. This utilized a three-group pretest- posttest quasi-experimental design with two delayed posttests that involved Grade 7 students. One group was exposed to CSCL with scripting while the other group was exposed to CSCL without scripting. On the other hand, the control group was exposed to a conventional teaching approach. Findings revealed that CSCL approaches significantly improved students' logical thinking, F(2, 113) = 5.616, p = .0025. Further, the delayed posttests consistently showed that CSCL with scripting significantly influenced the development of logical thinking. Notably, this research builds and extends on previous researches regarding the synergistic scaffolding of the inquiry-based approach, technology integration, collaborative learning, and question-asking activity. This innovation catalyzed learning and offered essential implications that provided opportunities for future research directions.
The research offers a synergistic approach that merges contextualized learning material and 21st century learning competencies. The research evaluated the effectiveness of the use of a DOST science courseware in a computer-supported collaborative learning (CSCL) approach through scripting activity. Its findings are useful on a national scale since the courseware is being used by numerous schools in the Philippines. The research provides valuable information which is not only beneficial towards its improvement, but also on how technology and collaborative inquiry can be effectively integrated towards a transformative learning process. Furthermore, the study pioneered research on DOST science courseware since all existing researches have focused on math courseware. Additionally, this research contributes to extensively examine how CSCL can be reinforced by student-generated questions to enhance logical thinking which is a necessary skill to comprehend and apply science concepts.
Link to the article: https://doi.org/10.1080/10494820.2019.1702063
Reginald Christian S. Bernardo1and Michael Francis Ian S. Vega II1
1National Institute of Physics, College of Science (UP Diliman)
Tailoring Cosmologies in Cubic Shift-symmetric Horndeski Gravity, Journal of Cosmology and Astroparticle Physics, doi: 10.1088/1475-7516/2019/10/058, 2019
Figure 1: Scale factor, a(t), predicted by general relativity with a radiation-baryon-vacuum fluid mixture. After the hot big bang at t = 0, the universe has been through the radiation and matter eras. Today, marked t = 1, we live in a universe dominated by an exotic fluid "dark energy" with negative pressure, thus, causing the cosmic expansion to accelerate.
Figure 2: The dark energy equation of state, wφ, as predicted by cubic Horndeski theory with the same expansion history as ΛCDM but sourced only by a single cosmic fluid (radiation, w = 1/3; baryon, w = 0; vacuum, w = – 1). Cosmological data pin down the dark energy equation of state, wφ, close to -1 today (t = 1).
The standard model of cosmology, known as the "ΛCDM model" is the best existing theory of the observable Universe. It beautifully supports the observed late-time cosmic inflation and the formation of galaxies and makes the most exciting predictions such as gravitational waves generated during the hot big bang. However, ΛCDM model brings with it the inevitable conclusion that ninety-five percent of the Universe is made up of invisible exotic fluid. Also, the observed cosmological constant Λ, ΛCDM model's stand in explaining "dark energy", clashes with its theoretically predicted quantum-fluctuations value by a colossal fifty orders of magnitude. This cosmological puzzle, among others, makes room for the enthusiastic set of contenders, generically called "alternative theories of gravity", to challenge the general theory of relativity on which ΛCDM model is strongly anchored. In the paper "Tailoring cosmologies in cubic shift-symmetric Horndeski gravity" we show how to link a dynamical universe's expansion to alternative theories known as cubic Horndeski gravity. This cosmology-designing algorithm tunes away dark energy by relieving its connection with the cosmological constant. Most importantly, this work can be used to narrow down the space of viable alternative theories of gravity to cubic Horndeski theory provided expansion history data.
"Dark energy", the invisible exotic fluid which permeates most of the Universe, can be explained by introducing the cosmological constant Λ into the Einstein field equations of general relativity. This approach, however, glances over the colossal fifty orders of magnitude-disagreement between the observed cosmological constant and its theoretically predicted quantum-fluctuations value. Scalar-tensor theories relieve this tension by associating dark energy with a gravitational scalar field. In this work, we have explicitly shown how to link expansion histories, such as the dark energy-sourced inflation, to scalar-tensor theories known as cubic Horndeski gravity. This result can be used to pin down the effective field theory of gravity to a unique cubic Horndeski theory provided expansion history data.
Link to the article: https://iopscience.iop.org/article/10.1088/1475-7516/2019/10/058
Impact factor: (2018/2019) 5.524
Rosalie C. Mendoza1, Vivian C. Daracan1, Ronniel D. Manalo1, Chelle Hennessy R. Batallones1, Arlene D. Romano1 and Willie P. Abasolo1
1Department of Forest Products and Paper Science, College of Forestry and Natural Resources (UP Los Baños)
Anatomical and Physico-mechanical Characterization of Narra (Pterocarpus indicus Willd.) Branchwood Collected in Mount Makiling Forest Reserve, Laguna, Philippines, Philippine Journal of Science, 148(4): 705-713, 2019
Fig. 1. Cross-section of narra branchwood.
Fig. 2. Isolated fibers of narra branchwood.
Narra wood is one of the best raw materials to support the wood-based industries. This species is known for its numerous uses such as fine furniture, cabinets, cartwheels, musical instruments, and other specialty items. Since the demand for log production grows as the population proliferates, coupled with the scarcity of wood brought about by the imposition of logging ban policies including EO 23, the Philippines has started to address the issue on developing a sustainable source of wood raw materials. Utilization of branch wood of High Value Forest Crops (HVFC) could be one of the potential sources of wood raw materials and could be used as an alternative to stem wood. Narra (Pterocarpus indicus Willd.) is one of the high value forest crops and most commercially important timber species in the Philippines.
The anatomical properties and physico-mechanical properties of narra branchwood collected from Mount Makiling Forest Reserve (MMFR), Los Baños, Laguna, Philippines were studied. These properties could provide information that could be used as indicators of wood quality and utilization. These branchwood properties were also compared with narra stemwood's experimental and published properties. Results showed that narra branchwood exhibits similar anatomical and physico-mechanical features to narra stemwood. Thus, narra branchwood may be used as a substitute for narra stemwood in various uses such as for high-grade furniture and cabinetry, musical instruments, pulp and paper, production of novelty items, and wood parquet.
Narra (Pterocarpus indicus Willd.) is one of the high-value forest crops (HVFC) and commercially-important timber species in the country. The high demand for its stem wood is continuously increasing however its supply declines due to the scarcity of wood raw materials imposed by logging ban policies. Hence, the utilization of Narra branchwood could be one of the potential sources of wood raw materials and could be used as an alternative to its stem wood. The study of the anatomical and physico-mechanical properties of Narra branchwood could provide baseline information that can be used as indicators of wood quality and utilization.
Link to the article: http://philjournalsci.dost.gov.ph/publication/regular-issues/current-issue/95-vol-148-no-4-december-2019/1143-anatomical-and-physico-mechanical-characterization-of-narra-pterocarpus-indicus-willd-branchwood-collected-in-mount-makiling-forest-reserve-laguna-philippines
Impact factor: Not yet available
Ian Jasper A. Agulo
Department of Physical Sciences
UP Baguio
High Concentration Bolometric System with Single-walled Carbon Nanotubes (SWCNT) Absorber, Nanotechnology, 31: 125202, doi: 10.1088/1361-6528/ab5dd4, 2020
Schematic diagram of the carbon nanotube bolometer. The components are the electrode (yellow) and the carbon nanotube film (black circle). The geometry of the electrode is specifically designed to focus the incident radiation to the carbon nanotube film in order to enhance the thermal and optical properties of the device.
We demonstrate that a planar single-walled carbon nanotube (SWCNT) film bolometer can exhibit enhanced thermal and optical properties. The SWCNT film were ink-printed on an oxidized silicon substrate between two pointed-tip Au electrodes across a gap of approximately 10 μm. We obtained a bolometer figure-of-merit temperature coefficient of resistance of greater than –3.0% at room temperature. An optical response of 1000 V W−1 was obtained from a 786 nm laser with an output power of 5 mW. The corresponding thermal time constant of 1.8 ms was estimated through the optical response by modulating the laser over a frequency range of 1 Hz–1 kHz. The optical noise equivalent power and optical detectivity of $4.5\times {10}^{-11}\,{\rm{W}}/\sqrt{{\rm{Hz}}}$ and $4.9\times {10}^{8}\,{\rm{cm}}\,\sqrt{{\rm{Hz}}}\,{{\rm{W}}}^{-1},$ respectively, were estimated from the responsivity, the spectral density, and area of the cell of the absorber, 4.9 × 10−4 cm2. We attribute the exceptional performance of the SWCNT microbolometer to the film nature of the absorber and to the high concentration of the incident electromagnetic radiation and localized heating between the tips of the electrode.
In this work, we developed a very sensitive infrared sensor. The sensing element is made from a carbon nanotube (CNT) film. The film has enhanced thermal and optical properties, better than the currently commercially available vanadium oxide. The enhancement is attributed to the electrode geometry, focusing the incident radiation directly to the CNT film.
Link to the article: https://iopscience.iop.org/article/10.1088/1361-6528/ab5dd4/pdf
Highly Efficient Photocatalysis by Zinc Oxide-Reduced Graphene Oxide (ZnO-rGO) Composite Synthesized via One-Pot Room-Temperature Chemical Deposition Method, Journal of Nanotechnology, https://doi.org/10.1155/2019/1895043, 2019
Process of photocatalysis by zinc oxide and reduced graphene oxide (rGO/graphite) composites. UV light provides the energy needed to create electrons in the zinc oxide. These electrons are transferred to the rGO. The remaining holes in the zinc oxide degrade the water, whose by product degrade methylene blue.
We synthesized zinc oxide-reduced graphene oxide (ZnO-rGO) composites using a one-pot chemical deposition method at room temperature. Zinc powder and graphene oxide (GO) of different mass ratios (1 : 1, 1 : 2, 1 : 5, 1 : 10, and 1 : 20 GO to Zn) were used as precursors in a mildly alkaline solution. UV-Vis spectroscopy was used to study the photocatalytic efficiency of the samples through the photodegradation of methylene blue (MB). UV-Vis measurements show the fast decomposition of methylene blue under UV light illumination with the best degradation efficiency of 97.7% within one hour, achieved with sample ZG2 (1 GO : 2 Zn mass ratio). The corresponding degradation rate was kZG2 = 0.1253 min−1, which is at least 5.5 times better than other existing works using hydrothermal methods. We argue that the excellent photodegradation of MB by ZG2 is due to the efficient charge separation brought about by the electronic interaction of the rGO with the ZnO and the formation of a Zn-O-C bond, as supported by XRD and Raman spectroscopy measurements.
In this work, we have found a new composite material that is able to degrade a specific pollutant, called methylene blue, with almost 100% efficiency within one hour upon illumination by ultraviolet light. The novel material is graphene-based and is called zinc oxide-reduced graphene oxide (ZnO-rGO). The excellent efficiency is due to the bonding formed between zinc and carbon through the oxygen atom.
Link to the article: https://www.hindawi.com/journals/jnt/2019/1895043/
Marisol P. Martinez1, Geleena A. Gestiada1, Allen L. Nazareno1, Ranzivelle Marianne Roxas-Villanueva1 and Marie Joy F. Lopez2
1Institute of Mathematical Sciences and Physics, College of Arts and Sciences (UP Los Baños)
2Institute of Statistics, College of Arts and Sciences (UP Los Baños)
Network Approach on Characterizing Floral Diversity in the Agroforestry Zone of Mount Makiling Forest Reserve, Philippines, Journal of Physics: Conference Series, 1245 (1): 012033, 2019
Complex networks have been used to characterize real world systems. The network structure may signify important relationships which may not be evident in other methods of analysis. In this study, we characterize the floral diversity in three study sites in the agroforestry zone of Mount Makiling Forest Reserve using network analysis. Plant species found in each study site are considered as nodes (N). Edges (E) are established to connect species with the same alternate role and habit. The dataset includes N = 157 and E = 4279 for Bagong Silang site, N = 145 and E = 3740 for the Karay site, and N= 122 and E = 2429 for the Magnetic Hill site. Network parameters such as degree, path length, clustering coefficient, modularity and number of connected components were calculated. Obtained values were compared to published diversity index. Results show that lower clustering coefficient and higher average path length signify higher diversity. A higher number of disconnected components also indicates diversity.
In this study, we used network analysis to quantify the floral diversity of three sites in the agroforestry zone of Mount Makiling Forest Reserve, namely, Bagong Silang, Sitio Karay, and Magnetic Hill. Calculated network parameters for the three sites were compared to their corresponding published diversity indices. It was found out that low number of disconnected components characterizes low diversity index. For networks with the same number of disconnected components, high degree, high clustering coefficient, low path length, and high modularity corresponds to low diversity index. For further study, other network parameters can be investigated. Network analysis can be used to study the diversity of more ecological sites to somehow generalize the trend of the network parameters against the diversity indices.
Link to the article: https://iopscience.iop.org/article/10.1088/1742-6596/1245/1/012033
Allen L. Nazareno1, Geleena A. Gestiada1, Marisol P. Martinez1, Ranzivelle Marianne Roxas-Villanueva1 and Marie Joy F. Lopez2
An Artificial Neural Network Approach in Predicting Career Strand of Incoming Senior High School Students, Journal of Physics: Conference Series, 1245(1): https://doi.org/10.1088/1742-6596/1245/1/012005, 2019
The basic architecture of ANN
The K to 12 program has been implemented in the Philippines by the Department of Education which implicated an additional two years in the students' basic education. These ancillary years allow senior high school students to take courses under the core curriculum and the track of choice. Each student must select one track to pursue that can equip him/her with skills to prepare for the future. Prediction of choice of a career track in senior high school is advantageous for educational institutions since it gives insights that can help them develop vital programs beneficial for students' learning in school. In this study, we applied an artificial neural network (ANN) to predict the career strand based on the students' grades in five major subjects. Different ANN models have been considered and compared. In training and testing the models, a sample of 293 student data information was used. The highest accuracy recorded among all the models was 74.1 %.
Determining the career track that a student may want to pursue in SHS is favorable to the school administrators, teachers, and students. With this information, it is possible to project future enrollees in a particular track. Further, it can help DepEd and other private education institutions in managing resources and in executing their plans and policies. The study can also help in identifying important factors relating to career track decision which can guide teachers in developing instructional materials and pedagogical strategies that might truly help students to perform well in their chosen track.
Ambrocio Melvin A. Matias
Institute of Biology
Asymmetric Dispersal is a Critical Element of Concordance Between Biophysical Dispersal Models and Spatial Genetic Structure in Great Barrier Reef Corals, Diversity and Distributions, 25: 1684-1696, 2019
Figure 1: Sampling locations and main attributes of biophysical model. A) Coastal Queensland and the Great Barrier Reef where bathymetry is shown by grey shading and 120 m depth reflects the approximate land mass exposure at lowest Pleistocene sea level stands. Designated regions correspond to management areas. Major offshore currents are shown (NVJ and SVJ: north and south Vanuatu jets, NQC: north Queensland current, EAC: east Australia current; modified from Coukroun et al. 2010; Mao & Luick 2014). B & C) Summary of top 50 percentile predicted connections based on relative path probabilities for Acropora tenuis (B) and Acropora millepora (C). Sampling locations are color coded by latitude with northern low latitude sites shown in reds (warm) and southern higher latitude sites shown in blues (cool). Vectors show predicted dispersal probabilities with thicker lines indicating higher probabilities and colored by source population.
Understanding the connections between coral populations is critical in developing management plans in preserving them. These connections specifically help us determine where corals in a population come from and where do their larvae go. While knowledge on these connections is valuable in management, determining them is not a trivial task. Here, we investigated the connections between populations of two Acropora species across the Great Barrier Reef (GBR), which is presently being threatened by events such mass bleaching. In particular, we employed biophysical modelling to determine likely connections between populations. This model takes into account not only the oceanographic currents, but also the biological attributes of the coral larvae, which are known to greatly influence their dispersal capacity. Results of biophysical modelling predicted a prevalence of north to south connections (southward dispersal). We then validated these results against genetic data (gene flow) of the two species, which is the evolutionary outcome of these connections. However, because genetic diversity is possibly influenced by other historical events, we first showed, through demographic modelling, that the genetic diversity observed in the two species is due to stepping-stone connections and not because of past divergence between the northern and southern populations of corals. With this affirmation, we next examined the alignment of biophysical model and genetic data. Here, we showed that the biophysical model significantly predicts the genetic data, especially when asymmetric dispersal is considered. Overall, these results suggest a considerable local recruitment and lack of long‐distance gene flow from south to north of GBR.
Coral reef ecosystems, particularly foundational coral species, are presently threatened by various factors such as coral bleaching and crown-of-thorns outbreaks. Recovery from these threats can be facilitated by dispersal of larval propagules, which supply new settlers to degraded reefs. Because of this role of larval dispersal to coral reefs' recovery, an understanding of mechanism underlying dispersal can be key in devising strategies in alleviating the threats to coral reefs and/or to promote reef recovery. However, documenting and predicting spatial connections resulting from larval dispersal in marine species remains a difficult task. For example, tagging and recovery of planktonic larvae is not trivial, especially when considering the spatial extent of coral reef ecosystem. Similarly, parentage analysis will be difficult because of the number of parents and new settlers needed to be sampled to gain an overview of the extent of dispersal. Whereas biophysical models present possible spatial connections between reefs, their results need to be validated with other empirical data before using them for practical applications such as management. In this work, we examined how well biophysical model can predict the genetic diversity of the coral Acropora tenuis and A. millepora across the Great Barier Reef, while taking into the possible asymmetry in dispersal and historical factors that can influence the genetic structuring in these species.
Link to the article: https://onlinelibrary.wiley.com/doi/10.1111/ddi.12969
Marjorie D. delos Angeles1, Inocencio E. Buot Jr.1, Virginia C. Cuevas1 and Pearl B. Sanchez2
1Institute of Biological Sciences, College of Arts and Sciences (UP Los Baños)
2Agricultural Systems Institute, College of Agriculture and Food Science (UP Los Baños)
Phytoremediating Capacity of Copper Tolerant Plants in Mine Tailing Soil Materials with Compost Amendment in Mankayan Benguet, Philippines, EnvironmentAsia 13(1): 86-98, 2020
Figure 1. A) Tailing pond 4 (TP4) and B) Tailing pond 3 (TP3) containing copper-contaminated soil and municipal wastes.
Figure 2. Changes in f A) Final bulk density, B) Mean final CEC and C) Mean final pH in treatments where materials from TP3 was used as growth medium.
Fifteen species representing 12 genera and 8 families were potential phytoremediators. The most dominant species different growth media with highly variable soil physical and chemical characteristics, was Cynodon dactylon (L.) Pers. This plant can be utilized as a phytostabilizing agent in different media types contaminated with copper. The addition of 4% and 8% compost application increased cover to more than double from 18.33% of 0% compost application to 47.33% at 8% application by the end of the experiment. Increased application of compost resulted to the formation of soluble copper complexes which are available for plant uptake.
The agricultural soil had an initial acidic pH of 4.9 and the soil materials from TP4 had an initial acidic pH of 5. Materials from TP3 had a strongly acidic pH of 2.5 and had the densest media with a bulk density of 1.42Mg m-3 and the lowest CEC with 3.79 cmol kg-1. TP3 has a very harsh environment making it difficult for plants to thrive in. This study also verified that addition of organic matter in phytoremediation illustrated a significant increase in pH. This study suggests exploring different plant species that are capable of tolerating copper-contaminated sites in the area. There will be a faster reclamation of these copper-contaminated sites once additional plant species, as well as its phytoremediating mechanism are determined.
This study identified fifteen species representing 12 genera and 8 families that were potential phytoremediators. These plant species were collected from copper-contaminated soils in abandoned tailings ponds in barangay Paco, Mankayan, Benguet, Philippines. It also examined and verified the role of organic matter in phytoremediation by utilizing growth media with varying concentrations of copper. Agricultural soil, tailings pond overlaid with tops soil and municipal biowaste (TP4), and abandoned tailings pond without rehabilitation (TP3) were amended with 4%, 8%, and 16% compost. This study verified that addition of organic matter in phytoremediation illustrated a significant increase in pH. This study suggests to explore different plant species that are capable of tolerating copper-contaminated sites in the area. There will be a faster reclamation of these copper-contaminated sites once additional plant species as well as its phytoremediating mechanism are determined.
Link to the article: http://www.tshe.org/ea/pdf/EA11(2)_08.pdf
Katrina Hannah D. Ignacio1, Marjorie Anne C. Bagnas1, Adrian I. Espiritu1 and Jose Paciano Baltazar T. Reyes1
1Department of Neurosciences, College of Medicine (UP Manila)
Secondary Hypokalemic Paralysis with Bulbar Weakness and Reversible Electrophysiologic Abnormalities: A Case Report and Systematic Review, Journal of Clinical Neuroscience, 70: 254-257, 2019
Figure 1. PRIMSA Flow Diagram
Hypokalemic periodic paralysis is a condition that presents with generalized weakness, at times to a severe degree. The exact mechanism by which this happens is not yet clear. It is rare for nerve conduction studies (studies that assess the integrity of nerves) to be done in this condition. This article reports a rare case of hypokalemic paralysis that presented with symptoms not typical of the condition. It also presents results of nerve conduction studies done that mimicked a nerve disease called Guillain Barre Syndrome. The study also summarized reports from the literature on cases similar to the one presented.
Secondary hypokalemic periodic paralysis from distal renal tubular acidosis is a rare cause of weakness. Nerve conduction abnormalities in this condition are seldom documented. We report a case presenting as acute flaccid quadriplegia associated with bulbar symptoms in which serial nerve conduction studies at various potassium levels were conducted. A systematic review was also conducted to make clinicians aware of the clinical presentation and electrophysiologic findings in hypokalemic paralysis. The article also stresses the importance of early recognition and appropriate management of HPP is important so that potassium replacement can be instituted.
Link to the article: https://www.sciencedirect.com/science/article/pii/S0967586819312354
Maria Patricia V. Azanza1, Rowena Grace R. Sanchez1, Una Grace M. Dollete1 and Bernard Niño Q. Membrere1
1Department of Food Science and Nutrition, College of Home Economics (UP Diliman)
Foodborne Disease Outbreaks in the Philippines (2005-2018), Philippine Journal of Science, 148 (2): 317-336, 2019
The study detailed reported Philippine foodborne disease outbreaks (FBDOs) for the period 2005 – June 2018 based on secondary data from web portals, electronic archives of local news agencies, electronic archives of local newspapers, government websites, and printed reports from the Epidemiology Bureau of the Philippine Department of Health (DOH). Data were categorized and evaluated in terms of associated food vehicles, etiological agents, venues of outbreaks, and morbidity and mortality cases. The specific food vehicle was unidentified in most FBDO cases in this study, however, for those that were identified, it was found that meat-containing dishes were the most common causative foods in the evaluated outbreaks. Food service eating facilities (such as canteens) and households were found more prone to outbreak occurrences. Although there were reported outbreaks with unidentified causative agents, Salmonella spp., Henipavirus, Entamoeba histolytica, and Vibrio parahaemolyticus were cited as primary causes of infections. On the other hand, staphylococcal enterotoxins, carbamate toxin, and paralytic shellfish poisoning (PSP) toxin were involved in human intoxications. Evidently, the Philippines has room for improvement in terms of monitoring, reporting, and laboratory testing for foodborne disease outbreaks.
This research presents a profile of the foodborne disease outbreaks in the Philippines for a period of 13 years. It provides data presented in different forms (such as classification of the food vehicles and location of outbreaks) as opposed to the data presented in the official national reports which combine the outbreaks from water and food. Regulatory health officials and food industry personnel can use the present information from the study for understanding the causes of Philippines FBDOs.
Link to the article: http://philjournalsci.dost.gov.ph/87-current-issue/vol-148-no-2-june-2019/1020-foodborne-disease-outbreaks-in-the-philippines-2005-2018
Rosalie C. Mendoza1, Ramon A. Razal1, Willie P. Abasolo1, Roberto G. Visco2 and Canesio D. Predo2
1Department of Forest Products and Paper Science, College of Forestry and Natural Resources
and Development (UP Los Baños)
2Institute of Renewable Natural Resources, College of Forestry and Natural Resources
Aboveground Biomass Characterization of a Young Kawayan Tinik Plantation (Bambusa blumeana J.A. & J.H. Schultes) in Nueva Ecija, Philippines for Bioenergy Production, Philippine Journal of Science, 148(4): 627-636, 2019
Kawayan tinik holds promise as an alternative biomass crop not only in the Philippines, but also in other countries where this species grows. Currently, kawayan tinik is being used as biofuels in the form of charcoal, briquettes or pellets. However, most of the raw materials come from mature kawayan tinik culms. This study assessed the potential of young kawayan tinik plantation development for energy production. Yearly determination of biomass and culm growth of kawayan tinik from branch cuttings was done for a young bamboo plantation established at Fort Magsaysay Military Reservation in Nueva Ecija. In general, results suggest that the kawayan tinik plantation studied is a potential source of raw materials for bioenergy production, provided that proper monitoring will be conducted and silvicultural treatments that are likely to increase the productivity of the clumps and improve its properties and suitability for bioenergy production will be done. Moreover, full documentation of these silvicultural treatments should be done by the plantation managers for easier tracking of what treatments are favoring the growth of the plantation.
This study can help promote kawayan tinik plantations as bioenergy sources. Kawayan tinik, being a "multi-purpose" bamboo species, would require a strategic plantation development and harvesting scheme. Studies looking into the properties of kawayan tinik as affected by clump age and information on the potential performance as an energy crop are still lacking. In order to more efficiently use kawayan tinik as raw material for energy production efficiently, it is essential to carry out a detailed characterization of its biomass.
Link to the article: http://philjournalsci.dost.gov.ph/publication/regular-issues/current-issue/95-vol-148-no-4-december-2019/1126-aboveground-biomass-characterization-of-a-young-kawayan-tinik-plantation-bambusa-blumeana-j-a-j-h-schultes-in-nueva-ecija-philippines-for-bioenergy-production
Bing Baltazar C. Brillo
Status, Governance and Development of Gunao Lake: The Little-Known Lake of Dolores, Quezon, Philippines, Asian Journal of Water, Environment and Pollution, 17(1): 27-33, 2020
Anchored on the scarcity of small lake studies in the country, the article explores the little-known small lake of Dolores, Quezon, Philippines— Gunao Lake; its existence virtually unheard of in scholarly literature. Specifically, the study documents the crater lake by delineating the status of the lake using seven basic governance and development parameters. Using data from interviews, site observations, and few existing documents on the lake, the study contends that Gunao Lake is deficient in key management and conservation enablers: the absence of a management council and a Master Development Plan, and the failure to institutionalise tourism, funding mechanisms and maintenance activities. These failings are attributable to the local government's limited funds and the failure to tie-up the small lake's development within its centerpiece tourism project in mount Banahaw area. In placing Gunao Lake on the literature map, overall, this study makes a small step in expanding the governance and development studies on small lakes in the country.
The assessment delineated the governance and development status of Gunao Lake as well as identify the areas of deficiency and their salient factors. The criteria utilised are the essential enablers of governance, development and conservation of a small lake, and improving on them is crucial if meaningful change is to be gained in Gunao Lake. The findings showed the areas for improvement: organising a management council, formulating an MDP, promoting tourism, establishing finance mechanisms, and institutionalising maintenance activities. Tying all together, it elucidates the failure of the Municipal Government to take serious action in Gunao Lake. This nonfeasance points to the little interest of the local government on the small lake due mainly to limited funds and failure to integrate its development within the Municipality's centerpiece tourism project in mount Banahaw area. As mentioned, the tourism funds and efforts of the Municipal Government are primarily dedicated to the improvement of the Mount Banahaw area. As a consequence, Gunao Lake has been perennially placed outside the priority of the administrative agency, and thus, failing to realise the development potentials and sustainability requirements the inland water resource. Conversely, this underscores the local government's commitment as critical to gain grounds in the management, conservation and development of Gunao Lake.
In closing, the article literally placed Gunao Lake on the map of scholarly literature by providing baseline-steering information about the inland water resource; particularly, by illustrating the governance-development circumstances and experiences of a little known small lake in the country. This undertaking is timely and consequential— considering the threatened condition of lakes in the country (see Aralar et al. 2005, Fernandez 2011, Aralar et al. 2013, GNF 2014) and the now acknowledged abundance (see Lehner and Doll 2004, Downing et al. 2006, Oertli et al. 2009, Brillo 2015a) and ecological significance of small lakes (see Kelly et al. 2001, Smith et al., 2002, Scheffer et al. 2006, Hanson et al. 2007, Downing 2010). Under this context, the study hopes to instigate more studies on Gunao Lake, in particular, and small lakes, in general, as they are numerous in the country (and in the world).
Link to the article: https://www.iospress.nl/journal/asian-journal-of-water-environment-and-pollution/
Bing Baltazar C. Brillo1, Rolando T. Bello1and Evelie P. Serrano1
1Institute for Governance and Rural Development, College of Public Affairs and Development (UP Los Baños)
The Administrative Performance of the Laguna Lake Development Authority on the Small Lakes of the Laguna de Bay Region, Philippines, Asia-Pacific Social Science Review, 19(4): 29-43, 2019
Few studies have dealt with small lakes in the Philippines, particularly aspects of their governance, which translate to information deficit on the status of administration of many lakes in the country. At the core of governance in any lake is its administrative agency, and in the eight crater lakes of San Pablo City, it is the Laguna Lake Development Authority (LLDA). Under this context, this article examines the administrative performance of the LLDA on the eight small lakes (i.e., Sampaloc Lake, Bunot Lake, Palakpakin Lake, Calibato Lake, Mohicap Lake, Pandin Lake, Yambo Lake, and Tadlac Lake) of the Laguna de Bay Region. The study evaluates the agency using four criteria deemed fundamental in managing, conserving, and developing small lakes: (1) having an approved management and development plan (MDP); (2) regulating fish pens and cages; (3) implementing the shoreline easement; and (4) conducting maintenance activities. Using data from interviews, site observations, documents, reports, and other secondary sources, the study contends that the LLDA's performance is ambivalent because its management of the small lakes can be characterized as slow and lacking in follow-through in the MDP issue; unsatisfactory in the regulation of aquastructure and shoreline easement; but decent in water quality monitoring, clean-up operations, and fingerlings dispersal. On the whole, the assessment exemplifies the inconsistent actions of the LLDA and underscores the long-term commitment and accountability of the agency in governing the crater lakes.
Properly managing the small lakes in the Philippines is important because the inland water resources are abundant in the country, and many of them are a potential catalyst for the development of the surrounding impoverished communities. This rationale is aligned with the concept of development as an improvement that is shared and sustainable (Global Monitoring Report, 2015), which in small lake development simply means improving the situation of the local people (making development inclusive) and ensuring the conservation of the water resource (making development sustainable).
Small lakes (together with major lakes in the country) are contemporarily ecologically threatened. The First National Congress on Philippine Lakes held in 2003 and the Second National Congress on Philippine Lakes held in 2011 have recognized that many lakes in the country, despite incremental improvements, remain at risk of environmental degradation due mainly to indiscriminate utilization and increasing demands of economic growth (Aralar et al., 2005; Fernandez, 2011; Aralar et al., 2013; Global Nature Fund, 2014). Against this backdrop, studies on Philippine lakes have been incrementally increased over the years. However, the concentration of scholarly works is on the abiotic and biotic features of the major lakes in the country (Brillo, 2015a; see also Guerrero, 2001, 2005). Presently, few studies have dealt with small lakes, particularly on the aspects of their governance (see International Lake Environment Committee, 2005; Downing, 2010; United Nations Development Programme-Water Governance Facility, 2015; Brillo, 2015a, 2017a). This reality translates to information deficit in small lakes as well as in the status of their administration. Governance is fundamental because enforcing the key regulations and implementing the many scientific findings on lakes are contingent in it (see Nowlan & Bakker, 2007; Simms & de Loë, 2010; Melnychuk, Murray, & de Loë, 2012). At the core of governance in any lake is its administrative agency, and in the eight small lakes, it is the LLDA.
Predicated on the preceding discussions, this study addresses the literature deficit by examining the LLDA's performance in managing, conserving, and developing the eight crater lakes under its jurisdiction. Consistent with the gap in the literature, governance and administration studies on the eight crater lakes are deficient, as the great majority of existing scholarly works are under limnology and aquaculture aspects (see Brillo, 2015b, 2015c, 2016a, 2016b, 2016c, 2016d, 2017b). Overall, the study argues that the LLDA's performance is ambivalent because its administration of the eight small lakes can be characterized as slow and lacking in follow-through in the management and development plan (MDP) issue; unsatisfactory in regulating aquastructure and shoreline easement; but satisfactorily in water quality monitoring, clean-up operations, and fingerlings dispersal. Moreover, the findings illustrate the lack of consistency in the actions of the LLDA and underscore two key features— the long-term commitment and accountability of the agency in governing the eight crater lakes.
Link to the article: https://apssr.com/publications/
Bing Baltazar C. Brillo1, Hadji C. Jalotjot2, Agnes C. Rola1
1Institute for Governance and Rural Development, College of Public Affairs
2Center for Strategic Planning and Policy Studies, College of Public Affairs
Impact on Income and Livelihood of Fisheries Workers: Closed Fishing Season Policy for Sardines in Zamboanga Peninsula, Philippines, Journal of Coastal Conservation, 23(6): 1057-1067, 2019
The closed fishing season policy for sardines in the Zamboanga Peninsula is intended to conserve the sardine species and sustain long-term operations for the sardine industry in the region. As the fishing regulation entails work suspension, it could inevitably pose serious repercussion on the fisheries workers who are highly dependent on sardine production for livelihood. This study assesses the impact of the three-month fishing ban on the income and livelihood of the fisheries workers. Utilizing a survey research design, the article shows that income loss as the first and immediate consequence following the implementation of the policy in 2011. This is consequential to the economic well-being of the affected fisheries workers, especially to those among low-income households. While income loss seems to cast a shadow over the favourability of the fishing regulation, the fisheries workers still managed to alleviate the impact by obtaining a replacement job or substitute livelihood; being rehired by the canning factories and bottling companies after the fishing ban; having multiple sources of income, and staying in the workforce for scaled-down operations in the sardine processing companies. These factors mitigate income loss, translate to a guaranteed re-employment, and offer some security to the fisheries workers and their households during the closed fishing season.
A conservation area in the coastal waters around Zamboanga Peninsula was established in 2011. Under the policy, there will be a closed fishing season for sardines and other related species for three months within a 12-month. The policy limited access to the livelihood of various stakeholders. Serious concern was raised on the possible effect on the affected stakeholders especially those in the fishing and sardine processing industry. This study assessed the impact of the closed fishing season policy on fisheries workers, specifically those in canneries. By assessing the impacts of fisheries policy, the study was able to determine the effect of the policy on income and livelihood of the workers and also identify possible policy actions to mitigate the adverse policy impacts.
Link to the article: https://link.springer.com/article/10.1007/s11852-019-00713-y
Jamie R. Chua1 and Albert Jr. B. Albay1
1Department of Medicine, College of Medicine (UP Manila)
Body Composition of Filipino Chronic Obstructive Pulmonary Disease (COPD) Patients in Relation to Their Lung Function, Exercise Capacity and Quality of Life, International Journal of Chronic Obstructive Pulmonary Disease, 14: 2759-2765, 2019
Chronic Obstructive Pulmonary Disease (COPD) is a disease caused by excessive smoking or exposure to harmful smoke that damages the lungs and airways. Many studies showed that COPD creates systemic inflammation which affects other organs such as the muscles, heart, brain; thus, we often see COPD patients that were undernourished and with other conditions (hypertension, coronary artery disease). Our study intends to know how COPD patients' body mass (muscle mass) can affect their lung symptoms, function, muscle strength and exercise performance. Our study recruited 41 Filipino COPD patients. After consent, they were interviewed, asked to answer questionnaires, perform several exercises, and underwent body composition analysis. We divide them according to the low and normal muscle mass (fat-free mass index). Our study showed that 32% of them are underweight, those with low muscle mass have weaker lung strength, lower lung function, weaker upper arm muscle strength. However, exercise measurements and lung symptoms were the same for both groups.
The paper asserts that poverty among poor, elderly women adds to their caring work which further reinforces their marginalization. The family as a social construct needs to be reexamined especially the power relations between women and men that disadvantage women in terms of the benefits and costs of care work.
To determine how body composition (muscle mass) of Filipino COPD patients can affect COPD-related clinical variables such as lung function, exercise capacity, muscle strength, etc. so that accurate nutritional assessment and management shall be advocated by physicians.
Link to the article: https://www.dovepress.com/articles.php?article_id=50171
Pablito M. Magdalita1 and Alangelico O. San Pascual1
1Institute of Crop Science, College of Agriculture and Food Science (UP Los Baños)
Somatic Embryogenesis, Regeneration, Phenotypic and Cytological Evaluation of Selected Philippine Papaya (Carica papaya L.) Genotypes, Philippine Journal of Crop Science, 44(3): 20-30, 2019
This paper presents the somatic embryogenesis and production of artificial seeds of different Philippine papaya genotypes, its regeneration and the results of phenotypic and cytological evaluation of somatic embryogenesis derived Papaya plants.
This study shows successful production of multiple artificial seeds (embryos) using only one embryo using somatic embryogenesis in vitro. The study also investigated on the fruit and tree characters of plants derived from somatic embryogenesis which were planted in the field. Cytological evaluation of trees were also done to check for somaclonal variations and changes that have been caused by in vitro culture.
This protocol shows promising results as to how to mass produce Papaya especially those from the Philippines using only one embryo. This study also proved low somaclonal variation on the plants derived from somatic embryogenesis as verified by phenotypic and cytological evaluation.
The Paper shows the production of artificial seeds using somatic embryo induced from zygotic embryos of Philippine Papaya genotypes. The paper showed a rapid increase in the production of artificial seeds to increase plant production in vitro. Selected lines showed promising performance in terms of somatic embryos produced and full plant produced. Also, the study explored the different exposure techniques of the apical dome to somatic embryogenesis and found that complete exposure of apical dome promotes somatic embryogenesis. In addition, squashing also promoted somatic embryogenesis.
The peak of regeneration and germination of secondary somatic embryos were observed 7-8 months after initial explanting. Field planting confirmed that fruiting habits of somatic embryogenesis derived plants and those seed derived are uniforms.
However, one inbred line 4172, a typical hermaphrodite line produced sexually ambivalent males as this can be a form of somaclonal, variation, only a minimal level was observed. SAM had chromosomal aberrations but despite these abnormalities, pollen viability and germination is high indicating fertility
Link to the article: https://www.cabdirect.org/cabdirect/abstract/20203113780
Allen L. Nazareno
Institute of Mathematical Sciences and Physics
Linear Conjugacy of Chemical Kinetic Systems, Mathematical Biosciences and Engineering, 16(6): 8322-8355, 2019
Chemical reaction network corresponding to the R. Schimtz's pre-industrial carbon cycle model
The graphs of the trajectories for the original carbon cycle model and sparse linearly conjugate realization
In this study, we developed an algorithm that can be used to establish linear conjugacy between two chemical kinetic systems. Linear conjugacy is a property that takes the flow of one system to the other via a linear transformation. This concept can be used primarily to analyze the behavior of a system. The algorithm is divided into two major procedures: 1) transforming a given system into another system (a complex factorizable) and 2) implementing an optimization process to find a linearly conjugate realization of the transformed system.
This research work can be generally used to model the behavior of chemical and biological systems. Also, this study can be used by other researchers to develop further the chemical reaction network theory.
Link to the article: https://www.aimspress.com/fileOther/PDF/MBE/mbe-16-06-421.pdf
Darwin B. Putungan
First-principles Investigation of the Hydrogen Evolution Reaction on Different Surfaces of Pyrites MnS2, FeS2, CoS2, NiS2, Physical Chemistry Chemical Physics, 21: 21561-21567, 2019
The whole world currently relies on the use of fossil fuels in producing energy. Energy is very indispensable, as everything can be traced back to it, whether it's about economics, politics and the environment. Burning fossil fuels releases greenhouse gases, in particular CO2, that is the main culprit behind global warming and subsequently climate change. To alleviate these concerns, alternative ways of producing energy must be considered, and the best possible alternative to fossil fuel is hydrogen. Hydrogen as an energy carrier does not produce any emissions, thus it is considered as clean. If used in a fuel cell, the only outputs are energy and water.
In this work, we theoretically investigated the hydrogen evolution reaction (HER) on those XRD observed (100), (110), (111), and (210) surfaces of pyrite structure CoS2 . The random structure searching method was employed in this work to thoroughly and less-biased identify the active sites for each considered surface. We calculated the free energy of hydrogen adsorption and found that (110) and (210) surfaces are more active than conventionally assumed (100) facet. While the lowest energy active site on the (100) and (210) surface is the five-coordinated transition metal site that is commonly seen in other HER catalysts, the lowest energy active site on the (110) surface is the two-coordinated S site, which is a S tetrahedron with two corners missing. Besides those lowest energy active sites, both (110) and (210) have more than one species of active sites on the surface, including not fully coordinated transition metals and sulfur. We further explored the reaction for MnS2, FeS2, and NiS2 and analyzed the density of states. Our results showed both CoS2 and NiS2 (110) and (210) surface are catalytically reactive for HER
Energy concerns will be going to rise in the years to come, and as such, it is paramount to study materials that could be utilized in generating clean and renewable energy. One such example is hydrogen, and the most renewable way of getting it is by splitting water into its oxygen and hydrogen components. Hydrogen gas can then be produced by using an appropriate catalyst material, such as transition metal pyrites. In this work, we probed the different surfaces of CoS2, MnS2, FeS2 and NiS2 for hydrogen evolution reaction using density functional theory calculations. Our results are of significance to experimentalists as these could serve as a guide in the fabrication and cleaving of appropriate surface for hydrogen evolution.
Link to the article: https://pubs.rsc.org/en/content/articlelanding/2019/cp/c9cp03893k/unauth#!divAbstract
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Effects of prescription restrictive interventions on antibiotic procurement in primary care settings: a controlled interrupted time series study in China
Yuqing Tang1,
Chaojie Liu2,
Zinan Zhang1 &
Xinping Zhang1
Cost Effectiveness and Resource Allocation volume 16, Article number: 1 (2018) Cite this article
The overuse of antibiotics has been identified as a major challenge in regard to the rational prescription of medicines in low and middle income countries. Extensive studies on the effectiveness of persuasive interventions, such as guidelines have been undertaken. There is a dearth of research pertaining to the effects of restrictive interventions. This study aimed to evaluate the impacts of prescription restrictions in relation to types and administration routes of antibiotics on antibiotic procurement in primary care settings in China.
Data were drawn from the monthly procurement records of medicines for primary care institutions in Hubei province over a 31-month period from May 2011 to November 2013. We analyzed the monthly procurement volume and costs of antibiotics. Interrupted time series analyses with a difference-in-difference approach were performed to evaluate the effect of the restrictive intervention (started in August 2012) on antibiotic procurement in comparison with those for cardiovascular conditions. Sensitivity tests were performed by replacing outliers using a simple linear interpolation technique.
Over the entire study period, antibiotics accounted for 33.65% of the total costs of medicines procured for primary care institutions: mostly non-restricted antibiotics (86.03%) and antibiotics administered through parenteral routes (79.59%). On average, 17.14 million defined daily doses (DDDs) of antibiotics were procured per month, with the majority (93.09%) for non-restricted antibiotics and over half (52.38%) for parenteral administered antibiotics. The restrictive intervention was associated with a decline in the secular trend of costs for non-restricted oral antibiotics (− 0.36 million Yuan per month, p = 0.029), and for parenteral administered restricted antibiotics (− 0.28 million Yuan per month, p = 0.019), as well as a decline in the secular trend of procurement volume for parenteral administered non-restricted antibiotics (− 0.038 million DDDs per month, p = 0.05).
Restrictive interventions are effective in reducing the procurement of antibiotics. However, the effect size is relatively small and antibiotic consumptions remain high, especially parenteral administered antibiotics.
The overuse of antibiotics has been identified as a major global challenge, especially in low and middle income countries (LMIC) [1, 2]. It has been proved to be associated with the development and spread of antimicrobial resistance (AMR). Rising AMR levels, in combination with a lack of new effective antibiotics, increases the morbidity and mortality of infectious diseases [3]. The overuse of antibiotics also drives inflation related to healthcare costs.
Empirical evidence shows that irrational antibiotic prescriptions are most prevalent in primary care settings [4]. The overuse of antibiotics for upper respiratory tract infections in primary care, for instance, was observed worldwide [5]. Antibiotic abuse has also been identified as a serious problem in China [6]. The primary health network in China includes community/township health centres and outreach stations/clinics. They provide essential medical care services (covering outpatient, inpatient and rehabilitation care in line with the Essential Medicines List policy) and essential public health services (including personal health records, health education, planned immunization, child (0–6 years) health care, maternity care, aged care, management of chronic conditions (hypertension and diabetes), management of severe mental disorders, management of tuberculosis, use of Chinese medicines for health promotion, reporting and emergency response to infectious diseases, supporting health inspection activities, free supply of contraception, and improving the health literacy of consumers) [7]. A growing body of literature has revealed a very high level of use of antibiotics in primary care settings in China [8]. The direct cost associated with the overuse of antibiotics in China is estimated to be around 2.91–13.93 billion yuan ($0.42–2.02 billion USD) per year [9]. A recent national survey shows that 52.9% of patients visiting primary care institutions in China were prescribed antibiotics, but only 39.4% of those who received antibiotics needed them based on their clinical condition [10].
Increased AMR triggered a surge of interventions on antibiotic prescribing practices [11]. Clinical guidelines are perhaps the most commonly used instrument for promoting rational prescriptions. Guidelines alone may play a limited role in changing prescribing practices [12, 13]. In a systematic review, Ivers and colleagues found that audit and feedback can bring about 70% compliance with prescription guidelines, leading to a 16% reduction in antibiotic prescriptions [14]. Nonetheless, the current intervention strategies have been heavily biased towards persuasive measures (such as guidelines), and restrictive interventions (such as administrative rules on prescribers) are rare both in practice and in research [15]. Some researchers argue that restrictive interventions may have great potential for curbing antibiotic abuse [14, 16].
In China, both persuasive and restrictive measures have been used to address antibiotic over-prescriptions [17]. In the recent round of health system reform launched in 2009, access to antibiotics in primary care facilities has been restricted to medicines listed in the Essential Medicines List (EML) and these medicines have to be sold with zero-markup [18]. Unfortunately, limited effects on antibiotic prescribing practices have been observed after a few years of implementing these policies [19,20,21]. In 2012, the Chinese government issued ''administrative rules for the clinical use of antibiotics'', which are considered the most rigid regulatory control over antibiotic prescriptions to date [22, 23].
This study aimed to evaluate the effects of the "administrative rules" on antibiotic prescriptions in primary care in Hubei province. Hubei's "administrative rules for clinical use of antibiotics" involve three major components [24]: (1) antibiotics are categorized into three groups—non-restricted, restricted and controlled. The EML for primary care contains non-restricted and restricted antibiotics only; (2) administrative restrictions are imposed on health facilities and medical practitioners in relation to prescriptions of restricted and controlled antibiotics, as well as intravenous infusion of antibiotics; (3) penalties apply to those who violate the rules (Box 1).
Box 1 Administrative rules for the clinical use of antibiotics
1. Antibiotic categorization
Antibiotics are categorized into three groups: non-restricted, restricted, and controlled. Non-restricted antibiotics can be used for the prevention and treatment of mild infections. Restricted antibiotics can be used for severe infections, infections in patients with immune dysfunction, and infected pathogens that are sensitive only to restricted antibiotics. Controlled antibiotics can only be used in special circumstances. Detailed guidelines were issued by the government in relation to the type of antibiotics that applies to various clinical conditions and evidence that is required to justify antibiotic prescriptions.
2. Prescribing authorization
Prescribing privileges are conditional to qualification, professional title, and training of prescribers. Only doctors with a middle or senior professional title are allowed to prescribe controlled antibiotics. Medical practitioners without a professional title (such as assistant doctors) are not allowed to prescribe restricted antibiotics. Health workers in village clinics can only prescribe non-restricted antibiotics. The intravenous infusion of antibiotics in village/community clinics is subject to approval from county health bureaux.
3. Pharmaceutical management committee
Secondary and tertiary hospitals are required to establish a pharmaceutical management committee, consisting of representatives of physicians, pharmacists, microbiologists, and managers. Primary care institutions are required to establish an antimicrobial working group.
4. Monitoring and evaluation of antibiotic prescriptions
Antibiotic prescriptions should be audited on a regular basis in line with the prescribing guidelines.
5. Penalty
Institutions that violate the rules are subject to penalties imposed by the health authorities, which include downgraded accreditation and dismissal of managers. Medical practitioners involved may lose permission to prescribe antibiotics, have their medical registration revoked, or prosecuted.
This study was conducted in Hubei province. Hubei is located in central China, with a population of over 61 million across a geographic area of 185,900 km2. The annual average income per capita in Hubei ranks in the middle range of all provinces: 6898 yuan ($1000 USD) for rural residents and 18,374 yuan ($2659 USD) for urban residents (2012).
Data used in this study were extracted from the procurement database for urban community health centers and rural township health centers. There are 1430 community/township health centers in Hubei. The procurement of medicines for the 1235 state-owned community/township health centers is made through a provincial tendering system managed by the Hubei Medical Procurement Administrative Agency (HMPA). Primary care institutions are only allowed to stock and dispense medicines listed in the EML at zero markup. The procurement system covers all medicines listed in the EML, including 32 generic non-restricted antibiotics and 4 generic restricted antibiotics.
The procurement system recorded volumes and prices of medicines delivered to community/township health centers. The medicines were coded using the Anatomical Therapeutic Chemical (ATC) coding system. We compared the changes in volumes and costs of antibiotics for systemic use (ATC code, J01) with those of medicines used for the cardiovascular system (ATC code, C).
We adopted a controlled interrupted time series design with a difference-in-difference approach. An interrupted time series is a strong quasi-experimental design in which data are collected at multiple time points before and after the intervention. The advantage of this design is that it can detect a possible underlying secular trend which occurs after the intervention. By adding a control group, it is possible to separate the intervention effect from other confounding effects that may have occurred at the same time [25,26,27]. In this study, the ATC "C" group of medicines served as a control group because it contained large volumes of orders and it is not subject to the influence of the restrictive intervention tested in this study.
The design of this study was further strengthened by adopting a difference-in-difference approach, which enables us to estimate the effect size of the restrictive intervention, adjusting for pre-existing differences and the confounding influence of other factors.
Data collection and management
Data were extracted from the HMPA procurement system, which contained monthly records in relation to the unit strength, pack size, price, procurement volume, and total cost of each delivered medicine. Procured medicines in the ATC "J01" group included 32 products classified as non-restricted antibiotics and 4 products classified as restricted antibiotics, compared with 39 products in the ATC "C" group.
The procurement records over a 31-month period (from May 2011 to November 2013) were collected. The administrative restrictive rules on antibiotic prescriptions were introduced in August 2012. This resulted in a final sample of 15 months of pre-intervention records and 16 months of post-intervention records. We discontinued the data collection in December 2013 because a new version of EML was introduced at that time.
Two indicators were used to examine the outcomes of the restrictive intervention: volume and costs of procured medicines. The cost of the procured medicines was calculated based on the price and volume of each product, without adjustment for present values. The volume of procured medicines was measured using defined daily dose (DDD, the average maintenance dose per day for a drug used for its main indication in adults), a measurement developed by the World Health Organization (WHO) to compare drug consumptions. According to the WHO Collaborative Centre for Drug Statistics Methodology [28], DDD equivalence per package (DPP) of medicines was calculated in DDD units [DPP = (unit strength × pack size/DDD)]. The total volume for each group of procured medicines (DDDs) was estimated as the summed DPPs of all-inclusive products [29].
$$ DDDs = \sum\limits_{i = 1}^{n} {\left( {DPP_{i} \times N_{i} } \right)} $$
N i represents the number of packages of certain product (i) delivered to the community/township health centers.
To estimate the effect of the intervention on the outcome variables, the following segmented linear regression model was developed [25]:
$$ {Y}_{t} = \,\beta_{0} + \,\beta_{1} \cdot {T}_{t} +\beta_{2} \cdot {I}_{t} +\beta_{3} \cdot {T \;{\text{after}}\; I}_{t} +\beta_{4} \cdot {G} +\beta_{5} \cdot {G} \cdot {T}_{t} +\beta_{6} \cdot {G} \cdot {I}_{t} +\beta_{7} \cdot {G} \cdot {T \;{\text{after}}\; I}_{t} +\beta_{8} \cdot \sin (2\pi{T}_{t} /12) + \,\beta_{9} \cdot \cos (2\pi{T}_{t} /12)\varepsilon_{t} $$
In this model, Y t is the outcome indicator in month t; T t is a continuous variable indicating the months passed at month t since the start of the observation period; I t represents the two periods before (value = 0) and after (value = 1) the intervention; T is a continuous variable indicating months passed since the intervention (time prior to the intervention is coded 0); G represents the two groups (0 for the control group and 1 for antibiotic group); β6 estimates the mean difference in pre-post (intervention) changes between the two groups in relation to the outcome indicators (the effect size of the intervention); β7 estimates the difference in secular trend changes in time series between the two groups (the change in trend due to the intervention). β8 and β9 were used to correct for a potential seasonality effect [30]. To correct for dependency of time series data, New-West standard errors were calculated in these models [31].
To better understand the changing pattern of prescribing practices, we also estimated the effects of the intervention on the consumption of non-restricted antibiotics, restricted antibiotics, oral antibiotics, and antibiotics administered through the parenteral route, respectively.
We observed two obvious outliners (at time point 21 and 22) which were caused by the holiday season (Chinese New Year). Sensitivity tests were performed by replacing the outliers using a simple linear interpolation technique [32].
All of the analyses were performed using Stata 15.0 (Stata Corp LP, College Station, TX, USA).
Overall, antibiotics accounted for 33.65% of the total procurement costs for all drugs (36.10% before intervention and 32.11% after intervention). On average, ¥47.97 million yuan per month were spent on antibiotics (¥44.31 million per month before intervention and ¥51.41 million after intervention). The percentage increase in costs over time was smaller for antibiotics than for cardiovascular medicines (Fig. 1).
Time trend of monthly costs for procured medicines: antibiotics vs medicines for cardiovascular system
Non-restricted antibiotics accounted for 86.03% of the total cost of antibiotics (17.43% oral and 68.6% parenteral), while restricted antibiotics accounted for 13.97% of the total cost of antibiotics (2.98% oral and 10.99% parenteral). As a group, parenteral antibiotics accounted for almost 80% (79.59%) of the total cost of antibiotics (Fig. 2).
Time trend of monthly costs for antibiotic subgroup
On average, 16.79 million DDDs of antibiotics were procured per month (13.60 million before intervention and 19.78 million after intervention), compared with 10.01 million DDDs of medicines for the cardiovascular system (8.24 million before intervention and 11.66 million after intervention). The percentage increase in DDDs over time was smaller for antibiotics than for cardiovascular medicines (Fig. 3).
Time trend of monthly DDDs for procured medicines: antibiotics vs medicines for the cardiovascular system
Non-restricted antibiotics accounted for 93.09% of the DDDs for antibiotics (46.11% oral and 46.98% parenteral), while restricted antibiotics accounted for 6.91% of the total cost of antibiotics (1.52% oral and 5.39% parenteral). As a group, parenteral antibiotics accounted for more than half (52.38%) of the DDDs for antibiotics (Fig. 4), despite their much higher contribution to the cost of antibiotics.
Time trend of monthly antibiotic DDDs by subgroups
The segmented linear regression models revealed that the intervention was associated with a decline in the secular trend of costs for non-restricted oral antibiotics (− 0.36 million Yuan per month, p = 0.029) compared to the control. Prior to the intervention, there was an increasing secular trend (average increase per month 0.25 million Yuan, p = 0.021). The intervention was associated with a 27.84% reduction in the cost of non-restricted oral antibiotics. The intervention was also associated with a decline in the secular trend of costs for restricted antibiotics administered through the parenteral route (− 0.28 million Yuan per month, p = 0.019) compared to the control. Prior to the intervention, there was not a significant increasing secular trend (p = 0.114). The interventions were associated with a 33.64% reduction in the cost of restricted antibiotics administered through the parenteral route. No significant trend changes were observed for the costs of non-restricted antibiotics administered through the parenteral route or oral restricted antibiotics. Despite the secular trend changes, the mean differences in the magnitude of changes remained statistically insignificant (Table 1).
Table 1 Effects of the intervention on the cost of antibiotics in comparison with controls: findings from the segmented linear regression models
The intervention was associated with a decline in the secular trend of the procurement volume of restricted antibiotics administered through the parenteral route (− 0.038 million DDDs per month, p = 0.05) compared to the control. Prior to the intervention, there was not a significant increasing secular trend (p = 0.128). The interventions were associated with a 26.82% reduction in the volume of restricted antibiotics administered through the parenteral route. No significant trend changes were observed for the volumes of non-restricted antibiotics, oral restricted antibiotics, and oral restricted antibiotics administered through the parenteral route. Again, the mean differences in the magnitude of volume changes remained statistically insignificant (Table 2).
Table 2 Effects of the intervention on procurement volumes of antibiotics in comparison with controls: findings from the segmented linear regression models
The results of the regression models using data with and without replacing outliers were consistent, with similar coefficients for the change in the secular trend of antibiotic prescriptions: − 1.58 vs − 1.56 for total cost; − 0.51 vs − 0.51 for total DDDs.
The restrictive intervention on antibiotic prescriptions is associated with some positive changes. This study demonstrated that the restrictive intervention resulted in a 26.82% reduction in the procurement volume of parenteral administered restricted antibiotics, a 33.64% reduction in the cost of parenteral administered restricted antibiotics, and a 27.84% reduction in the procurement cost of non-restricted oral antibiotics. This is understandable because the administrative rules set up a very strong control over the use of restricted antibiotics (details in Box 1). Although there is a paucity in the literature documenting the effectiveness of restrictive measures on prescribing practices [15], a recent study shows that a financial penalty for violating the existing guidelines can decrease subsequent antibiotic prescriptions and associated costs [33]. There was no change in the price of medicines over the study period. Medicines for primary care facilities were procured through a provincial tendering system. The price of the procured medicines was fixed until the next round of the tendering process. Over the study period, there was no new round of tendering. Therefore, the decline in cost reflects a reduction in the volume of prescriptions, not a reduction in price.
The cost-saving effect of the restrictive interventions on the use of antibiotics in primary care settings should not be interpreted as an effect on the entire health delivery system for several reasons. First, the restrictive intervention may encourage more referrals from primary care workers, increasing the cost associated with doctors' time. Second, the restrictive intervention involves additional administrative costs. Third, it is also likely that some patients may bypass primary care and seek more expensive care from hospitals. In China, primary care is delivered in both primary care facilities and hospitals, and patients enjoy the freedom to choose their preferred primary care providers [34]. In 2011, China established a medication review system for antibiotic prescriptions, where a medication review team involving physicians and pharmacists is required to provide advice on the rational use of antibiotics [35]. But there is a lack of mechanisms for action. The auditing and penalty strategies were supposed to serve as an instrument for the medication review team to introduce actions. No additional investment is required.
Restrictive intervention strategies may have the potential to complement persuasive intervention strategies. A few studies reported the limited effects of guidelines on antibiotic prescribing practices. For instance, a national guideline recommended no initial antibiotic therapy on acute otitis media and adult sinusitis. But the rate of encounters at which no antibiotics was prescribed for these clinical conditions had not changed since the publication of the guideline [12, 13, 36]. However, an audit and feedback plus the distribution of a pocket version of the guidelines increased the prescribing compliance in a Norwegian hospital in terms of the right choice of empirical antibiotics, appropriate treatment duration, and decreased use of high-dose benzyl penicillin [37]. Public reporting may also help enhance the effects of practice guidelines, albeit in a small effect size [38, 39].
The intervention strategies tested in this study comprise multifaceted measures, including prescribing guidelines, audit and feedback, administrative rules and penalties, The multifaceted approach is particularly important in a health system where prescribers have varied qualifications and financial incentives are not always well aligned with the quality of care [33], as is often the case in low and middle income countries.
No significant impact on the overall cost or volume of antibiotics was detected in this study, although the results were numerically lower in all cases except the change in volume of non-restricted orals and the trend in the volume of restricted orals. Clearly, the overconsumption of antibiotics remains a significant challenge in China, despite the enormous efforts made by the government. Empirical evidence shows that economic motivation plays a crucial role in physicians' prescribing practices [40]. The Chinese government has tried to decouple the link between the income of physicians and the sale of prescribed medicines through its EML and zero markup policies. However, it has resulted in a substantial loss of revenue for primary care facilities [41, 42] due to a shortage of government subsidies. These facilities have to turn their attention to other avenues to compensate for the loss, including user charges for the parenteral administration of medicines [38]. The regional centralized procurement arrangement does not forfeit the autonomy of health facilities to decide what and how much they can spend on medicines [43, 44]. In such a system context, persuasive measures alone barely have any significant impact on prescribing practices. The Antibacterial Use in Clinical Practice (2004) was proved to be fragmented and incomplete and has made only limited progress in containing antibiotic resistance [3, 45]. A coordinated systems approach may further tackle the issue of the irrational use of antibiotics.
The findings of this study have significant policy implications: administrative restrictive measures have the potential to lower antibiotic consumptions. However, it is important to note that the effect size of the tested intervention is rather small and antibiotic consumption remains high, especially parenteral administered antibiotics. Overall, antibiotics accounted for 33.65% of the total cost of procured medicines for primary care institutions in Hubei, much higher than the average level (11–18%) across all healthcare settings in Shanghai [46] or children's hospitals (17.1%) in the US [47]. Antibiotics administered through the parenteral route still comprise over half of DDDs of antibiotics, contributing to almost 80% of the total cost of antibiotics. Since the 2009 health system reform, primary care institutions in China are no longer able to make a profit from dispensing medicines. However, they are allowed to charge a fee for services (injections) and consumables (syringes). This has added complications to the efforts to curb the overuse of injections. There is a need to organize a coordinated systems approach to tackle this issue.
This study has several strengths. The SMPA dataset includes the procurement records for almost all the primary care institutions in Hubei, covering all essential medicines. The institutionalization of longitudinal data reporting and a control group in this study avoids much of the bias of sampling. The use of cardiovascular medicines was unlikely to involve any significant changes over the study period, because primary care facilities were only allowed to dispense medicines listed in the EML which remained unchanged over the study period and the age structure and disease pattern of the population had limited, if any, changes in such a small time window (31 months). We also used more sophisticated methods, "interrupted time series", which can make a more precise estimation of the policy impact compared with a simple pre-post comparison [48]. The sensitivity tests further enhanced the reliability of the results.
There are several limitations in this study. First, the data used in this study were drawn from procurement records, which do not directly reflect the actual use of medicines. We were unable to evaluate the appropriateness of antibiotic usage at the individual patient level. Second, prescriptions in private primary care facilities and the use of non-prescribed antibiotics (e.g. over-the-counter and leftover antibiotics at home) were excluded in this study. However, the impact of such exclusion is anticipated to be minimal. In 2012, primary care institutions received 68.51% of total outpatient visits in Hubei province. The participating community/township health centers in this study covered 86.36% of all primary care institutions in Hubei. Patients who visit health facilities usually fill their prescriptions at the same facility [44, 49]. We used multiple models and multiple tests to ensure the robustness of the study findings. However, some significant differences might arise due to chance where multiple tests were performed.
Administrative restrictive regulations on antibiotic prescriptions are effective in reducing both the cost and volume of parenteral administered restricted antibiotics (26.82% reduction in volume and 33.64% reduction in cost). However, the restrictive interventions have failed to have a significant impact on overall cost and volume of antibiotic procurement. It is also important to note that costs may shift to other professionals and providers as a result of restrictions on primary care. Antibiotic usage remains high in Hubei, China, especially parenteral administered antibiotics. A coordinated systems approach is needed to further tackle the issue of the irrational use of antibiotics.
LMIC:
low and middle income countries
AMR:
EML:
Essential Medicines List
ATC:
Anatomical Therapeutic Chemical
HMPA:
Hubei Medical Procurement Administrative Agency
defined daily dose
DPP:
defined daily dose equivalence per package
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YT made contributions to the study design, acquisition, analysis and interpretation of data, and drafted the manuscript. CL made significant contributions to the analysis and interpretation of data and writing of the manuscript. XZ made substantial contributions to the project design, acquisition, and interpretation of data. ZZ participated in the acquisition and interpretation of data. All authors agreed to be accountable for the accuracy and integrity of the work. All authors read and approved the final manuscript.
We would like to thank Liping Ye for her hard work in data cleansing and data analysis.
The data from which these findings were drawn is available from the corresponding author on reasonable request.
This study was funded by the Fundamental Research Funds for the Central Universities of Ministry of Education of China (No. 2017WKYXQY003). The funding body plays no roles in study design, collection, analysis, and interpretation of data, writing of the manuscript, or the decision to submit the manuscript for publication.
School of Medicine and Health Management, Tongji Medical College, Huazhong University of Science and Technology, Wuhan, 430030, People's Republic of China
Yuqing Tang
, Zinan Zhang
& Xinping Zhang
School of Psychology and Public Health, La Trobe University, Kingsbury Drive, Melbourne, VIC, 3086, Australia
Chaojie Liu
Search for Yuqing Tang in:
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Correspondence to Xinping Zhang.
Tang, Y., Liu, C., Zhang, Z. et al. Effects of prescription restrictive interventions on antibiotic procurement in primary care settings: a controlled interrupted time series study in China. Cost Eff Resour Alloc 16, 1 (2018) doi:10.1186/s12962-018-0086-y
Administrative regulation
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Applied Water Science
September 2018 , 8:120 | Cite as
Assessments of seasonal groundwater recharge and discharge using environmental stable isotopes at Lower Muda River Basin, Malaysia
Mohd Khairul Nizar Shamsuddin
Wan Nor Azmin Sulaiman
Mohammad Firuz Ramli
Faradiella Mohd Kusin
Kamarudin Samuding
An accurate estimation of groundwater recharge is required to properly manage aquifers, especially for riverbank filtration method (RBF) purposes. The isotopes correlations and differences in different water bodies were studied to assess the sources of groundwater recharge and preliminary tools in understanding of the surface water and groundwater interactions in the Lower Muda River Basin. The environmental isotope and hydrochemical sampling results had emphasised that the area near Lower Muda River Basin had a connection with the river and was actively recharging the near-river shallow alluvial aquifer, via RBF method. Furthermore, the shallow groundwater that was close to Muda River from groundwater signatures had indicated the recharge of the shallow aquifer system by Muda River based on the plots along LMWL on a δ2H versus δ18O. The comparisons between like δ2H and δ18O isotopes in the rainwater revealed the variations in the rainfall amount and the 18O-depleted water of those isotopes for wet season precipitation as compared to dry seasons. Furthermore, the groundwater δ2H and δ18O isotopes exhibited a slight deviation from the δ2H and δ18O isotopic meteoric water line in Lower Muda River. Therefore, in this basin, the groundwater could be a combination of river water and precipitation, which had led to the recharge of river water being more than the recharge of rainfall infiltration.
Water isotopes Unsaturated zone Groundwater recharge Muda River Basin Malaysia
Assessment of groundwater recharge is important in the management of groundwater resources (Hooji et al. 2011; Scanlon et al. 2006) because recharge is not a continuous process and is instead a sporadic event, observing and analysing it is extremely difficult. That has been a difficult challenge for hydrogeologists, made more difficult by the dual influence that human activities and climate change have had during the last three decades. Recent years have seen a growing trend in water consumption which corresponded to the continuous economic development (Dalin et al. 2017). Overexploitation of groundwater resources has led to water depletion, declining of groundwater levels, and deterioration of water quality (Custodio 2002; Blasch and Bryson 2007; Kumar 2007; Brunner et al. 2014; Unsal et al. 2014; Tamez-Meléndez et al. 2016). All those conditions are the threats to reliable water supplies and economic development. This is a serious situation, especially in semi-arid and arid areas because they often rely on groundwater as their main source for their irrigation, industrial, and domestic supply (Taylor et al. 2013). The role that it plays in the development of regional economy is so important that it becomes an indispensable source of water in those areas. Therefore, obtaining information about recharge mechanism is vital in being able to rationally use the groundwater resource.
Estimating recharge with the use of traditional hydrogeological methods (e.g. water balance methods, lysimeters, and Darcy's method) is influenced by methodological difficulties (Grindley 1969; Hough and Jones 1998; Chapman and Malone 2002; Yeh et al. 2004). On the other hand, a better understanding of aquifer recharge can be gained by monitoring isotope changes. Isotope methods are often considered as unique tools that offer insights into processes of groundwater recharge and the systems of groundwater flow. Furthermore, they have been in extensive use in various regions (Bhattacharya et al. 1985; Krishnamurthy and Bhattacharya 1991; Zhang et al. 2005; Blasch and Bryson 2007; Li et al. 2008; Yuan et al. 2011), especially when conditions are arid and semi-arid for several years (IAEA 2011). Specifically, δ2H and δ18O are considered ideal conservative tracers that offer insights into the groundwater system's recharge and flow since they make up the actual water molecule and their compositions remain the same unless the flow path has phase changes or fractionation (Clark and Fritz 1997). Therefore, groundwater would retain its isotopic fingerprint, which a reflection of its origin and history prior to infiltration. The groundwater isotopic content is dependent on the recharge isotopic content. Groundwater that goes through direct infiltration from precipitation possesses precipitation's isotopic signature. On the other hand, the groundwater that is recharged using other sources like rivers and lakes will reveal the contributing river or lake mean isotopic content. Furthermore, their signature is expected to vary from that of local precipitation (IAEA 2011). That makes it possible to trace the recharge water source and flow path by examining the differences in the isotopic signature of various different waters (Craig 1961; Onodera et al. 1995; Lee and Lee 1999; Coplen et al. 2000; Vandenschrick et al. 2002; Ma et al. 2007; Mukherjee et al. 2007; Kyle and Jeannie 2007; Palmer et al. 2007; Yeh et al. 2009, 2011; Mandal et al. 2011).
In the Lower River Muda Basin, groundwater and surface waters are the major water supply source for paddy fields and they are also a part of an area that is more popularly known as the rice bowl of Malaysia. This area measures 4150 km2. The Lower Muda Basin's water sources are essential in various important key economic sectors such as industry, services, agriculture, tourism, and paddy planting for Seberang Perai, central and south Kedah, and also for Pulau Pinang. Malaysia's northern states form the key portion of the Northern Corridor Development Region (NCDR). Next to the central region, they also have the second highest growth domestic product (GDP). Furthermore, this area possesses Malaysia's largest granary IADA (Integrated Agriculture Development Area) scheme, which measures about 100,000 ha and is responsible for approximately 40% of the country's total paddy production. The Lower River Muda Basin is a vital river because it serves as the main source of water for the potable water and irrigation of southern Perlis, Kedah, and Pulau Pinang. Thus, it is right for the river to be given due recognition and priority in the appropriate management of water resources to supply enough water resource and support the development of the regional corridor. The rise in water demand, which is mainly for potable water supply and agriculture, has led to water stress in the area which is made more apparent, especially during dry seasons. Water allocation conflicts have also become more pronounced among the different users and water use sectors from different states that depend on the same source of water. The issue gains more complexity since the dams and river infrastructures are being managed and developed by several various agencies from the federal to the state levels. In order to address these water issues that are expected to become more acute, the Riverbank Filtration (RBF) programme has started the development of tools to aid stakeholders in making informed decisions (Shamsuddin et al. 2014). The tool includes the conjunctive use of water resources between the ground and surface waters. The tool was also developed specifically for the basin. Thus, gaining an understanding of the current recharge to aquifer is a pressing need for water managers and hydrogeologists. This issue also involves some previous studies. Majority of these works utilised traditional methods of numerical modelling or hydrogeological investigations (e.g. Shao et al. 2006; Yang et al. 2009). Shao's (1989) study presented the first discussion about the mechanism and origin of groundwater recharge with the use of environmental tracers. However, the sources and processes of the current recharge in the Lower River Muda basin have not had any good demonstrations yet. Previous works suggest strong connections between groundwater aquifers and rivers (Sear et al. 1999; Jasechko et al. 2016). Thus, to further plan sustainable groundwater management, it is important to identify the groundwater recharge's sources and their relative contributions to the present hydrogeological situation. The aims are to determine (1) the possible groundwater sources, (2) the seasonal differences in groundwater recharge, and (3) processes of recharge to aquifer with the use of stable isotopes (δ2H and δ18O) and the hydrochemistry of Lower Muda River area's surface and groundwaters. The results offer vital information about hydrological processes like interaction among the river water, precipitation, and groundwater.
The study area is the Muda River Basin that is situated in the northwestern portion of Peninsular Malaysia. The basin's upper and middle reaches are part of the State of Kedah, and the downstream of the river forms a boundary between the States Penang and Kedah. The Muda River has undergone development as one of the most vital water resources for water supply and agriculture in the states of Kedah and Penang. Kedah and Penang both have the rights to utilise the water from the Muda River. The area of the study is found between 100°29′0″E and 100°33′30″E east longitudes and 5°31′30″ and 5°35′30″ north latitude and covers an area that measures 150 sq.km. The sub-basin has a tropical climate and receives rain due to the influence of two typical monsoons, namely the southwest monsoon (May to August) and the northeast monsoon (November to February). During the period of transition between the two monsoons, westerly winds are dominant from September to November and they lead to the heaviest amount of annual rainfall precipitation within the study area. Therefore, there is a tendency for the study area to have two rainy seasons in 1 year: one that happens from April to May and another one that takes place from September to November. In the two states, the average air temperature is approximately 27 °C. Agriculture is the most important economic activity, with paddy being the chief crops being raised. Kedah and a portion of Penang state serve are considered a vital important Malaysian agricultural area, popularly known as the rice bowl. It possesses the largest double cultivation paddy field (97,000 ha) that serves as the Malaysians' main food source.
Geology and hydrogeology
The study area's quaternary stratigraphy is divided into the Gula Formation and the Beruas Formation. The Beruas Formation is the uppermost layer and is made up of Holocene terrestrial sediments that are brownish in colour. The Gula Formation is underlying the Beruas Formation. It is made up of silt, clay, and sand with shells. The Gula Formation serves as an impermeable layer, which confines the other formations that are situated beneath it. The classifications of these units are based on age, lithology, and environment of deposition as defined by Bosch (1986), Suntharalingam and Teoh (1985), and Kamaludin (1990). Gula Formation unit is made up of mainly grey to greenish grey that are deposited within marine to subordinate sand and estuarine clay. The formation's thickness differs from 1 to 25 m. Beruas Formation is considered the younger formation within the study area and is deposited in the fluviatile–estuarine–lacustrine that consists of clay, sandy gravel, sandy clay, peat, and silt. Lithologically, the Gula Formation is made up of silt, sand, clay, some gravel, and peat that often have shell fragments. It is thought that this Holocene unit is deposited within a shallow marine environment and an estuarine. The Holocene Beruas Formation is considered a fluvial deposit that is made up of sand, clay, gravel, silt, and the occasional peat (Kamaludin 1989). This formation in the study area differs in thickness from a metre to less than 15 m. The potential of the groundwater aquifers could be seen around Kepala Batas and have a significant amount of gravel and sand layers with a thickness that ranges between 12 and 27 m (Kamaludin 1990). A large part of the Seberang Perai area is underlain by the sedimentary rocks and pre-Quaternary (Courtier 1974). Furthermore, the Seberang Perai coastal area is also underlain by Gula Formation, Simpang Formation, and Beruas Formation, which are quaternary in age. Additionally, the Pleistocene Simpang Formation, which consists of gravel, clay, sand, and local peat and silt, has been considered a fluvial, terrestrial deposit. Generally, the top layer for all boreholes logging possessed a semi-permeable layer such as clayey sand that measures 1–9 m in thickness from ground level (Fig. 1). Permeable material with thickness of 9–24 m was also found underneath the semi-permeable material. All the boreholes demonstrated the same compositions for the permeable materials, which are made up of sandy gravel. The hard layer marine clay is found at a depth of 36–40 m from ground level. The values of permeability vary from 13.65 to 24.36 m/day. The values of transmissivity range from 64 to 125 m2/h. The aquifer specific capacity ranged from 43.92 to 128 m3/h. The values of the piezometer head during pre-monsoon varied between 2.44 and 4.47 m below ground level, while it varied from 1.6 to 3.64 m below ground level after the monsoon seasons.
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a Location of the study area within states of Penang and Kedah and b schematic diagram of Riverbank Filtration (RBF) project at Penang and Kedah states
δ2H–δ18O serve as sources of groundwater recharge. Stable isotopes (δ18O and δ2H) in water that contain their origin's fingerprint characteristics are influenced by meteorological processes, which means that they can therefore be used by means to trace their origin (Clark and Fritz 1997). River water, groundwater, and precipitation samples were collected for the analyses of δ2H and δ18O isotopic from the periods of 2014–2016. Sampling was performed during both dry and wet periods. The study also conducted sampling for δ2H and δ18O isotopic compositions for the 25 locations using samples of surface water (rivers), groundwater (pumping wells and monitoring wells), and precipitation (rainfall). Table 1 and Fig. 1 illustrate the sampling locations. Approximately 120 samples were taken from the test wells and monitoring wells, and seven samples were obtained from the rivers (Muda River). The samples that were taken from the surface water were labelled SW (Muda River). The samples of the groundwater were obtained from the BH1, BH2, BH3, BH 4, BH5, BH6, BH7, BH8, BH10, BH12, BH13, BH15, BH16, BH17, BH18, BH19, BH20, BH21, BH22, TW1, TW2, TW3, and TW4 wells, which were considered to be from an aquifer. Furthermore, the rainwater samples were labelled as (RW). Measurement of the stable isotope contents for the samples of groundwater, surface water, and rainwater was taken at the Isotope Hydrology Laboratory of the Malaysian Nuclear Agency. All the water samples collected for the stable isotopes were measured for δ2H and δ18O with the use of the SerCon GEO 20–20® Continuous Flow Isotope Ratio Mass Spectrometer (CF-IRMS). For the sampling of δ2H and δ18O, the water samples were made to fully fill 100-mL HDPE bottles to make sure that there were no air bubbles. The bottles were then capped tightly. The samples for the analyses of δ2H and δ18O underwent treatment in the SerCon Water Equilibration System (WES) before they were analysed using the isotope ratio mass spectrometer (IRMS). Both values of the δ2H and δ18O were measured in terms of the internal/secondary laboratory standard (MTW). Furthermore, calibration of the values of − 45.00 and − 7.23‰, respectively, was done with the use of the international standard (VSMOW). For 8O and 2H, the expression of their stable isotopic composition is per mil deviation (δ ‰) of the ratio 18O/16O or 2H/1H based on the Standard Mean Ocean Water. δ18O and δ2H had ± 0.1‰ and ± 1.0‰ analytical errors, respectively. The samples for the isotope analysis were measured in triplicates for every analytical run so that more conclusive results can be obtained. Equation 1 is used to compute δ ‰:
$$\permille = \left[{\frac{{R_{\text{sample}} - R_{\text{standard}}}}{{R_{\text{standard}}}}} \right] \times 1000$$
Sampling 24 locations including surface water and groundwater of the study area
BH1
Muda River
Generally, the groundwater isotopic composition is managed by meteorological processes and is meant for the δ2H and δ18O analyses in groundwater system. From the isotope exchange process using the δ18O–δ2H diagram, the result deviates from the meteoric water line that is found along a line having a lower slope, which is affected by the relative humidity. However, some extreme geological environments exist where the meteoric signature of the water can be changed due to the reaction between groundwater and the subsurface gases or the aquifer matrix (Cartwright et al. 2012).
Isotope techniques work especially well in identifying the groundwater recharge's source and the interaction that takes place between the groundwater and the surface water. The mechanisms and sources of recharge were identified using the fundamental relationship between δ2H and δ18O. Figure 2 demonstrates the results for all isotope samples found within the study area. The rainfall (RW) composition revealed that the isotopic signature ranged from − 43.43 to − 8.57 for δ2H and from − 7.19 to − 0.6 for δ18O. The surface water's (river water) composition also revealed that the range of the isotopic signature was from − 48.79 to − 34.14 for δ2H and − 7.06 to − 6.05 for δ18O. Both the groundwater and surface water stable isotope compositions were plotted together along with those for the rainfall that was gathered at the study area (Fig. 2). The Malaysian Meteoric Water Line (MMWL; δ2H = 8 δ18O +13.255; Ayub 2005) and the Global Meteoric Water Line (GMWL; δ2H = 8 δ18O +10; Craig 1961) served as reference lines. The plotted points that were nearer or on the MMWL and GMWL lines were likely to undergo direct recharging from local precipitation with minimal evaporation. The overall stable isotopic composition data that were gathered from river water and groundwater were spread over a range that is relatively small. Groundwater was generally more isotopically 18O enriched. The result revealed that the groundwater samples in the area had isotopic compositions with a narrow range, from − 53.18 to − 38.79 for δ2H and − 8.25 to − 2.80 for δ18O. Furthermore, evaporation affected the some of the samples. This narrow isotopic variation indicated that all the groundwater samples came from the same recharge area or the same water regime, mainly from rainfall (non-evaporated and close to the main river (Muda River). This is indicative to the fact that the groundwater is recharged by Muda River in the confined aquifer and that evaporation takes place before the lower aquifer system is recharged.
The isotopic data for precipitation (rainwater), groundwater, and surface water collected during October, November, March, May, August 2014–2016
The δ2H and δ18O data in rainwater were gathered from the portion of the Muda River Basin that were seen as the representative for the study area and as a provider of isotopic information that was related to the principal recharge source, i.e. precipitation. The isotopic data for groundwater, precipitation (rainwater), and surface water were gathered during October, November, March, May, and August during the years 2014–2016. These data are plotted in Fig. 2 and also presented as part of supplementary Table 1.
Variations in the components' stable isotope composition in a water catchment are due to (a) natural variations in rainfall's isotopic composition, (b) combination with groundwater, and (c) evaporation (Kendall and McDonell 1998; Guglielmi et al. 1998; Hunt et al. 1998; Huddard et al. 1999). Furthermore, stable isotopes δ18O and δ2H are important tools that make it possible to trace water movement.
Comparison of δ18O and chloride (Cl−)
The relationship between δ18O and chloride (Cl−) concentration can also be useful in identifying the groundwater recharge mechanism (Shao 1989; Chen et al. 2006). As demonstrated in Fig. 3, comparing δ18O and Cl− data offers more understanding of the scale of the interactions between groundwater and surface water in the Lower Muda River Basin. It is suggested by the chloride–δ18O plot (Fig. 3) that there are four types of groundwaters found in the study area: (1) some groundwater that comes from the upstream Lower Muda River Basin and is characterised by relatively 18O enriched and low Cl−, representing areas that are recharged more frequently by high rainfall and less frequently by the river; (2) some groundwater located upstream Lower Muda River Basin and characterised by relatively 18O-depleted water and low Cl−; (3) majority of the Lower Muda River Basin's groundwater, characterised by relatively 18O-depleted water and by low Cl−, which signifies that it is most often recharged by river water or that the groundwater is recharging the river water; and (4) highly saline groundwater that is found further downstream of Lower Muda River Basin and is characterised by 18O-depleted water and very high Cl−, which means that it rarely or never receives any surface water recharge. It has been observed that relatively 18O-depleted water signatures recharging and low Cl− characterise recharging by surface water through bank infiltration, whereas diffuse recharge would have a tendency to be 18O enriched in both Cl− and 18O.
Chloride versus δ18O concentration for river water and groundwater in the Lower Muda River Basin
Isotopic compositions of precipitation (rainfall)
For this study, an analysis of a total of five precipitation (rainfall) samples in the Lower Muda River Basin was performed to discuss the rainfall's characteristic isotopic signatures. The precipitation's δ2H was between − 43.83 and − 8.57‰. The range of the δ18O was between − 7.19 and − 0.62‰. This study also examined precipitation's isotopic composition during the dry and wet seasons in Muda River Basin. In wet seasons, the δ2H ranged between − 24.76 and − 8.57‰. On the other hand, the δ18O was found to range between − 3.82 and − 0.62‰. In dry seasons, the δ2H ranged between − 43.43 and 40.30‰. On the other hand, the δ18O ranged between − 7.19 and − 6.90‰ (Fig. 4). Compared to the wet seasons, a more 18O (or D)-depleted water composition and 2H isotopes were observed in the dry seasons. In Malaysia, compared to the northeast monsoon's rainfall in the wet season, the composition of hydrogen isotopes experiences more 18O enriched in the rainfall of the southwest monsoon during dry seasons. Based on the effects of temperature in the atmosphere, the signatures of 18O and 2H isotopes in precipitation may have been enhanced by low temperature. In seasons that experience great rainfall, the isotopes of 18O and 2H in precipitation are significantly lower value in many tropical area due to the rainfall amount effect (Dansgaard 1964), which is brought about by heavy precipitation amounts over a relatively short period of time. Thus, isotopes are also affected by rainfall. The amount of rainfall and temperature has opposite effects on the 18O and 2H isotopes' fractionation. Moreover, the isotopes found in the Muda River Basin experience more 18O enriched during rain seasons than during dry seasons. The 18O and 2H isotopes' signatures can be attributed to the mutual influence that rainfall amount and temperature have, with the effect of the former being stronger than the latter. During seasons of rain in the dry season, there are often heavier and greater amounts of rainfall during a specific period of time, leading to higher precipitation rates despite low temperatures. Consequently, at this time, the 2H and 18O isotopes' composition is depleted in terms of 18O and 2H. Yurtesever and Gat (1981) emphasised that generally the effect of temperature is normally more noticeable in high-latitude continental regions, while the effect of amount is more noticeable in tropical regions. It is commonly known that the precipitation's heavy content of 2H and 18O isotopes decreases when the altitude is increased. For this study, the sites of precipitation are found in the basin's lowest area. Thus, in this study, discussing the range of the effect of altitude in the basin is difficult. Precipitation sites may not be an adequate representation of the lower basin's average precipitation. If the effect of altitude is large, groundwater may also derive from the mountainous area's precipitation with 18O-depleted water isotopic compositions that is akin to those in streams.
Plot of δ2H versus δ18O for precipitation samples for wet and dry seasons. LMWL represents the local meteoric water line
Isotopic compositions of river water
For the river water in the Muda River Basin, the δ2H was between − 48.93 and − 34.14‰. On the other hand, the range of δ18O was from − 7.06 to − 5.34‰. The comparison of the compositions of δ2H and δ18O isotopes for the rainwater and river water revealed that the δ2H and δ18O isotopes from the river water had compositions that matched those from the local meteoric water that are found across the Muda River Basin. This is indicative of the fact that in the river water, rainfall is the primary source. Furthermore, the δ2H and δ18O isotopes found in the river water had compositions that were more 18O-depleted water compared to the precipitation that was observed in the valley, which that the river water is made up of rainfall in the upstream catchment. Thus, the precipitation in the valley had more minimal influence on the river water. During wet seasons, the isotopes of δ2H and δ18O possessed a more 18O-enriched composition in comparison with dry seasons, a phenomenon that is similar to the rainwater. Thus, the seasons influence the composition of the rainwater. In short, one can conclude that seasonal rainfall significantly affects the Muda River Basin's water recharge. Furthermore, as previously mentioned, the compositions of the δ2H and δ18O isotopes of the rainfall that is brought about by the southwest monsoon during the dry season experience more depletion compared to the northeast monsoon during the wet season.
This study also compared the signatures of the δ2H and δ18O isotopes for the river water in the Muda River main stream during dry and rainy seasons. Figure 5 illustrates that there are significant differences in the signatures of the δ2H and δ18O isotopes in the mainstream of the Muda River. The river leaks water into the groundwater system during rainy seasons. Alternatively, in dry seasons, water may be discharged from the groundwater to surface waters. Thus, the compositions of the groundwater were similar to the compositions of the river water, which is an indication that in this area, the source of the groundwater may be linked to the rivers water. During both dry and rainy seasons, the river of the Muda River had isotope signatures having more depleted interflow. Interflow refers to the water's lateral movement in the vadose zone that first enters a river or returns to the surface before becoming groundwater. The interflow mixes with groundwater before joining the Muda River. Consequently, those two had significantly different compositions. Verification of these results may need to be performed by gathering related data.
Plot of δ2H versus δ18O for river water samples during a dry season and b wet seasons
Isotopic compositions of groundwater
This research involved groundwater analysis in the Muda River Basin. Figure 6 illustrates the compositions of δ2H and δ18O isotopes in the groundwater samples that were gathered from the monitoring wells situated along the Lower Muda River Basin. In the Muda River Basin, the δ2H of the groundwater was between − 53.18 and − 20.10‰, having a mean of − 36.5‰. The range of δ18O was from − 8.25 to − 2.80‰, having a mean of − 5.52‰. The river water isotopic composition is managed by the mixing rates for the three main components: interflow, surface run-off, and groundwater (base flow). The rainfall and river water has recharged the basin groundwater. For this study, the compositions of groundwater are similar to those of the river water, which is an indication that in this area, the source of groundwater may be linked to the river water (interaction). Instead of a simple rainfall recharge, the primary groundwater source could be river water. Furthermore, the isotopes of hydrogen and oxygen in the groundwater that is close to the Muda River exhibited a slight deviation from the isotopes of δ2H and δ18O of the local meteoric water line found in the Lower Muda River. Thus, the groundwater could be a combination of river water and rainwater, which could be the reason why the river water recharge had a greater effect compared to the rainfall infiltration. For the wells, the δ18O and δ2H isotopic concentration ranged from − 8.39 to − 2.42 and had a mean of − 6.78 and − 53.18 to − 20.10 along with a mean of − 43.61, respectively. Based on the plot (Fig. 6), it is observable that the sampled monitoring found along the meteoric water lines (MMWL and GMWL) offers evidence of a meteoric origin and the isotopic enhancement through evaporation in the unsaturated zone or on the surface prior to recharge. It is also suggested by the plot (Fig. 6) that direct infiltration of rainwater is likely recharging the groundwater. The monitoring wells' plot also seemed to be grouped within a narrow range, which indicates a mixed system that is well balanced and has a relatively constant isotopic composition.
Relationship between δ2H and δ18O for groundwater during a wet seasons and b dry season
Deuterium excess factor
The value of deuterium excess (d-excess) can be computed using the equation that Dansgaard (1964) formulated as: d = δ2H–8δ18O. The value of the d-excess can be more than 10‰, and it can also range around 9.823‰ in Muda River. Figure 7 demonstrates the relationship between the 18O and the d-excess for the study area. The range of the value of the d-excess is situated between 25.25 and − 9.67‰. This broad range in the observed d-excess values is a reflection of the interaction of recent recharge waters, which were 18O enriched in δ18O values that were taken from rainfall that had low mineralisation, and also for those 18O-depleted water in δ18O values (Maduabuchi et al. 2006; Hoefs 2009; Wassenaar et al. 2011). Based on the findings of Samir (2011), the low values for the d-excess (≤ 6‰) that were observed in the study area's waters were the indications that there was a significant rainwater evaporation, which meant that the residual groundwater was left behind and that they had lower 'd-excess' values. The d-excess value that is > 10‰ is an indication that the source of the recharge is from mixed continental and oceanic vapour (Samir 2011). The d-excess constituent that Dansgaard (1964) defined makes it possible to relate any water sample's isotopic composition to the meteoric water line. Thus, the distribution of values possesses a meteorological significance (Onugba et al. 1990). Generally, the d-excess values exhibit a deviation from 10‰ (Fig. 7). The high values of d-excess are indicative of a evaporative flux from continental waters. It is also considered as a main contributor to the air mass' total moisture balance (Onugba et al. 1990). In this instance, high values of d-excess (> 10‰) are the indication that the evaporative flux from west Malaysia has more significant contributions to the precipitation that is responsible for recharging the groundwater.
Relationship between δ18O (‰) and d-excess for Lower Muda River Basin aquifer
The study has evaluated the isotope hydrology of rivers, rainfall, and groundwater within the Lower River Muda Basin. Gaining a comprehension of the sources of recharge, their mechanisms, and their changes are vital in managing groundwater resources to sustainably satisfy current and future requirements in the Lower Muda River Basin. In terms of the characteristics of Cl− concentrations and δ2H and δ18O compositions, the recharge sources were successfully determined. The composition of the stable isotopes δ2H and δ18O was also successfully determined in the studied waters (river, rainfall, and groundwater). Comparisons between the isotopes of δ2H and δ18O in precipitation revealed that variations in the rainfall amount were resulted the enrichment of δ2H and δ18O isotopes for precipitation in wet seasons compared to precipitation in dry seasons. Compared to precipitation, river water had more 18O-depleted water isotopes of δ2H and δ18O, which signified that the river water was mostly sourced from the upstream catchment. Based on the river, rainfall, and groundwater samples gathered, one can conclude that in the hydrology system groundwater and surface water interact between each other and river water recharges discharge to groundwater depending on rainy and dry season.
The results from the environmental isotope and hydrochemical sampling from the Lower Muda River area are the indications that the perennial river as well as the shallow alluvial aquifer that is near the river in the Lower Muda River Basin area has a relationship that can be called hydraulic interaction. In the Lower Muda River Basin during wet season, the river is mainly losing and recharging the shallow aquifer. With the use of data for the environmental isotope, the alluvial aquifer's recharge by surface water takes place via riverbank filtration and diffuses recharge during events of high rainfall. Although data from the hydrochemical (Cl−) and environmental isotope indicate a relationship between the river and shallow groundwater, one needs more time-series data so that the seasonal change to connectivity can be defined. This is the useful information in gaining a better understanding of the hydrogeological processes taking place at the river aquifer interface and how they are connected to the biogeochemical processes and the policies for water allocation.
The present study was funded as part of the Ministry of Natural Resources and Environment (NRE) to this study under National Water Resources Council (P23101110117300). Many colleagues have contributed to this study. We would like to thank the staffs of the National Hydraulic Research Institute of Malaysia (NAHRIM), for their technical assistance and analysing the samples.
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
1.Hydrogeology Research CentreNational Hydraulic Research Institute of MalaysiaSeri KembanganMalaysia
2.Faculty of Environmental StudiesUniversiti Putra MalaysiaSerdangMalaysia
3.Environmental DivisionMalaysian Nuclear AgencyBangi, KajangMalaysia
Shamsuddin, M.K.N., Sulaiman, W.N.A., Ramli, M.F. et al. Appl Water Sci (2018) 8: 120. https://doi.org/10.1007/s13201-018-0767-x
Received 01 November 2017
Accepted 05 July 2018
DOI https://doi.org/10.1007/s13201-018-0767-x
King Abdulaziz City for Science and Technology
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\begin{document}
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\title{Global well-posedness of large scale moist atmosphere system with only horizontal viscosity in the dynamic equation } \author{Shenyang Tan$^{1,2}$,\ \ Wenjun Liu$^{1}$\footnote{Email address: [email protected] (W. Liu), [email protected] (S. Tan). }\ \
\\$^{1}$School of Mathematics and Statistics, Nanjing University of Information Science \\
and Technology, Nanjing 210044, China\\ $^{2}$Taizhou Institute of Sci. $\&$ Tech. NJUST, Taizhou 225300, China}
\date{} \maketitle
\begin{abstract}
In order to find a better physical model to describe the large-scale cloud-water transformation and rainfall, we consider a moist atmosphere model consisting of the primitive equations with only horizontal viscosity in the dynamic equation and a set of humidity equations describing water vapor, rain water and cloud condensates. To overcome difficulties caused by the absence of vertical viscosity in the dynamic equation, we get the local existence of $v$ in $H^{1}$ space by combining the viscous elimination method and the $z-$weak solution method and using the generalized Bihari-Lasalle inequality. And then, we get the global existence of $v$ under higher regularity assumption of initial data. In turn, the existence of quasi-strong and strong solutions to the whole system is obtained. By introducing two new unknown quantities appropriately and utilizing the monotone operator theory to overcome difficulties caused by the Heaviside function in the source terms, we get the uniqueness of solutions. \end{abstract}
\noindent {\bf 2010 Mathematics Subject Classification:} 35Q35, 35Q86, 35B65. \\ \noindent {\bf Keywords:} primitive equations, multi-phase, well-posedness.
\maketitle
\section{Introduction } The main factors that determine cloud formation and precipitation are the thermal and dynamic processes of atmospheric movement, the content of water vapor, and the microphysical factors of cloud and precipitation. In a noninertial coordinate system, under the $(x,y,p)$ coordinates, the large-scale moist atmosphere system is formed by coupling the primitive equations \begin{align} \label{pe1} &\partial_{t}v+(v\cdot\nabla)v +\omega\partial_{p}v+\nabla\Phi+f v^{\bot}+\mathcal{A}_{v}v=S_{v},\\
\label{pe2}&\partial_{p}\Phi=-\frac{RT}{p},\\ \label{pe3}&\nabla\cdot v+\partial_{p}\omega=0,\\ &\partial_{t}T+v\cdot\nabla T+\omega\partial_{p}T-\frac{RT}{c_{p}p}\omega+\mathcal{A}_{T}T=S_{T},\label{pe4} \end{align} and the conservation equation of water in the air \begin{align} \frac{d}{dt}q+\mathcal{A}_{q}q=S_{q}.\label{pe5} \end{align} Here $v,\omega,T,q$ are unknown functions, where $v=(v_{1},v_{2})$ is the horizontal velocity vector, $v^{\bot}=(-v_{2},v_{1})$, $w$ is the vertical velocity under the $(x,y,p)$ coordinates, $\nabla=(\partial_{x},\partial_{y})$, $T$ is the temperature, $\Phi$ is the geopotential, $f$ is the Coriolis force parameter, $R$ is the gas constants for dry air, $c_{p}$ is the specific heat of air at constant pressure, $p$ is the pressure, $S_{T}$ represents the sum of the heat increased or reduced by solar heating and condensation or evaporation, $S_{q}$ represents the amount of water added or removed by condensation or evaporation, $S_{v}$ is a forcing term added for mathematical generality which does not exist in reality, $\mathcal{A}_{v},\mathcal{A}_{T},\mathcal{A}_{q}$ are the viscosity terms, $\mathcal{A}_{\ast}=-\mu_{\ast}\Delta-\nu_{\ast}\frac{\partial}{\partial p}\left(\left(\frac{gp}{R\bar{T}}\right)^{2}\frac{\partial}{\partial p}\right), \ \ast=v,T,q,$ where $\Delta$ is the horizontal Laplacian, $\bar{T}$ is a given temperature distribution.
Equations (\ref{pe1})-(\ref{pe4}) is the well-known primitive equations which is widely used in numerical weather forecasting of large-scale atmosphere. It was first proposed systematically in mathematics by Lions, Temam and Wang \cite{Lions} in 1992. Since then, the mathematical theory about the primitive equations has been studied by many mathematicians, see \cite{CaoTiti,CaoTiti2,CaoTiti3,CaoTiti4,CaoTiti5,CaoTiti6,Li2016,CLT2020,Ju,Ju2,kukavica1,kukavica2,You,Zhou,Gao1,Gao2,Temambook} and references therein. Recently, the primitive equations with only partial dissipation or partial viscosity have been also studied extensively and deeply. Cao and Titi\cite{CaoTiti2} studied the primitive equations with only vertical dissipation in the thermal equation and full viscosity in the dynamic equation. They obtained the global existence for strong solutions. Cao, Li and Titi \cite{CaoTiti6,CLT2020} considered the primitive equations with only horizontal eddy viscosity in the momentum equation and only horizontal diffusion in the temperature equation. They got the global existence of strong solutions with near $H^{1}$ initial data under the periodic boundary conditions. Hussein, Saal and Wrona\cite{Hussein} further studied the primitive equations with only horizontal viscosity in momentum equation with physical boundary conditions.
They got the local existence of $z-$weak solutions and the global existence of strong solutions as well as uniqueness of solutions through the way of Galerkin approximation. For other cases with partial viscosity or non-viscosity, we refer readers to \cite{CaoTiti3,CaoTiti4,CaoTiti5,Li2022} and \cite{Ibrahimnoviscosity,Ghoul,CaoTiti7,Saal} respectively.
As to the moist atmosphere system, inspired by the work in \cite{Lions}, Guo and Huang \cite{Guo1,Guo2} proposed a new mathematical formulation of the large scale moist atmosphere in 2006. They obtained the global existence of weak solutions. In the early study on moist atmospheric system, only one humidity equation was included, neither water vapor saturation nor microphysical factors were considered. In fact, when the concentration of water vapor in the air reaches a certain saturation concentration $q_{vs}$, the phase transition will occur. In the last decade, Coti Zelati and Temam et al. \cite{Zelati,Zelati2,Zelati3,Bousquet,Temam-wu,Temam-Wang} studied the saturation phenomenon and used the multi-Heaviside function $H(q_{v}-q_{vs})$ to describe the phase transition of water vapor. Hittmeir, Li, Titi, et al. \cite{Hittmeir2017,Hittmeir} and Cao, Temam, et al. \cite{Cao1} respectively studied more detailed moist atmosphere models, in which not only multi-phases but also the microphysics factors were taken into account.
Inspired by the work of \cite{Cao1} and \cite{Hittmeir}, we also considered a multi-phase moist atmospheric system in \cite{TanLiu}, and get the well-posedness of strong solutions under the assumption that $q_{vs}$ is a constant.
In large-scale atmospheric dynamics, strong horizontal turbulent mixing results in much higher horizontal viscosity than vertical viscosity. However, the velocity field $v$ in the above models is either a given velocity field or is controlled by the primitive equations with full viscosity.
Little work has been done about the study of humid atmospheric system with partial diffusion or partial viscosity. Inspired by the work mentioned above, in this paper we consider the following multi-phase moist atmosphere equations with only horizontal viscosity in the velocity equation and full dissipation in other equations: \begin{align}\label{original system} &\partial_{t}v+(v\cdot\nabla)v+\omega\partial_{p}v +\nabla\Phi+fv^{\bot}-\mu_{v}\Delta v=0,\\ &\partial_{p}\Phi=-\frac{RT}{p},\\ &\nabla\cdot v+\partial_{p}w=0,\\ &\partial_{t}T+v\cdot\nabla T+w\partial_{p}T-\frac{RT}{c_{p}p}w+\mathcal{A}_{T}T=f_{T}+\frac{L}{c_{p}}(W_{cd}-W_{ev}) ,\\ &\partial_{t}q_{v}+v\cdot\nabla q_{v}+w\partial_{p}q_{v}+\mathcal{A}_{q_{v}} q_{v} =W_{ev}-W_{cd},\\ &\partial_{t}q_{c}+v\cdot\nabla q_{c}+w\partial_{p}q_{c}+\mathcal{A}_{q_{c}}q_{c} =W_{cd}-W_{ac}-W_{cr},\\ &\partial_{t}q_{r}+v\cdot\nabla q_{r}+w\partial_{p}q_{r}+\mathcal{A}_{q_{r}} q_{r} =g\frac{\partial}{\partial p}(\rho q_{r}V_{t})+W_{ac}+W_{cr}-W_{ev},\label{original system 11} \end{align} where $L$ is the latent heat of vaporization, $W_{ev}$ is the rate of evaporation of rain water, $W_{cd}$ is the condensation of water vapor to cloud water, $W_{ac}$ is the auto-conversion of cloud water vapor into rain water by accumulation of microscopic droplets, $W_{cr}$ is the collection of cloud water by falling rain, $V_{t}$ is the terminal velocity of the falling rain, $f_{T}$ is a source term for temperature variation.
Due to the lack of vertical viscosity in the dynamic equation as well as the characteristics of microphysical factors in the source term for the temperature equation, the classical methods for dealing with the $H^{1}-$regularity of velocity fields $v$ in references \cite{CaoTiti6,CLT2020,Zelati} do not work. In order to overcome this difficulty, we combine the viscous elimination method and the idea of $z-$weak solution, and use the generalized Bihari-Lasalle inequality to get the local existence of $v$ in $H^{1}$ space. And then, under higher regularity assumption of initial data, we obtain the global existence of strong solutions as in \cite{CaoTiti6}. The absence of vertical viscosity in dynamic equation makes the term $\|\partial_{p}^{2}v\|_{L^{2}(0,t;L^{2})}$ unbounded. This term must be avoided in the a priori estimates. Fortunately, we can solve this problem with the helpful inequalities introduced in \cite{CaoTiti6,CLT2020}. In addition, the emergence of the Heaviside function greatly increases the difficulty of estimating the source terms. During the proof of uniqueness of solutions, we take advantage of the monotonicity of the Heaviside function and introduce two new unknowns $Q=q_{v}+q_{c}$ and $H=T+\frac{L}{c_{p}}q_{v}$ to circumvent difficulties caused by the Heaviside function.
The rest of this paper is organized as follows. In section 2, we give the mathematical formulation of moist atmosphere system with only horizontal viscosity in the dynamic equation and the main results about the existence of quasi-strong and strong solutions. In section 3, we introduce an approximated system. By finding uniform boundness of approximate solutions, we get the local and global existence of quasi-strong and strong solutions. In section 4, by introducing two new unknown quantities, we get the uniqueness of solutions.
\section{Mathematical formulation and main results} By introducing the potential temperature \cite{Zelati} \begin{align*} \theta=T\left(\frac{p_{0}}{p}\right)^{R/c_{p}}-\theta_{h}, \end{align*} where $\theta_{h}$ is a reference temperature satisfying $\theta_{h},\partial\theta_{h}/\partial p\in L^{\infty}\left((0,t)\times\mathcal{M}\right)$, the equation for $T$ becomes \begin{align*} \partial_{t}\theta+v\cdot\nabla\theta+w\partial_{p}\theta=Q_{\theta}, \end{align*} where $Q_{\theta}$ is the corresponding source term for $\theta$.
In this paper, we introduce the multi-Heaviside function to describe the transient phase transitions as in \cite{Cao1,Zelati,Temam-wu}. In addition, we use expressions in \cite{Hernandez,Deng,Majda,klemp} to describe the source terms. Namely, \begin{align}\label{equationwithoutv} \partial_{t}\theta+v\cdot\nabla\theta+w\partial_{p}\theta+\mathcal{A}_{\theta}\theta &\in f_{\theta}(q_{v},q_{c},q_{r},\theta) +\frac{L}{c_{p}\Pi}w^{-}\tilde{F}\mathcal{H}(q_{v}-q_{vs}),\nonumber\\ \partial_{t}q_{v}+v\cdot\nabla q_{v}+w\partial_{p}q_{v}+\mathcal{A}_{q_{v}}q_{v}&\in f_{q_{v}}(q_{v},q_{c},q_{r},\theta)-w^{-}F\mathcal{H}(q_{v}-q_{vs}),\nonumber\\ \partial_{t}q_{c}+v\cdot\nabla q_{c}+w\partial_{p}q_{c}+\mathcal{A}_{q_{c}}q_{c}&\in f_{q_{c}}(q_{v},q_{c},q_{r},\theta)+w^{-}F\mathcal{H}(q_{v}-q_{vs}),\nonumber\\ \partial_{t}q_{r}+v\cdot\nabla q_{r}+w\partial_{p}q_{r}+\mathcal{A}_{q_{r}}q_{r}&= f_{q_{r}}(q_{v},q_{c},q_{r},\theta). \end{align} Here $w^{-}=\max\{-w,0\}$, $\Pi=(p/p_{0})^{\gamma}$, and \begin{align} \mathcal{A}_{\theta}=-\mu_{\theta}\Delta-\nu_{\theta}\left(p_{0}/p\right)^{R/c_{p}} \partial_{p}\left(\left(gp/R\bar{\theta}\right)^{2}\partial_{p}\left(p_{0}/p\right)^{R/c_{p}}\right), \end{align} where $\gamma$ is the ratio of specific heats at constant pressure and at constant volume, $\bar{\theta}$ is the given potential temperature profile. For the convenience of mathematical processing, we take all the viscosity coefficients as constant $1$ in this paper (since $\mathcal{A}_{q_v}=\mathcal{A}_{q_c}=\mathcal{A}_{q_r}$, for simplicity, we use $\mathcal{A}_{q}$ to represent them in section 4).
$F$ can be expressed as \begin{equation}\label{F} F=F(T,p)=\frac{q_{vs}\phi(T)}{p}\left( \frac{LR-c_{p}R_{v}\phi(T)}{c_{p}R_{v}\phi(T)^{2}+q_{vs}L^{2}}\right), \end{equation} where \begin{eqnarray*} \phi(T) \begin{cases} =T, & T_{\ast}\leq T\leq T^{\ast}, \\ \geq T_{\ast}/2, & T\leq T_{\ast}, \\ 0, & T\geq 2T_{\ast}. \end{cases} \end{eqnarray*} Here $T_{\ast}>0$ is a given temperature smaller than any temperature on earth and $T^{\ast}$ is also a given temperature larger than any temperature on earth. It was verified in \cite{Zelati} that $F$ is uniformly bounded and Lipschitz continuous with respect to $T$. $\mathcal{H}$ is the classical multi-valued Heaviside function. And \begin{equation}\label{f-theta} f_{\theta}(q_{v}, q_{c}, q_{r})=-\frac{gp}{R\Pi\theta_{h}}\frac{\partial\theta_{h}}{\partial p}w-\frac{L}{c_{p}\Pi}k_{3}\tau(q_{r})(q_{vs}-q_{v})^{+}+w\frac{\partial\theta_{h}(p)}{\partial p}+f_{\theta}^{1}, \end{equation} \begin{equation}\label{f-qv} f_{q_{v}}(q_{v}, q_{c}, q_{r})=k_{3}\tau(q_{r})(q_{vs}-q_{v})^{+}, \end{equation} \begin{equation}\label{f-qc} f_{q_{c}}(q_{v}, q_{c}, q_{r})=-k_{1}(q_{c}-q_{crit})^{+}-k_{2}q_{c}\tau(q_{r}), \end{equation} \begin{align}\label{f-qr} f_{q_{r}}(q_{v}, q_{c}, q_{r})=&V_{t}\partial_{p}(\frac{p}{R\bar{\theta}}q_{r}) +k_{1}(q_{c}-q_{crit})^{+}+k_{2}q_{c}\tau(q_{r})-k_{3}\tau(q_{r})(q_{vs}-q_{v})^{+}, \end{align} where $k_{1}, k_{2}, k_{3}$ are some dimensionless rate constants, $f_{\theta}^{1}\in L^{2}(\mathcal{M})$ is a source term, $q_{crit}$ is a given constant representing the threshold of the cloud-water mixing ratio. Considering the physical meaning of $q_{r}(0\leq q_{r}\leq 1),$ we define $\tau(q_{r})$ as \begin{eqnarray}\label{tau} \tau(q_{r})= \begin{cases} 0,&q_{r}<0,\\ q_{r},&0\leq q_{r}\leq1,\\ 1,&q_{r}>1. \end{cases} \end{eqnarray}
In summary, we mainly consider the following multi-phase moist atmosphere system in this paper: \begin{equation}\label{e1} \partial_{t}v-\Delta v+(v\cdot\nabla)v+w\partial_{p}v+\nabla\Phi+fv^{\bot}=0, \end{equation} \begin{equation}\label{e2} \partial_{p}\Phi+\frac{RT}{p}=0, \end{equation} \begin{equation}\label{e3} \nabla\cdot v+\partial_{p}w=0, \end{equation} \begin{equation}\label{e4} \partial_{t}\theta+\mathcal{A}_{\theta}\theta+v\cdot\nabla\theta+w\partial_{p}\theta\in f_{\theta}(q_{v},q_{c},q_{r},\theta)+\frac{L}{c_{p}\Pi}w^{-}\tilde{F}\mathcal{H}(q_{v}-q_{vs}), \end{equation} \begin{equation}\label{e5} \partial_{t}q_{v}+\mathcal{A}_{q_{v}}q_{v}+v\cdot\nabla q_{v}+w\partial_{p}q_{v}\in f_{q_{v}}(q_{v},q_{c},q_{r},\theta)-w^{-}F\mathcal{H}(q_{v}-q_{vs}), \end{equation} \begin{equation}\label{e6} \partial_{t}q_{c}+\mathcal{A}_{q_{c}}q_{c}+v\cdot\nabla q_{c}+w\partial_{p}q_{c}\in f_{q_{c}}(q_{v},q_{c},q_{r},\theta)+w^{-}F\mathcal{H}(q_{v}-q_{vs}), \end{equation} \begin{equation}\label{e7} \partial_{t}q_{r}+\mathcal{A}_{q_{r}}q_{r}+v\cdot\nabla q_{r}+w\partial_{p}q_{r}= f_{q_{r}}(q_{v},q_{c},q_{r},\theta). \end{equation}
We assume $\mathcal{M}=\mathcal{M}'\times (p_{0},p_{1})$, where $\mathcal{M}'$ is a smooth bounded domain in $\mathbb{R}^{2}$, and $p_{0}<p_{1}$ are positive constants. The boundary of $\mathcal{M}$ is composed of $\Gamma_{i}, \Gamma_{u}, \Gamma_{l}$, where \begin{align} \Gamma_{i}&=\{(x,y,p)\in\bar{\mathcal{M}}:p=p_{1}\},\nonumber\\ \Gamma_{u}&=\{(x,y,p)\in\bar{\mathcal{M}}:p=p_{0}\},\nonumber\\ \Gamma_{l}&=\{(x,y,p)\in\bar{\mathcal{M}}:(x,y)\in\partial\mathcal{M}',p_{0}\leq p\leq p_{1}\}.\nonumber \end{align}
The boundary conditions are:
\begin{align}\label{boundary condition}
&{\rm on}\ \Gamma_{i}:\ \partial_{p}v=0,\ \ w=0,\ \partial_{p}\theta=\theta_{\ast}-\theta,\
\partial_{p}q_{j}=q_{j\ast}-q_{j}, j\in \{v, c, r\};\nonumber\\
&{\rm on}\ \Gamma_{u}:\ \partial_{p}v=0,\ w=0,\ \partial_{p}w=0,\ \partial_{p}\theta=0,\ \partial_{p}q_{j}=0,\ j\in \{v, c, r\};\\
&{\rm on}\ \Gamma_{l}:\ v=0,\ \ \partial_{\textbf{n}}v=0,\ \ \partial_{\textbf{n}}\theta=\theta_{bl}-\theta,\ \partial_{\textbf{n}}q_{j}=q_{blj}-q_{j},\ j\in \{v, c, r\}.\nonumber
\end{align}
where $\textbf{n}$ is the unit normal vector of $\Gamma_{l}$,
$\theta_{bl}, q_{blv}, q_{blc}, q_{blr}$ are given nonnegative and sufficiently smooth functions, $\theta_{\ast}$ is a typical potential temperature, $q_{v*}, q_{c*}, q_{r*}$ are given humidity distribution, which are given nonnegative and sufficiently smooth functions.\\ In addition, The Initial conditions: \begin{align}\label{initial condition} v(x,y,p,0)&=v_{0}(x,y,p),\nonumber\\ \theta(x,y,p,0)&=\theta_{0}(x,y,p),\\ q_{j}(x,y,p,0)&=q_{j0}(x,y,p),\ j\in\{v, c, r\}.\nonumber \end{align}
We denote
$\|(f_{1},\cdots,f_{n})\|_{L^{2}(\mathcal{M})}=\sum_{j=1}^{n}\|f_{j}\|_{L^{2}(\mathcal{M})}^{2},
$ and $\|f\|_{w}=\left\|\left(gp/R\bar{\theta}\right)f\right\|_{L^{2}(\mathcal{M})}. $
Obviously, $\|f\|_{w}$ is equivalent to $\|f\|_{L^{2}}$. For simplicity, we use $H^{s}$ to represent the classical Sobolev spaces $H^{s}(\mathcal{M})$, and use $L^{p}$ to represent the classical Lebesgue space $L^{p}(\mathcal{M}),1\leq p\leq\infty$.
As usual, we introduce the space \begin{align*} \mathcal{V}&=\left\{v\in C^{\infty}(\mathcal{M};\mathbb{R}^{2}):\nabla\cdot\int_{p_{0}}^{p_{1}}v(x,y,p')dp'=0,v\ \text{satisfies}\ (\ref{e1})\right\},\\ \mathbb{H}&=\text{The closure of}\ \mathcal{V}\ \text{with respect to the norm of}\ (L^{2})^{2}, \\ \mathbb{V}&=\text{The closure of}\ \mathcal{V}\ \text{with respect to the norm of}\ (H^{1})^{2}. \end{align*} \begin{definition} Let $v_{0}\in\mathbb{V}$, $\theta_{0},q_{v0},q_{r0},q_{c0}\in L^{2}$. If for some $h_{q}\in L^{\infty}(\mathcal{M}\times (0,t_{1}))$ satisfying the variational inequality \begin{align} ([\tilde{q}_{v}-q_{vs}]^{+},1)-([q_{v}-q_{vs}]^{+},1)\geq \langle h_{q},\tilde{q}_{v}-q_{v}\rangle, \ \text{a.e.}\ t\in[0,t_{1}],\forall \tilde{q}_{v}\in H^{1}, \end{align} a solution $(v,\theta,q_{v},q_{c}.q_{r})$ to equations (\ref{e1})-(\ref{e7}) with boundary condition (\ref{boundary condition}) and initial data (\ref{initial condition}) satisfies that \begin{align*} &v\in C(0,t_{1};\mathbb{V}), \ \ \Delta v,\nabla\partial_{p}v\in L^{2}(0,t_{1};L^{2}),\nonumber\\ &(\theta,q_{v},q_{r},q_{c})\in C(0,t_{1};(L^{2})^{4})\cap L^{2}(0,t_{1}; (H^{1})^{4}),\nonumber\\ &\partial_{t}v\in L^{2}(0,t_{1};\mathbb{H}),\ \ \partial_{t}(\theta,q_{v},q_{c},q_{r})\in L^{2}(0,t_{1}; (H^{-1})^{4}). \end{align*} Then we call $(v,\theta,q_{v},q_{c},q_{r})$ a quasi-strong solution to equations (\ref{e1})-(\ref{e7}) with boundary condition (\ref{boundary condition}) and initial data (\ref{initial condition}). \end{definition} \begin{definition} Let $v_{0}\in\mathbb{V},\theta_{0}, q_{v0},q_{r0},q_{c0}\in H^{1}$. A quasi-strong solution $(v,\theta,q_{v},q_{c},q_{r})$ to equations (\ref{e1})-(\ref{e7}) with boundary condition (\ref{boundary condition}) and initial data (\ref{initial condition}) is called a strong solution if \begin{align*} &(\theta,q_{v},q_{r},q_{c})\in C(0,t_{1};(H^{1})^{4})\cap L^{2}(0,t_{1}; (H^{2})^{4}), \nonumber\\ &\partial_{t}(v,\theta,q_{v},q_{c},q_{r})\in L^{2}(0,t_{1};\mathbb{H}\times (L^{2})^{4}). \end{align*} \end{definition}
We state our main results as follows. \begin{theorem}\label{quasistrong existence global}
Let $v_{0}\in\mathbb{V}\cap L^{\infty}$, $\partial_{p}v_{0}\in L^{m}$ for some $m>2$, $\theta_{0}, q_{v0},q_{r0},q_{c0}\in L^{2}$, and $\|v_{0}\|_{H^{1}}+\|\partial_{p}v_{0}\|_{L^{m}}+
\|v_{0}\|_{L^{\infty}}<\infty$. Then there exist at least one global in time quasi-strong solution to equations (\ref{e1})-(\ref{e7}) associated with the initial data (\ref{initial condition}) and boundary condition (\ref{boundary condition}). When $\theta_{0}, q_{v0},q_{r0},q_{c0}\geq 0$ in $\mathcal{M}$ and $F$ is replaced by its positive part $F^{+}$, the solution is unique. \end{theorem} \begin{theorem}\label{strong existence global} Let $v_{0}\in \mathbb{V}\cap L^{\infty}$, $\partial_{p}v_{0}\in L^{m}$ for some $m>2$, $\theta_{0},q_{v},q_{c},q_{r}\in H^{1}$, and
$\|v_{0}\|_{H^{1}}+\|\partial_{p}v_{0}\|_{L^{m}}+
\|v_{0}\|_{L^{\infty}}<\infty$. Then equations (\ref{e1})-(\ref{e7}) associated with the initial data (\ref{initial condition}) and boundary condition (\ref{boundary condition}) has at least one global in time strong solution $(v,\theta,q_{v},q_{c},q_{r})$. When $\theta_{0}, q_{v0},q_{r0},q_{c0}\geq 0$ in $\mathcal{M}$ and $F$ is replaced by its positive part $F^{+}$, the solution is unique.
\end{theorem}
Proofs of theorems are naturally divided into two parts: existence and uniqueness, which will be given in Section 3 and Section 4 respectively.
\section{Existence of quasi-strong and strong solutions} In this section, we mainly consider the existence of solutions. We first propose a regularized approximated system that approximates equations (\ref{e1})-(\ref{e7}) in a suitable sense. By finding uniform a priori bounds for solutions and taking limit of approximated solutions, we get the existence of quasi-strong and strong solutions. \subsection{An approximated problem } In order to deal with the differential inclusion, as in \cite{Zelati,Temam-wu}, we select a single-valued Heaviside function $h_{q_{v}}\in \mathcal{H}(q_{v}-q_{vs})$ satisfying that \begin{align}\label{variational inequality} ([\tilde{q}_{v}-q_{vs}]^{+},1)-([q_{v}-q_{vs}]^{+},1)\geq \langle h_{q_{v}},\tilde{q}_{v}-q_{v}\rangle, \ \text{a.e.}\ t\in[0,t_{1}],\forall \tilde{q}_{v}\in H^{1}. \end{align} As in \cite{Cao1,Zelati,Zelati3}, we define \begin{eqnarray}\label{H-epsilon2} \mathcal{H}_{\epsilon_{2}}(r)= \begin{cases} 0, & r\leq 0, \\ r/\epsilon_{2}, & r\in (0,\epsilon_{2}], \\ 1, & r>\epsilon_{2}. \end{cases} \end{eqnarray} Thus $\mathcal{H}_{\epsilon_{2}}$ can overcome the discontinuity caused by the Heaviside function $h_{q_{v}}$. Additionally, \begin{align}\label{H-epsilon21}
|\mathcal{H}_{\epsilon_{2}}(r)|\leq 1,\qquad |\mathcal{H}_{\epsilon_{2}}(r_{1})-\mathcal{H}_{\epsilon_{2}}(r_{2})|\leq\frac{1}{\epsilon_{2}}|r_{1}-r_{2}|, \quad\forall r_{1}, r_{2}\in \mathbb{R}. \end{align}
In this section we consider the following approximated problem: \begin{equation}\label{ae1} \partial_{t}v^{\epsilon}-\Delta v^{\epsilon}-\epsilon_{1}\partial_{p}\left(\left(\frac{gp}{R\bar{\theta}}\right)^{2}\partial_ {p}v^{\epsilon}\right)+(v^{\epsilon}\cdot\nabla)v^{\epsilon}+w^{\epsilon}\partial_{p}v^{\epsilon}+\nabla\Phi^{\epsilon} +f{v^{\epsilon}}^{\bot}=0, \end{equation} \begin{equation}\label{ae2} \partial_{t}\theta^{\epsilon}+\mathcal{A}_{\theta}\theta^{\epsilon}+v^{\epsilon}\cdot\nabla\theta^{\epsilon} +w^{\epsilon}\partial_{p}\theta^{\epsilon}= f_{\theta^{\epsilon}}+\frac{L}{c_{p}\Pi}{w^{\epsilon}}^{-}\tilde{F}\mathcal{H}_{\epsilon_{2}}, \end{equation} \begin{equation}\label{ae3} \partial_{t}q_{v}^{\epsilon}+\mathcal{A}_{q_{v}}q_{v}^{\epsilon}+v^{\epsilon}\cdot\nabla q_{v}^{\epsilon}+w^{\epsilon}\partial_{p}q_{v}^{\epsilon} = f_{q_{v}^{\epsilon}}-{w^{\epsilon}}^{-}F\mathcal{H}_{\epsilon_{2}}, \end{equation} \begin{equation}\label{ae4} \partial_{t}q_{c}^{\epsilon}+\mathcal{A}_{q_{c}}q_{c}^{\epsilon}+v^{\epsilon}\cdot\nabla q_{c}^{\epsilon}+w^{\epsilon}\partial_{p}q_{c}^{\epsilon}= f_{q_{c}^{\epsilon}}+{w^{\epsilon}}^{-}F\mathcal{H}_{\epsilon_{2}}, \end{equation} \begin{equation}\label{ae5} \partial_{t}q_{r}^{\epsilon}+\mathcal{A}_{q_{r}}q_{r}^{\epsilon}+v^{\epsilon}\cdot\nabla q_{r}^{\epsilon}+w^{\epsilon}\partial_{p}q_{r}^{\epsilon}= f_{q_{r}^{\epsilon}}. \end{equation} In addition, we still supplement the approximated system with boundary condition (\ref{boundary condition}) and initial condition (\ref{initial condition}).
Obviously, through similar argument as in \cite{TanLiu}, we can have the following well-posedness result:
\begin{proposition}
For any given $\epsilon=(\epsilon_{1},\epsilon_{2})$, and given $h_{q}\in L^{\infty}(\mathcal{M}\times (0,t_{1}))$ satisfying (\ref{variational inequality}),
\begin{align*} &(i)\ \text{If}\ v_{0}\in\mathbb{V},\theta_{0}, q_{v0},q_{r0},q_{c0}\in L^{2},\ \text{then equations (\ref{ae1})-(\ref{ae5})}\ \text{with boundary condition (\ref{boundary condition}) and}\\ \
&\text{ initial condition (\ref{initial condition})}\ \text{has at least one global in time quasi-strong solution }
(v^{\epsilon},\theta^{\epsilon},q_{v}^{\epsilon},q_{c}^{\epsilon},q_{r}^{\epsilon}). \\ &(ii)\ \text{If}\ v_{0}\in\mathbb{V},\theta_{0}, q_{v0},q_{r0},q_{c0}\in H^{1},\
\text{then there exists at least one global in time strong solution}\ \\
&(v^{\epsilon},\theta^{\epsilon},q_{v}^{\epsilon},q_{c}^{\epsilon},q_{r}^{\epsilon})\ \text{to equations (\ref{ae1})-(\ref{ae5})}\ \text{with conditions (\ref{boundary condition}) and (\ref{initial condition})}\\ \
\end{align*}
\end{proposition}
In subsequent subsections, we focus on finding uniform a priori bounds for solutions respect to $\epsilon$. \subsection{A priori $L^{2}$-estimates}
The following two lemmas are useful in the estimation of trilinear terms. \begin{lemma}\label{HHP}(\cite[Lemma 6.2]{Hussein}) There exists a constant $C>0$ such that \begin{align}
\int_{\mathcal{M}}|f||g||h|d\mathcal{M}\leq C\|\nabla f\|_{L^{2}}^{\frac{1}{2}}
\| f\|_{L^{2}}^{\frac{1}{2}}\|\nabla g\|_{L^{2}}^{\frac{1}{2}}
\| g\|_{L^{2}}^{\frac{1}{2}}\left(\|h\|_{L^{2}}^{\frac{1}{2}}
\| \partial_{p}h\|_{L^{2}}^{\frac{1}{2}}+ \|h\|_{L^{2}}\right) \end{align} for $f,g\in L^{2}(p_{0},p_{1};H_{0}^{1}(G)),h\in H^{1}(p_{0},p_{1};L^{2}(G))$. \end{lemma}
\begin{lemma}(\cite[Lemma 2.1]{CaoTiti6})\label{trilinear term lemma} For any $f,g,h \in H^1$, the following inequality holds true: \begin{align*}
\int_{\mathcal{M}'} \int_{p_{0}}^{p_{1}} fdp\int_{p_{0}}^{p_{1}} ghdpd\mathcal{M}'
\leq& {\rm{min}}\left\{C \|f\|_{L^2}^{\frac{1}{2}}\left(\|f\|_{L^{2}}^{\frac{1}{2}} + \|\nabla f\|_{L^{2}}^{\frac{1}{2}}\right)
\|g\|_{L^{2}} \|h\|_{L^{2}}^{\frac{1}{2}}\left(\|h\|_{L^{2}}^{\frac{1}{2}}+ \|\nabla h\|_{L^{2}}^{\frac{1}{2}}\right),\right.\\
&\left.C\|f\|_{L^2}\|g\|_{L^{2}}^{\frac{1}{2}}
\left(\|g\|_{L^{2}}^{\frac{1}{2}} + \|\nabla g\|_{L^{2}}^{\frac{1}{2}}\right) \|h\|_{L^{2}}^{\frac{1}{2}}\left(\|h\|_{L^{2}}^{\frac{1}{2}}+ \|\nabla h\|_{L^{2}}^{\frac{1}{2}}\right)\right\}. \end{align*} \end{lemma}
Next we introduce the generalized Bihari-Lasalle inequality, which is useful during the proof of local existence of solutions. \begin{lemma}(\cite[Theorem 28]{Dragomir})\label{nonlinearGronwall} Let $\beta(t):[t_{0},\infty)\rightarrow [0,\infty)$ be continuous function, $f(t)$ be nonnegative differentiable function, $\alpha(t)$ be nonnegative and nonincreasing differentiable functions, $g(s):[0,\infty)\rightarrow (0,\infty)$ be strictly increasing function, $u(t):[t_{0},\infty)\rightarrow [0,\infty)$ be continuous function. Suppose that \begin{align} u(t)\leq f(t)+\alpha(t)\int_{t_{0}}^{t}\beta(s)g(u(s))ds \end{align} and $f'(t)\left[\frac{1}{g(\gamma(t))}-1\right]\leq 0 $ on $[t_{0},\infty)$ for every nonnegative continuous function $\gamma$. Then \begin{align}\label{nonlinearGronwall1} u(t)\leq \mathcal{G}^{-1}\{\mathcal{G}(f(t_{0})) +\int_{t_{0}}^{t}[\alpha(s)\beta(s)+f'(s)]ds\} \end{align} where \begin{align} \mathcal{G}(r)=\int_{r_{0}}^{r}\frac{1}{g(s)}ds,r\geq r_{0}\geq0, \end{align} and (\ref{nonlinearGronwall1}) holds for all values of $t$ for which the function \begin{align} r(t)=\mathcal{G}[f(t_{0})]+\int_{t_{0}}^{t}[\alpha(s)\beta(s)+f'(s)]ds \end{align} belongs to the domain of the inverse function $\mathcal{G}^{-1}$. \end{lemma}
Fix $\epsilon>0$, by virtue of the similar $L^{2}$ a priori estimates carried out in \cite{Cao1,Zelati}, we can immediately conclude the following result: \begin{lemma}\label{l2prior} Let $\epsilon>0$, $t_{1}>0$, $\theta_{0},q_{v0},q_{c0},q_{r0}\in L^{2},v_{0}\in\mathbb{H}$. Then there exists at least one solution $(v^{\epsilon}, \theta^{\epsilon}, q_{v}^{\epsilon}, q_{c}^{\epsilon}, q_{r}^{\epsilon})$ to equations (\ref{ae1})-(\ref{ae5}) with initial data (\ref{initial condition}) and boundary condition (\ref{boundary condition}). Moreover \begin{align}\label{L2-v-theta-qj}
\sup_{t\in [0,t_{1}]}\|(v^{\epsilon},\theta^{\epsilon},q_{j}^{\epsilon})\|_{L^{2}}^{2}
+\int_{0}^{t_{1}}\|(\nabla v^{\epsilon},\nabla\theta^{\epsilon},\nabla q_{j}^{\epsilon})\|_{L^{2}}^{2}+\|(\epsilon\partial_{p}v^{\epsilon},\partial_{p}\theta^{\epsilon}
,\partial_{p}q_{j}^{\epsilon})\|_{w}^{2}dt\leq C_{1},
\end{align}
for any $t_{1}\geq0$, $j\in \{v,c,r\}$, where $C_{1}$ is independent of $\epsilon$
and depends on the initial value. \end{lemma} \begin{proof}
By taking the inner product of equation (\ref{ae1}) with $v$ in $L^{2}(\mathcal{M})$ and using the integration by parts, we have
\begin{align*}
\frac{1}{2}\frac{d}{dt}\|v^{\epsilon}\|_{L^{2}}^{2}+\|\nabla v^{\epsilon}\|_{L^{2}}^{2}
+\epsilon_{1}\|\partial_{p}v^{\epsilon}\|_{w}^{2} =-\int_{\mathcal{M}}\nabla\Phi^{\epsilon}\cdot v^{\epsilon}d\mathcal{M},
\end{align*}
where we have used
\begin{align*} \int_{\mathcal{M}}\left[(v^{\epsilon}\cdot\nabla)v^{\epsilon}+\omega^{\epsilon}\partial_{p}v^{\epsilon}\right]\cdot v^{\epsilon} d\mathcal{M}=0, \end{align*} and \begin{align*} \int_{\mathcal{M}}f\textbf{\emph{k}}{v^{\epsilon}}^{\bot}\cdot v^{\epsilon}d\mathcal{M}=0. \end{align*} Considering the relation between $\Phi$ and $T$ in (\ref{e2}), through a similar argument as in \cite{CaoTiti}, we have \begin{align*} -\int_{\mathcal{M}}\nabla\Phi^{\epsilon}\cdot v^{\epsilon}d\mathcal{M}
\leq C\|\theta^{\epsilon}\|_{L^{2}}^{2}+\frac{1}{5}\|\nabla v^{\epsilon}\|_{L^{2}}^{2}. \end{align*} Then we can deduce that \begin{align}\label{L2-v}
\frac{1}{2}\frac{d}{dt}\|v^{\epsilon}\|_{L^{2}}^{2}+\frac{4}{5}\|\nabla v^{\epsilon}\|_{L^{2}}^{2}
+\epsilon_{1}\|\partial_{p}v\|_{w}^{2}
\leq C\|v^{\epsilon}\|_{L^{2}}^{2}+C\|\theta^{\epsilon}\|_{L^{2}}^{2}.
\end{align}
By taking the inner product of equation (\ref{ae2}) with $\theta^{\epsilon}$, we get that
\begin{align*}
\frac{1}{2}\frac{d}{dt}\|\theta^{\epsilon}\|_{L^{2}}^{2}
+\int_{\mathcal{M}}\theta^{\epsilon}\mathcal{A}_{\theta}\theta^{\epsilon}d\mathcal{M} =\int_{\mathcal{M}}f_{\theta^{\epsilon}} \theta^{\epsilon}d\mathcal{M}+\int_{\mathcal{M}}\frac{L}{c_{p}\Pi}{w^{\epsilon}}^{-}\tilde{F}\mathcal{H}_{\epsilon_{2}} \theta^{\epsilon}d\mathcal{M}.
\end{align*}
Through a direct calculation and using integration by parts, we have
\begin{align}\label{L2-theta-0}
\int_{\mathcal{M}}\theta^{\epsilon}\mathcal{A}_{\theta}\theta^{\epsilon}d\mathcal{M}
\geq& \|\nabla\theta^{\epsilon}\|_{L^{2}}^{2}+\|\partial_{p}\theta^{\epsilon}\|_{w}^{2}
+\int_{\mathcal{M}}\left(\frac{gp}{R\bar{\theta}}\right)^{2}\partial_{p}\left(\frac{p_{0}}{p}\right)^{R/c_{p}}|\theta^{\epsilon}|^{2}d\mathcal{M}
+\nonumber\\
&2\int_{\mathcal{M}}\left(\frac{gp}{R\bar{\theta}}\right)^{2}\left(\frac{p_{0}}{p}\right)^{R/c_{p}}\theta^{\epsilon}\partial_{p}\theta^{\epsilon}d\mathcal{M} -\int_{\Gamma_{l}}\theta_{bl}\theta^{\epsilon}d\Gamma_{l}- \int_{\Gamma_{i}}\theta_{\ast}\theta^{\epsilon}d\Gamma_{i}.
\end{align}
It is obviously that
\begin{align}\label{L2-theta-1}
\int_{\mathcal{M}}\left(\frac{gp}{R\bar{\theta}}\right)^{2}\partial_{p}\left(\frac{p_{0}}{p}\right)^{R/c_{p}}|\theta^{\epsilon}|^{2}d\mathcal{M}
\leq C\|\theta^{\epsilon}\|_{L^{2}}^{2}.
\end{align}
Utilizing the H\"older inequality and the Young inequality, we get
\begin{align}\label{L2-theta-2}
2\int_{\mathcal{M}}\left(\frac{gp}{R\bar{\theta}}\right)^{2}\left(\frac{p_{0}}{p}\right)^{R/c_{p}}
\theta^{\epsilon}\partial_{p}\theta^{\epsilon}d\mathcal{M}\leq C\| \theta^{\epsilon}\|_{L^{2}}^{2}+\frac{1}{4}\|\partial_{p}\theta^{\epsilon}\|_{L^{2}}^{2},
\end{align}
and
\begin{align}\label{L2-theta-3}
\int_{\Gamma_{l}}\theta_{bl}\theta^{\epsilon}d\Gamma_{l}+ \int_{\Gamma_{i}}\theta_{\ast}\theta^{\epsilon}d\Gamma_{i}
\leq& C\|\theta\|_{L^{2}(\partial\mathcal{M})}\leq C(\|\theta^{\epsilon}\|_{L^{2}}+\|\nabla\theta^{\epsilon}\|_{L^{2}}+\|\partial_{p}\theta^{\epsilon}\|_{L^{2}}) \nonumber\\
\leq& C(1+\|\theta^{\epsilon}\|_{L^{2}}^{2})+\frac{1}{2}\|\nabla\theta^{\epsilon}\|_{L^{2}}^{2}
+\frac{1}{4}\|\partial_{p}\theta^{\epsilon}\|_{L^{2}}^{2},
\end{align}
where we have used the trace inequality in the second step of (\ref{L2-theta-3}).
Then substituting (\ref{L2-theta-1})-(\ref{L2-theta-3}) into (\ref{L2-theta-0}), we can get
\begin{align}
\int_{\mathcal{M}}\theta^{\epsilon}\mathcal{A}_{\theta}\theta^{\epsilon}d\mathcal{M}
\geq\frac{1}{2}\|\nabla\theta^{\epsilon}\|_{L^{2}}^{2}+\frac{1}{2}\|\partial_{p}\theta^{\epsilon}\|_{w}^{2}
-C(\|\theta^{\epsilon}\|_{L^{2}}^{2}+1).
\end{align}
Considering the definition of $f_{\theta}$, we have
\begin{align*}
\int_{\mathcal{M}}f_{\theta^{\epsilon}}\theta^{\epsilon}d\mathcal{M}
\leq&\left\|-\frac{gp}{R\Pi\theta_{h}}\frac{\partial\theta_{h}}{\partial p}w^{\epsilon}-\frac{L}{c_{p}\Pi}k_{3}\tau(q_{r}^{\epsilon})(q_{vs}-q_{v}^{\epsilon})^{+}+w^{\epsilon}\frac{\partial\theta_{h}(p)}{\partial p}+f_{\theta}^{1}\right\|_{L^{2}}\|\theta^{\epsilon}\|_{L^{2}}\nonumber\\
\leq& C(\|w^{\epsilon}\|_{L^{2}}+\|q_{v}^{\epsilon}\|_{L^{2}}+1)\|\theta^{\epsilon}\|_{L^{2}}
\leq C(1+\|q_{v}^{\epsilon}\|_{L^{2}}^{2}+\|\theta^{\epsilon}\|_{L^{2}}^{2})+\frac{1}{20}\|\nabla v^{\epsilon}\|_{L^{2}}^{2},
\end{align*}
where we have used the uniform boundness of $\tau(q_{r})$.
Considering the definition of $\tilde{F}$ and the uniform boundness of $F$ and $\mathcal{H}_{\epsilon_{2}}$, we get
\begin{align*}
\int_{\mathcal{M}}\frac{L}{c_{p}\Pi}{w^{\epsilon}}^{-}\tilde{F}\mathcal{H}_{\epsilon_{2}} \theta^{\epsilon}d\mathcal{M}\leq C\|w^{\epsilon}\|_{L^{2}}\|\theta^{\epsilon}\|_{L^{2}}
\leq C\|\theta^{\epsilon}\|_{L^{2}}^{2}+\frac{1}{20}\|\nabla v^{\epsilon}\|_{L^{2}}^{2}.
\end{align*}
Thus
\begin{align}\label{L2-theta}
\frac{d}{dt}\|\theta^{\epsilon}\|_{L^{2}}^{2}
+\|\nabla\theta^{\epsilon}\|_{L^{2}}^{2}+\|\partial_{p}\theta^{\epsilon}\|_{w}^{2}
\leq C(1+\|q_{v}^{\epsilon}\|_{L^{2}}^{2}+\|\theta^{\epsilon}\|_{L^{2}}^{2})+\frac{1}{10}\|\nabla v^{\epsilon}\|_{L^{2}}^{2}.
\end{align}
By similar calculations as for $\theta^{\epsilon}$, we can deal with the $L^{2}$ a priori estimates for $q_{j}^{\epsilon},j\in\{v,c,r\}$, and obtain that
\begin{align}\label{L2-qv}
\frac{d}{dt}\|q_{v}^{\epsilon}\|_{L^{2}}^{2}
+\|\nabla q_{v}^{\epsilon}\|_{L^{2}}^{2}+\|\partial_{p}q_{v}^{\epsilon}\|_{w}^{2}
\leq C(\|q_{v}^{\epsilon}\|_{L^{2}}^{2}+1)+\frac{1}{10}\|\nabla v^{\epsilon}\|_{L^{2}}^{2},
\end{align}
\begin{align}\label{L2-qc}
\frac{d}{dt}\|q_{c}^{\epsilon}\|_{L^{2}}^{2}
+\|\nabla q_{c}^{\epsilon}\|_{L^{2}}^{2}+\|\partial_{p}q_{c}^{\epsilon}\|_{w}^{2}
\leq C(\|q_{c}^{\epsilon}\|_{L^{2}}^{2}+1)+\frac{1}{10}\|\nabla v^{\epsilon}\|_{L^{2}}^{2},
\end{align}
and
\begin{align}\label{L2-qr}
\frac{d}{dt}\|q_{r}^{\epsilon}\|_{L^{2}}^{2}
+\|\nabla q_{r}^{\epsilon}\|_{L^{2}}^{2}+\|\partial_{p}q_{r}^{\epsilon}\|_{w}^{2}
\leq C(\|q_{v}^{\epsilon}\|_{L^{2}}^{2}+\|q_{c}^{\epsilon}\|_{L^{2}}^{2}+\|q_{r}^{\epsilon}\|_{L^{2}}^{2}+1).
\end{align}
Combining (\ref{L2-v}), (\ref{L2-theta})-(\ref{L2-qr}), we can deduce that, for $j\in\{v,c,r\}$
\begin{align*}
\frac{d}{dt}\|(v^{\epsilon},\theta^{\epsilon},q_{j}^{\epsilon})\|_{L^{2}}^{2}
+\|(\nabla v^{\epsilon},\nabla\theta^{\epsilon},\nabla q_{j}^{\epsilon})\|_{L^{2}}^{2}+\|(\epsilon\partial_{p}v^{\epsilon},\partial_{p}\theta^{\epsilon},\partial_{p}q_{j}^{\epsilon})\|_{w}^{2}
\leq C\|(v^{\epsilon},\theta^{\epsilon},q_{v}^{\epsilon},q_{c}^{\epsilon},q_{r}^{\epsilon})\|_{L^{2}}^{2}+C.
\end{align*}
Utilizing the Gronwall inequality, we can complete the proof. \end{proof} \subsection{A priori $H^{1}$-estimates for $v^{\epsilon}$} In order to get the $H^{1}$ a priori estimate for solutions, we first consider the $H^{1}$ a priori estimate for the velocity $v^{\epsilon}$. Due to the arising of ${w^{\epsilon}}^{-}F\mathcal{H}_{\epsilon}(q_{v^{\epsilon}}-q_{vs})$ in the source term of $\theta^{\epsilon}$ and $q_{v}^{\epsilon},q_{c}^{\epsilon}$, we have to
first study the $H^{1}$ a priori estimate for $v^{\epsilon}$ with $\theta^{\epsilon}\in L^{\infty}(0,t_{1};L^{2})\cap L^{2}(0,t_{1};H^{1})$. Based on the a priori estimate for weak solutions, we will first improve the regularity of $v^{\epsilon}$ in $p-$direction, and then the horizontal direction. \begin{lemma}\label{vpL2} Let $\epsilon>0$, $t_{1}>0$, $\partial_{p}v_{0}^{\epsilon}\in L^{2}(0,t_{1};\mathbb{H})$, $\theta^{\epsilon}\in L^{\infty}(0,t_{1};L^{2})\cap L^{2}(0,t_{1};H^{1})$. Then there exists $t_{\ast}\in (0,t_{1}]$ and at least one solution $v^{\epsilon}$ to equation (\ref{ae1}) with boundary condition (\ref{boundary condition}) and initial condition (\ref{initial condition}). Moreover, \begin{align}
\sup_{0\leq t\leq t_{\ast}}\|\partial_{p}v^{\epsilon}\|_{L^{2}}^{2}+\int_{0}^{t_{\ast}}\|\nabla \partial_{p}v^{\epsilon}\|_{L^{2}}^{2}
+\epsilon_{1}\|\partial_{p}^{2}v^{\epsilon}\|_{w}^{2}ds \leq C_{2}, \end{align} where $C_{2}$ is independent of $\epsilon$. \end{lemma} \begin{proof} Set $u^{\epsilon}=\partial_{p}v^{\epsilon}$. Then it satisfies that \begin{align*} &\partial_{t}u^{\epsilon}+(u^{\epsilon}\cdot\nabla)v^{\epsilon}+v^{\epsilon}\cdot\nabla u^{\epsilon}+(\nabla\cdot v^{\epsilon})u^{\epsilon}+w^{\epsilon}\partial_{p}u^{\epsilon}-\Delta u^{\epsilon}-\epsilon_{1}\partial_{p}^{2}\left(\left(\frac{gp}{R\bar{\theta}}\right)^{2}u^{\epsilon}\right) +f{u^{\epsilon}}^{\bot} =\frac{R}{p}\nabla T^{\epsilon}. \end{align*} Taking the inner product of the above equation with $u^{\epsilon}$ in $L^{2}$ space, we can obtain that \begin{align}\label{vpL20}
&\frac{1}{2}\frac{d}{dt}\|u^{\epsilon}\|_{L^{2}}^{2}+\|\nabla u^{\epsilon}\|_{L^{2}}^{2}
+\epsilon_{1}\|\partial_{p}u^{\epsilon}\|_{w}^{2}\nonumber\\ =&\int_{\mathcal{M}}\left[u^{\epsilon}\cdot\nabla v^{\epsilon}+(\nabla\cdot v^{\epsilon})u^{\epsilon}\right]\cdot u^{\epsilon}d\mathcal{M} +\int_{\mathcal{M}}\frac{R}{p}\nabla T^{\epsilon}\cdot u^{\epsilon}d\mathcal{M},
\end{align}
where we have used
\begin{align*}
\int_{\mathcal{M}}\left[(v^{\epsilon}\cdot\nabla)u^{\epsilon}+w^{\epsilon}\partial_{p}u^{\epsilon}\right]\cdot u^{\epsilon}d\mathcal{M}=0,
\end{align*}
and
\begin{align*}
\int_{\mathcal{M}}f{u^{\epsilon}}^{\bot}\cdot u^{\epsilon}d\mathcal{M}=0.
\end{align*}
Using the inequality in Lemma \ref{HHP}, we can obtain that
\begin{align*}
&\int_{\mathcal{M}}\left[u^{\epsilon}\cdot\nabla v^{\epsilon}+(\nabla\cdot v^{\epsilon})u^{\epsilon}\right]\cdot u^{\epsilon}d\mathcal{M}
\leq C\int_{\mathcal{M}}|u^{\epsilon}||\nabla v^{\epsilon}||u^{\epsilon}|d\mathcal{M}\nonumber\\
\leq& C\|u^{\epsilon}\|_{L^{2}}\|\nabla u^{\epsilon}\|_{L^{2}}\left(\|\nabla v^{\epsilon}\|_{L^{2}}+\|\nabla v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}\|\nabla u^{\epsilon}\|_{L^{2}}^{\frac{1}{2}} \right)\nonumber\\
\leq& C\|\nabla v^{\epsilon}\|_{L^{2}}^{2}\left(\|u^{\epsilon}\|_{L^{2}}^{4}
+\|u^{\epsilon}\|_{L^{2}}^{2}\right)+\frac{1}{2}\|\nabla u^{\epsilon}\|_{L^{2}}^{2}.
\end{align*}
By the H\"older inequality and the Young inequality, we have
\begin{align*}
\int_{\mathcal{M}}\frac{R}{p}\nabla T^{\epsilon}\cdot u^{\epsilon}d\mathcal{M}
\leq C\|u^{\epsilon}\|_{L^{2}}^{2}+\|\nabla\theta^{\epsilon}\|_{L^{2}}^{2}.
\end{align*}
Substituting the above inequalities into (\ref{vpL20}), we can deduce that
\begin{align}
&\frac{d}{dt}\|u^{\epsilon}\|_{L^{2}}^{2}+\|\nabla u^{\epsilon}\|_{L^{2}}^{2}
+\epsilon_{1}\|\partial_{p}u^{\epsilon}\|_{w}^{2}\nonumber\\ \leq
&C\left(\|\nabla v^{\epsilon}\|_{L^{2}}^{2}+1\right)\left(\|u^{\epsilon}\|_{L^{2}}^{4}
+\|u^{\epsilon}\|_{L^{2}}^{2}+1\right)+\|\nabla\theta^{\epsilon}\|_{L^{2}}^{2}. \end{align} Then integrating on time from $0$ to $t$, we get
\begin{align*}
&\|u^{\epsilon}\|_{L^{2}}^{2}+\int_{0}^{t}\|\nabla u^{\epsilon}\|_{L^{2}}^{2}
+\epsilon_{1}\|\partial_{p}u^{\epsilon}\|_{w}^{2}ds\nonumber\\ \leq
&C\int_{0}^{t}\left(\|\nabla v^{\epsilon}\|_{L^{2}}^{2}+1\right)\left(\|u^{\epsilon}\|_{L^{2}}^{4}
+\|u^{\epsilon}\|_{L^{2}}^{2}+1\right)ds+\int_{0}^{t}\|\nabla\theta^{\epsilon}\|_{L^{2}}^{2}
ds+\|u^{\epsilon}(0)\|_{L^{2}}^{2}. \end{align*} Set \begin{align*}
&f(t)=\int_{0}^{t}\|\nabla\theta^{\epsilon}\|_{L^{2}}^{2}
+\|u^{\epsilon}(0)\|_{L^{2}}^{2},\ \ \alpha(t)=1,\nonumber\\
&\beta(t)=\|\nabla v^{\epsilon}\|_{L^{2}}^{2}+1,\ \ g(u)=1+u^{2}+u^{4}.
\end{align*}
Utilizing the generalized Bihari-Lasalle inequality (\ref{nonlinearGronwall1}), we can deduce that
\begin{align*}
\|u^{\epsilon}\|_{L^{2}}^{2}+\int_{0}^{t}\|\nabla u^{\epsilon}\|_{L^{2}}^{2}
+\epsilon_{1}\|\partial_{p}u^{\epsilon}\|_{w}^{2}ds \leq
\mathcal{G}^{-1}\left\{\int_{0}^{t}\|\nabla v^{\epsilon}\|_{L^{2}}^{2}+\|\nabla\theta^{\epsilon}\|_{L^{2}}^{2}
+1ds\right\} \end{align*} for $t\in[0,t_{\ast}]$, where $t_{\ast}\in(0,t_{1}]$ is sufficiently small and the function $\mathcal{G}$ is defined as
\begin{align*}
\mathcal{G}(r)=\int_{0}^{r}\frac{1}{1+s+s^{2}}ds
=\frac{2}{\sqrt{3}}{\rm artan}\left(\frac{1+2r}{\sqrt{3}}\right)-\frac{\pi}{3\sqrt{3}},\ \ \forall r>0.
\end{align*} Then considering the $L^{2}$ a priori estimate (\ref{L2-v-theta-qj}), we have \begin{align}\label{uL2}
\sup_{0\leq t\leq t_{\ast}}\|u^{\epsilon}\|_{L^{2}}^{2}+\int_{0}^{t_{\ast}}\|\nabla u^{\epsilon}\|_{L^{2}}^{2}
+\epsilon_{1}\|\partial_{p}u^{\epsilon}\|_{w}^{2}ds \leq C_{2}, \end{align} where $C_{2}$ is independent of $\epsilon$. \end{proof}
Next, we consider the $H^{1}$ a priori estimate for $v^{\epsilon}$. Based on the result in the above lemma, we only need to prove the $H^{1}$ estimate in the horizontal direction.
\begin{lemma}
Let $\epsilon>0$, $t_{1}>0$, $\theta^{\epsilon}\in L^{\infty}(0,t_{1};L^{2})\cap L^{2}(0,t_{1};H^{1})$, $v_{0}^{\epsilon}\in\mathbb{V}$. Then there exists $0< t_{\ast}\leq t_{1}$ and a solution $v^{\epsilon}$ to equation (\ref{ae1}) with boundary condition (\ref{boundary condition}) and initial condition (\ref{initial condition}) in time interval $(0,t_{\ast})$. Moreover, \begin{align}
\sup_{0\leq t\leq t_{\ast}}\|v^{\epsilon}\|_{H^{1}}^{2}+\int_{0}^{t_{\ast}}\|\nabla \partial_{p}v^{\epsilon}\|_{L^{2}}^{2}+\|\Delta v^{\epsilon}\|_{L^{2}}^{2}
+\epsilon_{1}\|\partial_{p}^{2}v^{\epsilon}\|_{w}^{2}ds \leq C_{3}, \end{align} where $C_{3}$ is independent of $\epsilon$.
\end{lemma} \begin{proof}
By taking the inner product of equation (\ref{ae1}) with $-\Delta v$ in $L^{2}(\mathcal{M})$, using integration by parts, we obtain that \begin{align*}
&\frac{1}{2}\frac{d}{dt}\|\nabla v^{\epsilon}\|_{L^{2}}^{2}+\|\Delta v^{\epsilon}\|_{L^{2}}^{2}
+\epsilon_{1}\|\nabla\partial_{p}v^{\epsilon}\|_{w}^{2}\nonumber\\ =&\int_{\mathcal{M}}\left[(v^{\epsilon}\cdot\nabla) v^{\epsilon}+w^{\epsilon}\partial_{p}v^{\epsilon}\right]\cdot \Delta v^{\epsilon}d\mathcal{M} +\int_{\mathcal{M}}\nabla\Phi^{\epsilon}\cdot\Delta v^{\epsilon}d\mathcal{M} +\int_{\mathcal{M}}f{v^{\epsilon}}^{\bot}\cdot\Delta v^{\epsilon}d\mathcal{M}.
\end{align*}
Noting the fact that
\begin{align}
f(p)\leq C\int_{p_{0}}^{p_{1}}|f|dp+C\int_{p_{0}}^{p_{1}}|\partial_{p}f|dp,
\end{align}
and utilizing the inequality in Lemma \ref{trilinear term lemma}, we can deduce that
\begin{align*}
&\int_{\mathcal{M}}(v^{\epsilon}\cdot\nabla)v^{\epsilon}\cdot\Delta v^{\epsilon}d\mathcal{M}\leq\int_{\mathcal{M}'} \int_{p_{0}}^{p_{1}} |v^{\epsilon}|+|\partial_{p}v^{\epsilon}|dp\int_{p_{0}}^{p_{1}} |\nabla v^{\epsilon}||\Delta v^{\epsilon}|dp\mathcal{M}'\nonumber\\
\leq&C \left(\|v^{\epsilon}\|_{L^2}^{\frac{1}{2}}+\|u^{\epsilon}\|_{L^2}^{\frac{1}{2}}\right)
\left(\|v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}} + \|\nabla v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}+\|u^{\epsilon}\|_{L^2}^{\frac{1}{2}}+\|\nabla u^{\epsilon}\|_{L^2}^{\frac{1}{2}}\right)\cdot\nonumber\\
&\|\Delta v^{\epsilon}\|_{L^{2}} \|\nabla v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}\left(\|\nabla v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}+ \|\nabla^{2} v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}\right)\nonumber\\
\leq&C\left(\|v^{\epsilon}\|_{L^{2}}^{2}+\|\nabla v^{\epsilon}\|_{L^{2}}^{2}+\|v^{\epsilon}\|_{L^{2}}^{4}
+\|v^{\epsilon}\|_{L^{2}}^{2}\|\nabla v^{\epsilon}\|_{L^{2}}^{2}+\|u^{\epsilon}\|_{L^{2}}^{2}+\|\nabla u^{\epsilon}\|_{L^{2}}^{2}\right.\nonumber\\
&\left.+\|u^{\epsilon}\|_{L^{2}}^{4}+\|u^{\epsilon}\|_{L^{2}}^{2}\|\nabla u^{\epsilon}\|_{L^{2}}^{2}\right)\|\nabla v^{\epsilon}\|_{L^{2}}^{2}+\frac{1}{8}\|\Delta v^{\epsilon}\|_{L^{2}}^{2}.
\end{align*}
Considering the inequality in Lemma \ref{trilinear term lemma} again, we have \begin{align*}
&\int_{\mathcal{M}}w^{\epsilon}\partial_{p}v^{\epsilon}\cdot\Delta v^{\epsilon}d\mathcal{M}\leq\int_{\mathcal{M}'}\int_{p_{0}}^{p_{1}}|\nabla v^{\epsilon}|dp\int_{p_{0}}^{p_{1}}|\partial_{p}v^{\epsilon}||\Delta v^{\epsilon}|dpd\mathcal{M}'\nonumber\\
\leq& C\|\Delta v^{\epsilon}\|_{L^{2}}\|\nabla v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}\left(
\|\nabla v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}+\|\nabla^{2} v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}\right)\|u^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}\left(
\|u^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}+\|\nabla u^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}\right)\nonumber\\
\leq& C\left(\|u^{\epsilon}\|_{L^{2}}^{2}+\|u^{\epsilon}\|_{L^{2}}^{4}+\|u^{\epsilon}\|_{L^{2}}^{2}
\|\nabla u^{\epsilon}\|_{L^{2}}^{2}+\|\nabla u^{\epsilon}\|_{L^{2}}^{2}\right)\|\nabla v^{\epsilon}\|_{L^{2}}^{2}+\frac{1}{8}\|\Delta v^{\epsilon}\|_{L^{2}}^{2}.
\end{align*} Using the H\"older inequality and the Young inequality, as well as the Minkowski inequality in integral form, we have \begin{align*} \int_{\mathcal{M}}\nabla\Phi^{\epsilon}\cdot\Delta v^{\epsilon}d\mathcal{M} &=\int_{\mathcal{M}}\nabla\Phi^{\epsilon}_{s}\cdot\Delta v^{\epsilon}d\mathcal{M} +\int_{\mathcal{M}}\int_{p_{0}}^{p_{1}}\frac{R}{p}\nabla T^{\epsilon}dp'\cdot\Delta v^{\epsilon}d\mathcal{M}\nonumber\\
\leq& C\|\nabla\theta^{\epsilon}\|_{L^{2}}^{2}+\frac{1}{8}\|\Delta v^{\epsilon}\|_{L^{2}}^{2}, \end{align*} where we have used the fact that \begin{align*} \int_{\mathcal{M}}\nabla\Phi^{\epsilon}_{s}\cdot\Delta v^{\epsilon}d\mathcal{M} =\int_{\mathcal{M}'}\nabla\Phi^{\epsilon}_{s}\cdot\Delta \bar{v^{\epsilon}}d\mathcal{M}'=0. \end{align*} Here $\bar{v^{\epsilon}}$ stands for the mean value of $v^{\epsilon}$ from $p_{0}$ to $p_{1}$, and $\Phi_{s}$ is the geopotential at $p=p_{1}$. Similarly, \begin{align*} \int_{\mathcal{M}}f{v^{\epsilon}}^{\bot}\cdot\Delta v^{\epsilon}d\mathcal{M}
\leq C\|v^{\epsilon}\|_{L^{2}}^{2}
+\frac{1}{8}\|\Delta v^{\epsilon}\|_{L^{2}}^{2}. \end{align*} Thus \begin{align*}
\frac{d}{dt}\|\nabla v^{\epsilon}\|_{L^{2}}^{2}+\|\Delta v^{\epsilon}\|_{L^{2}}^{2}
+\epsilon_{1}\|\nabla\partial_{p}v^{\epsilon}\|_{w}^{2}
\leq A_{1}(t)\|\nabla v^{\epsilon}\|_{L^{2}}^{2}+A_{2}(t),
\end{align*}
where
\begin{align*}
A_{1}(t)=C(&\|v^{\epsilon}\|_{L^{2}}^{2}+\|\nabla v^{\epsilon}\|_{L^{2}}^{2}+\|v^{\epsilon}\|_{L^{2}}^{4}
+\|v^{\epsilon}\|_{L^{2}}^{2}\|\nabla v^{\epsilon}\|_{L^{2}}^{2}+\|u^{\epsilon}\|_{L^{2}}^{2}+\|\nabla u^{\epsilon}\|_{L^{2}}^{2}+\nonumber\\
&\|u^{\epsilon}\|_{L^{2}}^{4}
+\|u^{\epsilon}\|_{L^{2}}^{2}\|\nabla u^{\epsilon}\|_{L^{2}}^{2}),
\end{align*}
and
\begin{align*}
A_{2}(t)=C\left(\|v^{\epsilon}\|_{L^{2}}^{2}+
\|\nabla\theta^{\epsilon}\|_{L^{2}}^{2}\right).
\end{align*}
Considering regularities of $v^{\epsilon},\nabla v^{\epsilon},\nabla\theta^{\epsilon},u^{\epsilon},\nabla u^{\epsilon}$ in (\ref{L2-v-theta-qj}) and (\ref{uL2}), we know that $A_{1}$ and $A_{2}$ are $L^{2}$ integrable in $(0,t_{\ast})$. Then using the Gronwall inequality, we can obtain that
\begin{align}
\sup_{0\leq t\leq t_{\ast}}\|\nabla v^{\epsilon}\|_{L^{2}}^{2}+\int_{0}^{t_{\ast}}\|\Delta v^{\epsilon}\|_{L^{2}}^{2}
+\epsilon_{1}\|\nabla\partial_{p}v^{\epsilon}\|_{w}^{2}ds \leq C_{3},
\end{align}
where $C_{3}$ is independent of $\epsilon$. \end{proof}
Next, we consider the time regularity of solutions to equations (\ref{ae1})-(\ref{ae5}). We first consider the time regularity of $v$.
\begin{lemma}\label{trv}
Let $\epsilon>0$, $t_{1}>0$, $\theta^{\epsilon}\in L^{\infty}(0,t_{1};L^{2})\cap L^{2}(0,t_{1};H^{1})$, $v_{0}^{\epsilon}\in\mathbb{V}$ and $v^{\epsilon}$ be the solution to equation (\ref{ae1}) with boundary condition (\ref{boundary condition}) and initial condition (\ref{initial condition}). Then
\begin{align*}
\partial_{t}v^{\epsilon}\in L^{2}(0,t_{\ast};\mathbb{H}).
\end{align*}
\end{lemma}
\begin{proof}
Choosing a test function $\tilde{v}\in L^{2}(0,t_{\ast};\mathbb{H})$ satisfying that $\|\tilde{v}\|_{L^{2}(0,t_{\ast};\mathbb{H})}\leq 1$, we can refer from equation (\ref{e1}) that \begin{align*}
\int_{\mathcal{M}}\partial_{t}v^{\epsilon}\cdot\tilde{v}d\mathcal{M}
=&\int_{\mathcal{M}}\Delta v^{\epsilon}\cdot\tilde{v}d\mathcal{M}
-\int_{\mathcal{M}}(v^{\epsilon}\cdot\nabla)v^{\epsilon}\cdot\tilde{v}+w^{\epsilon}\partial_{p}v^{\epsilon}\cdot\tilde{v}d\mathcal{M}\nonumber\\
&-\int_{\mathcal{M}}\nabla\Phi^{\epsilon}\cdot\tilde{v}d\mathcal{M} -\int_{\mathcal{M}}f{v^{\epsilon}}^{\bot}\cdot\tilde{v}d\mathcal{M}.
\end{align*}
It is easy to verify that
\begin{align*}
\int_{\mathcal{M}}\Delta v^{\epsilon}\cdot\tilde{v}d\mathcal{M}\leq C\|\Delta v^{\epsilon}\|_{L^{2}}\|\tilde{v}\|_{L^{2}},
\end{align*}
\begin{align*}
-\int_{\mathcal{M}}\nabla\Phi^{\epsilon}\cdot\tilde{v}d\mathcal{M}\leq C\|\theta^{\epsilon}\|_{H^{1}}\|\tilde{v}\|_{L^{2}},
\end{align*}
and
\begin{align*}
-\int_{\mathcal{M}}f{v^{\epsilon}}^{\bot}\cdot\tilde{v}d\mathcal{M}\leq C\|v^{\epsilon}\|_{L^{2}}\|\tilde{v}\|_{L^{2}}.
\end{align*}
Then we consider the trilinear term. Through similar arguments as in (\ref{qvH1h1}) and (\ref{qvH1h2}), we can deduce that
\begin{align*}
&\int_{\mathcal{M}}(v^{\epsilon}\cdot\nabla)v^{\epsilon}\cdot\tilde{v}d\mathcal{M}\leq\int_{\mathcal{M}'} \int_{p_{0}}^{p_{1}} |v^{\epsilon}|+|\partial_{p}v^{\epsilon}|dp\int_{p_{0}}^{p_{1}} |\nabla v^{\epsilon}||\tilde{v}|dp\mathcal{M}'\nonumber\\
\leq& C \left(\|v^{\epsilon}\|_{L^2}^{\frac{1}{2}}+\|u^{\epsilon}\|_{L^2}^{\frac{1}{2}}\right)
\left(\|v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}} + \|\nabla v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}+\|u^{\epsilon}\|_{L^2}^{\frac{1}{2}}+\|\nabla u^{\epsilon}\|_{L^2}^{\frac{1}{2}}\right)\|\nabla v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}\left(\|\nabla v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}+\right.\nonumber\\
&\left.\|\nabla^{2} v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}\right)\|\tilde{v}\|_{L^{2}}
\leq C\left(\|v^{\epsilon}\|_{H^{1}}^{2}+\|v^{\epsilon}\|_{H^{1}}\|\Delta v^{\epsilon}\|_{L^{2}}\right)\|\tilde{v}\|_{L^{2}},
\end{align*}
and \begin{align*}
&\int_{\mathcal{M}}w^{\epsilon}\partial_{p}v^{\epsilon}\cdot\tilde{v}d\mathcal{M}
\leq\int_{\mathcal{M}'}\int_{p_{0}}^{p_{1}}|\nabla v^{\epsilon}|dp\int_{p_{0}}^{p_{1}}|\partial_{p}v^{\epsilon}||\tilde{v}|dpd\mathcal{M}'\nonumber\\
\leq& C\|\tilde{v}\|_{L^{2}}\|\nabla v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}\left(
\|\nabla v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}+\|\nabla^{2} v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}\right)\|\partial_{p}v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}\left(
\|\partial_{p}v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}+\|\nabla \partial_{p}v^{\epsilon}\|_{L^{2}}^{\frac{1}{2}}\right)\nonumber\\
\leq& C\left(\|v^{\epsilon}\|_{H^{1}}^{2}+\|v^{\epsilon}\|_{H^{1}}^{3}+\|\Delta v^{\epsilon}\|_{L^{2}}+\|\nabla\partial_{p}v^{\epsilon}\|_{L^{2}}+\|v^{\epsilon}\|_{H^{1}}\|\Delta v^{\epsilon}\|_{L^{2}}+ \|v^{\epsilon}\|_{H^{1}}\|\nabla\partial_{p}v^{\epsilon}\|_{L^{2}}\right)\|\tilde{v}\|_{L^{2}}.
\end{align*}
Combining the above inequalities, we have
\begin{align*}
\int_{\mathcal{M}}\partial_{t}v^{\epsilon}\cdot\tilde{v}d\mathcal{M}\leq A_{3}(t)\|\tilde{v}\|_{L^{2}}\leq A_{3}(t),
\end{align*}
where
\begin{align*}
A_{3}(t)=&\|\Delta v^{\epsilon}\|_{L^{2}}+\|\theta^{\epsilon}\|_{H^{1}}+\|v^{\epsilon}\|_{L^{2}}+
\|v^{\epsilon}\|_{H^{1}}^{2}+\|v^{\epsilon}\|_{H^{1}}\|\Delta v^{\epsilon}\|_{L^{2}}+\|v^{\epsilon}\|_{H^{1}}^{2}+\|v^{\epsilon}\|_{H^{1}}^{3}\nonumber\\
&+\|\Delta v^{\epsilon}\|_{L^{2}}+\|\nabla\partial_{p}v^{\epsilon}\|_{L^{2}}+\|v^{\epsilon}\|_{H^{1}}\|\Delta v^{\epsilon}\|_{L^{2}}+ \|v^{\epsilon}\|_{H^{1}}\|\nabla\partial_{p}v^{\epsilon}\|_{L^{2}}.
\end{align*}
Considering regularities of $v^{\epsilon}$ and $\theta^{\epsilon}$, we can know that $A_{3}(t)$ is $L^{2}$ integrable in $(0,t_{\ast})$. Then we can deduce that
$\partial_{t}v^{\epsilon}\in L^{2}(0,t_{\ast};\mathbb{H})$. \end{proof}
As for the time regularities of $\theta,q_{v},q_{c},q_{r}$, we can deal with them in a similar way. For more detailed procedures, we refer readers to \cite{Zelati} or \cite{TanLiu}. It should be noted that the treatment of trilinear terms can be referred to (\ref{qvH1h1}) and (\ref{qvH1h2}). As a conclusion, we can get that
\begin{align}
\partial_{t}\theta,\partial_{t}q_{j}\in L^{2}(0,t_{\ast};H^{-1}),j\in \{v,c,r\}.
\end{align} So far, we have obtained the local existence of quasi-strong solutions to the approximated system.
\begin{proposition}
Let $v_{0}\in\mathbb{V}$, $\theta_{0}, q_{v0},q_{r0},q_{c0}\in L^{2}$, and $\epsilon>0$ be fixed. Then there exists $t_{\ast}>0\ (t_{\ast}\leq t_{1}, \text{independent of $\epsilon$})$, and a quasi-strong solution $(v^{\epsilon},\theta^{\epsilon},q_{v}^{\epsilon},q_{c}^{\epsilon},q_{r}^{\epsilon})$ to equations (\ref{ae1})-(\ref{ae5}) with boundary condition (\ref{boundary condition}) and initial condition (\ref{initial condition}) satisfying that \begin{align*} &v^{\epsilon}\in L^{\infty}(0,t_{\ast};\mathbb{V}),\ \Delta v^{\epsilon},\nabla\partial_{p}v^{\epsilon}\in L^{2}(0,t_{\ast}; (L^{2})^{2}), \nonumber\\ &(\theta_{\epsilon},q_{v}^{\epsilon},q_{c}^{\epsilon},q_{r}^{\epsilon})\in C(0,t_{\ast};(L^{2})^{4})\cap L^{2}(0,t_{\ast}; (H^{1})^{4}). \end{align*} Moreover, \begin{align}\label{quasistrong bound}
&\|v^{\epsilon}\| _{L^{\infty}(0,t_{\ast};\mathbb{V})}
+\|\left(\theta^{\epsilon},q_{v}^{\epsilon},q_{c}^{\epsilon},q_{r}^{\epsilon}\right)\|_{L^{\infty}(0,t_{\ast};L^{2})}
+\|(\Delta v^{\epsilon},\nabla\partial_{p}v^{\epsilon},\epsilon\partial_{p}^{2}v^{\epsilon})\|_{L^{2}(0,t_{\ast};L^{2})}\nonumber\\
&+\|\left(\theta^{\epsilon},q_{v}^{\epsilon},q_{c}^{\epsilon},q_{r}^{\epsilon}\right)\|_{L^{2}(0,t_{\ast};H^{1})}
+\|\partial_{t}v^{\epsilon}\| _{L^{2}(0,t_{\ast};\mathbb{H})}
+\|\partial_{t}(\theta^{\epsilon},q_{v}^{\epsilon},q_{c}^{\epsilon},q_{r}^{\epsilon})\|_{L^{2}\left(0,t_{\ast};H^{-1}\right)} \leq \mathcal{C}, \end{align} where $\mathcal{C}$ is monotonically increasing positive function with respect to $t_{\ast}$, which is independent of $\epsilon$, and depends on the initial data. \end{proposition} \subsection{The local existence of quasi-strong solutions} In this subsection, we obtain the local existence of quasi-strong solutions to equations (\ref{e1})-(\ref{e7}) by passing to the limit in the approximated equations (\ref{ae1})-(\ref{ae5}) as $\epsilon\rightarrow 0$. As usual, we denote strong, weak, and weak-$\ast$ convergence as $\epsilon\rightarrow 0$ by $\rightarrow,\rightharpoonup,\stackrel{\ast}{\rightharpoonup}$ respectively. Considering inequality (\ref{quasistrong bound}) and Aubin-Lions compactness theorem \cite{Aubin,Lions2}, we can deduce the existence of a subsequence, still denoted by $v^{\epsilon},\theta^{\epsilon},q_{v}^{\epsilon},q_{c}^{\epsilon},q_{r}^{\epsilon},$ and \begin{align} &v\in C(0,t_{\ast};\mathbb{V}), \ \Delta v,\nabla\partial_{p}v\in L^{2}(0,t_{\ast};L^{2}),\ \partial_{t}v\in L^{2}(0,t_{\ast};\mathbb{H}),\nonumber\\ &\theta,q_{v},q_{c},q_{r}\in C(0,t_{\ast};L^{2})\cap L^{2}(0,t_{\ast};H^{1}),\ \partial_{t}\theta,\partial_{t}q_{v},\partial_{t}q_{c},\partial_{t}q_{r}\in L^{2}(0,t_{\ast};H^{-1}) \end{align} such that, as $\epsilon\rightarrow 0$, \begin{align*} &\ v^{\epsilon}\stackrel{\ast}{\rightharpoonup} v\ \text{in}\ C(0,t_{\ast};\mathbb{V}), \\
&(\Delta v^{\epsilon},\nabla\partial_{p}v^{\epsilon}){\rightharpoonup} (\Delta v,\nabla\partial_{p}v) \ \text{in}\ L^{2}(0,t_{\ast};(L^{2})^{2}),\\ &\partial_{t}v^{\epsilon}\rightharpoonup\partial_{t} v\ \text{in}\ L^{2}(0,t_{\ast};\mathbb{H}), \end{align*} and \begin{align*} &\left(\theta^{\epsilon},q_{v}^{\epsilon},q_{c}^{\epsilon},q_{r}^{\epsilon}\right)\rightharpoonup\left(\theta,q_{v},q_{c},q_{r}\right) \ \text{in}\ \ L^{2}(0,t_{\ast};(H^{1})^{4}),\\ &\left(\theta^{\epsilon},q_{v}^{\epsilon},q_{c}^{\epsilon},q_{r}^{\epsilon}\right)\stackrel{\ast}{\rightharpoonup}\left(\theta,q_{v},q_{c},q_{r}\right) \ \text{in}\ \ L^{\infty}(0,t_{\ast};(L^{2})^{4}),\ \nonumber\\ &\partial_{t}(\theta^{\epsilon},q_{v}^{\epsilon},q_{c}^{\epsilon},q_{r}^{\epsilon})\rightharpoonup \partial_{t}(\theta,q_{v},q_{c},q_{r})\ \text{in}\ L^{2}\left(0,t_{\ast};(H^{-1})^{4}\right),\\ &\left(\theta^{\epsilon},q_{v}^{\epsilon},q_{c}^{\epsilon},q_{r}^{\epsilon}\right)\rightarrow\left(\theta,q_{v},q_{c},q_{r}\right) \ \text{in}\ \ L^{2}(0,t_{\ast};(L^{2})^{4}). \end{align*}
As to the proof of convergence between the approximated system (\ref{ae1})-(\ref{ae5}) and the original system (\ref{e1})-(\ref{e7}), we refer readers to \cite{CaoTiti6}, \cite{Cao1} and \cite{Zelati}. In fact, we have proved the local existence of quasi-strong solutions of system (\ref{e1})-(\ref{e7}). \begin{proposition}\label{quasistrong existence local}
Let $v_{0}\in\mathbb{V}$, $\theta_{0}, q_{v0}, q_{r0},q_{c0}\in L^{2}(\mathcal{M})$. Then there exists $t_{\ast}>0\ (t_{\ast}\leq t_{1}$), and a quasi-strong solution $(v,\theta,q_{v},q_{c},q_{r})$ to equations (\ref{e1})-(\ref{e7}) with boundary condition (\ref{boundary condition}) and initial condition (\ref{initial condition}) in time interval $(0,t_{\ast})$. Moreover, \begin{align*} &v\in C(0,t_{\ast};\mathbb{V}),\ \ \Delta v,\nabla\partial_{p}v\in L^{2}(0,t_{\ast}; L^{2}), \nonumber\\ &(\theta,q_{v},q_{c},q_{r})\in C(0,t_{\ast};(L^{2})^{4})\cap L^{2}(0,t_{\ast}; (H^{1})^{4}),\nonumber\\ &\partial_{t}v\in L^{2}(0,t_{\ast};\mathbb{H}),\ \ \partial_{t}(\theta,q_{v},q_{c},q_{r})\in L^{2}(0,t_{\ast}; (H^{-1})^{4}). \end{align*} \end{proposition} \subsection{The local existence of strong solutions}
Compared with the case where the velocity equation is full viscosity, the most difficulty during the proof of existence of strong solution is caused by the absence of vertical dissipation in the velocity fields, which makes that $\|\partial_{p}^{2}v\|_{L^{2}(0,t;L^{2})}$ can not be bounded. Therefore, our main work is to ensure that the a priori estimates does not include this item. Here, we take $q_{v}$ as an example to illustrate how to get the $H^{1}$ regularity. As for other quantities, one can get the $H^{1}$ regularity by combining calculations in literatures \cite{Zelati,TanLiu,Hittmeir} and this paper. \begin{lemma}\label{H1} Let $v_{0}\in\mathbb{V}$, $\theta_{0}, q_{v0}, q_{r0},q_{c0}\in H^{1}$. Then there exists $t_{\ast}\in (0,t_{1})$ and a solution $q_{v}$ to equation (\ref{e5}) with boundary condition (\ref{boundary condition}) and initial condition (\ref{initial condition}). Moreover \begin{align}
\sup_{0\leq t\leq t_{\ast}}\|q_{v}\|_{H^{1}}^{2}
+\int_{0}^{t_{\ast}}\|q_{v}\|_{H^{2}}^{2}dt\leq C_{4}, \end{align} where $C_{4}$ depends on the initial data and $t_{\ast}$. \end{lemma} \begin{proof} By taking the inner product of equation (\ref{ae3}) with $-\Delta q_{v}$ in the $L^{2}$ space, we can infer that \begin{align}\label{qvH1H} &-\int_{\mathcal{M}}\partial_{t}q_{v}\Delta q_{v}d\mathcal{M} -\int_{\mathcal{M}}\mathcal{A}_{q_{v}}q_{v}\Delta q_{v}d\mathcal{M}\nonumber\\ = &\int_{\mathcal{M}}\left(v\cdot\nabla q_{v}+w\partial_{p}q_{v}\right)\Delta q_{v}d\mathcal{M} -\int_{\mathcal{M}}\left(f_{q_{v}}-w^{-}Fh_{q_{v}}\right)\Delta q_{v}d\mathcal{M}. \end{align} Utilizing similar arguments as in \cite{Hittmeir} and \cite{TanLiu}, we can obtain that \begin{align*} -\int_{\mathcal{M}}\partial_{t}q_{v}\Delta q_{v}d\mathcal{M}
\geq\frac{d}{dt}\left(\frac{1}{2}\|\nabla q_{v}\|_{L^{2}}^{2}+ \alpha_{lv}\int_{\Gamma_{l}}\left(\frac{1}{2}(q_{v})^{2}-q_{v}q_{blv}\right)d\Gamma_{l}\right)
-C(1+\|q_{v}\|_{L_{2}}^{2}+\|\nabla q_{v}\|_{L_{2}}^{2}), \end{align*} and \begin{align*} -\int_{\mathcal{M}}\mathcal{A}_{q_{v}}q_{v}\Delta q_{v}d\mathcal{M}
\geq\|\Delta q_{v}\|_{L^{2}}^{2}+\|\nabla\partial_{p}q_{v}\|_{w}^{2}-C. \end{align*} For the source term, recalling definitions of $F,f_{q_{v}}$ and $h_{q_{v}}$, we have \begin{align}\label{qvH1050} &\int_{\mathcal{M}}\left(f_{q_{v}}-{w}^{-}Fh_{q_{v}}\right)\Delta q_{v}d\mathcal{M}
\leq\|f_{q_{v}}-{w}^{-}Fh_{q_{v}}\|_{L^{2}}\|\Delta q_{v}\|_{L^{2}} \nonumber\\
\leq&\left(\left\|k_{3}\tau(q_{r})(q_{vs}-q_{v})^{+}\right\|_{L^{2}}
+\left\|{w}^{-}Fh_{q_{v}}\right\|_{L^{2}}\right)
\|\Delta q_{v}\|_{L^{2}}. \end{align} Using the fact that $q_{vs}$ is a constant and $\tau(q_{r})$ is uniformly bounded, we have \begin{align}\label{fqv1}
\left\|k_{3}\tau(q_{r})(q_{vs}-q_{v})^{+}\right\|_{L^{2}}\leq C(1+\|q_{v}\|_{L^{2}}^{2}). \end{align} As verified in \cite{Zelati}, the function $F$ and $h_{q}$ are uniformly bounded. Then \begin{align}\label{F1}
\left\|w^{-}Fh_{q_{v}}\right\|_{L^{2}}
\leq C\|w\|_{L^{2}}\leq C\|\nabla v\|_{L^{2}}, \end{align} where we have used $\partial_{p}w=-\nabla\cdot v$. Thus \begin{align} \int_{\mathcal{M}}\left(f_{q_{v}}-{w}^{-}Fh_{q_{v}}\right)\Delta q_{v}d\mathcal{M}
\leq C(1+\|q_{v}\|_{L^{2}}^{2}+\|\nabla v\|_{L^{2}}^{2})+\frac{1}{8}\|\Delta q_{v}\|_{L^{2}}^{2}. \end{align}
Then we consider the trilinear term \begin{align*} \int_{\mathcal{M}}\left(v\cdot\nabla q_{v}+w\partial_{p}q_{v}\right)\Delta q_{v}d\mathcal{M}. \end{align*} Noting the fact that \begin{align*}
f(p)\leq C\int_{p_{0}}^{p_{1}}|f|dp+C\int_{p_{0}}^{p_{1}}|\partial_{p}f|dp
\end{align*}
and using the inequality in Lemma \ref{trilinear term lemma}, we can deduce that
\begin{align}\label{qvH1h1}
&\int_{\mathcal{M}}(v\cdot\nabla)q_{v}\cdot\Delta q_{v}d\mathcal{M}\leq\int_{\mathcal{M}'} \int_{p_{0}}^{p_{1}} |v|+|\partial_{p}v|dp\int_{p_{0}}^{p_{1}} |\nabla q_{v}||\Delta q_{v}|dp\mathcal{M}'\nonumber\\
\leq& C \left(\|v\|_{L^2}^{\frac{1}{2}}+\|u\|_{L^2}^{\frac{1}{2}}\right)
\left(\|v\|_{L^{2}}^{\frac{1}{2}} + \|\nabla v\|_{L^{2}}^{\frac{1}{2}}+\|u\|_{L^2}^{\frac{1}{2}}+\|\nabla u\|_{L^2}^{\frac{1}{2}}\right)\|\Delta q_{v}\|_{L^{2}}\cdot\nonumber\\
&\ \ \ \|\nabla q_{v}\|_{L^{2}}^{\frac{1}{2}}\left(\|\nabla q_{v}\|_{L^{2}}^{\frac{1}{2}}+ \|\nabla^{2} q_{v}\|_{L^{2}}^{\frac{1}{2}}\right)\nonumber\\
\leq&C\left(\|v\|_{L^{2}}^{2}+\|u\|_{L^{2}}^{2}+\|\nabla v\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{2}
+\|v\|_{L^{2}}^{4}+\|\nabla v\|_{L^{2}}^{2}\|v\|_{L^{2}}^{2}+\|u\|_{L^{2}}^{2}\|\nabla v\|_{L^{2}}^{2}\right.\nonumber\\
&\left.+\|u\|_{L^{2}}^{4}+\|v\|_{L^{2}}^{2}\|\nabla u\|_{L^{2}}^{2}+
\|u\|_{L^{2}}^{2}\|\nabla u\|_{L^{2}}^{2}\right)\|\nabla q_{v}\|_{L^{2}}^{2}+\frac{1}{8}\|\Delta q_{v}\|_{L^{2}}^{2}.
\end{align}
Similarly, \begin{align}\label{qvH1h2}
&\int_{\mathcal{M}}w\partial_{p}q_{v}\cdot\Delta q_{v}d\mathcal{M}\leq\int_{\mathcal{M}'}\int_{p_{0}}^{p_{1}}|\nabla v|dp\int_{p_{0}}^{p_{1}}|\partial_{p}q_{v}||\Delta q_{v}|dpd\mathcal{M}'\nonumber\\
\leq& C\|\Delta q_{v}\|_{L^{2}}\|\nabla v\|_{L^{2}}^{\frac{1}{2}}\left(
\|\nabla v\|_{L^{2}}^{\frac{1}{2}}+\|\nabla^{2} v\|_{L^{2}}^{\frac{1}{2}}\right)\|\partial_{p}q_{v}\|_{L^{2}}^{\frac{1}{2}}\left(
\|\partial_{p}q_{v}\|_{L^{2}}^{\frac{1}{2}}+\|\nabla \partial_{p}q_{v}\|_{L^{2}}^{\frac{1}{2}}\right)\\
\leq& C\left(\|\nabla v\|_{L^{2}}^{2}+\|\nabla v\|_{L^{2}}^{4}+\|\Delta v\|_{L^{2}}^{2}+
\|\nabla v\|_{L^{2}}^{2}\|\Delta v\|_{L^{2}}^{2}\right)\|\partial_{p}q_{v} \|_{L^{2}}^{2}+\frac{1}{8}\|\Delta q_{v}\|_{L^{2}}^{2}+\frac{3}{4}\|\nabla \partial_{p}v\|_{L^{2}}^{2}.\nonumber
\end{align}
Combining all the above inequalities, we can infer that
\begin{align}\label{qvH1h3}
&\frac{d}{dt}\left(\frac{1}{2}\|\nabla q_{v}\|_{L^{2}}^{2}+ \alpha_{lv}\int_{\Gamma_{l}}\left(\frac{1}{2}(q_{v})^{2}-q_{v}q_{blv}\right)d\Gamma_{l}\right)
+\frac{5}{8}\|\Delta q_{v}\|_{L^{2}}^{2}+\frac{1}{4}\|\nabla\partial_{p}q_{v}\|_{w}^{2}\nonumber\\
\leq &A_{4}(t)(\|\nabla q_{v}\|_{L^{2}}^{2}+\|\partial_{p}q_{v} \|_{L^{2}}^{2})+A_{5}(t), \end{align} where \begin{align*}
A_{4}(t)=&C\left(\|\nabla v\|_{L^{2}}^{2}+\|\nabla v\|_{L^{2}}^{4}+\|\Delta v\|_{L^{2}}^{2}+
\|\nabla v\|_{L^{2}}^{2}\|\Delta v\|_{L^{2}}^{2}+\|v\|_{L^{2}}^{2}+\|u\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{2}+\right.\nonumber\\
&\left.\|v\|_{L^{2}}^{4}+\|\nabla v\|_{L^{2}}^{2}\|v\|_{L^{2}}^{2}+\|u\|_{L^{2}}^{2}\|\nabla v\|_{L^{2}}^{2}+\|u\|_{L^{2}}^{4}+\|v\|_{L^{2}}^{2}\|\nabla u\|_{L^{2}}^{2}+\|u\|_{L^{2}}^{2}\|\nabla u\|_{L^{2}}^{2}+1\right)\nonumber \end{align*} and \begin{align*}
A_{5}(t)=C(1+\|q_{v}\|_{L_{2}}^{2}+\|\nabla v\|_{L^{2}}^{2}). \end{align*}
By taking the inner product of equation $(\ref{ae3})$ with $-\partial_{p}^{2}q_{v}$ in the $L^{2}$ space, we have \begin{align}\label{qvH10} &-\int_{\mathcal{M}}\partial_{t}q_{v}\partial_{p}^{2}q_{v}d\mathcal{M} -\int_{\mathcal{M}}\mathcal{A}_{q_{v}}q_{v}\partial_{p}^{2}q_{v}d\mathcal{M} \nonumber\\ = &\int_{\mathcal{M}}\left(v\cdot\nabla q_{v}+w\partial_{p}q_{v}\right)\partial_{p}^{2}q_{v}d\mathcal{M} -\int_{\mathcal{M}}\left(f_{q_{v}}-w^{-}Fh_{q_{v}}\right)\partial_{p}^{2}q_{v}d\mathcal{M}. \end{align} Through similar arguments as in \cite{TanLiu}, we can get that \begin{align} -\int_{\mathcal{M}}\partial_{t}q_{v}\partial_{p}^{2}q_{v}d\mathcal{M}
\geq&\frac{d}{dt}\left(\frac{1}{2}\|\partial_{p}q_{v}\|_{L^{2}}^{2} +\beta_{v}\int_{\Gamma_{i}}\left(\frac{1}{2}(q_{v})^{2}-q_{v_{\ast}}q_{v}\right)d\Gamma_{i}\right)\nonumber\\
&-C\left(\|\partial_{p}q_{v}\|_{L^{2}}^{2}+\|q_{v}\|_{L^{2}}^{2}+1\right), \end{align} \begin{align}\label{qvH103} -\int_{\mathcal{M}}\mathcal{A}_{q_{v}}q_{v}\partial_{p}^{2}q_{v}d\mathcal{M}
\geq \frac{3}{4}\left(\|\nabla\partial_{p}q_{v}\|_{L^{2}}^{2}
+\left\|\partial_{p}^{2}q_{v}\right\|_{w}^{2} \right)-C\left(\|\partial_{p}q_{v}\|_{L^{2}}^{2}+1\right), \end{align} and \begin{align}\label{qvH1051} \int_{\mathcal{M}}\left(f_{q_{v}}-{w}^{-}Fh_{q_{v}}\right)\partial_{p}^{2}q_{v}d\mathcal{M}
\leq C(1+\|q_{v}\|_{L^{2}}^{2}+\|\nabla v\|_{L^{2}}^{2})+\frac{1}{12}\|\partial_{p}^{2}q_{v}\|_{L^{2}}^{2}. \end{align} Then we consider the most problematic term \begin{align*} \int_{\mathcal{M}}\left(v\cdot\nabla q_{v}+w\partial_{p}q_{v}\right)\partial_{p}^{2}q_{v}d\mathcal{M}. \end{align*} In fact, through similar arguments as in (\ref{qvH1h1}) and (\ref{qvH1h2}), we have \begin{align*}
&\int_{\mathcal{M}}(v\cdot\nabla)q_{v}\cdot\partial_{p}^{2} q_{v}d\mathcal{M}\leq\int_{\mathcal{M}'} \int_{p_{0}}^{p_{1}} |v|+|\partial_{p}v|dp\int_{p_{0}}^{p_{1}} |\nabla q_{v}||\partial_{p}^{2} q_{v}|dp\mathcal{M}'\nonumber\\
\leq& C \left(\|v\|_{L^2}^{\frac{1}{2}}+\|u\|_{L^2}^{\frac{1}{2}}\right)
\left(\|v\|_{L^{2}}^{\frac{1}{2}} + \|\nabla v\|_{L^{2}}^{\frac{1}{2}}+\|u\|_{L^2}^{\frac{1}{2}}+\|\nabla u\|_{L^2}^{\frac{1}{2}}\right)\|\partial_{p}^{2} q_{v}\|_{L^{2}}\times\nonumber\\
&\ \ \ \|\nabla q_{v}\|_{L^{2}}^{\frac{1}{2}}\left(\|\nabla q_{v}\|_{L^{2}}^{\frac{1}{2}}+ \|\nabla^{2} q_{v}\|_{L^{2}}^{\frac{1}{2}}\right)\nonumber\\
\leq&C\left(\|v\|_{L^{2}}^{2}+\|u\|_{L^{2}}^{2}+\|\nabla v\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{2}
+\|v\|_{L^{2}}^{4}+\|\nabla v\|_{L^{2}}^{2}\|v\|_{L^{2}}^{2}+\|u\|_{L^{2}}^{2}\|\nabla v\|_{L^{2}}^{2}\right.\nonumber\\
&\left.+\|u\|_{L^{2}}^{4}+\|v\|_{L^{2}}^{2}\|\nabla u\|_{L^{2}}^{2}+
\|u\|_{L^{2}}^{2}\|\nabla u\|_{L^{2}}^{2}\right)\|\nabla q_{v}\|_{L^{2}}^{2}+\frac{1}{8}\|\Delta q_{v}\|_{L^{2}}^{2}
+\frac{1}{12}\|\partial_{p}^{2}q_{v}\|_{L^{2}}^{2}.
\end{align*}
Similarly, \begin{align*}
&\int_{\mathcal{M}}w\partial_{p}q_{v}\partial_{p}^{2} q_{v}d\mathcal{M}\leq\int_{\mathcal{M}'}\int_{p_{0}}^{p_{1}}|\nabla v|dp\int_{p_{0}}^{p_{1}}|\partial_{p}q_{v}||\partial_{p}^{2} q_{v}|dpd\mathcal{M}'\nonumber\\
\leq& C\|\partial_{p}^{2} q_{v}\|_{L^{2}}\|\nabla v\|_{L^{2}}^{\frac{1}{2}}\left(
\|\nabla v\|_{L^{2}}^{\frac{1}{2}}+\|\nabla^{2} v\|_{L^{2}}^{\frac{1}{2}}\right)\|\partial_{p}q_{v}\|_{L^{2}}^{\frac{1}{2}}\left(
\|\partial_{p}q_{v}\|_{L^{2}}^{\frac{1}{2}}+\|\nabla \partial_{p}q_{v}\|_{L^{2}}^{\frac{1}{2}}\right)\nonumber\\
\leq& C\left(\|\nabla v\|_{L^{2}}^{2}+\|\nabla v\|_{L^{2}}^{4}+\|\Delta v\|_{L^{2}}^{2}+
\|\nabla v\|_{L^{2}}^{2}\|\Delta v\|_{L^{2}}^{2}\right)\|\partial_{p}q_{v} \|_{L^{2}}^{2}+\frac{1}{12}\|\partial_{p}^{2} v\|_{L^{2}}^{2}+\frac{3}{4}\|\nabla \partial_{p}q_{v}\|_{L^{2}}^{2}.
\end{align*}
Thus,
\begin{align}\label{qvH1p22}
&\frac{d}{dt}\left(\frac{1}{2}\|\partial_{p}q_{v}\|_{L^{2}}^{2} +\int_{\Gamma_{i}}\left(\frac{1}{2}(q_{v})^{2}-q_{v_{\ast}}q_{v}\right)d\Gamma_{i}\right)
+\frac{1}{4}\|\nabla\partial_{p}q_{v}\|_{L^{2}}^{2}
+\frac{1}{2}\left\|\partial_{p}^{2}q_{v}\right\|_{w}^{2}\nonumber\\
\leq&A_{1}(t)(\|\nabla q_{v}\|_{L^{2}}^{2}+\|\partial_{p}q_{v} \|_{L^{2}}^{2})+A_{2}(t)+\frac{1}{8}\|\Delta q_{v}\|_{L^{2}}^{2}.
\end{align}
Combining (\ref{qvH1h3}) and (\ref{qvH1p22}), we can infer that
\begin{align}
&\frac{d}{dt}\left(\frac{1}{2}\|q_{v}\|_{H^{1}}^{2} +\int_{\Gamma_{i}}\left(\frac{1}{2}(q_{v})^{2}-q_{v_{\ast}}q_{v}\right)d\Gamma_{i} + \int_{\Gamma_{l}}\left(\frac{1}{2}(q_{v})^{2}-q_{v}q_{blv}\right)d\Gamma_{l}\right)
+\frac{1}{2}\|q_{v}\|_{H^{2}}^{2}\nonumber\\
\leq&A_{4}(t)\|q_{v}\|_{H^{1}}^{2}+A_{5}(t).
\end{align}
Utilizing the H\"older inequality and the Young inequality, we can obtain that
\begin{align*} G=&\int_{\Gamma_{i}}\left(\frac{1}{2}(q_{v})^{2}-q_{v_{\ast}}q_{v}\right)d\Gamma_{i} +\int_{\Gamma_{l}}\left(\frac{1}{2}(q_{v})^{2}-q_{v}q_{blv}\right)d\Gamma_{l}\nonumber\\ \geq
&-C(\|q_{v_{\ast}}\|_{L^{2}(\Gamma_{i})}^{2}+\|q_{blv}\|_{L^{2}(\Gamma_{l})}^{2}):=-C_{\ast}.
\end{align*}
Thus
\begin{align*}
\frac{d}{dt}\left(\frac{1}{2}\|q_{v}\|_{H^{1}}^{2}+G+C_{\ast}\right)
+\frac{1}{2}\|q_{v}\|_{H^{2}}^{2} \leq A_{4}(t)(G+C_{\ast}+\|q_{v}\|_{H^{1}}^{2})+A_{5}(t).
\end{align*}
Considering regularities of $v,u,q_{v}$ in (\ref{L2-v-theta-qj}) and (\ref{uL2}) respectively, we know the $L^{2}$ integrability of $A_{4}$ and $A_{5}$. Then using the Gronwall inequality, we can complete the proof of this lemma.
\end{proof}
Thus, we get the following result about the local existence of strong solutions:
\begin{proposition} Let $v_{0}\in\mathbb{V}$, $\theta_{0}, q_{v0}, q_{r0},q_{c0}\in H^{1}$. Then there exists $t_{\ast}\in (0,t_{1}]$ and a solution $(v,\theta,q_{v},q_{c},q_{r})$ to equations (\ref{e1})-(\ref{e7}) with boundary condition (\ref{boundary condition}) and initial condition (\ref{initial condition}) satisfying that \begin{align}
\sup_{0\leq t\leq t_{\ast}}\|(v,\theta,q_{v},q_{c},q_{r})\|_{H^{1}}^{2}
+\int_{0}^{t_{\ast}}\|\Delta v\|_{L^{2}}^{2}+\|\nabla\partial_{p}v\|_{L^{2}}^{2}+\|(\theta,q_{v},q_{c},q_{r})\|_{H^{2}}^{2}dt\leq \mathcal{C}_{1}, \end{align} where $\mathcal{C}_{1}$ depends on the initial data and $t_{\ast}$. \end{proposition}
\subsection{The time regularity of strong solutions} In this subsection, we consider the time regularity of strong solutions. In fact, we have obtained the time regularity of velocity field $v$ in space $L^{2}(0,t_{\ast};\mathbb{H})$. Thus we mainly talk about the time regularities of $\theta,q_{v},q_{c},q_{r}$. Here we give a detailed proof of the time regularity of $q_{c}$. The proof of time regularity of other variables can be obtained similarly. \begin{lemma}\label{H1t} Let $v_{0}\in\mathbb{V}$, $\theta_{0}, q_{v0}, q_{r0},q_{c0}\in H^{1}$, and $q_{c}$ be the solution to equation (\ref{e6}) with boundary condition (\ref{boundary condition}) and initial condition (\ref{initial condition}). Then \begin{align*} \partial_{t}q_{c}\in L^{2}(0,t_{\ast};L^{2}). \end{align*} \end{lemma} \begin{proof}
Choosing a test function $\tilde{q}_{c}\in L^{2}(0,t_{\ast};L^{2})$ with $\|\tilde{q}_{c}\|_{L^{2}(0,t_{\ast};L^{2})}\leq 1$, we can infer from equation (\ref{e6}) that
\begin{align}\label{qct1}
|\langle\partial_{t}q_{c},\tilde{q}_{c}\rangle|\leq
|\langle\mathcal{A}_{q_{c}}q_{c},\tilde{q}_{c}\rangle|+ |\langle v\cdot\nabla q_{c}+w\partial_{p}q_{c},\tilde{q}_{c}\rangle|
+|\langle f_{q_{c}}+w^{-}Fh_{q_{v}},\tilde{q}_{c}\rangle|.
\end{align}
Using the H\"older inequality, we obtain that
\begin{align}\label{qct2}
|\langle\mathcal{A}_{q_{c}}q_{c},\tilde{q}_{c}\rangle|\leq C\|q_{c}\|_{H^{2}}\|\tilde{q}_{c}\|_{L^{2}}.
\end{align} By the inequality in Lemma \ref{trilinear term lemma}, we can infer that
\begin{align}\label{qct3}
&|\langle(v\cdot\nabla)q_{c},\tilde q_{c}\rangle|\leq\int_{\mathcal{M}'} \int_{p_{0}}^{p_{1}} |v|+|\partial_{p}v|dp\int_{p_{0}}^{p_{1}} |\nabla q_{c}||\tilde q_{c}|dp\mathcal{M}'\nonumber\\
\leq &C \left(\|v\|_{L^2}^{\frac{1}{2}}+\|u\|_{L^2}^{\frac{1}{2}}\right)
\left(\|v\|_{L^{2}}^{\frac{1}{2}} + \|\nabla v\|_{L^{2}}^{\frac{1}{2}}+\|u\|_{L^2}^{\frac{1}{2}}+\|\nabla u\|_{L^2}^{\frac{1}{2}}\right)\|\tilde q_{c}\|_{L^{2}}\|\nabla q_{c}\|_{L^{2}}^{\frac{1}{2}}\cdot\nonumber\\
&\ \ \left(\|\nabla q_{c}\|_{L^{2}}^{\frac{1}{2}}+ \|\nabla^{2} q_{c}\|_{L^{2}}^{\frac{1}{2}}\right)\nonumber\\
\leq &C\left(\|v\|_{L^{2}}\|\nabla q_{c}\|_{L^{2}}+\|\nabla v\|_{L^{2}}\|\nabla q_{c}\|_{L^{2}}+\|u\|_{L^{2}}\|\nabla q_{c}\|_{L^{2}}+\|\nabla u\|_{L^{2}}\|\nabla q_{c}\|_{L^{2}}+\|v\|_{L^{2}}\|\Delta q_{c}\|_{L^{2}}\right.\nonumber\\
&\left.+\|\nabla v\|_{L^{2}}\|\Delta q_{c}\|_{L^{2}}+\|u\|_{L^{2}}\|\Delta q_{c}\|_{L^{2}}\right)\|\tilde q_{c}\|_{L^{2}}.
\end{align}
Similarly, \begin{align}\label{qct4}
&|\langle w\partial_{p}q_{c},\tilde q_{c}\rangle|\leq\int_{\mathcal{M}'}\int_{p_{0}}^{p_{1}}|\nabla v|dp\int_{p_{0}}^{p_{1}}|\partial_{p}q_{c}||\tilde q_{c}|dpd\mathcal{M}'\nonumber\\
\leq &C\|\tilde q_{c}\|_{L^{2}}\|\nabla v\|_{L^{2}}^{\frac{1}{2}}\left(
\|\nabla v\|_{L^{2}}^{\frac{1}{2}}+\|\nabla^{2} v\|_{L^{2}}^{\frac{1}{2}}\right)\|\partial_{p}q_{c}\|_{L^{2}}^{\frac{1}{2}}\left(
\|\partial_{p}q_{c}\|_{L^{2}}^{\frac{1}{2}}+\|\nabla \partial_{p}q_{c}\|_{L^{2}}^{\frac{1}{2}}\right)\\
\leq &C\left(\|\nabla v\|_{L^{2}}\|\partial_{p}q_{c}\|_{L^{2}}+\|\nabla v\|_{L^{2}}\|\nabla\partial_{p}q_{c}\|_{L^{2}}+\|\Delta v\|_{L^{2}}\|\partial_{p}q_{c}\|_{L^{2}}+
\|\nabla v\|_{L^{2}}\|\Delta v\|_{L^{2}}\right.\nonumber\\
&\left.+\|\partial_{p}q_{c}\|_{L^{2}}\|\nabla\partial_{p}q_{c}\|_{L^{2}}
\right)\|\tilde q_{c}\|_{L^{2}}^{2}.
\end{align}
Recalling definitions of $F$ and $f_{q_{c}}$, we have
\begin{align}\label{qct5}
\left|\langle f_{q_{c}}
+w^{-}Fh_{q_{v}},\tilde{q}_{c}\rangle\right|
\leq &\left\|-k_{1}(q_{c}-q_{crit})^{+}-k_{2}q_{c}\tau(q_{r})+w^{-}Fh_{q_{v}}\right\|_{L^{2}}\|\tilde{q}_{c}\|_{L^{2}} \nonumber\\ \leq &
C (1+\|q_{c}\|_{L^{2}}+\|\nabla v\|_{L^{2}})\|\tilde{q}_{c}\|_{L^{2}},
\end{align}
where we have used the uniform boundness of $\tau(q_{r})$ in the last step.
Combining (\ref{qct1})-(\ref{qct5}), we can obtain that
\begin{align*}
\|\partial_{t}q_{c}\|_{L_{2}} \leq A_{6}(t) \end{align*} where \begin{align*}
A_{6}(t)&=C\left(1+\|q_{c}\|_{H^{2}}+\|q_{c}\|_{L^{2}}+\|\nabla v\|_{L^{2}}+\|v\|_{L^{2}}\|\nabla q_{c}\|_{L^{2}}+\|\nabla v\|_{L^{2}}\|\nabla q_{c}\|_{L^{2}}+\|u\|_{L^{2}}\|\nabla q_{c}\|_{L^{2}}\right.\nonumber\\
&\left.+\|\nabla u\|_{L^{2}}\|\nabla q_{c}\|_{L^{2}}+\|v\|_{L^{2}}\|\Delta q_{c}\|_{L^{2}}
+\|\nabla v\|_{L^{2}}\|\Delta q_{c}\|_{L^{2}}+\|u\|_{L^{2}}\|\Delta q_{c}\|_{L^{2}}+\|\nabla v\|_{L^{2}}\|\partial_{p}q_{c}\|_{L^{2}}\right.\nonumber\\
&\left.+\|\nabla v\|_{L^{2}}\|\nabla\partial_{p}q_{c}\|_{L^{2}}+
\|\Delta v\|_{L^{2}}\|\partial_{p}q_{c}\|_{L^{2}}+
\|\nabla v\|_{L^{2}}\|\Delta v\|_{L^{2}}
+\|\partial_{p}q_{c}\|_{L^{2}}\|\nabla\partial_{p}q_{c}\|_{L^{2}}\right).
\end{align*}
Considering regularities for $v$ and $q_{c}$, it is obviously that $A_{6}(t)$ is $L^{2}$ integrable on $(0,t_{\ast})$. Then integrating in time from $0$ to $t_{\ast}$, we can infer that
\begin{align*}
\int_{0}^{t_{\ast}}\|\partial_{t}q_{c}\|_{L_{2}}^{2}dt
\leq &\int_{0}^{t_{\ast}}A_{6}^{2}(t)dt<\infty,
\end{align*}
which shows that $q_{c}\in L^{2}(0,t_{\ast};L^{2})$.
\end{proof} \subsection{The global existence of strong solutions}
Checking the proof in previous sections, we find that the arising of $t_{\ast}$ is due to the $L^{2}$ a priori estimate for $\partial_{p}v$. If we want to obtain the global existence of strong solutions, the $L^{\infty}(0,t_{1};L^{2}(\mathcal{M}))$ estimate for $\partial_{p}v$ should rebuilt. In fact, through a similar argument as in \cite{CaoTiti6} and \cite{Hussein}, we know that, if $\|v_{0}\|_{H^{1}}+\left\|\partial_{p}v_{0}\right\|_{L^{m}}+
\|v_{0}\|_{L^{\infty}}<\infty$ for some $m>2$, then for any time $0\leq t\leq t_{1}$, \begin{align}\label{Global-v}
\sup\limits_{s\in [0,t]}\left(\|v\|_{H^{1}}^{2}
+\left\|\partial_{p}v\right\|_{L^{m}}^{m}\right)+
\int_{0}^{t}\|\Delta v\|_{L^{2}}^{2}+\big\|\nabla\partial_{p}v\big\|_{w}^{2}ds \leq C, \end{align}
where $C$ is a constant depending on $m,t$ and $\|v_{0}\|_{H^{1}}+\|\partial_{p}v_{0}\|_{L^{m}}+
\|v_{0}\|_{L^{\infty}}$.
Combining the $L^{2}$ estimates for $\theta,q_{v},q_{c},q_{r}$ in Lemma \ref{l2prior} and the $H^{1}$ estimate for $v$ in (\ref{Global-v}), and testifying their time regularities as in Lemma \ref{trv}, we can get the global existence of quasi-strong solutions. \begin{proposition}\label{Globalquasi-strong} Let $(\theta_{0},q_{v},q_{c},q_{r})\in (L^{2})^{4}$, $v_{0}\in L^{\infty}\cap\mathbb{V}$, $\partial_{p}v_{0}\in L^{m}$ for some $m>2$,
$\|v_{0}\|_{H^{1}}+\|\partial_{p}v_{0}\|_{L^{m}}+
\|v_{0}\|_{L^{\infty}}<\infty$. Then there exists a global quasi-strong solution $(v,\theta,q_{v},q_{c},q_{r})$ to equations (\ref{e1})-(\ref{e7}) with boundary condition (\ref{boundary condition}) and initial condition (\ref{initial condition}). Moreover \begin{align*}
\sup_{0\leq t\leq t_{1}}\left(\|v\|_{\mathbb{V}}^{2}+\|(\theta,q_{j})\|_{L^{2}}^{2}+\big\|\partial_{p}v\big\|_{L^{m}}^{m}\right)
+\int_{0}^{t_{1}}\|\Delta v\|_{L^{2}}^{2}+\|\nabla\partial_{p} v\|_{w}^{2}+\|(\theta,q_{j})\|_{H^{1}}^{2}dt\leq \mathcal{C}_{2}, \end{align*} for $j\in\{v,c,r\}$, where $\mathcal{C}_{2}$ depends on the initial data, $m$ and $t_{1}$. \end{proposition}
With the uniform estimate \eqref{Global-v} for $v$ at hand, proceeding exactly as in the proof of Lemma \ref{H1} to seek uniform $H^{1}$-estimates for $\theta$ and $q_{v},q_{c},q_{r}$, we are able to obtain that, for $j\in\{v,c,r\}$ \begin{align*}
\sup\limits_{t\in [0,t_{1}]}\left(\|(v,\theta,q_{j})\|_{H^{1}}^{2}
+\big\|\partial_{p}v\big\|_{L^{m}}^{m}\right)+
\int_{0}^{t_{1}}\|\Delta v\|_{L^{2}}^{2}+\|\nabla\partial_{p}v\big\|_{w}^{2}+\|(\theta,q_{j})\|_{H^{2}}^{2}ds \leq C, \end{align*}
where $C$ is a constant depending on $m,t_{1}$, $\|(\theta_{0},q_{j0})\|_{H^{1}}$ as well as $\|v_{0}\|_{H^{1}}+\|\partial_{p}v_{0}\|_{L^{m}}+
\|v_{0}\|_{L^{\infty}}$. At the same time, following the proof in Lemma \ref{H1t}, we can also obtain that \begin{align*} \left(\partial_{t}v,\partial_{t}\theta,\partial_{t}q_{v},\partial_{t}q_{c},\partial_{t}q_{r}\right)\in L^{2}(0,t_{1};\mathbb{H}\times L^{2}(\mathcal{M})^{4}). \end{align*} Then using the Aubin-lions compactness theorem, we deduce that \begin{align*} \left(v,\theta,q_{v},q_{c},q_{r}\right)\in C(0,t_{1};\mathbb{V}\times(H^{1})^{4}). \end{align*}
As summary, we get the global existence of strong solutions.
\begin{proposition}\label{Global whole} Let $(v_{0},\theta_{0},q_{v},q_{c},q_{r})\in \mathbb{V}\times (H^{1})^{4}$, $v_{0}\in L^{\infty}$, $\partial_{p}v_{0}\in L^{m}$ for some $m>2$,
$\|v_{0}\|_{H^{1}}+\|\partial_{p}v_{0}\|_{L^{m}}+
\|v_{0}\|_{L^{\infty}}<\infty$. Then there exists a global strong solution $(v,\theta,q_{v},q_{c},q_{r})$ to equations (\ref{e1})-(\ref{e7}) with boundary condition (\ref{boundary condition}) and initial condition (\ref{initial condition}). Moreover \begin{align*}
\sup_{0\leq t\leq t_{1}}\left(\|(v,\theta,q_{v},q_{c},q_{r})\|_{H^{1}}^{2}+\big\|\partial_{p}v\big\|_{L^{m}}^{m}\right)
+\int_{0}^{t_{1}}\|\Delta v\|_{L^{2}}^{2}+\|\nabla\partial_{p}v\|_{L^{2}}^{2}+\|(\theta,q_{v},q_{c},q_{r})\|_{H^{2}}^{2}dt\leq \mathcal{C}_{3}, \end{align*} where $\mathcal{C}_{3}$ depends on the initial data, $m$ and $t_{1}$. \end{proposition}
\section{The uniqueness of solutions} In this section, we prove the uniqueness of quasi-strong solution to equations (\ref{e1})-(\ref{e7}). Then the uniqueness of strong solution naturally holds. In this section, we return to consider the $(v,T,q_{v},q_{c},q_{r})$ system. Considering the relationship between $T$ and $\theta$, we know that the existence results of quasi-strong and strong solution for $\theta$ still hold for $T$. In order to overcome the difficulty caused by the Heaviside function, two new unknown quantities are introduced to substitute $T$ and $q_{c}$ as in \cite{Hittmeir,Hittmeir2017}, while the monotone operator theory is used in dealing with $q_{v}$ as in \cite{Zelati}.
Recalling the relationship between $\theta$ and $T$, we suppose that $T$ satisfies the following boundary conditions: \begin{align*} {\rm on}\ \Gamma_{i}:\ \partial_{p}T=T_{\ast}-T,\
{\rm on}\ \Gamma_{u}:\ \partial_{p}T=0,\
{\rm on}\ \Gamma_{l}:\ \partial_{\textbf{n}}T=T_{bl}-T. \end{align*} where $T_{\ast}$ and $T_{bl}$ are given sufficiently smooth temperature distribution.
Combining the Stampacchia method and De Giorgi iterations as in \cite{Zelati} and \cite {Hittmeir2017}, we can get the following uniform boundness result: \begin{lemma}\label{unform boundness} Let $(v_{0},T_{0},q_{v0},q_{c0},q_{r0})\in\mathbb{V}\times (H^{1})^{4}\cap L^{\infty}
(\mathcal{M})^{6}$, $\|v_{0}\|_{H^{1}}+\|\partial_{p}v_{0}\|_{L^{m}}+
\|v_{0}\|_{L^{\infty}}<\infty$ for some $m>2$ and $T_{0},q_{v0},q_{c0},q_{r0}$ be nonnegative initial data. Then for any $t\in[0,t_{1}]$, \begin{align} 0\leq q_{v}\leq q_{v}^{\ast},\ \ 0\leq q_{c}\leq q_{c}^{\ast},\ \ 0\leq q_{r}\leq q_{r}^{\ast},\ \ 0\leq T\leq T^{\ast}, \end{align} where \begin{align*}
q_{v}^{\ast}&={\rm max}\{\|q_{v0}\|_{L^{\infty}},\|q_{v\ast}\|_{L^{\infty}(0,t;\Gamma_{i})},
\|q_{blv}\|_{L^{\infty}(0,t;\Gamma_{l})}\},\nonumber\\
q_{c}^{\ast}&={\rm max}\{\|q_{c0}\|_{L^{\infty}},\|q_{c\ast}\|_{L^{\infty}(0,t;\Gamma_{i})},
\|q_{blc}\|_{L^{\infty}(0,t;\Gamma_{l})}\},\nonumber\\
q_{r}^{\ast}&={\rm max}\{\|q_{r0}\|_{L^{\infty}},\|q_{r\ast}\|_{L^{\infty}(0,t;\Gamma_{i})},
\|q_{blr}\|_{L^{\infty}(0,t;\Gamma_{l})}\},\nonumber\\
T^{\ast}&={\rm max}\{\|T_{0}\|_{L^{\infty}},\|T_{\ast}\|_{L^{\infty}(0,t;\Gamma_{i})},
\|T_{bl}\|_{L^{\infty}(0,t;\Gamma_{l})}\}. \end{align*} \end{lemma} \begin{remark} During the De Giorgi iterations process as in \cite{Zelati}, one should note that
$\|w\|_{H^{1}}^{2}$ can be bounded by $\|\Delta v\|_{L^{2}}^{2}+\|\nabla\partial_{p}v\|_{L^{2}}^{2}$, whose $L^{2}$ integrability in time interval $[0,t_{1}]$ can be ensured by the result in Proposition \ref{Global whole}. \end{remark}
\begin{proposition}\label{uniqueness} Assume $(v_{1},T_{1},q_{v1},q_{c1},q_{r1})$ and $(v_{2},T_{2},q_{v2},q_{c2},q_{r2})$ are two quasi-strong solutions to equations (\ref{e1})-(\ref{e7}) corresponding to the same initial data $(v_{0},T_{0},q_{v0},q_{c0},q_{r0})$, with the function $F$ in source terms replaced by its positive part $F^{+}$. Then \begin{align*} v_{1}=v_{2},T_{1}=T_{2},q_{v1}=q_{v2},q_{c1}=q_{c2},q_{r1}=q_{r2}, \end{align*} in the sense of $L^{2}$. \end{proposition} \begin{remark} It is worth noting that the uniqueness holds under the assumption that the function $F$ is replaced by its positive part $F^{+}$. This assumption is in line with physical reality. For more detailed explanation, we refer readers to \cite{Zelati}. In addition, due to the uniform boundness results in Lemma \ref{unform boundness}, we can take $\tau(q_{r})=q_{r}$ in this section. \end{remark} \begin{proof} Let $(v_{1},T_{1},q_{v1},q_{c1},q_{r1})$ and $(v_{2},T_{2},q_{v2},q_{c2},q_{r2})$ be two global quasi-strong solutions corresponding to initial data $(v_{1}^{0},T_{1}^{0},q_{v1}^{0},q_{c1}^{0},q_{r1}^{0})$ and $(v_{2}^{0},T_{2}^{0},q_{v2}^{0},q_{c2}^{0},q_{r2}^{0})$ respectively.
Set \begin{align*} \hat{v}=v_{1}-v_{2},\ \hat{T}=T_{1}-T_{2},\ \hat{q}_{j}=q_{j1}-q_{j2}, j\in\{v,c,r\}. \end{align*} Then $\hat{v}$ satisfies that \begin{align}\label{hat{v0}} \partial_{t}\hat{v}-\Delta\hat{v}+(v_{1}\cdot\nabla)\hat{v}+w_{1}\partial_{p}\hat{v}+ (\hat{v}\cdot\nabla)v_{2}+\hat{w}\partial_{p}v_{2}+f\hat{v}^{\bot}+\nabla\hat{\Phi}_{s}+\nabla\int_{p}^{p_{1}}\frac{R}{p'}\hat{T}dp'=0, \end{align} where $\hat{w}=w_{1}-w_{2}$ and $\hat{\Phi}_{s}=\Phi_{s1}-\Phi_{s2}$. And the corresponding boundary conditions are
\begin{align}
{\rm on}\ \Gamma_{i}:\ \partial_{p}\hat{v}=0,\ \hat{w}=0,\
{\rm on}\ \Gamma_{u}:\ \partial_{p}\hat{v}=0,\ \hat{w}=0,\ \partial_{p}\hat{w}=0,\
{\rm on}\ \Gamma_{l}:\ \hat{v}=0,\ \partial_{\textbf{n}}\hat{v}=0.\nonumber
\end{align}
Taking the inner product of equation (\ref{hat{v0}}) with $\hat{v}$ in $L^{2}$ space, using integration by parts, we can deduce that \begin{align*}
\frac{1}{2}\frac{d}{dt}\|\hat{v}\|_{L^{2}}^{2}+\|\nabla \hat{v}\|_{L^{2}}^{2} =&-\int_{\mathcal{M}}\left[(\hat{v}\cdot\nabla)v_{2}+\hat{w}\partial_{p}v_{2}\right]\cdot\hat{v}d\mathcal{M} -\int_{\mathcal{M}}\left[(v_{1}\cdot\nabla)\hat{v}+w_{1}\partial_{p}\hat{v}\right]\cdot\hat{v}d\mathcal{M}\nonumber\\ &-\int_{\mathcal{M}}f\hat{v}^{\bot}\cdot\hat{v}d\mathcal{M}- \int_{\mathcal{M}}\nabla\hat{\Phi}_{s}\cdot\hat{v}d\mathcal{M} -\int_{\mathcal{M}}\nabla\int_{p}^{p_{1}}\frac{R}{p'}\hat{T}dp'\cdot\hat{v}d\mathcal{M}. \end{align*} It is easy to verify that \begin{align*} \int_{\mathcal{M}}f\hat{v}^{\bot}\cdot\hat{v}d\mathcal{M}=0. \end{align*} By integration by parts, we can obtain that \begin{align*} \int_{\mathcal{M}}\nabla\hat{\Phi}_{s}\cdot\hat{v}d\mathcal{M}=0, \end{align*} and \begin{align*} \int_{\mathcal{M}}\left[(v_{1}\cdot\nabla)\hat{v}+w_{1}\partial_{p}\hat{v}\right]\cdot\hat{v}d\mathcal{M}=0. \end{align*} Using integration by parts, the H\"older inequality and the Young inequality, we can infer that \begin{align*} \int_{\mathcal{M}}\nabla\int_{p}^{p_{1}}\frac{R}{p'}\hat{T}dp'\cdot\hat{v}d\mathcal{M}
\leq C\|\hat{T}\|_{L^{2}}^{2}+\frac{1}{8}\|\nabla\hat{v}\|_{L^{2}}^{2}. \end{align*} Utilizing the inequality in Lemma \ref{HHP}, we can get that \begin{align*} \int_{\mathcal{M}}(\hat{v}\cdot\nabla)v_{2}\cdot\hat{v}d\mathcal{M}
\leq& C\|\hat{v}\|_{L^{2}}\|\nabla\hat{v}\|_{L^{2}}(\|\nabla v_{2}\|_{L^{2}}+\|\nabla v_{2}\|_{L^{2}}^{\frac{1}{2}}\|\nabla\partial_{p}v_{2}\|_{L^{2}}^{\frac{1}{2}})\nonumber\\
\leq& C(\|\nabla v_{2}\|_{L^{2}}^{2}+\|\nabla\partial_{p}v_{2}\|_{L^{2}}^{2})\|\hat{v}\|_{L^{2}}^{2}
+\frac{1}{8}\|\nabla\hat{v}\|_{L^{2}}^{2}. \end{align*} Considering the inequality in Lemma \ref{trilinear term lemma}, we have \begin{align*} &\int_{\mathcal{M}}\hat{w}\partial_{p}v_{2}\cdot\hat{v}d\mathcal{M}
\leq\int_{\mathcal{M}'}\int_{p_{0}}^{p_{1}}|\nabla\hat{v}|dp
\int_{p_{0}}^{p_{1}}|\partial_{p}v_{2}||\hat{v}|dpd\mathcal{M}'\nonumber\\
\leq& C\|\nabla\hat{v}\|_{L^{2}}\|\partial_{p}v_{2}\|_{L^{2}}^{\frac{1}{2}}
(\|\partial_{p}v_{2}\|_{L^{2}}^{\frac{1}{2}}+\|\nabla\partial_{p}v_{2}\|_{L^{2}}^{\frac{1}{2}})
\|\hat{v}\|_{L^{2}}^{\frac{1}{2}}(\|\hat{v}\|_{L^{2}}^{\frac{1}{2}}+\|\nabla\hat{v}\|_{L^{2}}^{\frac{1}{2}})\nonumber\\
\leq& C\left(\|\partial_{p}v_{2}\|_{L^{2}}^{2}+\|\partial_{p}v_{2}\|_{L^{2}}^{4}+\|\nabla\partial_{p}v_{2}\|_{L^{2}}^{2}
+\|\partial_{p}v_{2}\|_{L^{2}}^{2}\|\nabla\partial_{p}v_{2}\|_{L^{2}}^{2} \right)\|\hat{v}\|_{L^{2}}^{2}
+\frac{1}{8}\|\nabla\hat{v}\|_{L^{2}}^{2}. \end{align*} Thus combining all the above inequalities, we can deduce that \begin{align}\label{hat{v}}
\frac{d}{dt}\|\hat{v}\|_{L^{2}}^{2}+2\|\nabla \hat{v}\|_{L^{2}}^{2}
\leq &C\left(\|\nabla v_{2}\|_{L^{2}}^{2}+\|\nabla\partial_{p}v_{2}\|_{L^{2}}^{2}+\|\partial_{p}v_{2}\|_{L^{2}}^{2}
+\|\partial_{p}v_{2}\|_{L^{2}}^{4}+\|\nabla\partial_{p}v_{2}\|_{L^{2}}^{2}\right.\nonumber\\ &\left.
+\|\partial_{p}v_{2}\|_{L^{2}}^{2}\|\nabla\partial_{p}v_{2}\|_{L^{2}}^{2} \right)\|\hat{v}\|_{L^{2}}^{2}+C\|\hat{T}\|_{L^{2}}^{2}+\frac{3}{4}\|\nabla\hat{v}\|_{L^{2}}^{2}. \end{align}
Set \begin{align*} Q=q_{v}+q_{c},\ \ H=T+\frac{L}{c_{p}\Pi}q_{v}. \end{align*} Then $\hat{Q}=Q_{1}-Q_{2}$ satisfies that \begin{align}\label{hat{Q0}} \partial_{t}\hat{Q}+v_{1}\cdot\nabla\hat{Q}+w_{1}\partial_{p}\hat{Q}+\hat{v}\cdot\nabla Q_{2}+\hat{w}\partial_{p} Q_{2}+A_{q}\hat{Q} =f_{q_{v1}}-f_{q_{v2}}+f_{q_{c1}}-f_{q_{c2}}, \end{align} with the boundary conditions \begin{align*} {\rm on}\ \Gamma_{i}:\ \partial_{p}\hat{Q}=-\hat{Q},\
{\rm on}\ \Gamma_{u}:\ \partial_{p}\hat{Q}=0, \
{\rm on}\ \Gamma_{l}:\ \partial_{\textbf{n}}\hat{Q}=-\hat{Q}. \end{align*} Taking the inner product of equation (\ref{hat{Q0}}) with $\hat{Q}$ in $L^{2}$ space, we can deduce that \begin{align*}
&\frac{1}{2}\frac{d}{dt}\|\hat{Q}\|_{L^{2}}^{2}+\int_{\mathcal{M}}\hat{Q}A_{q}\hat{Q}d\mathcal{M} +\int_{\mathcal{M}}\left[\hat{v}\cdot\nabla Q_{2}+\hat{w}\partial_{p} Q_{2}\right]\hat{Q}d\mathcal{M}\nonumber\\ =&\int_{\mathcal{M}}\left(f_{q_{v1}}-f_{q_{v2}}\right)\hat{Q}d\mathcal{M} +\int_{\mathcal{M}}\left(f_{q_{c1}}-f_{q_{c2}}\right)\hat{Q}d\mathcal{M}, \end{align*} where we have used \begin{align*} \int_{\mathcal{M}}(v_{1}\cdot\nabla\hat{Q}+w_{1}\partial_{p}\hat{Q})\hat{Q}d\mathcal{M}=0. \end{align*} Through integration by parts and considering the uniform boundness of $q_{v},q_{c}$, we can infer that \begin{align}\label{multiterm} \int_{\mathcal{M}}\left[\hat{v}\cdot\nabla Q_{2}+\hat{w}\partial_{p} Q_{2}\right]\hat{Q}d\mathcal{M} =&-\int_{\mathcal{M}}\left[\hat{v}\cdot\nabla \hat{Q}+\hat{w}\partial_{p}\hat{Q}\right]Q_{2}d\mathcal{M}\nonumber\\
\leq&\|Q_{2}\|_{L^{\infty}}\|\hat{v}\|_{L^{2}}\|\nabla\hat{Q}\|_{L^{2}}+
\|Q_{2}\|_{L^{\infty}}\|\hat{w}\|_{L^{2}}\|\partial_{p}\hat{Q}\|_{L^{2}}\nonumber\\
\leq& C\|\hat{v}\|_{L^{2}}^{2}+\frac{1}{8}\|\nabla\hat{Q}\|_{L^{2}}^{2}+\frac{1}{8}\|\partial_{p}\hat{Q}\|_{L^{2}}^{2}
+\frac{1}{8\delta^{2}}\|\nabla\hat{v}\|_{L^{2}}^{2}, \end{align} where $\delta$ is a sufficiently small fixed constant. Recalling the definition of $f_{q_{v}}$, considering the uniform boundness of $q_{r}$ and $q_{v}$, we can infer that \begin{align*} &\int_{\mathcal{M}}\left(f_{q_{v1}}-f_{q_{v2}}\right)\hat{Q}d\mathcal{M} =\int_{\mathcal{M}}\left[k_{3}q_{r1}(q_{vs}-q_{v1})^{+}-k_{3}q_{r2}(q_{vs}-q_{v2})^{+}\right]\hat{Q}d\mathcal{M}\nonumber\\ =&k_{3}\int_{\mathcal{M}}(q_{r1}-q_{r2})(q_{vs}-q_{v1})^{+}\hat{Q}d\mathcal{M} +k_{3}\int_{\mathcal{M}}q_{r2}\left[(q_{vs}-q_{v1})^{+}-(q_{vs}-q_{v2})^{+}\right]\hat{Q}d\mathcal{M}\nonumber\\
\leq& C\|q_{r1}-q_{r2}\|_{L^{2}}\|\hat{Q}\|_{L^{2}}+C\|q_{v1}-q_{v2}\|_{L^{2}}\|\hat{Q}\|_{L^{2}}\leq C(\|\hat{q}_{r}\|_{L^{2}}^{2}+\|\hat{q}_{v}\|_{L^{2}}^{2}+\|\hat{Q}\|_{L^{2}}^{2}). \end{align*} Similarly, \begin{align*} &\int_{\mathcal{M}}\left(f_{q_{c1}}-f_{q_{c2}}\right)\hat{Q}d\mathcal{M}\nonumber\\ =&\int_{\mathcal{M}}\left(-k_{1}(q_{c1}-q_{crit})^{+}-k_{2}q_{c1}q_{r1}+k_{1}(q_{c2}-q_{crit})^{+}+k_{2}q_{c2}q_{r2}\right)\hat{Q}d\mathcal{M}\nonumber\\ =&-k_{1}\int_{\mathcal{M}}\left[(q_{c1}-q_{crit})^{+}-(q_{c2}-q_{crit})^{+}\right]\hat{Q}d\mathcal{M} -k_{2}\int_{\mathcal{M}}\left(q_{c1}q_{r1}-q_{c2}q_{r2}\right)\hat{Q}d\mathcal{M}\nonumber\\
\leq& C\|q_{c1}-q_{c2}\|_{L^{2}}\|\hat{Q}\|_{L^{2}}+C\|q_{r1}-q_{r2}\|_{L^{2}}\|\hat{Q}\|_{L^{2}}\leq C(\|\hat{q}_{r}\|_{L^{2}}^{2}+\|\hat{q}_{c}\|_{L^{2}}^{2}+\|\hat{Q}\|_{L^{2}}^{2})\nonumber\\
\leq& C(\|\hat{q}_{v}\|_{L^{2}}^{2}+\|\hat{q}_{r}\|_{L^{2}}^{2}+\|\hat{Q}\|_{L^{2}}^{2}). \end{align*} Integrating by parts and considering the boundary condition of $\hat{Q}$, we have \begin{align*}
\int_{\mathcal{M}}\hat{Q}A_{q}\hat{Q}d\mathcal{M}=\|\nabla\hat{Q}\|_{L^{2}}^{2}
+\|\partial_{p}\hat{Q}\|_{w}^{2}-\int_{\Gamma_{l}}\hat{Q}\partial_{\textbf{n}}\hat{Q}d\Gamma_{l}
-\int_{\Gamma_{i}}\hat{Q}\partial_{p}\hat{Q}d\Gamma_{i}\geq\|\nabla\hat{Q}\|_{L^{2}}^{2}
+\|\partial_{p}\hat{Q}\|_{w}^{2}. \end{align*}
Thus combining the above inequalities, we can deduce that
\begin{align}\label{hat{Q}}
\frac{d}{dt}\|\hat{Q}\|_{L^{2}}^{2}+\|\nabla\hat{Q}\|_{L^{2}}^{2}
+\|\partial_{p}\hat{Q}\|_{w}^{2}
\leq C(\|\hat{v}\|_{L^{2}}^{2}+\|\hat{q}_{v}\|_{L^{2}}^{2}+\|\hat{q}_{r}\|_{L^{2}}^{2}+\|\hat{Q}\|_{L^{2}}^{2})
+\frac{1}{8\delta^{2}}\|\nabla\hat{v}\|_{L^{2}}^{2}. \end{align}
Next, we consider the $L^{2}$ estimate for $\hat{q}_{v}$. It is easy to check that \begin{align}\label{hat{q}} &\partial_{t}\hat{q}_{v}+v_{1}\cdot\nabla\hat{q}_{v}+w_{1}\partial_{p}\hat{q}_{v}+A_{q}\hat{q}_{v} +\hat{v}\cdot\nabla q_{v2}+\hat{w}\partial_{p} q_{v2}+A_{q}\hat{q}_{v}\nonumber\\ =&f_{q_{v1}}-f_{q_{v2}}-w_{1}^{-}F^{+}(T_{1})h(q_{v1}-q_{vs})+w_{2}^{-}F^{+}(T_{2})h(q_{v2}-q_{vs}). \end{align} Taking the inner product of equation (\ref{hat{q}}) with $\hat{q}_{v}$ in $L^{2}$ space, we have \begin{align*}
&\frac{1}{2}\frac{d}{dt}\|\hat{q}_{v}\|_{L^{2}}^{2}+\int_{\mathcal{M}}\hat{q}_{v}A_{q}\hat{q}_{v}d\mathcal{M} +\int_{\mathcal{M}}(\hat{v}\cdot\nabla q_{v2}+\hat{w}\partial_{p} q_{v2})\hat{q}_{v}d\mathcal{M}\nonumber\\ =&\int_{\mathcal{M}}\left(f_{q_{v1}}-f_{q_{v2}}\right)\hat{q}_{v}d\mathcal{M} +\int_{\mathcal{M}}\left[-w_{1}^{-}F^{+}(T_{1})h(q_{v1}-q_{vs})+w_{2}^{-}F^{+}(T_{2})h(q_{v2}-q_{vs})\right]\hat{q}_{v}d\mathcal{M}. \end{align*} Through integration by parts, we have \begin{align*}
\int_{\mathcal{M}}\hat{q}_{v}A_{q}\hat{q}_{v}d\mathcal{M}\geq \|\nabla\hat{q}_{v}\|_{L^{2}}^{2}+\|\partial_{p}\hat{q}_{v}\|_{w}^{2}. \end{align*} Through a similar calculation as in (\ref{multiterm}), we can infer that
\begin{align*} \int_{\mathcal{M}}(\hat{v}\cdot\nabla q_{v2}+\hat{w}\partial_{p} q_{v2})\hat{q}_{v}d\mathcal{M}=&-\int_{\mathcal{M}}(\hat{v}\cdot\nabla \hat{q}_{v}+\hat{w}\partial_{p}\hat{q}_{v})q_{v2}d\mathcal{M}\nonumber\\
\leq&\|q_{v2}\|_{L^{\infty}}\|\hat{v}\|_{L^{2}}\|\nabla\hat{q}_{v}\|_{L^{2}}+
\|q_{v2}\|_{L^{\infty}}\|\hat{w}\|_{L^{2}}\|\partial_{p}\hat{q}_{v}\|_{L^{2}}\nonumber\\
\leq& C\|\hat{v}\|_{L^{2}}^{2}+\frac{1}{8}\|\nabla\hat{q}_{v}\|_{L^{2}}^{2}+\frac{1}{8}\|\partial_{p}\hat{q}_{v}\|_{L^{2}}^{2}
+\frac{1}{8\delta^{2}}\|\nabla\hat{v}\|_{L^{2}}^{2}, \end{align*} where we have used the uniform boundness of $q_{v}$. Recalling the definition of $f_{q_{v}}$, we have \begin{align*} &\int_{\mathcal{M}}\left(f_{q_{v1}}-f_{q_{v2}}\right)\hat{q}_{v}d\mathcal{M} =\int_{\mathcal{M}}\left[k_{3}q_{r1}(q_{vs}-q_{v1})^{+}-k_{3}q_{r2}(q_{vs}-q_{v2})^{+}\right]\hat{q}_{v}d\mathcal{M}\nonumber\\ =&k_{3}\int_{\mathcal{M}}(q_{r1}-q_{r2})(q_{vs}-q_{v1})^{+}\hat{q}_{v}d\mathcal{M} +k_{3}\int_{\mathcal{M}}q_{r2}\left[(q_{vs}-q_{v1})^{+}-(q_{vs}-q_{v2})^{+}\right]\hat{q}_{v}d\mathcal{M}\nonumber\\
\leq& C\|\hat{q}_{r}\|_{L^{2}}\|\hat{q}_{v}\|_{L^{2}}+C\|\hat{q}_{v}\|_{L^{2}}\|\hat{q}_{v}\|_{L^{2}}\leq C(\|\hat{q}_{r}\|_{L^{2}}^{2}+\|\hat{q}_{v}\|_{L^{2}}^{2}). \end{align*} Through a direct calculation, we have \begin{align*} &\int_{\mathcal{M}}\left[-w_{1}^{-}F^{+}(T_{1})h(q_{v1}-q_{vs})+w_{2}^{-}F^{+}(T_{2})h(q_{v2}-q_{vs})\right]\hat{q}_{v}d\mathcal{M}\nonumber\\ =&\int_{\mathcal{M}}\left[-w_{1}^{-}F^{+}(T_{1})h(q_{v1}-q_{vs})+w_{2}^{-}F^{+}(T_{1})h(q_{v1}-q_{vs})\right]\hat{q}_{v}d\mathcal{M}\nonumber\\ &+\int_{\mathcal{M}}\left[w_{2}^{-}F^{+}(T_{2})h(q_{v1}-q_{vs})-w_{2}^{-}F^{+}(T_{1})h(q_{v1}-q_{vs})\right]\hat{q}_{v}d\mathcal{M}\nonumber\\ &+\int_{\mathcal{M}}\left[w_{2}^{-}F^{+}(T_{2})h(q_{v2}-q_{vs})-w_{2}^{-}F^{+}(T_{2})h(q_{v1}-q_{vs})\right]\hat{q}_{v}d\mathcal{M}\nonumber \end{align*} Utilizing the uniform boundness of $F$ and $h(q_{v}-q_{vs})$, we have \begin{align*} &\int_{\mathcal{M}}\left[-w_{1}^{-}F^{+}(T_{1})h(q_{v1}-q_{vs})+w_{2}^{-}F^{+}(T_{1})h(q_{v1}-q_{vs})\right]\hat{q}_{v}d\mathcal{M}\nonumber\\
\leq &C\|w_{1}^{-}-w_{2}^{-}\|_{L^{2}}\|\hat{q}_{v}\|_{L^{2}}\leq C\|\hat{q}_{v}\|_{L^{2}}^{2}+\frac{1}{8}\|\nabla\hat{v}\|_{L^{2}}^{2}. \end{align*} Considering the Lipschitz continuity of $F$ and the uniform boundness of $h(q_{v}-q_{vs})$, we can infer that \begin{align*} &\int_{\mathcal{M}}\left[w_{2}^{-}F^{+}(T_{2})h(q_{v1}-q_{vs})-w_{2}^{-}F^{+}(T_{1})h(q_{v1}-q_{vs})\right]\hat{q}_{v}d\mathcal{M}\nonumber\\
\leq& C\|w_{2}^{-}\|_{L^{2}}\|F^{+}(T_{2})-F^{+}(T_{1})\|_{L^{6}}\|\hat{q}_{v}\|_{L^{3}}
\leq C\|v_{2}\|_{H^{1}}\|\hat{T}\|_{H^{1}}\|\hat{q}_{v}\|_{L^{2}}^{\frac{1}{2}}
\|\hat{q}_{v}\|_{H^{1}}^{\frac{1}{2}}\nonumber\\
\leq& C\|v_{2}\|_{H^{1}}^{4}\|\hat{q}_{v}\|_{L^{2}}^{2}+\frac{1}{8}\|\hat{q}_{v}\|_{H^{1}}^{2}
+\frac{1}{8}\|\hat{T}\|_{H^{1}}^{2}. \end{align*} Considering the monotonicity of the heaviside function $q_{v}\rightarrow h(q_{v}-q_{vs})$ and the positivity of $F^{+}$, we get \begin{align*} \int_{\mathcal{M}}\left[w_{2}^{-}F^{+}(T_{2})h(q_{v2}-q_{vs})-w_{2}^{-}F^{+}(T_{2})h(q_{v1}-q_{vs})\right]\hat{q}_{v}d\mathcal{M}\leq 0. \end{align*} Therefore \begin{align*} &\int_{\mathcal{M}}\left[-w_{1}^{-}F^{+}(T_{1})h(q_{v1}-q_{vs})+w_{2}^{-}F^{+}(T_{2})h(q_{v2}-q_{vs})\right]\hat{q}_{v}d\mathcal{M}\nonumber\\
\leq & C(1+\|v_{2}\|_{H^{1}}^{4})\|\hat{q}_{v}\|_{L^{2}}^{2}+\frac{1}{8}\|\nabla\hat{v}\|_{L^{2}}^{2}
+\frac{1}{8}\|\hat{q}_{v}\|_{H^{1}}^{2}
+\frac{1}{8}\|\hat{T}\|_{H^{1}}^{2}\nonumber\\
\leq&C(1+\|v_{2}\|_{H^{1}}^{4})\|\hat{q}_{v}\|_{L^{2}}^{2}+\frac{1}{8\delta^{2}}\|\nabla\hat{v}\|_{L^{2}}^{2}
+\frac{1}{4}\|\hat{q}_{v}\|_{H^{1}}^{2}
+\frac{1}{8}\|\hat{H}\|_{H^{1}}^{2}. \end{align*} Thus \begin{align}\label{hat{q}_{v}}
&\frac{d}{dt}\|\hat{q}_{v}\|_{L^{2}}^{2}+\|\nabla\hat{q}_{v}\|_{L^{2}}^{2}+\|\partial_{p}\hat{q}_{v}\|_{L^{2}}^{2}\nonumber\\
\leq& C(1+\|v_{2}\|_{H^{1}}^{4})(\|\hat{v}\|_{L^{2}}^{2}+\|\hat{q}_{v}\|_{L^{2}}^{2}+\|\hat{q}_{r}\|_{L^{2}}^{2}+\|\hat{v}\|_{L^{2}}^{2})
+\frac{1}{4\delta^{2}}\|\nabla\hat{v}\|_{L^{2}}^{2}+\frac{1}{8}\|\hat{H}\|_{H^{1}}^{2}. \end{align}
Next we consider the estimate for $\hat{q}_{r}$, Through a similar argument to $\hat{q}_{v}$, we can obtain that \begin{align}\label{hat{q}_{r0}}
&\frac{1}{2}\frac{d}{dt}\|\hat{q}_{r}\|_{L^{2}}^{2}+\|\nabla\hat{q}_{r}\|_{L^{2}}^{2}+\|\partial_{p}\hat{q}_{r}\|_{L^{2}}^{2}\nonumber\\ \leq& \int_{\mathcal{M}}\left[\hat{v}\cdot\nabla q_{r2}+\hat{w}\partial_{p} q_{r2}\right]\hat{q}_{r}d\mathcal{M}+\int_{\mathcal{M}}\left(f_{q_{r1}}-f_{q_{r2}}\right)\hat{q}_{r}d\mathcal{M}. \end{align} Through a similar argument as in (\ref{multiterm}), we have \begin{align}\label{hat{q}_{r1}} \int_{\mathcal{M}}\left[\hat{v}\cdot\nabla q_{r2}+\hat{w}\partial_{p} q_{r2}\right]\hat{q}_{r}d\mathcal{M}=&-\int_{\mathcal{M}}\left[\hat{v}\cdot\nabla \hat{q}_{r}+\hat{w}\partial_{p}\hat{q}_{r}\right]q_{r2}d\mathcal{M}\nonumber\\
\leq&\|q_{r2}\|_{L^{\infty}}\|\hat{v}\|_{L^{2}}\|\nabla\hat{q}_{r}\|_{L^{2}}+
\|q_{r2}\|_{L^{\infty}}\|\hat{w}\|_{L^{2}}\|\partial_{p}\hat{q}_{r}\|_{L^{2}}\nonumber\\
\leq& C\|\hat{v}\|_{L^{2}}^{2}+\frac{1}{2}\|\nabla\hat{q}_{r}\|_{L^{2}}^{2}+\frac{1}{4}\|\partial_{p}\hat{q}_{r}\|_{L^{2}}^{2}
+\frac{1}{8\delta^{2}}\|\nabla\hat{v}\|_{L^{2}}^{2}. \end{align} Recalling the definition of $f_{q_{r}},$ we infer that \begin{align}\label{hat{q}_{r2}} &\int_{\mathcal{M}}\left(f_{q_{r1}}-f_{q_{r2}}\right)\hat{q}_{r}d\mathcal{M}\nonumber\\ \leq& C\int_{\mathcal{M}}\partial_{p}(\frac{p}{R\bar{\theta}}\hat{q}_{r})\hat{q}_{r}d\mathcal{M} +C\int_{\mathcal{M}}\left((q_{c1}-q_{crit})^{+}-(q_{c2}-q_{crit})^{+}\right)\hat{q}_{r}d\mathcal{M}\nonumber\\ &+C\int_{\mathcal{M}}\left(q_{c1}q_{r1}-q_{c2}q_{r2}\right)\hat{q}_{r}d\mathcal{M} +C\int_{\mathcal{M}}\left[q_{r1}(q_{vs}-q_{v1})^{+}-q_{r2}(q_{vs}-q_{v2})^{+}\right]d\mathcal{M}\nonumber\\
\leq& C(\|\hat{q}_{v}\|_{L^{2}}^{2}+\|\hat{q}_{r}\|_{L^{2}}^{2}+\|\hat{Q}\|_{L^{2}}^{2})
+\frac{1}{4}\|\partial_{p}\hat{q}_{r}\|_{L^{2}}^{2}. \end{align} Substituting (\ref{hat{q}_{r1}}), (\ref{hat{q}_{r2}}) into (\ref{hat{q}_{r0}}), we can deduce that \begin{align}\label{hat{q}_{r}}
\frac{d}{dt}\|\hat{q}_{r}\|_{L^{2}}^{2}+\|\nabla\hat{q}_{r}\|_{L^{2}}^{2}+\|\partial_{p}\hat{q}_{r}\|_{L^{2}}^{2}
\leq C\left(\|\hat{v}\|_{L^{2}}^{2}+\|\hat{q}_{v}\|_{L^{2}}^{2}+\|\hat{q}_{r}\|_{L^{2}}^{2}+\|\hat{Q}\|_{L^{2}}^{2} \right)+\frac{1}{8\delta^{2}}\|\nabla\hat{v}\|_{L^{2}}^{2}. \end{align}
Next, we consider the $L^{2}$ estimate for $H$.
Noting that $H=T+\frac{L}{c_{p}}q_{v}$, then $H$ satisfies \begin{align*} \partial_{t}H+v\nabla H+w\partial_{p}H+\mathcal{A}_{q}H-\frac{RT}{c_{p}p}w=f_{T}+\frac{L}{c_{p}}f_{q_{v}}. \end{align*} Thus \begin{align*} &\partial_{t}\hat{H}+v_{1}\nabla\hat{H}+w_{1}\partial_{p}\hat{H}+\hat{v}\cdot\nabla H_{2}+\hat{w}\partial_{p}H_{2}+\mathcal{A}_{q}\hat{H}\nonumber\\ =&\frac{R}{pc_{p}}\left(T_{1}\hat{w}+w_{2}\hat{T}\right)+f_{T_{1}}+\frac{L}{c_{p}}f_{q_{v1}}- f_{T_{2}}-\frac{L}{c_{p}}f_{q_{v2}}, \end{align*} with the boundary condition \begin{align*} {\rm on}\ \Gamma_{i}:\ \partial_{p}\hat{H}=-\hat{H},\
{\rm on}\ \Gamma_{u}:\ \partial_{p}\hat{Q}=0,\
{\rm on}\ \Gamma_{l}:\ \partial_{\textbf{n}}\hat{Q}=-\hat{H}. \end{align*} Taking the inner product of the equation for $\hat{H}$ with $\hat{H}$ in $L^{2}$ space, we can deduce that \begin{align*}
&\frac{1}{2}\frac{d}{dt}\|\hat{H}\|_{L^{2}}^{2}+\|\nabla\hat{H}\|_{L^{2}}^{2}
+\|\partial_{p}\hat{H}\|_{L^{2}}^{2}\nonumber\\ \leq& \int_{\mathcal{M}}\frac{R}{pc_{p}}\left(T_{1}\hat{w}+w_{2}\hat{T}\right)\hat{H}d\mathcal{M}+ \int_{\mathcal{M}}\left(\hat{v}\cdot\nabla H_{2}+\hat{w}\partial_{p} H_{2}\right)\hat{H}d\mathcal{M}\nonumber\\ &+\int_{\mathcal{M}}\left(f_{T_{1}}-f_{T_{2}}\right)\hat{H}d\mathcal{M} +\frac{L}{c_{p}}\int_{\mathcal{M}}\left(f_{q_{v1}}-f_{q_{v2}}\right)\hat{H}d\mathcal{M}. \end{align*} Through a similar argument as in (\ref{multiterm}), we have \begin{align*} &\int_{\mathcal{M}}\left(\hat{v}\cdot\nabla H_{2}+\hat{w}\partial_{p} H_{2}\right)\hat{H}d\mathcal{M}=-\int_{\mathcal{M}}(\hat{v}\cdot\nabla \hat{H}+\hat{w}\partial_{p}\hat{H})H_{2}d\mathcal{M}\nonumber\\
\leq&\|H_{2}\|_{L^{\infty}}\|\hat{v}\|_{L^{2}}\|\nabla\hat{H}\|_{L^{2}}+
\|H_{2}\|_{L^{\infty}}\|\hat{w}\|_{L^{2}}\|\partial_{p}\hat{H}\|_{L^{2}}\nonumber\\
\leq& C\|\hat{v}\|_{L^{2}}^{2}+\frac{1}{2}\|\nabla\hat{H}\|_{L^{2}}^{2}+\frac{1}{4}\|\partial_{p}\hat{H}\|_{L^{2}}^{2}
+\frac{1}{8\delta^{2}}\|\nabla\hat{v}\|_{L^{2}}^{2}, \end{align*} where we have used the uniform boundness of $H_{2}$ which can be ensured by the uniform boundness of $T$ and $q_{v}$. Recalling definitions of $f_{T},f_{q_{v}}$, we have \begin{align*} \int_{\mathcal{M}}\left(f_{T_{1}}-f_{T_{2}}\right)\hat{H}d\mathcal{M}
\leq C\left(\|\hat{q}_{r}\|_{L^{2}}^{2}+\|\hat{q}_{v}\|_{L^{2}}^{2}+
\|\hat{H}\|_{L^{2}}^{2}\right)+\frac{1}{8\delta^{2}}\|\nabla\hat{v}\|_{L^{2}}^{2} \end{align*} and \begin{align*} \frac{L}{c_{p}}\int_{\mathcal{M}}\left(f_{q_{v1}}-f_{q_{v2}}\right)\hat{H}d\mathcal{M}
\leq C(\|\hat{q}_{r}\|_{L^{2}}^{2}+\|\hat{q}_{v}\|_{L^{2}}^{2}+\|\hat{H}\|_{L^{2}}^{2}). \end{align*} Using the H\"older inequality and the Young inequality, we can infer that \begin{align*} &\int_{\mathcal{M}}\frac{R}{pc_{p}}\left(T_{1}\hat{w}+w_{2}\hat{T}\right)\hat{H}d\mathcal{M}\nonumber\\
\leq& C\|T_{1}\|_{L^{\infty}}\|\hat{w}\|_{L^{2}}\|\hat{H}\|_{L^{2}}
+C\|w_{2}\|_{L^{2}}\|\hat{T}\|_{L^{4}}\|\hat{H}\|_{L^{4}}\nonumber\\
\leq&C\|\hat{H}\|_{L^{2}}\|\nabla\hat{v}\|_{L^{2}}
+C\|v_{2}\|_{H^{1}}(\|\hat{H}\|_{L^{4}}+\|\hat{q}_{v}\|_{L^{4}})\|\hat{H}\|_{L^{4}}\nonumber\\
\leq&C\|\hat{H}\|_{L^{2}}\|\nabla\hat{v}\|_{L^{2}}
+C\|v_{2}\|_{H^{1}}\|\hat{H}\|_{L^{2}}^{\frac{1}{2}}\|\hat{H}\|_{H^{1}}^{\frac{3}{2}}
+C\|v_{2}\|_{H^{1}}\|\hat{q}_{v}\|_{L^{2}}^{\frac{1}{4}}\|\hat{q}_{v}\|_{H^{1}}^{\frac{3}{4}}
\|\hat{H}\|_{L^{2}}^{\frac{1}{4}}\|\hat{H}\|_{H^{1}}^{\frac{3}{4}}\nonumber\\
\leq &C\|\hat{H}\|_{L^{2}}^{2}
+C\|v_{2}\|_{H^{1}}^{4}(\|\hat{q}_{v}\|_{L^{2}}^{2}+\|\hat{H}\|_{L^{2}}^{2})
+\frac{1}{8}\|\hat{H}\|_{H^{1}}^{2}+\frac{1}{8}\|\hat{q}_{v}\|_{H^{1}}^{2}
+\frac{1}{8\delta^{2}}\|\nabla\hat{v}\|_{L^{2}}^{2}, \end{align*} where we have used the uniform boundness of $T$ in the second step and the Gagliardo-Nirenberg-Sobolev inequality in the third step. Combining the above inequalities, we can deduce that \begin{align}\label{hat{H}}
&\frac{d}{dt}\|\hat{H}\|_{L^{2}}^{2}+\|\nabla\hat{H}\|_{L^{2}}^{2}
+\|\partial_{p}\hat{H}\|_{L^{2}}^{2}\nonumber\\
\leq&C(1+\|v_{2}\|_{H^{1}}^{4})\left(\|\hat{v}\|_{L^{2}}^{2}+|\|\hat{q}_{r}\|_{L^{2}}^{2}+\|\hat{q}_{v}\|_{L^{2}}^{2}+
\|\hat{H}\|_{L^{2}}^{2}\right)+\frac{1}{4\delta^{2}}\|\nabla\hat{v}\|_{L^{2}}^{2}
+\frac{1}{8}\|\hat{q}_{v}\|_{H^{1}}^{2}. \end{align} Considering inequalities (\ref{hat{v}}), (\ref{hat{Q}}), (\ref{hat{q}_{v}}), (\ref{hat{q}_{r}}) and (\ref{hat{H}}), and setting \begin{align*}
\Psi(t)=\delta^{2}\left(|\|\hat{q}_{r}\|_{L^{2}}^{2}+\|\hat{q}_{v}\|_{L^{2}}^{2}+
\|\hat{H}\|_{L^{2}}^{2}+\|\hat{Q}\|_{L^{2}}^{2}\right)+\|\hat{v}\|_{L^{2}}^{2}, \end{align*}
we can deduce that
\begin{align*}
\frac{d}{dt}\Psi(t)\leq A_{7}(t)\Psi(t),
\end{align*}
where
\begin{align*}
A_{7}(t)=&C\left(1+\|v_{2}\|_{H^{1}}^{4}+\|\nabla v_{2}\|_{L^{2}}^{2}+\|\nabla\partial_{p}v_{2}\|_{L^{2}}^{2}+\|\partial_{p}v_{2}\|_{L^{2}}^{2}
+\|\partial_{p}v_{2}\|_{L^{2}}^{4}\right.\nonumber\\ &\left.+\|\nabla\partial_{p}v_{2}\|_{L^{2}}^{2}
+\|\partial_{p}v_{2}\|_{L^{2}}^{2}\|\nabla\partial_{p}v_{2}\|_{L^{2}}^{2}\right).
\end{align*} Considering the regularity of $v$, using the Gronwall inequality, we can obtain the uniqueness of $v,Q,H,q_{v},q_{r}$. \end{proof}
\textbf{Proofs of main results}
Combining the existence result in Proposition \ref{Globalquasi-strong} and the uniqueness result in Proposition \ref{uniqueness}, we can complete the proof of Theorem 2.1. Similarly, Theorem 2.2 can also be proved.
\subsection*{Acknowledgments}
This work was supported by the Natural Science Foundation of China (No. 12271261), the Key Research and Development Program of Jiangsu Province (Social Development) (No. BE2019725), the Qing Lan Project of Jiangsu Province and Postgraduate Research and Practice Innovation Program of Jiangsu Province (No. KYCX21\_0930).
\end{document} | arXiv |
\begin{definition}[Definition:Imperial/Volume]
The imperial units of volume are based on a binary system, in which each unit is a factor of $2$ larger than the next smaller unit.
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\begin{document}
\title{\Large \bf GREEN FUNCTIONS WITH SINGULARITIES
ALONG COMPLEX SPACES}
\begin{abstract} \noindent We study properties of a Green function $G_{A}$ with singularities along a complex subspace $A$ of a complex manifold $X$. It is defined as the largest negative plurisubharmonic function $u$
satisfying locally $u\leq \log|\psi|+C$, where $\psi=(\psi_1,\dots,\psi_m)$, $\psi_1,\dots,\psi_m$ are local generators for the ideal sheaf ${\cal I}_A$ of $A$, and $C$ is a constant depending on the function $u$ and the generators. A motivation for this study is to estimate global bounded functions from the sheaf ${\cal I}_A$ and thus proving a ``Schwarz Lemma'' for ${\cal I}_A$.
\par \noindent{\em Subject Classification (2000)}: Primary 32U35. Secondary 32C15, 32C25, 32H02, 32S45, 32U25, 32U40. \end{abstract}
\section{Introduction}\label{sec:intro}
If $\varphi$ is a bounded holomorphic function on a complex manifold $X$, then it is a natural problem to estimate $|\varphi|$ given some information on the location of the zeros of $\varphi$
and their multiplicities. If $|\varphi|\leq 1$ and the only given information is that $\varphi(a)=0$ for a single point $a$, then $$
\log|\varphi|\leq G_{X,a}=G_X(\cdot,a), $$ where $G_{X,a}$ is the {\it pluricomplex Green function with logarithmic pole at } $a$. It is defined as the supremum over the class ${\cal F}_{X,a}$ of all negative plurisubharmonic functions $u$
such that $u\leq \log|{\zeta}|+C$ near $a$, where ${\zeta}$ are local coordinates near $a$ with $\zeta(a)=0$ and $C$ is a positive constant depending on $u$ and $\zeta$. The function $G_{X,a}$ was introduced and studied by several authors \cite{Lem}, \cite{Zah}, \cite{Kl1}, \cite{PSh}, \cite{D0}, see also \cite{Kl}, \cite{D1}.
A generalization is to take $A=(|A|,\set{m_a}_{a\in |A|})$, where
$|A|$ is a finite subset of $X$, $m_a$ is a positive real number for every $a\in |A|$, and assume that $\varphi$ has a zero of multiplicity at least $m_a$ at every point $a$ in $|A|$. Then $$
\log|\varphi|\leq G_{A}, $$ where $G_{A}$ is the {\it Green function with several weighted logarithmic poles}. It is defined as the supremum over the class of all negative plurisubharmonic functions $u$ on $X$ satisfying
$u\leq m_a\log|\zeta_a|+C$ for every $a$ in $|A|$, where $\zeta_a$ are local coordinates near $a$ with $\zeta_a(a)=0$ and $C$ is a positive constant depending on $\zeta_a$ and $u$. The function $G_{A}$ was first introduced by Zaharyuta \cite{Zah} and independently by Lelong \cite{Le}.
The notion of multiplicity of a zero of an analytic function has a natural generalization as a Lelong number of a plurisubharmonic function. If $u$ is plurisubharmonic in some neighbourhood of the origin $0$ in ${\mathbb C}^n$, then the {\it Lelong number} $\nu_u(0)$ of $u$ at $0$ can be defined as $${\nu}_u(0)=\lim_{r\to 0}
\frac{\sup\,\{u(x);\: |x|\leq r\}}{\log r}$$ and if $u$ is plurisubharmonic on a manifold $X$ then the Lelong number $\nu_u(a)$ of $u$ at $a\in X$ is defined as $\nu_u(a)=\nu_{u\circ \zeta^{-1}}(0)$, where $\zeta$ are local coordinates near $a$ with $\zeta(a)=0$. It is clear that this definition is independent of the choice of local coordinates and that $\nu_u(a)$ equals the multiplicity of $a$ as a zero of the holomorphic function
$\varphi$ in the case $u=\log|\varphi|$. Note that the pluricomplex Green function $G_{X,a}$ with logarithmic pole at $a$ can be equivalently defined as the upper envelope of all negative plurisubharmonic functions $u$ on $X$ satisfying $\nu_u(a)\ge 1$ and, similarly, $\nu_u(a)\ge m_a$ for the Green functions with several weighted logarithmic poles.
For any non-negative function $\alpha$ on $X$, L\'arusson and Sigurdsson \cite{LarSig1}, \cite{LarSig2} introduced the Green function $\tilde G_\alpha$ as the supremum over the class $\tilde {\cal F}_\alpha$ of all negative plurisubharmonic functions with
$\nu_u\geq \alpha$. It is clear that if $\varphi$ is holomorphic on $X$, $|\varphi|\leq 1$, and every zero $a$ of $\varphi$ has multiplicity at least
$\alpha(a)$, then $$
\log|\varphi|\leq \tilde G_\alpha. $$ In this context it is necessary to note that we assume that the manifold $X$ is connected, we take the constant function $-\infty$ as plurisubharmonic, and set $\nu_{-\infty}=+\infty$. Hence $-\infty\in \tilde {\cal F}_\alpha$ for every $\alpha$. By \cite{LarSig1}, Prop.~5.1, $\tilde G_\alpha\in \tilde {\cal F}_\alpha$.
In the special case when $X$ is the unit disc ${\mathbb D}$ in ${\mathbb C}$, we have $$ \tilde G_\alpha(z)=\sum_{w\in {\mathbb D}} \alpha(w) G_{\mathbb D}(z,w), \qquad z\in {\mathbb D}, $$ where $G_{\mathbb D}$ is the Green function for the unit disc, $$
G_{\mathbb D}(z,w)=\log\bigg|\dfrac{z-w}{1-\bar w z}\bigg|, \qquad z,w\in {\mathbb D}. $$
If $X$ and $Y$ are complex manifolds, $\alpha$ is a non-negative function on $X$, and $\Phi:Y\to X$ is a holomorphic map, then the pullback $\Phi^*u=u\circ \Phi$ satisfies $\nu_{\Phi^*u}\geq \Phi^*\nu_u$, so $\Phi^*u\in \tilde{\cal F}_{\Phi^*\alpha}$ for every $u\in \tilde{\cal F}_\alpha$. This implies $\Phi^*\tilde G_\alpha\leq \tilde G_{\Phi^*\alpha}$, i.e., $\tilde G_\alpha(x)\leq \tilde G_{\Phi^*\alpha}(y)$ if $x=\Phi(y)$, and in particular $$\tilde G_\alpha(x)\leq \tilde G_{f^*\alpha}(0)=\sum_{w\in {\mathbb D}}
f^*\alpha(w)\log|w|, \qquad f\in {\cal O}({\mathbb D},X), \quad f(0)=x. $$ One of the main results of \cite{LarSig1} and \cite{LarSig3} is that for every manifold $X$ and every non-negative function $\alpha$ we have the formula \begin{equation} \tilde G_\alpha(x)=\inf\set{\tilde G_{f^*\alpha}(0) \,;\, f\in {\cal O}(\overline {\mathbb D},X), f(0)=x}, \qquad x\in X. \label{eq:disc} \end{equation} Here ${\cal O}({\mathbb D},X)$ is the family of all analytic discs in $X$ and ${\cal O}(\overline {\mathbb D},X)$ is the subclass of closed analytic discs, i.e., maps from ${\mathbb D}$ to $X$ that can be extended to holomorphic maps in some neighbourhood of the closed disc $\overline {\mathbb D}$. Results of this kind originate in Poletsky's theory of analytic disc functionals, started in \cite{PSh} and \cite{P0}.
A natural way of describing the zero set of a holomorphic function $\varphi$ is to state that its germs $(\varphi)_x$ are in the stalk ${\cal I}_{A,x}$ of a prescribed coherent ideal sheaf ${\cal I}_A=({\cal I}_{A,x})_{x\in X}$ of a closed complex subspace $A$ of $X$. Then, if $\psi_1,\dots,\psi_m$ are local generators of ${\cal I}_A$ near the point $a$, the function $\varphi$ can be represented as $\varphi=\varphi_1\psi_1+\cdots+\varphi_m\psi_m$
near $a$, which implies that $\log|\varphi|\leq \log|\psi|+C$ near
$a$, where $\psi=(\psi_1,\dots,\psi_m)$, $|\cdot|$ is the euclidean norm, and $C$ is a constant depending on $\varphi$ and the generators.
We define ${\cal F}_{A}$ as the class of all negative plurisubharmonic functions $u$ in $X$ satisfying $u\leq \log|\psi|+O(1)$ locally in $X$, and we define the function $G_{A}$, the {\it pluricomplex Green function with singularities along $A$}, as the supremum over the class ${\cal F}_{A}$.
It follows from the definition of $G_A$ that if $A'$ is the restriction of $A$ to a domain $X'$ in $X$, $\varphi$ is holomorphic function on $X$, and $(\varphi)_x\in {\cal I}_{A,x}$ for all $x\in X'$, then
$$|\varphi|\le e^{G_{A'}(x)}\sup_{X'}|\varphi|,\quad x\in X',$$ which is a variant of the Schwarz lemma for the ideal sheaves.
In order to relate $G_{A}$ to the Green functions $\tilde G_\alpha$ above, we define the function $\tilde\nu_A$ on $X$ by
$\tilde\nu_A(x)=\nu_{\log|\psi|}(x)$ if $\psi=(\psi_1,\dots,\psi_m)$ are local generators for ${\cal I}_A$ in some neigbourhood of $x$. It is easy to see that $\tilde\nu_A(x)$ is independent of the choice of the generators (actually, it equals the minimal multiplicity of the functions from ${\cal I}_{A,x}$ at $x$), so $\tilde\nu_A$ is a well defined function on $X$ and $\nu_u\geq \tilde\nu_A$ for all $u\in {\cal F}_{A}$. Hence, with $\tilde\nu_A$ in the role of $\alpha$ above, we have ${\cal F}_{A}\subseteq \tilde {\cal F}_{\tilde\nu_A}$ which implies $$ G_{A}\leq \tilde G_{\tilde\nu_A}. $$ In general, $G_{A}\neq\tilde G_{\tilde\nu_A}$ as seen from the example where $X={\mathbb D}^2$ and ${\cal I}_A$ has the global generators
$\psi=(\psi_1,\psi_2)$ with $\psi_1(z)=z_1^2$ and $\psi_2(z)=z_2$. Then $G_{A}(z)=\max\set{2\log|z_1|,\log|z_2|}$ and $\tilde G_{\tilde\nu_A}(z)=\max\set{\log|z_1|,\log|z_2|}$ for $z=(z_1,z_2)\in {\mathbb D}^2$. If, on the other hand, $A$ is an effective divisor generated by the function $\psi$ in an open subset $U$ of
$X$, then by \cite{LarSig2}, Prop.~3.2, the function $\tilde G_{\tilde\nu_A}-\log|\psi|$ on $U\setminus |A|$ can be extended to a plurisubharmonic function on $U$. This implies that $G_{A}=\tilde G_{\tilde\nu_A}$ for effective divisors $A$.
Now to the content of the paper. In Section~2 we present the main results, which are proved in later sections. Our first task is to prove that $G_{A}\in {\cal F}_{A}$. In Section~2 we show how this follows from the facts that $\tilde G_\alpha\in \tilde {\cal F}_\alpha$ for all $\alpha:X\to [0,+\infty)$, $G_A=\tilde G_{\tilde\nu_A}$ if $A$ is an effective divisor, and a variant of the Hironaka desingularization theorem. By the same desingularization technique we establish a representation of the Green function as the lower envelope of the analytic disc functional $f\mapsto G_{f^*A}(0)$. In Section~3 we study decomposition in ideal sheaves as a preparation for Section~\ref{sec:properties} where we prove that the estimates in the definition of the class ${\cal F}_A$ are locally uniform. This gives a direct proof of the relation $G_{A}\in {\cal F}_{A}$ (without referring to desingularization), which in turn implies certain refined maximality properties of the Green function. In Section~5 we get a representation for the current $(dd^cG_A)^p$ in the case when the ideal sheaf ${\cal I}_A$ has global generators, and in Section~6 we study the case when the space is reduced. In Section~7 we prove the product property of Green functions, and finally in Section~8 we give a few explicit examples.
\section{Definitions and main results}\label{sec:def}
We shall always let $X$ be a complex manifold and assume that $X$ is connected. We denote by ${\operatorname{PSH}}(X)$ the class of all plurisubharmonic functions on $X$ and by ${\operatorname{PSH}}^-(X)$ its subclass of all non-positive functions. We take $-\infty\in {\operatorname{PSH}}(X)$ and set $\nu_{-\infty}=+\infty$. We let ${\cal O}_X$ denote the sheaf of germs of locally defined holomorphic functions on $X$. We let $A$
be a closed complex subspace of $X$, ${\cal I}_A=({\cal I}_{A,x})_{x\in X}$ be the associated coherent sheaf of ideals in ${\cal O}_X$, and $|A|$ be the analytic variety in $X$ defined as the common set of zeros of the locally defined functions on $X$ with germs in ${\cal I}_A$. If $U$ is an open subset of $X$, then we let ${\cal I}_{A,U}$ denote the space of all holomorphic functions on $U$ with germs in ${\cal I}_A$. We let ${\mathbb D}$ denote the open unit disc in the complex plane ${\mathbb C}$ and ${\mathbb T}$ denote the unit circle. We let ${\cal O}(Y,X)$ denote the set of all holomorphic maps from a complex manifold $Y$ into $X$. A map in ${\cal O}({\mathbb D},X)$ is called an {\it analytic disc}, and if it can be extended to a holomorphic map in some neighbourhood of the closed disc $\overline {\mathbb D}$ then it is said to be {\it closed}. The collection of all closed analytic discs is denoted by ${\cal O}(\overline {\mathbb D},X)$.
\begin{definition} Given a complex subspace $A$ of a connected complex manifold $X$, the class ${\cal F}_A$ consists of all functions $u\in{\operatorname{PSH}}^-(X)$ such that for every point $a\in X$ there exist local generators $\psi_1,\dots,\psi_m$ for ${\cal I}_A$ near $a$ and a constant $C$ depending on $u$ and the generators with $u\leq
\log|\psi|+C$ near $a$. \end{definition}
Observe that $-\infty\in {\cal F}_{A}$ for every $A$.
\begin{definition} The {\it pluricomplex Green function $G_A$ with singularities along} $A$ is the upper envelope of all the functions from the class ${\cal F}_A$, i.e., $$ G_A(x) = \sup\set{u(x);\: u\in{\cal F}_A}, \qquad x\in X. $$ \end{definition}
The local estimate $u\leq \log|\psi|+C$ is independent of the choice of generators, i.e., if we have another set of generators
$\psi'=(\psi_1',\dots,\psi_k')$, then $u\leq \log|\psi'|+C'$ for some constant $C'$. Furthermore, in the definition of the class ${\cal F}_A$, $\psi=(\psi_1,\dots,\psi_m)$ can be replaced by any holomorphic $\xi=(\xi_1,\dots,\xi_l)$, defined near $a$ and satisfying $$
\log|\xi|+c_1\le \log|\psi|\le \log|\xi|+c_2, $$ which means precisely that the integral closure of the ideal generated by the germs of the functions $\xi_i$ at $x$ coincides with the integral closure of the ideal ${\cal I}_{A,x}$ for all $x$ in some neighbourhood of $a$. (See \cite{D3}, Ch.~VIII, Cor.~10.5.)
We occasionally write $\log|\xi| \asymp\log|\psi|$ when inequalities of this kind hold.
Let $X$ and $Y$ be complex manifolds and $\Phi:Y\to X$ be a holomorphic map. If $A$ is a complex subspace of $X$, then we have a natural definition of a pullback $\Phi^*A$ of $A$ as a complex subspace of $Y$. The ideal sheaf ${\cal I}_{\Phi^*A}$ is locally generated at a point $b$ by $\Phi^*\psi_1,\dots,\Phi^*\psi_m$ if $\psi_1,\dots,\psi_m$ are local generators for ${\cal I}_A$ at $\Phi(b)$. It is evident that $\Phi^*u\in {\cal F}_{\Phi^*A}$ for all $u\in {\cal F}_A$, so \begin{equation} \Phi^*G_A\leq G_{\Phi^*A}. \label{eq:2.1} \end{equation}
If $\Phi$ is proper and surjective and $v:Y\to {\mathbb R}\cup \set{-\infty}$ is an upper semi-continuous function, then the push-forward $\Phi_*v$ of $v$ to $X$ is well defined by the formula $$ \Phi_*v(x)=\max_{y\in \Phi^{-1}(x)}v(y), \qquad x\in X. $$
\begin{proposition}\label{propermap} Let $X$ and $Y$ be complex manifolds of the same dimension and $\Phi:Y\to X$ be a proper surjective holomorphic map (for example, a finite branched covering). Then $\Phi_*v\in{\operatorname{PSH}}(X)$ for all $v\in{\operatorname{PSH}}(Y)$. \end{proposition}
\begin{proof} In order to show that $\Phi_*v$ is upper semicontinuous, we need to prove the relation $\Phi_*v(a)\geq \limsup_{x\to a}\Phi_*v(x)$ for every $a\in X$. We take a sequence $a_j\to a$ such that $\Phi_*v(a_j)\to \limsup_{x\to a}\Phi_*v(x)$. Since $v$ is upper semicontinuous and $\Phi$ is proper, there exist $b_j\in \Phi^{-1}(a_j)$ such that $v(b_j)=\Phi_*v(a_j)$. By replacing $(b_j)$ by a subsequence we may assume that $b_j\to b\in Y$. Then $\Phi(b)=a$ and $$ \Phi_*v(a)\geq v(b)\geq \limsup_{j\to +\infty}v(b_j)=\lim_{j\to +\infty} \Phi_*v(a_j)=\limsup_{x\to a}\Phi_*v(x). $$ We let $V$ denote the set of all points $y$ in $Y$ for which $d_y\Phi$ is degenerate. Then $V$ is an analytic variety in $Y$ and Remmert's proper mapping theorem implies that $W=\Phi(V)$ is an analytic variety in $X$. It is sufficient to show that $\Phi_*v$ is plurisubharmonic in a neighbourhood of every point $a\in X\setminus W$, for the upper semicontinuity of $\Phi_*v$ then implies that $\Phi_*v\in {\operatorname{PSH}}(X)$.
Since $\Phi$ is a local biholomorphism on $Y\setminus \Phi^{-1}(W)$, it follows that the fiber $\Phi^{-1}(a)$ is discrete and compact, thus finite, say that it consists of the points $b_1,\dots,b_m$. We choose a neighbourhood $U$ of $a$ in $X\setminus W$ and biholomorphic maps $F_j:U\to F_j(U)\subseteq Y\setminus \Phi^{-1}(W)$ with $F_j(a)=b_j$. Then $\Phi_*v(x)=\sup_{1\leq j\leq m}v\circ F_j(x)$ for all $x\in U$, which shows that $\Phi_*v$ is plurisubharmonic in $U$. \end{proof}
It is obvious that $u=\Phi_*\Phi^*u$ for all $u\in {\operatorname{PSH}}(X)$ and $v\le \Phi^*\Phi_*v$ for all $v\in {\operatorname{PSH}}(Y)$.
\begin{proposition}\label{proper} Let $X$ and $Y$ be complex manifolds of the same dimension, $A$ be a closed complex subspace of $X$, and $\Phi:Y\to X$ be a proper surjective holomorphic map. Then $\Phi_*v\in {\cal F}_A$ for all $v\in {\cal F}_{\Phi^*A}$ and $$ \Phi^*G_A = G_{\Phi^*A}. $$ \end{proposition}
\begin{proof} If $a\in X$ and $\psi_1,\dots,\psi_m$ are local generators for
${\cal I}_A$ near $a$, then $v\leq \Phi^*\log|\psi|+C$ in some neighbourhood of the compact set $\Phi^{-1}(a)$, which implies
$\Phi_*v\leq \log|\psi|+C$ near $a$. Hence we conclude from Prop.~\ref{propermap} that $\Phi_*v\in {\cal F}_A$. Since $\Phi^*G_A\leq G_{\Phi^*A}$, it is sufficient to prove that $v\leq \Phi^*G_A$ for every $v\in {\cal F}_{\Phi^*A}$. We have $\Phi_*v\in {\cal F}_A$, so $v\leq \Phi^*\Phi_*v\leq \Phi^*G_A$. \end{proof}
Our first main result is
\begin{theorem}\label{GinF} If $X$ is a complex manifold and $A$ is a closed complex subspace of $X$, then $G_A\in {\cal F}_A$. \end{theorem}
Observe that in our definition of the class ${\cal F}_A$, the constant
$C$ in the local estimates $u\leq \log|\psi|+C$ is allowed to depend both on the function $u$ and the local generators. The main work in our proof of Theorem~\ref{GinF} in Sections~\ref{sec:decomposition} and \ref{sec:properties} is to prove that these estimates are indeed locally uniform, i.e., we show that if $U$ is the domain of definition of $\psi$ and $K$ is a compact subset of $U$, then there exists a constant $C_K$, only depending on $K$ and $\psi$, such that $u\leq \log|\psi|+C_K$ on $K$ for all $u\in {\cal F}_A$. (See Lemma \ref{lub-lemma}.)
Let us now show how Theorem~\ref{GinF} follows from the facts that $\tilde G_\alpha\in \tilde {\cal F}_\alpha$ for all $\alpha:X\to
[0,+\infty)$, $G_A=\tilde G_{\tilde\nu_A}$ if $A$ is an effective divisor, and the following variant of the Hironaka desingularization theorem. (See \cite{BM}, Theorems 1.10 and 13.4.)
\noindent {\em Given a closed complex subspace $A$ on a manifold $X$, there exists a complex manifold $\hat X$ and a proper surjective holomorphic map $\Phi:\hat X\to X$ which is an isomorphism outside
$\Phi^{-1}(|A|)$ and such that $\hat A=\Phi^*A$ is a normal-crossing principal ideal sheaf (i.e., generated locally by a monomial in suitable coordinates).}
\noindent If we let $\Phi$ denote the desingularization map, then $$ G_A=\Phi_*\Phi^*G_A=\Phi_*G_{\Phi^*A}=\Phi_*G_{\hat A} =\Phi_*\tilde G_{\nu_{\hat A}}. $$ Since $\tilde G_{\tilde\nu_{\hat A}}\in{\cal F}_{\hat A}$, Proposition~\ref{proper} gives $G_A\in {\cal F}_A$ and the theorem is proved.
If $X$ is one-dimensional, i.e., a Riemann surface, then ${\cal I}_{A}$ is a principal ideal sheaf. If ${\cal I}_A=0$, the zero sheaf, then
$\tilde\nu_A=+\infty$. If ${\cal I}_A\neq 0$, then $|A|$ is discrete, for $a\not\in |A|$ we have ${\cal I}_{A,a}={\cal O}_{X,a}$ and
$\tilde\nu_A(a)=0$, and for $a\in |A|$ the ideal ${\cal I}_{A,a}$ is generated by the germ of $\zeta_a^m$ at $a$, where $m=\tilde\nu_A(a)>0$ and $\zeta_a$ is a local generator for ${\cal I}_A$ near $a$ with $\zeta_a(a)=0$. We obviously have $$ G_A\geq \sum_{a\in {\mathbb D}}\tilde\nu_A(a)G_X(\cdot,a) \in {\cal F}_A, $$ where $G_X(\cdot,a)$ is the Green function on $X$ with single pole at $a$. In the special case $X={\mathbb D}$, every function $u\in {\cal F}_A\setminus\set{-\infty}$ can be represented by the Poisson--Jensen formula $$ u(z)=\dfrac 1{2\pi}\int_{\mathbb D} G_{\mathbb D}(z,\cdot)\, \Delta u +\int_{\mathbb T} P_{\mathbb D}(z,t)\, d\lambda_u(t), \qquad z\in {\mathbb D}, $$ where $P_{\mathbb D}$ is the Poisson kernel for the unit disc ${\mathbb D}$ and $\lambda_u$ is a nonpositive measure on the unit circle ${\mathbb T}$ (the boundary value of $u$). We have $\nu_u(a)=\Delta u(\set a)/2\pi$, so $\Delta u \geq 2\pi\sum_{a\in {\mathbb D}} \tilde\nu_A(a)\delta_a$, where $\delta_a$ is the Dirac measure at the point $a$. Thus the Poisson--Jensen formula implies $$ u(z)\leq \int_{\mathbb D} G_{\mathbb D}(z,\cdot)\,\bigg(\sum_{a\in {\mathbb D}} \tilde\nu_A(a)\delta_a\bigg)= \sum_{a\in {\mathbb D}}\tilde\nu_A(a) G_{\mathbb D}(z,a) $$ and we conclude that $$ G_A(z)=\sum_{a\in {\mathbb D}} \tilde\nu_A(a)G_{\mathbb D}(z,a)= \sum_{a\in {\mathbb D}}
\tilde\nu_A(a)\log\bigg|\dfrac{z-a}{1-\bar az}\bigg|, \qquad z\in {\mathbb D}, $$ for every closed complex subspace $A$ of ${\mathbb D}$.
Now we let $X$ be any manifold, $f\in {\cal O}({\mathbb D},X)$ be an analytic disc, and $a\in {\mathbb D}$. If $\psi_1,\dots,\psi_m$ are local generators for $A$ at $f(a)$, then $f^*\psi_1,\dots,f^*\psi_m$ are local generators for ${\cal I}_{f^*A}$ near $a$. If all these functions are zero in some neighbourhood of $A$, then ${\cal I}_{f^*A}=0$ and $\tilde\nu_{f^*A}=+\infty$. If one of them is not zero at $a$, then ${\cal I}_{f^*A,a}={\cal O}_{X,a}$ and $\tilde\nu_{f^*A}(a)=0$, and if they have a common isolated zero at $a$, then $\tilde\nu_{f^*A}(a)$ is the smallest positive multiplicity of them. Since $f^*G_A\leq G_{f^*A}$, we get $$
G_A(x)\leq G_{f^*A}(0)=\sum_{a\in {\mathbb D}}\tilde\nu_{f^*A}(a)\log|a|, \qquad f\in {\cal O}({\mathbb D},X),\quad f(0)=x. $$
\begin{theorem}\label{th:envelope} Let $X$ be a complex manifold and $A$ be a closed complex subspace of $X$. Then $$ G_A(x)=\inf\set{G_{f^*A}(0)\, ;\, f\in {\cal O}(\overline {\mathbb D},X), x=f(0)}, \qquad x\in X. $$ \end{theorem}
Let us show how the theorem follows from Hironaka's desingularization theorem. If we use the disc formula (\ref{eq:disc}) for $\tilde G_\alpha$ with $\alpha=\tilde\nu_{\hat A}$, the fact that $G_{\hat A}=\tilde G_\alpha$, and the desingularization map $\Phi$ above with $x=\Phi( \hat x)$, then \begin{align*} G_A(x)=\Phi^*G_{A}(\hat x)=G_{\hat A}(\hat x) &=\inf\set{\tilde G_{g^*\alpha}(0)\,;\, g\in{\cal O}(\overline {\mathbb D},\hat X), g(0)=\hat x} \\ &\geq \inf\set{G_{g^*\hat A}(0)\,;\, g\in{\cal O}(\overline {\mathbb D},\hat X), g(0)=\hat x} \\ &= \inf\set{G_{g^*\Phi^*A}(0)\,;\, g\in{\cal O}(\overline {\mathbb D},\hat X), g(0)=\hat x} \\ &= \inf\set{G_{(\Phi_*g)^*A}(0)\,;\, g\in{\cal O}(\overline {\mathbb D},\hat X), g(0)=\hat x} \\ &\geq \inf\set{G_{f^*A}(0)\,;\, f\in{\cal O}(\overline {\mathbb D}, X), f(0)=x}
\end{align*} and we have proved Theorem \ref{th:envelope}. We will prove this theorem without reference to desingularization or the disc formula for $\tilde G_\alpha$ in a separate paper.
Let $X_1$ and $X_2$ be complex manifolds, $A_1$ and $A_2$ be closed complex subspaces of $X_1$ and $X_2$, respectively, $X=X_1\times X_2$ be the product manifold of $X_1$ and $X_2$, and $A=A_1\times A_2$ be the product space of $A_1$ and $A_2$. If $a=(a_1,a_2)\in X$ and $\psi^1_1,\dots,\psi^1_k$ and $\psi^2_1,\dots,\psi^2_l$ are local generators for ${\cal I}_{A_1}$ and ${\cal I}_{A_2}$ near $a_1$ and $a_2$, respectively, then the functions $$ x=(x_1,x_2)\mapsto \psi^1_1(x_1),\dots,\psi^1_k(x_1),\psi^2_1(x_2),\dots,\psi^2_l(x_2). $$ are generators for ${\cal I}_A$ near $a$. This implies that $X\ni x=(x_1,x_2)\mapsto \max\set{u_1(x_1), u_2(x_2)}$ is in ${\cal F}_A$ for all $u_1\in {\cal F}_{A_1}$ and $u_2\in {\cal F}_{A_2}$ , so we obviously have $$ G_A(x) \geq \max\set{G_{A_1}(x_1), G_{A_2}(x_2)}, \qquad x=(x_1,x_2)\in X. $$ The following is called the {\it product property} for Green functions.
\begin{theorem}\label{th:product} Let $X_1$ and $X_2$ be complex manifolds, $A_1$ and $A_2$ be closed complex subspaces of $X_1$ and $X_2$, respectively, and $A$ be the product of $A_1$ and $A_2$ in $X=X_1\times X_2$. Then $$ G_A(x)=\max\set{G_{A_1}(x_1),G_{A_2}(x_2)}, \qquad x=(x_1,x_2)\in X. $$ \end{theorem}
We base our proof on Th.~\ref{th:envelope} and give it in Section \ref{sec:product}.
It was shown in \cite{LarSig2}, Prop.~3.2, that if $A$ is given by a single holomorphic function with effective divisor $Z_A$, then $G_A$ satisfies $dd^cG_A\ge Z_A$ and, moreover, it is the largest negative plurisubharmonic function with this property. (See the last statement of Th.~3.3 in \cite{LarSig2}). Here $d=\partial + \bar\partial$, $d^c= ( \partial -\bar\partial)/2\pi i$.
In the general case, a space $A$ generates holomorphic chains \begin{equation}\label{eq:reschain} Z_A^p=\sum_i m_{i,p}[A_i^p],\end{equation} where $A_i^p$ are
$p$-codimensional components of $|A|$ and $m_{i,p}\in{\mathbb Z}^+$. Namely, if on a domain $U\subset X$ the space $A$ is given by functions $\psi_1,\dots,\psi_m$ and ${\operatorname{codim}}\,|A|=p$ there, then by the King-Demailly formula (\cite{D1}, Th.~6.20), $$
(dd^c\log|\psi|)^p=\sum_i m_{i,p}[A_i^p] +R\quad{\rm on}\ U, $$ where $m_{i,p}$ is the generic multiplicity of $\psi$ along $A_i^p$ and $R$ is a positive closed current of bidegree $(p,p)$
on $U$, such that $\chi_{|A|}R=0$ and ${\operatorname{codim}}\,E_c(R)>p$ for every $c>0$. Here $\chi_S$ is the characteristic function of a set $S$,
$E_c(R)=\{x\,;\, \nu_R(x)\ge c\}$ and $\nu_R(x)$ is the Lelong number of the current $R$ at $x$. In other words, the holomorphic chain $Z_A^p$ given by (\ref{eq:reschain}) is the residual Monge-Amp\`ere current of $\log|\psi|$ on $|A|\cap U$.
\begin{theorem}\label{th:current} Let $A$ have bounded global generators $\psi$ in $X$. Then \begin{description}
\item{(i)} $G_A=\log|\psi|+O(1)$ locally near $|A|$. \item{(ii)}
If ${\operatorname{codim}}\,|A|=p$ on $U\subset X$, then $(dd^cG_A)^p=Z_A^p +Q$ on $U$, where $Q$ is a positive closed current of bidegree $(p,p)$
on $U$, such that $\chi_{|A|}Q=0$ and ${\operatorname{codim}}\,E_c(Q)>p$ for every
$c>0$. If $U\cap |A|\subset J^p$, then $Q$ has zero Lelong numbers; here the set $J^p$ consists of all points $a\in |A|$ such that $p$ is the minimal number of generators of a subideal of ${\cal I}_{A,a}$ whose integral closure is equal to the integral closure of ${\cal I}_{A,a}$. \end{description} \end{theorem}
A proof is given in Section \ref{sec:bounded} (and the sets $J^p$ are introduced and studied in Section~\ref{sec:decomposition}).
\section{Decomposition in ideal sheaves}\label{sec:decomposition}
In the case of complete intersection, i.e., when for every $a\in
|A|$ the local ideal ${\cal I}_{A,a}$ is generated by precisely
$p={\operatorname{codim}}_a|A|$ germs of holomorphic functions, the relation $G_A\in{\cal F}_A$ is in fact quite easy to prove without using the desingularization technique. The main result of this section, Prop.~\ref{decomp-theo}, gives a tool for the reduction of the general situation to the complete intersection case in Section~\ref{sec:properties}. Our approach develops a method from \cite{R}.
We recall some basics on complex Grassmannians. (See, e.g., \cite{Chirka}, A3.4-5.) The Grassmannian $G(k,m)$ is the set of all $k$-dimensional linear subspaces of ${\mathbb C}^m$ with the following complex structure. Let $S_{1\dots k}$ be the set of all $L\in G(k,m)$ whose projections to the coordinate plane ${\mathbb C}_{1\dots k}$ of the variables $z_1,\dots,z_k$ are bijective. Choosing a basis $\set{(e_j,w_j)}$ in $L\in S_{1\dots k}$ with $e_j$ the standard basis vectors in ${\mathbb C}^k$ and $w_j$ vectors in ${\mathbb C}^{m-k}$, we get a representation of $L$ as the $k\times m$-matrix $(E,W)$, where $E$ is the unit $k\times k$-matrix and $W$ is a $k\times (m-k)$-matrix. This gives a parametrization of $S_{1\dots k}$ by $k\times (m-k)$-matrices $W$. In a similar way we parametrize all the charts $S_I$, $I=(i_1,\dots,i_k)$. Since the neighbouring relations are holomorphic, this determines a complex structure on $G(k,m)$. It is easy to see that ${\operatorname{dim}}\, G(k,m)=k(m-k)$. The set $\set{(z,L)\,;\, z\in L}\subset {\mathbb C}^m\times G(k,m)$ is sometimes called the {\it incidence manifold}.
Let $\psi: \Omega\to{\mathbb C}^m$, $m>1$, be a holomorphic map on a domain $\Omega$ in ${\mathbb C}^n$ and $Z=\{x\in \Omega\,;\,\psi(x)=0\}$. If $U$ is a subdomain of $\Omega$, then the graph $\Gamma_U=\{(x,\psi(x))\,;\,x\in U\}$ of $\psi$ over $U$ is an $n$-dimensional complex manifold in ${\mathbb C}^n\times{\mathbb C}^m$. Given $k\le m-1$, let $\Gamma_U^k$ be the pullback of $\Gamma_U$ to the incidence variety in $\Gamma_U\times G(k,m)$. Namely, $\Gamma_U^k$ is the closure of the set $$ \{(x,\psi(x),L)\,;\, x\in U\setminus Z,\ L\in G(k,m),\ \psi(x)\in L\}. $$ By $\rho_k$ we denote the projection from $\Gamma^k_\Omega$ to $G(k,m)$, and by $\pi_k$ its projection to $\Omega$.
For $x\in \Omega\setminus Z$, the fiber $\rho_k\circ\pi_k^{-1}(x)$ consists of all $L\in G(k,m)$ passing through $\psi(x)\neq 0$ and thus is isomorphic to $G(k-1,m-1)$. Therefore ${\operatorname{dim}}\, \Gamma_\Omega^k=n+(k-1)(m-k)$.
Let $I^k=I^k(\psi)$ be the collection of all points $x$ in $\Omega$ such that $\rho_k(\Gamma_U^k)=G(k,m)$ for every neighbourhood $U$ of $x$, i.e., $\rho_k\circ\pi_k^{-1}(x)=G(k,m)$. Evidently, $I^1\subseteq I^2\subseteq\ldots\ \subseteq I^{m-1}\subseteq Z$.
\begin{lemma}\label{dimind-lemma} $I^k$ is an analytic set of dimension at most $n-m+k-1$. \end{lemma}
\begin{proof} We have \begin{equation}\label{eq:prod} \pi_k^{-1}(I^k)=I^k\times\{0\}\times G(k,m), \end{equation} so $I^k\subset \pi_k\circ\rho_k^{-1}(L)$ for each $L\in G(k,m)$. On the other hand, for every $x\not\in I^k$ there exists $L\in G(k,m)$ such that $x\not\in\pi_k\circ\rho_k^{-1}(L)$. Thus $$I^k=\bigcap_{L\in G(k,m)} \pi_k\circ\rho_k^{-1}(L). $$ Each $\rho_k^{-1}(L)$ is an analytic set in $\Gamma^k_\Omega$. Since the map $\pi_k$ is proper, Remmert's theorem implies that $\pi_k\circ\rho_k^{-1}(L)$ is an analytic subset of $\Omega$ for any $L$, and so is $I^k$.
The set $\pi_k^{-1}(I^k)$ is a nowhere dense analytic subset of $\Gamma^k_\Omega$, and thus ${\operatorname{dim}}\,\pi_k^{-1}(I^k)<{\operatorname{dim}}\,\Gamma^k_\Omega=n+(k-1)(m-k)$. By (\ref{eq:prod}), ${\operatorname{dim}}\,\pi_k^{-1}(I^k)={\operatorname{dim}}\,I^k+k(m-k)$. Therefore ${\operatorname{dim}}\,I^k<n+(k-1)(m-k)-k(m-k)=n-m+k$. \end{proof}
\begin{corollary}\label{cor:empty} If $m>n$, then $I^k=\emptyset$ for all $k\le m-n$. \end{corollary}
\begin{lemma}\label{reduce-lemma} For any $a\in Z\setminus I^k$ there exist a neighbourhood $U$ of $a$ and holomorphic functions $\xi_1,\ldots,\xi_{m-k}$ (linear combinations of $\psi_1,\ldots,\psi_m$) such that
$\log|\psi|\asymp \log|\xi|$ in $U$. \end{lemma}
\begin{proof} Given $a\in Z\setminus I^k$, one can find a neighbourhood $U$ of $a$ such that $\rho_k(\Gamma_U^k)\neq G(k,m)$. Since the set $G(k,m)\setminus \rho_k(\Gamma_U^k)$ is open, there exists $L_0$ in the chart $S_{1\ldots k}$ of $G(k,m)$ such that \begin{equation}\label{eq:avoid} \psi(x)\cap\omega=\emptyset \end{equation}
for some neighbourhood $\omega\subset S_{1\ldots k}$ of $L_0$ and all $x\in U\setminus Z$.
Let $(E,W_0)$ be the canonical representation of $L_0$. For every $y=(y',y'')\in {\mathbb C}^k\times{\mathbb C}^{m-k}$ with $y'\neq 0$, the map $y\mapsto (y',y'W_0)$ is the projection to the space $L_0$. By elementary linear algebra arguments (see Lemma~\ref{linear-lemma} below), relation (\ref{eq:avoid}) implies existence of $r>0$ such that \begin{equation}\label{eq:elbound}
|\psi''(x)-\psi'(x)W_0|\ge r|\psi'(x)|, \quad x\in U. \end{equation} We define a map $\xi:U\to{\mathbb C}^{m-k}$ by $\xi(x)=\psi''(x)-\psi'(x)W_0$. Then $$
|\xi(x)|\le C|\psi(x)|,\quad x\in U. $$ Furthermore, inequality (\ref{eq:elbound}) implies $$
|\psi|^2\le |\psi'|^2+2|\psi''-\psi'W_0|^2+2|\psi'W_0|^2\le C|\psi''-\psi'W_0|^2=C|\xi|^2, $$ and the assertion follows. \end{proof}
\begin{lemma}\label{linear-lemma} Let $W_0$ be a complex $k\times (m-k)$-matrix and a set $S\subset{\mathbb C}^k\times{\mathbb C}^{m-k}$ be such that
$|y''-y'W|>0$ for all $y=(y',y'')\in S$ and all matrices $W\in
{\mathbb C}^{k(m-k)}$ with $|W-W_0|<\delta$ (all the norms $|\cdot|$ are the Euclidean norms in the corresponding linear spaces). Then
$$|y''-y'W|\ge\frac\delta{k}|y'|,\quad y\in S,\ |W-W_0|<\delta.$$ \end{lemma}
\begin{proof} Suppose there exists $y\in S$ and $W$ in the $\delta$-neighbourhood of $W_0$ such that
$|y''-y'W|<\frac\delta{k}|y'|$. For the vector $z=(z',z''):=
(y',y''-y'W_0)$ this means $|z''|<\tfrac{\delta} {k}|z'|$.
We choose $l\in [1,k]$ such that $|z_l|=\max\{|z_i|\,;\, 1\le i\le k\}$ and consider the $k\times (m-k)$-matrix $V$ with the entries $V_{lj}=z_{k+j}/z_l$ for $1\le j\le m-k$, and $V_{ij}=0$ for all $i\neq l$ and $1\le j\le m-k$. Then $$
|V|=\frac{|z''|}{|z_l|}\le \frac{|z''|}{k|z'|}<\delta $$ and $z'V=z''$. The latter relation is equivalent to $y''-y'W=0$
with $W=W_0+V$. Since $|W-W_0|=|V|<\delta$, this contradicts the hypothesis of the lemma. \end{proof}
We recall that the {\it analytic spread} of an ideal ${\cal I}$ equals the minimal number of generators of a subideal of ${\cal I}$ whose integral closure coincides with the integral closure of ${\cal I}$, see \cite{NR}.
\begin{proposition}\label{decomp-theo} Let $A$ be a closed complex subspace of a manifold $X$, ${\operatorname{dim}}\,X=n$. Then the set $|A|$ can be decomposed into the disjoint union of local analytic varieties $J^k$, $1\le k\le n$, such that \begin{description} \item{(i)} ${\operatorname{codim}}\,J^k\ge k$ and \item{(ii)} for each $a\in J^k$, the ideal ${\cal I}_{A,a}$ has analytic spread at most $k$. \end{description} \end{proposition}
\begin{proof} Let $\psi=(\psi_1,\ldots,\psi_m)$ be generators of ${\cal I}_A$ on a domain $\Omega\subset X$. Set $N=\min\,\{n,m\}$,
$Z=|A|\cap\Omega$, $J^1=Z\setminus I^{m-1}$, $J^k=I^{m-k+1}\setminus I^{m-k}$ for $k=2,\ldots,N-1$, and
$J^{N}=I^{m-N+1}$ (some of them can be empty). The sets $J^k$ are pairwise disjoint, ${\operatorname{dim}}\,J^k\le n-k$ (Lemma~\ref{dimind-lemma}), and $\cup_k J^k=Z$. On a neighbourhood of each point of $J^k$, the singularity of the function $\log|\psi|$ is equivalent to one defined by the function $\log|\xi|$ with $\xi=(\xi_1,\ldots,\xi_k)$ (this follows from Lemma~\ref{reduce-lemma}, if $m\leq n$, and Corollary~\ref{cor:empty}, in the case $m>n$). This means that the ideal generated by the germs of $\psi_i$ at $a\in J^k$ has analytic spread at most $k$.
Let $\psi'=(\psi_1',\ldots,\psi_{m'}')$ be other generators of ${\cal I}_A$ on $\Omega$; by adding some identically zero components to either $\psi$ or $\psi'$ we can assume $m'=m$. For any point $a\in Z\setminus I^k(\psi)$, relation (\ref{eq:avoid}) implies existence of a neighbourhood $U'$ of $a$ and a plane $L_0'\in G(k,m)$ such that $\psi'(x)\cap\omega'=\emptyset$ for some neighbourhood $\omega'$ of $L_0'$ and all $x\in U'\setminus Z$, so $a\in Z\setminus I^k(\psi')$. This shows that the sets $J^k$ are independent of the choice of generators of ${\cal I}_{A,\Omega}$. Therefore each $J^k$ is well defined as a local (not necessarily closed) analytic variety in $X$ with properties (i) and (ii). \end{proof}
\begin{example} Let $A$ be generated by $\psi(x)=(x_1^2x_2,x_1^2x_3,x_1x_2x_3)$ in
${\mathbb C}^3$. Then $|A|={\mathbb C}_{23}\cup {\mathbb C}_1$; here ${\mathbb C}_{23}$ is the coordinate plane of the variables $x_2$ and $x_3$, i.e., ${\mathbb C}_{23}=\set{x_1=0}$, and ${\mathbb C}_1=\set{x_2=x_3=0}$. The variety
$|A|$ has the decomposition $|A|=J^1\cup J^2\cup J^3$ with $J^1={\mathbb C}_{23}\setminus ({\mathbb C}_2\cup{\mathbb C}_3)$, $J^2={\mathbb C}_1^*\cup{\mathbb C}_2^*\cup {\mathbb C}_3^*$, and $J^3=\set{0}$. Near points of $J^1$ we have
$\log|\psi|\asymp \log|x_1|$. As to $J^2$, the relation
$\log|\psi|\asymp \log|\xi|$ is satisfied with $\xi=(x_2,x_3)$ near points of ${\mathbb C}_1^*$, and we can take $\xi=(x_1^2,x_1x_3)$ near points of ${\mathbb C}_2^*$ and $\xi=(x_1^2,x_1x_2)$ near points of ${\mathbb C}_3^*$. \end{example}
\section{Upper bounds and maximality}\label{sec:properties}
We recall that a function $u\in {\operatorname{PSH}}(X)$ is called {\it maximal} in $X$ if for every relatively compact subset $U$ of $X$ and for each upper semicontinuous function $v$ on $\overline U$ such that $v\in {\operatorname{PSH}}(U)$ and $v\leq u$ on $\partial U$, we have $v\leq u$ in $U$. An equivalent form is that for any $v\in {\operatorname{PSH}}(X)$ the relation $\set{v>u}\Subset X$ implies $v\leq u$ on $X$.
We will use the following variant of the maximum principle for unbounded plurisubharmonic functions.
\begin{lemma}\label{maxprin-lemma} Let $D\subset{\mathbb C}^k$ be a bounded domain and $u, v\in{\operatorname{PSH}}(D)$ such that
\begin{description} \item{(i)} $v$ is bounded above,
\item{(ii)} the set $S:=v^{-1}(-\infty)$ is closed in $D$,
\item{(iii)} $v$ is locally bounded and maximal on $D\setminus S$,
\item{(iv)} for any $\epsilon>0$ there exists a compact $K_\epsilon \subset D$ such that $u(z)\le v(z)+\epsilon$ on $D\setminus K_\epsilon$, and
\item{(v)} $\displaystyle \limsup_{z\to a,\:z\not\in S}(u(z)-v(z))<\infty$ for each $a\in S$. \end{description} \noindent Then $u\le v$ in $D$. \end{lemma}
\begin{proof} By (i) we may assume that $v$ is negative in $D$. Take any $\epsilon>0$ and $\delta>0$. Then it is sufficient to prove that $u_1=(1+\delta)(u-\epsilon)\leq v$. By (v) we conclude that each point $a\in S$ has a neighbourhood $U_a\Subset D$ where $u_1\le v$ and by (iv) that there is a domain $D_1\Subset D$ such that $u_1\le v$ on $D\setminus D_1$. By (ii) $S\cap \overline D_1$ is compact, so we can take a finite covering of $S\cap \overline D_1$ by $U_{a_j}$, $1\leq j\leq N$. Then $D_2=D_1\setminus\bigcup_j\overline U_{a_j}$ is an open subset of $D$ on which $v$ is bounded and $u_1\le v$ holds on $\partial D_2$. By (iii) $v$ is maximal on $D_2$, so $u_1\leq v$ on $D_2$ and thus on $D$. \end{proof}
The next statement is the crucial point in the proof that $G_A\in{\cal F}_A$.
\begin{lemma}\label{lub-lemma} Let $\psi=(\psi_1,\ldots,\psi_m)$ be a holomorphic map on a domain $\Omega\subset{\mathbb C}^n$ and $Z$ be its zero set. Then for every
$K\Subset \Omega$ there exists a number $C_K$ such that any function $u\in PSH^-(\Omega)$ which satisfies $u\le \log|\psi| + O(1)$ locally near points of $Z$ has the bound $u(x)\le
\log|\psi(x)| + C_K$ for all $x\in K$. \end{lemma}
\begin{proof} What we need to prove is that each point $a\in Z$
has a neighbourhood $U$ where $u\le\log|\psi|+C$ with $C$ independent of the function $u$.
Let ${\operatorname{codim}}_a\,Z=p$. Then, by Prop.~\ref{decomp-theo}, $a\in J^{k}$ for some $k\in [p,n]$ and thus there exist $k$ holomorphic functions $\xi_1,\ldots,\xi_k$ such that $\log|\xi|\asymp
\log|\psi|$ near $a$. We will argue by induction in $k$ from $p$ to $n$.
Let $a\in J^{p}$; this means that there is a neighbourhood $V$ of $a$ such that $Z\cap V$ is a complete intersection given by the functions $\xi_1,\ldots,\xi_{p}$. By Thie's theorem \cite{Thie}, (see also \cite{D1}, Th.~5.8), there exist local coordinates $x=(x',x'')$, $x'=(x_1,\ldots,x_p)$, $x''=(x_{p+1},\ldots,x_n)$, centered at $a$ and balls ${\mathbb B}'\subset {{\mathbb C}}^p$, ${\mathbb B}''\subset {{\mathbb C}}^{n-p}$ such that ${\mathbb B}'\times {\mathbb B}''\Subset V$, $Z\cap({\mathbb B}'\times
{\mathbb B}'')$ is contained in the cone $\{|x'|\le \gamma |x''|\}$ with some constant $\gamma>0$, and the projection of $Z\cap({\mathbb B}'\times {\mathbb B}'')$ onto ${\mathbb B}''$ is a ramified covering with a finite number of sheets. Let $r_1=2\gamma r_2$ with a sufficiently small $r_2>0$ so that ${\mathbb B}_{r_1}'\subset {\mathbb B}'$ and ${\mathbb B}_{r_2}''\subset {\mathbb B}''$, then for some $\delta>0$
$$|\xi(x)|\ge\delta,\qquad x\in \partial {\mathbb B}_{r_1}'\times {\mathbb B}_{r_2}''. $$
Given $x_0''\subset {\mathbb B}_{r_2}''$, denote by $Z(x_0'')$ and ${\operatorname{Sing}}\,Z(x_0'')$ the intersections of the set ${\mathbb B}_{r_1}'\times\{x_0''\}$ with the varieties $Z$ and ${\operatorname{Sing}}\,Z$, respectively. Since the projection is a ramified covering, $Z(x_0'')$ is finite for any $x_0''\in {\mathbb B}_{r_2}''$, while ${\operatorname{Sing}}\,Z(x_0'')$ is empty for almost all $x_0''\in {\mathbb B}_{r_2}''$ because ${\operatorname{dim}}\, {\operatorname{Sing}}\,Z\le n-p-1$; we denote the set of all such generic $x_0''$ by $E$.
Fix any $x_0''\in E$ and consider the function
$$v(x')=\log(|\xi(x',x_0'')|/\delta).$$ It is plurisubharmonic on ${\mathbb B}_{r_1}'$, nonnegative on $\partial {\mathbb B}_{r_1}'$ and maximal on ${\mathbb B}_{r_1}'\setminus Z(x_0'')$, since the map $\xi(\cdot,x_0''):{\mathbb B}_{r_1}'\to{{\mathbb C}}^p$ has no zeros outside $Z(x_0'')$.
For any function $u\in PSH^-(Y)$ which satisfies $u\le \log|\xi| + O(1)$ locally near regular points of $Z$, we have, by Lemma~\ref{maxprin-lemma}, $u(x',x_0'')<v(x')$ on the whole ball ${\mathbb B}_{r_1}'$.
Since $x_0''\in E$ is arbitrary, this gives us $u\le
\log|\xi|-\log\delta$ on ${\mathbb B}_{r_1}'\times E$. The continuity of the function $\log|\xi|$ extends this relation to the whole set $U={\mathbb B}_{r_1}'\times {\mathbb B}_{r_2}''$, which proves the claim for $k=p$.
Now we make a step from $k-1$ to $k$. Since ${\operatorname{dim}}\, J^{k}\le n-k$, we use Thie's theorem to get a coordinate system centered at $a\in J^{k}$ such that the projection of $J^{k}\cap ({\mathbb B}'\times{\mathbb B}'')$ to ${\mathbb B}''\subset{\mathbb C}^{n-k}$ is a finite map and $(\partial{\mathbb B}'\times\overline{{\mathbb B}''})\cap J^i=\emptyset$ for all
$i\ge k$. Therefore, by the induction assumption and a compactness argument, $u\le\log|\xi|+C$ near $\partial{\mathbb B}'\times\overline{{\mathbb B}''}$, where the constant $C$ is independent of $u$.
Now for any $x_0''\subset {\mathbb B}''$ we consider the function
$v(x')=\log|\xi(x',x_0'')|+C$. Then Lemma~\ref{maxprin-lemma}
gives us $u(x',x_0'')<v(x')$ on ${\mathbb B}'$ and hence $u\le\log|\xi|+C$ on ${\mathbb B}'\times{\mathbb B}''$. \end{proof}
{\it Remark.} Note that the uniform bound $u\le\log|\psi|+C$ near points $a\in J^p$, ${\operatorname{codim}}_a Z =p$, was deduced from the local bounds only near regular points of $Z$.
\begin{prooftx}{Proof of Theorem~\ref{GinF}}
The relation $G_A\le\log|\psi|+O(1)$ follows from Lemma~\ref{lub-lemma}. This implies that its upper semicontinuous regularization $G_A^*$ is in ${\cal F}_A$ and thus $G_A^*=G_A$. \end{prooftx}
One of the most important properties of the ``standard'' pluricomplex Green function $G_{X,a}$ with logarithmic pole at $a\in X$ is that it satisfies the homogeneous Monge-Amp\`ere equation $(dd^cG_{X,a})^n=0$ outside the point $a$; in other words, $G_{X,a}$ is a maximal plurisubharmonic function on $X\setminus\{a\}$. In our situation, one can say more.
\begin{theorem}\label{max-theo} The function $G_A$ is maximal on
$X\setminus |A|$ and locally maximal outside a discrete subset of
$|A|$ (actually, the set $J^n$ from Prop.~\ref{decomp-theo}). If $A$ has $k<n$ global generators on $X$, then $G_A$ is maximal on the whole $X$. \end{theorem}
\begin{proof}
Take any point $a\not\in J^n$. By Proposition~\ref{decomp-theo}, there exist functions $\xi_1,\ldots,\xi_{k}\in{\cal I}_{A,U}$, $k<n$, generating an ideal whose integral closure coincides with the integral closure of ${\cal I}_{A,U}$, and so $G_A\le\log|\xi|+C$ on $U$. The function $\log|\xi|$ is maximal on $U$, which follows from the fact that it is the limit of the decreasing sequence of maximal plurisubharmonic functions $u_j=\frac 12\log(|\xi|^2+{\frac 1j})$. (See \cite{R}, Example~1.) Take any domain $W\Subset U$. Given a function $v\in PSH(U)$ with $v\le G_A$ on $U\setminus W$, we have to show that $v\le G_A$ on $U$. Consider the function $w$ such that $w=G_A$ on $X\setminus W$ and $w=\max\{G_A,v\}$ on $W$. Since
$G_A\le\log|\xi|+C$ on $U$, we have $w\le \log|\xi|+C$ on
$U\setminus W$, and the maximality of $\log|\xi|$ on $U$ extends this inequality to the domain $W$. Therefore, $w\in{\cal F}_A$ and thus $w\le G_A$ on $U$.
When $a\not\in |A|$, we can take $U=X\setminus |A|$ and $\xi\equiv 1$, which gives us maximality of $G_A$ on $U=X\setminus |A|$.
Finally, if $A$ has $k<n$ global generators on $X$, then the same arguments with $U=X$ show the maximality of $G_A$ on the whole $X$. \end{proof}
{\it Remark.} If $J^n=\emptyset$, the Green function is locally maximal on the whole $X$. We don't know if this implies its maximality on $X$.
\section{Complex spaces with bounded global generators}\label{sec:bounded}
If $A$ has bounded generators $\psi$, which we can choose such that $|\psi|<1$, then $\log|\psi|\in{\cal F}_A$. This gives immediately
\begin{proposition}\label{th:asymp} Let $A$ be a closed complex subspace of a manifold $X$ and assume that $A$ has bounded global generators $\{\psi_i\}$ (for example, $X$ is a relatively compact domain in a Stein manifold $Y$ and $A$ is a restriction to $X$ of a complex space $B$ on $Y$), then \begin{equation}\label{eq:asymp}
G_A=\log|\psi|+O(1) \end{equation}
locally near $|A|$. \end{proposition}
To describe the boundary behaviour of $G_A$, we recall the notion of strong plurisubharmonic barrier. Let $X$ be a domain in a complex manifold $Y$, and let $p\in \partial X$. A plurisubharmonic function $v$ on $X$ is called a {\it strong plurisubharmonic barrier at} $p$ if $v(x)\to 0$ as $x\to p$, while $\sup_{X\setminus V}v <0$ for every neighbourhood $V$ of $p$ in $Y$. By standard arguments (see, e.g., \cite{LarSig2}, Proposition~2.4) we get
\begin{proposition} Let $X$ be a domain in a complex manifold $Y$, and let a closed complex subspace of $X$ have bounded global generators. If $X$ has a strong plurisubharmonic barrier at $p\in\partial
X\setminus |A|$, then $G_A(x)\to 0$ as $x\to p$. \end{proposition}
A uniqueness theorem for the Green function is similar to that for the divisor case in \cite{LarSig2}, but the proof is different
(since the function $u-\log|\psi|$ need not be plurisubharmonic) and follows from Lemma~\ref{maxprin-lemma} and Proposition~\ref{th:asymp}.
\begin{theorem} Let a complex space $A$ have bounded global generators $\psi_i$ on $X$, and let a function $u\in PSH^-(X)$ have the properties \begin{description}
\item{(i)} $u$ is locally bounded and maximal on $X\setminus |A|$. \item{(ii)} For any $\epsilon>0$ there exists a compact subset $K$ of $X$ such that $u\ge G_A-\epsilon$ on $X\setminus K$;
\item{(iii)} $u=\log|\psi|+O(1)$ locally near $|A|$. \end{description} Then $u=G_A$. \end{theorem}
Relation (\ref{eq:asymp}) allows us to derive the properties of the Monge-Amp\'ere current $(dd^cG_A)^p$.
\begin{prooftx}{Proof of Theorem \ref{th:current}}
Since $G_A$ is locally bounded on $U\setminus |A|$ and
${\operatorname{codim}}\,|A|=p$, the current $(dd^cG_A)^p$ is well defined on $U$. Moreover, Siu's structural formula for positive closed currents \cite{Siu} (see also \cite{D1}, Theorem~6.19) gives us a (unique) representation for the current $(dd^c G_A)^p$ as $$(dd^c G_A)^p=\sum_j\lambda_j[B_j]+Q, $$ where $B_j$ are some irreducible analytic varieties of codimension $p$, $\lambda_j$ are the generic Lelong numbers of $(dd^c G_A)^p$ along $B_j$, i.e., $$ \lambda_j=\inf\{\nu((dd^c G_A)^p,a):a\in B_j\}, $$ and $Q$ is a positive closed current such that $ {\operatorname{codim}}\{x:\nu(Q,x)\ge c\}>p$ for each $c>0$.
As $G_A$ has asymptotics (\ref{eq:asymp}) near points of the set
$|A|$, Demailly's Comparison Theorem for Lelong numbers (\cite{D1}, Theorem~5.9) implies
$$\nu((dd^c G_A)^p,a)=\nu((dd^c \log|\psi|)^p,a)$$
at every point $a\in |A|\cap J^p\cap U$. In particular, the generic Lelong number of $(dd^c G_A)^p$ along each variety $A_i^p$
equals the multiplicity of this component in $|A|$. Besides,
$\nu((dd^c G_A)^p,a)=0$ for any $a\not\in |A|$. This shows that
$\{B_j\}_j$ are exactly the $p$-codimensional components of the variety $|A|$ in $U$ and $\sum\lambda_j[B_j]=Z_A^p$ on $U$.
Finally, if $U\cap |A|\subset J^p$, then $U\cap|A|$ can be given locally by $p$ holomorphic functions $\xi_i$ with
$\log|\xi|\asymp\log|\psi|$. By King's formula,
$(dd^c\log|\xi|)^p=Z_A^p$, which means, in particular, that
$(dd^c\log|\xi|)^p$ has zero Lelong numbers outside $\cup_i A_i^p$. Since the currents $(dd^c G_A)^p$ and $(dd^c\log|\xi|)^p$ have the same Lelong numbers, this proves the last statement. \end{prooftx}
So the Green function satisfies, as in the divisor case, the relation $(dd^cG_A)^p\ge Z_A^p$, but for $p>1$ it is not the largest negative plurisubharmonic function with this property (even for reduced spaces that are complete intersections).
For example, let $X$ be the unit polydisc in ${{\mathbb C}}^3$ and $A$ be generated by
$\psi(z)=(z_1,z_2)$. Then $G_A=\max\{\log|z_1|,\log|z_2|\}$ and, moreover, $(dd^cG_A)^2=Z_A=[A]$. But the functions
$u_N=\max\{N\log|z_1|,N^{-1}\log|z_2|\}$, $N>0$, also satisfy
$(dd^c u_N )^2=[A]$, although they are not dominated by $G_A$. It is easy to see that the upper envelope of all such functions equals $0$ outside $|A|$ and $-\infty$ on $|A|$. Therefore, in the case ${\operatorname{codim}}\,|A|>1$ there is no counterpart for the description of the Green function in terms of the current $Z_A$.
\section{Reduced spaces}\label{sec:reduced}
Now we return to relations between the functions $G_A$ and $\tilde G_{\tilde\nu_A}$ (see Introduction). As was already mentioned, one has always $G_A\le\tilde G_{\tilde\nu_A}$ and $G_A<\tilde G_{\tilde\nu_A}$ for 'generic' spaces $A$, however $G_A=\tilde G_{\tilde\nu_A}$ for effective divisors $A$. Here we show that the equality holds also in the case of {\sl reduced} complex spaces.
When $A$ is a reduced space, it can be identified with the analytic variety $|A|$. Its generators $\psi_1,\ldots,\psi_m$ on $U$ have the property: if a holomorphic function $\varphi$ vanishes on $A\cap U$, then $\varphi=\sum h_i\psi_i$ with $h_i\in{\cal O}(U)$.
Since $\tilde\nu_A=1$ at all regular points of $A$, it is natural to consider the class $$ \tilde{\cal F}_A^1=\{u\in PSH^-(X);\: \nu_u(a)\ge 1 \text{ for all } a\in Reg\,A\}. $$ Note that upper semicontinuity of the Lelong numbers implies $\nu_u\ge 1$ on the whole $A$.
We evidently have $\tilde{\cal F}_A^1 \subseteq \tilde{\cal F}_{\tilde\nu_A}\subseteq {\cal F}_A$.
\begin{theorem} If $A$ is a reduced subspace of $X$, then $\tilde{\cal F}_A^1 = \tilde{\cal F}_{\tilde\nu_A}= {\cal F}_A$ and consequently $$ G_A(x)=\tilde G_{\tilde\nu_A}(x)=\sup\set{u(x)\,;\,\ u\in \tilde{\cal F}_A^1}. $$ \end{theorem}
\begin{proof} It suffices to show that for any function $u\in\tilde{\cal F}_A^1$ and every point $a\in A$ there is a neighbourhood $U$ of $a$ and a constant $C$ such that \begin{equation}\label{eq:regloc}
u(x)\le \log|\psi(x)|+C, \qquad x\in U. \end{equation}
We will use induction on the dimension of $X$. The case ${\operatorname{dim}}\,X=1$ is evident. Assume it proved for all $X$ with ${\operatorname{dim}}\,X<n$ and take any $u\in \tilde{\cal F}_A^1$. When ${\operatorname{dim}}_a A=0$, relation (\ref{eq:regloc}) follows easily from the fact that
$\log|\psi(x)|=\log|\zeta(x)|+O(1)$ near $a\in A$, where $\zeta$ are local coordinates near $a$ with $\zeta(a)=0$. So we assume ${\operatorname{dim}}_a A>0$. We first treat the case when $a$ is a regular point of $A$, ${\operatorname{codim}}_a A=p<n$. Since the problem is local, we may then assume that $X\subset{\mathbb C\sp n}$ and contains the unit polydisc ${{\mathbb D}}^n$, $a=0$, and the restriction $A'$ of $A$ to ${{\mathbb D}}^n$ is given by $\psi(x)=(x_1,\ldots,x_p)$. Then the restriction of $u$ to
${{\mathbb D}}^n$ is dominated by the Green function $\tilde G_{\tilde\nu_{A'}}$. By the product property for this type of Green function (\cite{LarSig2}, Theorem~2.5), $\tilde G_{\tilde\nu_{A'}}(x)=\max\{\log|x_j|,\ 1\le j\le p\}$. This implies (\ref{eq:regloc}) for $a\in Reg\,A$.
For $a\in Sing\,A$ we will argue similarly to the proof of Lemma~\ref{lub-lemma}. There is a neighbourhood $V$ of $a$ such that $V\cap Sing\,A\subset J^p\cup J^{p+1}\cup\ldots J^n$. The proof for $a\in J^{k}$, $p\le k\le n$, is then by induction in $k$.
For $a\in J^p\cap V$ relation (\ref{eq:regloc}) follows directly from the remark after Lemma~\ref{lub-lemma}.
Assuming (\ref{eq:regloc}) proved for $a\in J^p\cup\ldots\cup J^{k}$, we take $a\in J^{k+1}$. We choose coordinates $x=(x',x'')\in{\mathbb C}^{k+1}\times{\mathbb C}^{n-k-1}$ such that $a=0$, the projection of $J^{k+1}\cap {\mathbb B}$ to ${\mathbb B}''$ is a finite map and $\partial{\mathbb B}'\times{\mathbb B}''\cap J^i=\emptyset$ for all $i\ge k+1$, so the $k$-induction assumption gives \begin{equation}\label{eq:Lbound}
u(x)\le \log|\psi(x)|+C,\quad x\in\partial{\mathbb B}'\times{\mathbb B}''. \end{equation}
Take any $b=(b',b'')\in {\mathbb B}'\times{\mathbb B}''$ and consider the $(k+1)$-dimensional plane $L=\{x\,;\,x''=b''\}$. Then the restriction $u_L$ of $u$ to the plane $L$ (in the same way we will use the denotation $\psi_L$, ${\mathbb B}_L$, $A_L$, etc.) has Lelong numbers at least $1$ at all points of $A_L$, so $u_L\in\tilde{\cal F}_{A_L,{\mathbb B}_L}^1$. Since ${\operatorname{dim}}\,{\mathbb B}_L<n$ and the components of $\psi_L$ generate $A_L$, the $n$-induction assumption implies $u_L\in{\cal F}_{A_L,{\mathbb B}_L}$. Therefore, $u_L\le
\log|\psi_L|+O(1)$ locally near points of $A_L$.
Since $a\in J^{k+1}$, we can find functions
$\xi_1,\ldots,\xi_{k+1}$ such that $\log|\xi|\asymp\log|\psi|$ on
${\mathbb B}$. Therefore $u_L\le \log|\xi_L|+O(1)$ locally near all points of $A_L$, and, by (\ref{eq:Lbound}), $u_L\le \log|\psi_L|+C_1$ on a neighbourhood of $\partial {\mathbb B}_L$ with $C_1$ independent of $L$. The function $\xi_L$ is maximal on ${\mathbb B}_L\setminus A_L$, so by Lemma~\ref{maxprin-lemma}, $u_L\le \log|\xi_L|+C_1$ everywhere on ${\mathbb B}_L$. Since the plane $L$ was chosen arbitrary, this gives us (\ref{eq:regloc}) for $a\in J^{k+1}$.
This proves the inductive step in the induction in $k$ and, at the same time, in the induction in $n$. \end{proof}
Theorem~\ref{max-theo} for reduced spaces has the following form (compare with the remark after the proof of Theorem~\ref{max-theo}).
\begin{theorem} The Green function of a reduced space $A$ is maximal on $X\setminus A_0$, where $A_0$ is the collection of $0$-dimensional components of $A$. \end{theorem}
\begin{proof} We need to show that for every domain $U\Subset X':=X\setminus A_0$ and a function $u\in PSH(X')$ the condition $u\le G_A$ on $X'\setminus U$ implies $u\le G_A$ on $U$.
Consider the set $E_1(u)=\{x\in X\,;\, \nu_u(x)\ge 1\}$. Since $u\le G_A$ on $X'\setminus U$, we have $E_1(u)\setminus U\supset A\setminus U$. By Siu's theorem, $E_u$ is an analytic variety in $X$, so it must contain the whole $A$. This means that $u\in\tilde{\cal F}_A^1$ and thus is dominated by $\tilde G_{\tilde\nu_A}=G_A$ on $X$. \end{proof}
\section{The product property}\label{sec:product}
Our proof of Th.~\ref{th:product} in this section is based on Th.~\ref{th:envelope}. It is a modification of the proof of Th.~2.5 in \cite{LarSig2} which in turn generalizes a proof of Edigarian \cite{E} of the product property for the single pole Green function. For the sake of completeness we have repeated some arguments from \cite{E}, \cite{LarSig2}, and \cite{LarLasSig1}.
We introduce the following notation: If the function $\varphi$ is holomorphic in some neighbourhood of the point $a$ in ${\mathbb C}$, then we set $m_a(\varphi)=0$ if $\varphi(a)\neq 0$, $m_a(\varphi)=+\infty$ if $\varphi=0$ in some neighbourhood of $a$, and let $m_a(\varphi)$ be the multiplicity of $a$ if it is an isolated zero of $\varphi$.
\begin{lemma}\label{var2} Let $x\in X$, ${\alpha} \in (-\infty,0)$ and assume that $g\in {\cal O}(\overline {\mathbb D},X)$, $g(0)=x$, and $G_{g^*A}(0)<\alpha$. Then there exist $f\in {\cal O}(\overline {\mathbb D},X)$ and finitely many different points $a_1,\dots,a_k\in {\mathbb D}\setminus\set 0$ such that $f(0)=x$ and \begin{equation}
-\infty <\sum_{j=1}^k \tilde\nu_{f^*A}(a_j)\log|a_j|<\alpha. \label{eq:var0} \end{equation} \end{lemma}
\begin{proof} We have $G_{g^*A}(0)=\sum_{a\in
{\mathbb D}}\tilde\nu_{g^*A}(a)\log|a|<\alpha$, so we can choose finitely many points $a_1,\dots,a_k\in {\mathbb D}\setminus\set 0$ such that \begin{equation}
\sum_{j=1}^k \tilde\nu_{g^*A}(a_j)\log|a_j|<\alpha. \label{eq:var1} \end{equation}
If the sum in (\ref{eq:var1}) is finite, we take $f=g$. If the sum is equal to $-\infty$ and $g({\mathbb D})$ is not contained in $|A|$, then $a_j=0$ and $0<\tilde\nu_{g^*A}(a_j)<+\infty$ for some $j$. We choose $a \in {\mathbb D}\setminus\set{0}$ so close to $0$ that
$\log|a|<{\alpha}$ and $g$ is holomorphic in a neighbourhood of the image of $h:\overline{\mathbb D}\to {\mathbb C}$, $h(z)=z(z-a)$. If $\psi_1,\dots,\psi_m$ are local generators for ${\cal I}_A$ near $x$, then $m_a(\psi_j\circ g\circ h)=m_0(\psi_j\circ g)$ for all $j$, which implies $\tilde\nu_{g^*A}(a)=\tilde\nu_{(g\circ h)^*A}(a)$. If we set $f=g\circ h$, $k=1$, and $a_1=a$, then $f(0)=x$ and (\ref{eq:var0}) holds.
If the sum in (\ref{eq:var1}) equals $-\infty$ and $g({\mathbb D})$ is contained in $|A|$, then we may replace $g$ by the constant disc $z\mapsto x=g(0)$. We choose a neighbourhood $U$ of $x$ in $X$
and a biholomorphic map $\Phi:U\to{\mathbb D}^n$ such that $\Phi(x)=0$. We take $v\in {\mathbb C}^n$ with $|v|<1$ such that the disc $\overline {\mathbb D}\to X$, $z\mapsto \Phi^{-1}(zv)$ is not contained in $|A|$ and choose
$a\in {\mathbb D}\setminus\set{0}$ so small that $\log |a|<{\alpha}$ and $z(z-a)v\in {\mathbb D}^n$ for all $z\in \overline {\mathbb D}$. If we take $k=1$,
$a_1=a$, and let $f$ be the map $z\mapsto \Phi^{-1}(z(z-a)v)$, then $f(0)=f(a)=x\in |A|$, $0< \nu_{f^*A}(a)<+\infty$, and (\ref{eq:var0}) holds. \end{proof}
\begin{prooftx}{Proof of Theorem \ref{th:product}} We need to prove that $G_A(x)\leq \max\set{G_{A_1}(x_1),G_{A_2}(x_2)}$. Take $\alpha\in (-\infty,0)$ larger than the right hand side of this inequality. It is then sufficient to show that $G_A(x)<{\alpha}$.
By Theorem~\ref{th:envelope} and Lemma \ref{var2} we have $f_j\in {\cal O}(\overline{\mathbb D},X_j)$ with $f_j(0)=x_j$ and $a_{jk}\in {\mathbb D}\setminus\set{0}$, $k=1,\dots,l_j$, $j=1,2$, such that \begin{equation}
-\infty<\sum_{k=1}^{l_j}\tilde\nu_{f_j^*A_j}(a_{jk})\log|a_{jk}|<\alpha,\qquad j=1,2. \label{eq:9.1} \end{equation} We choose $f_j$ so that $l_j$ becomes as small as possible. Then $0<\tilde\nu_{f_j^*A_j}(a_{jk})<+\infty$ and $a_{jk}\neq 0$ for all $j$ and $k$. We define the Blaschke products $B_j$ by $$ B_j(z)=\prod_{k=1}^{l_j}\bigg(\dfrac{a_{jk}-z}{1-\bar a_{jk}z}\bigg)^{\mu_{jk}}, \qquad \text{ where } \ \mu_{jk}=\nu_{f_j^*A_j}(a_{jk}). $$
Then (\ref{eq:9.1}) implies $|B_j(0)|<e^\alpha$. We set
$b_j=B_j(0)$ and $\mu_j=\sum_{k=1}^{l_j}\mu_{jk}$ and we may assume that $|b_1|\geq |b_2|$. We have
$B_j'(0)=B_j(0)\sum_{k=1}^{l_j}\mu_{jk}(|a_{jk}|^2-1)/a_{jk}$. If
$B_1'(0)=0$ we precompose $f_1$ with a map ${\mathbb D}\to {\mathbb D}$ which fixes the origin and makes a slight change of the points $a_{1k}$ so that $B_1'(0)\neq 0$. By Schwarz Lemma this operation increases the value of $|b_1|$, so we still have $|b_1|\geq|b_2|$. By precomposing $f_1$ by a rotation, we may assume that $B_1(0)=b_1$ is not a critical value of $B_1$.
If $c_j$ is one of the points $a_{jk}$ having largest absolute value, then $|c_j|e^\beta\leq |b_j|$. For proving this inequality we assume the reverse inequality $|b_j|<|c_j|e^\beta$ and for simplicity enumerate the points so that
$|a_{j1}|\leq|a_{j2}|\leq\cdots$. Then $$
\prod_{k=1}^{m_j}\bigg|\dfrac{a_{jk}}{c_j}\bigg|^{{\mu}_{jk}}<e^\beta $$ where $m_j<l_j$ is the smallest natural number with
$|a_{jk}|=|c_j|$ for $k>m_j$. Hence (\ref{eq:9.1}) holds with $f_j$ replaced by $z\mapsto f_j(c_jz)$, $a_{jk}$ replaced by $a_{jk}/c_j$, and $l_j$ by $m_j$, which contradicts the fact that $l_j$ is minimal.
We may assume that $b_1=b_2$. Indeed, if $|b_1|>|b_2|$, we choose
$t\in (0,1)$ with $t^{-{\mu}_2}|b_2|=|b_1|$. Then $|a_{2k}|<t$, for $$
|a_{2k}|^{{\mu}_2}\leq |c_2|^{{\mu}_2}\leq |b_2|e^{-\beta}
<|b_2/b_1|=t^{{\mu}_2}. $$
Replacing $f_2$ by $z\mapsto f_2(tz)$ and $a_{2k}$ by $a_{2k}/t$, we get $|b_1|=|b_2|$. Finally, replacing $f_2$ by $z\mapsto f_2(e^{i\theta}z)$, where $e^{i\theta{\mu}_2}=b_2/b_1$, and replacing $a_{2k}$ by $e^{-i\theta}a_{2k}$, we get $b_1=b_2$.
We let $C$ denote the set of all critical values of $B_1$. We have $B_1(0)=B_2(0)$, so we can take $\varphi_2:{\mathbb D}\to {\mathbb D}\setminus B_2^{-1}(C)$ as the universal covering map with $\varphi(0)=0$. A theorem of Frostman, see \cite{Nos}, p.~33, states that an inner function on ${\mathbb D}$ omitting $0$ as a non-tangential boundary value is a Blaschke product. It is easy to show, see \cite{LarLasSig1}, p.~272, that since $0\not\in B_2^{-1}(C)$, $\varphi_2$ satisfies the assumption in Frostman's theorem and is thus a Blaschke product. The restriction of $B_1$ to ${\mathbb D}\setminus B_1^{-1}(C)$ is a finite covering over ${\mathbb D}\setminus C$, so by lifting $B_2\circ \varphi_2$ we conclude that there exists a function $\varphi_1:{\mathbb D}\to B_1^{-1}(C)$ with $\varphi_1(0)=0$ and $B_1\circ\varphi_1=B_2\circ\varphi_2$ and Frostman's theorem implies again that $\varphi_1$ is a Blaschke product. Since
$|B_j\circ\varphi_j|=1$ almost everywhere on ${\mathbb T}$ and $B_j(0)=b_j$, we can choose $r\in (0,1)$ such that $$
\log|B_j\circ\varphi_j(0)|-\dfrac 1{2\pi}\int_0^{2\pi}
\log|B_j\circ\varphi_j(re^{i\theta})|\, d\theta <\alpha. $$
We set $\sigma(z)=B_1\circ \varphi_1(rz)=B_2\circ\varphi_2(rz)$. By the Poisson--Jensen representation formula, the left hands side of this inequality equals $\sum_{i=1}^n\nu_i\log|z_i|$, where $z_i$ are the zeros of $\sigma$ in ${\mathbb D}$ with multiplicities $\nu_i$ for $i=1,\dots,n$.
We define $g_j\in {\cal O}(\overline {\mathbb D},X_j)$ by $g_j(z)=f_j\circ\varphi_j(rz)$ and $f\in {\cal O}(\overline {\mathbb D},X)$ with $f(0)=(x_1,x_2)$ by $f=(g_1,g_2)$. If $\sigma(z_i)=0$, then ${\varphi}_j(rz_i)=a_{jk_j}$ for some $k_j$, and $$ \nu_i=m_{z_i}(\sigma) =\mu_{jk_j}m_{z_i}(a_{jk_j}-\varphi_j(r\cdot)) =\tilde\nu_{f_j^*A_j}(a_{j,k_j})m_{z_i}(a_{jk_j}-\varphi_j(r\cdot)) =\tilde\nu_{g_j^*{A_j}}(z_i) $$ Since the left hand side of this equation is independent of $j$, we get $$ \tilde\nu_{f^*A}(z_i)=\min_j\set{\tilde\nu_{g_j^*{A_j}}(z_i)} =\nu_i. $$ Hence $$
G_A(x)\leq G_{f^*A}(0)=\sum_{a\in {\mathbb D}}\tilde\nu_{f^*A}(a)\log|a|
\leq \sum_{i=1}^n\tilde\nu_{f^*A}(z_i)\log|z_i|
=\sum_{i=1}^n\nu_i\log|z_i|<\alpha. $$ \end{prooftx}
\section{Examples}
\begin{example} Let $X$ be the unit polydisc ${\mathbb D}^n$ in ${\mathbb C\sp n}$, $1\leq p\leq n$, and let $A$ be generated by $\psi_k(z)=z_k^{\nu_k}$ for $1\leq k\leq p$ and positive integers $\nu_k$. Then the product property gives $$
G_A(z)=\max_{1\leq k\leq p}\nu_k\log|z_k|. $$ Furthermore, we have $$
\big(dd^cG_A\big)^p=\nu_1\cdots\nu_p\, [|A|]. $$ \end{example}
\begin{example} Let $X={\mathbb D}^n$, $n\ge 2$, and let $A$ be generated by $\psi_1(z)=z_1^2$, $\psi_2(z)=z_1z_2$. Then
$$G_A(z)=v(z):=\log|z_1|+\max\set{\log|z_1|,\log|z_2|}, \qquad z=(z_1,z_2,z'')\in {\mathbb D}^n. $$
First we take any $z\in {\mathbb D}^n\setminus\set{z_1=0}$ with $|z_1|\ge
|z_2|$ and consider the disc $$ f(\zeta)=(\zeta,\frac{z_2}{z_1}\zeta,z''),\quad \zeta\in{\mathbb D}. $$ Then for any $u\in{\cal F}_A$ we have
$f^*u(\zeta)\le\log|f^*\psi(\zeta)| +C=2\log|\zeta|+C$ and so, since $u\le 0$, $f^*u(\zeta)\le 2\log|\zeta| =f^*v(\zeta)$. As $f(z_1)=z$, this gives us $u(z)\le v(z)$.
For $z\in {\mathbb D}^n\setminus\set{z_1=0}$ with $|z_1|< |z_2|$, we take the disc $$ g(\zeta)=(\frac{z_1}{z_2}\zeta,\zeta,z''),\quad \zeta\in{\mathbb D}. $$
Then for any $u\in{\cal F}_A$ we have again $g^*u(\zeta)\le 2\log|\zeta|+C$ near the origin and, since $u(z)\le\log|z_1|$
(which is the Green function for the polydisc with the poles along the space $z_1=0$), $g^*u(\zeta)\le \log|z_1/z_2|$ near
$\partial{\mathbb D}$. Therefore, $g^*u(\zeta)\le 2\log|\zeta|+\log|z_1/z_2|=g^*v(\zeta)$ everywhere in ${\mathbb D}$. Since $g(z_2)=z$, this shows $u(z)\le v(z)$ at all such $z$ as well.
Note that $f$ is, up to a M\"obius transformation, an extremal disc for the disc functional $f\mapsto G_{f^*A}(0)$, while $g$ is not. Note also that we have $dd^cG_A=[z_1=0]+Q$, where the current
$Q=dd^c\max\set{\log|z_1|,\log|z_2|}$ has the property $Q^2=[z_1=z_2=0]$. \end{example}
\begin{example}
Consider the variety $|A|=\{z_1=z_2=0\}\cup \{z_2=z_3=0\}\cup\{z_1=z_3=0\} $ in the unit polydisk ${\mathbb D}^3$ of
${\mathbb C}^3$. It is easy to see that the corresponding reduced complex space $A$ is generated by $\psi(z)=(z_1z_2,z_2z_3,z_1z_3)$ and that $|A|$ has the decomposition (in the sense of Prop.~\ref{decomp-theo}) $|A|=J^2\cup J^3$ with $J^3=\set{0}$. We claim that
$$G_A(z)=v(z):=\max\,\{\log|z_1z_2|,\log|z_2z_3|,\log|z_1z_3|\}. $$ It suffices to check the relation $u(z)\le v(z)$ for any function
$u\in{\cal F}_A$ and each point $z\in{\mathbb D}^3$ with $|z_1|\ge |z_2|\ge
|z_3|$, $z_2\neq 0$. We take first any $z$ with $|z_1|= |z_2|\ge
|z_3|$ and consider the disc $f(\zeta)=\zeta z/|z_1|$, $\zeta\in{\mathbb D}$. Then $f^*u\in SH^-({\mathbb D})$ and, since
$u\le\log|\psi|+C_1$ near the origin, $f^*u(\zeta)\le
\log|f^*\psi(\zeta)|+C_1=2\log|\zeta|+C_2$ when
$|\zeta|<\epsilon$. Therefore, $f^*u(\zeta)\le 2\log|\zeta|=f^*v(\zeta)$ and, in particular, $u(z)=f^*u(|z_1|)\le f^*v(|z_1|)=v(z)$. The disc $f$ is, up to a M\"obius transformation, an extremal disc for the disc functional $f\mapsto G_{f^*A}(0)$ at such a point $z$.
Now we can take any $z$ with $|z_1|> |z_2|\ge |z_3|$, $|z_2|\neq 0$, and consider the analytic disc $g(\zeta)=(z_1, \zeta z_2,\zeta z_3)$, $\zeta\in D_R$ with $R=|z_1|/|z_2|>1$. We have
$|g_1(\zeta)|= |g_2(\zeta)|\ge |g_3(\zeta)|$ when $|\zeta|=R$ and thus $g^*u\le g^*v$ on $\partial D_R$. Furthermore,
$g^*u(\zeta)\le \log|g^*\psi(\zeta)|+C_3\le \log|\zeta|+C_4$ near the origin. Since $g^*v(\zeta)=\log|\zeta z_1z_2|$, this shows that $g^*u\le g^*v$ on $D_R$. Hence we get $u(z)=g^*u(1)\le f^*v(1)=v(z)$, which proves the claim.
The current $(dd^c G_A)^2$ has Lelong numbers equal $1$ at each point $a\in J^2=|A|\setminus\{0\}$. The point $0$ is exceptional:
the Lelong number $\nu(dd^cG_A,0)=2$, so $\nu((dd^cG_A)^2,0)\ge 4$, while $\nu([|A|],0)=3$. \end{example}
\begin{example} Let $X$ be the unit ball ${\mathbb B}_n$ in ${\mathbb C\sp n}$, $1\leq p\leq n$, and let $A$ be generated by $\psi_k(z)=z_k$ for all $1\leq k\leq p$. In the notation $z=(z',z'')$ with $z'\in{\mathbb C}^p$ and $z''\in{\mathbb C}^{n-p}$, the Green function $$
G_A(z)=\log\frac{|z'|}{\sqrt{1-|z''|^2}}, $$ because its restriction to every plane $z''=c\in{\mathbb B}_{n-p}$ is the pluricomplex Green function for the ball of radius
$\sqrt{1-|c|^2}$ in ${\mathbb C}^p$ with simple pole at the origin. \end{example}
\begin{example} The Green function $G_A$ for the unit ball ${\mathbb B}_n$ in ${\mathbb C\sp n}$, $n\ge 2$, with respect to $A$ generated by $(\psi_1,\psi_2)=(z_1^2,z_2)$ is given by
$$G_A(z)=\dfrac 1 2\log\bigg( \frac{|z_1|^4}{(1-|z''|^2)^2}
+ \frac{2|z_2|^2}{1-|z''|^2} + \frac{|z_1|^2}{1-|z''|^2} \sqrt{
\frac{|z_1|^4}{(1-|z''|^2)^2} + \frac{4|z_2|^2}{1-|z''|^2} }\bigg)
- \dfrac 12\log 2,$$ for $z=(z_1,z_2,z'')\in {\mathbb B}_n$ (compare with the formula for the pluricomplex Green function with two poles in the ball \cite{Coman}).
For proving this we let $v$ denote the function defined by the right hand side. Then $v\in {\operatorname{PSH}}({\mathbb B}_n)\cap C(\overline{\mathbb B}_n
\setminus |A|)$ and satisfies $v(z)\le
\max\{\log|z_1|^2,\log|z_2|\}+C$ locally near
$|A|=\set{z_1=z_2=0}$.
Let us show that its boundary values on $\partial {\mathbb B}_n\setminus
|A|$ are zero. Take any $z\in\partial {\mathbb B}_n\setminus |A|$, then
$|z_1|^2=a$, $|z_2|^2=b$, $|z''|^2=1-a-b$ with $a,b\ge 0$, $0<a+b\le 1$. We get $$ v(z)= \dfrac 12\log\left[ \frac{a^2}{(a+b)^2} + \frac{2b}{a+b} + \frac{a}{a+b}\left(2-\frac{a}{a+b}\right) \right]
- \dfrac 12\log 2=0. $$
Finally we show that $v(z)\ge G_A(z)$ for almost all $z\in {\mathbb B}_n$
(which implies $v\equiv G_A$). Take any $z\in {\mathbb B}_n$ with $z_1\neq 0$
and consider the analytic curve
$$f(\zeta)=(\zeta, \frac{z_2}{z_1^2}\zeta^2, z'').
$$
Note that $f(z_1)=z$. We have $f^*v(\zeta) =2\log(|\zeta|/R(z))$, while $f^*G_A$ is a negative subharmonic function in the disc
$|\zeta|<R(z)$ with the singularity $2\log|\zeta|$. So $f^*G_A \le f^*v$ and, in particular, $G_A(z)=f^*G_A(z_1)\le f^*v (z_1)=v(z)$.
This shows also that $f$ is, up to a M\"obius transformation, an extremal disc for the disc functional $f\mapsto G_{f^*A}(0)$ at $z$ with $z_1\neq 0$. A corresponding extremal curve for $z=(0,z_2,z'')$ is $f(\zeta)= (0,\zeta,z'')$. \end{example}
{\small{\it Acknowledgments.} The authors thank Daniel Barlet and Alain Yger for valuable discussions. Part of the work was done during Alexander's visit to the University of Iceland and Ragnar's visit to H\o gskolen i Stavanger, and the authors thank the both institutions for their kind hospitality.}
Alexander Rashkovskii
Tek/nat, H\o gskolen i Stavanger, POB 8002, 4068 Stavanger, Norway
E-mail: [email protected]
Ragnar Sigurdsson
Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland
E-mail: [email protected]
\end{document} | arXiv |
Hierarchical structural component modeling of microRNA-mRNA integration analysis
Yongkang Kim1,
Sungyoung Lee2,
Sungkyoung Choi2,
Jin-Young Jang3 &
Taesung Park1,2
BMC Bioinformatics volume 19, Article number: 75 (2018) Cite this article
Identification of multi-markers is one of the most challenging issues in personalized medicine era. Nowadays, many different types of omics data are generated from the same subject. Although many methods endeavor to identify candidate markers, for each type of omics data, few or none can facilitate such identification.
It is well known that microRNAs affect phenotypes only indirectly, through regulating mRNA expression and/or protein translation. Toward addressing this issue, we suggest a hierarchical structured component analysis of microRNA-mRNA integration ("HisCoM-mimi") model that accounts for this biological relationship, to efficiently study and identify such integrated markers. In simulation studies, HisCoM-mimi showed the better performance than the other three methods. Also, in real data analysis, HisCoM-mimi successfully identified more gives more informative miRNA-mRNA integration sets relationships for pancreatic ductal adenocarcinoma (PDAC) diagnosis, compared to the other methods.
As exemplified by an application to pancreatic cancer data, our proposed model effectively identified integrated miRNA/target mRNA pairs as markers for early diagnosis, providing a much broader biological interpretation.
Presently, numerous types of "omics" data are generated by many accurate and cost-effective methods. For instance, next-generation sequencing (NGS) technology is used to find DNA or RNA variations, bisulfite sequencing is used to find DNA-methylated variants, and multiple reaction monitoring (MRM) is applied to measure protein abundances [1,2,3]. These efficient omics data platforms allow researchers to use multi-omics data, obtained from the same subjects, for analyzing huge numbers of variants. As a result, efficient multi-omics data analysis is becoming more important in integrating large-scale data sets, making it possible to interpret fundamental biological systems [4].
MicroRNAs (miRNAs) are noncoding RNAs having a length less than 25 base pairs, regulating the expression of specific genes by mRNA degradation or blocking translation by binding to the 3′ regions of their "target" mRNAs. Many recent studies have now implicated miRNAs in the pathogenesis of cancer, including triggering cancer initiation and progression. MiRNAs have been shown to have tissue-specific and disease-specific expression patterns [5,6,7,8]. Intensive investigation is now underway for using applying miRNAs' inhibitory information to mRNAs. For example, Nam et al. developed "miRNA and mRNA integrated analysis" (MMIA) to examine biological functions of miRNA expression [9]. Moreover, Buffa et al. used pathway information to independently validate miRNAs significant for breast cancer [10], while Cho et al. performed network analysis, and hierarchical clustering, to find biological "signatures" of interstitial lung diseases [11]. Most miRNA and mRNA integration analyses focus on first identifying miRNAs significantly associated with the phenotype of interest, and then experimentally validating those miRNAs' phenotype involvement by inhibiting or ectopically overregulating their expression [9,10,11]. Although these approaches are effective at validating significant miRNAs, they do not provide information on how they regulate expression of their target mRNAs, as relevant to the pathway level.
In this work, we propose a structured component-based analysis, for integrating omics data for identifying multiple accurate biomarkers. It is well known that miRNAs affect phenotypes indirectly, by regulating mRNA expression or protein translation [8]. Herein, we propose hierarchical structured component analysis of miRNA-mRNA integration (HisCoM-mimi) analysis, which models biological relationships as structured components, to efficiently yield integrated markers. Our proposed model is based on generalized structured component analysis (GSCA), which tests hypothesized relationships between observed and latent variables [12]. GSCA is a component-based method whereby each component represents a latent variable. Extending GSCA, we previously developed Pathway-based approach using hierarchical components of collapsed rare variants (PHARAOH) [13]. PHARAOH uses a hierarchial structure of rare variants, genes, and pathways. The advantage of such hierarchical structural component models is their generation of (unobservable) latent variables, such as genes and pathways, which are inferred by observed variables, such as rare variants. Using latent variables, we can collapse unstructured data into a structured form, providing less ambiguous biological explanations of the results. In this current work, mRNAs, inhibited by miRNAs, can be merged into latent variables.
Accordingly, our proposed HisCoM-mimi model can efficiently account for biological relationships between miRNA and mRNA, in the structured component, and effectively provide integrated (e.g., miRNA-to-target-mRNA) markers. As an illustration, we tried HisCoM-mimi for identifying biomarkers for the early diagnosis of pancreatic cancer (PC). Note that PC is one of the most fatal diseases in the world, having a mere 8% five-year survival rate in the USA and a 9.4% survival rate in the Republic of Korea [14,15,16]. In particular, the tumor heterogeneity in PC patients' tumors makes early diagnosis harder than cancers of most other organs [17]. To adjust for heterogeneity among tumor cells, we need a more robust and complex statistical model which can interpret and integrate several causes of cancer altogether. Although many bioinformatics research studies have been performed to find diagnostic markers for PC, to date, no clinically approved prognostic markers exist [18].
Here, we applied HisCoM-mimi to computationally identify diagnostic markers of pancreatic ductal adenocarcinoma (PDAC), the most common type of PC. By applying the HisCoM-mimi approach to miRNA and mRNA microarray data from PDAC patients, at Seoul National University Hospital (SNUH), we identified numerous cognate miRNA-mRNA partners, as markers for diagnosis of PDAC. Finally, our HisCoM-mimi provided integrated marker sets, with more biological and intuitive interpretation, than other existing methods.
Pancreatic ductal adenocarcinoma (PDAC) samples
Between the years 2009 and 2012, 200 pancreatic ductal adenocarcinoma (PDAC) samples were collected by the Department of Hepatobiliary and Pancreas Surgery of Seoul National University Hospital. The study protocol was approved by the Institutional Review Board of Seoul National University Hospital (IRB H-0901-010-267) and written, informed consent was obtained from each patient or legally authorized representative.
Of the 200 tumors, 96 were excluded because of RNA degradation or insufficient RNA content, leaving 104 samples valid for microarray analysis. After quality control, 97 PDAC samples remained for microarray assessment. The PDAC patients' average age was 64.3 years (standard deviation (SD): 9.7). Twenty-nine patients were male, and 31 female. For the normal groups, 17 benign pancreatic tissues were used. Subsequently, we built and implemented our mini model, using the 97 PDAC and 17 normal tissues, respectively.
HisCoM-mimi model
To perform the integration analysis of miRNA and mRNA data, we developed and implemented our HisCoM-mimi approach. This model analyzes multiple subnetworks simultaneously, with specific regard to inverse correlations between mRNA and miRNA. Figure 1 shows the flowchart of the method. First, for a given miRNA, a miRNA-mRNA subnetwork, consisting of one miRNA and multiple potential target mRNAs, is constructed if the following two conditions are satisfied: (i) the mRNAs are reported as target of the miRNA by TargetScan 7.1 (targetscan.org) [19], and the negative correlation coefficients between the mRNA and miRNAs are significant (p-value < 0.05). Second, for all entities deemed significant, we derived our hierarchical structural component model by using all miRNA-mRNA subnetworks.
Flow chart for analyzing mRNA-miRNA integration
As shown in Fig. 2, there are three structures to consider: miRNA-mRNA structure, miRNA integration latent structure, and phenotype-latent structure. Each structure can be represented as a generalized linear model, similar to PHARAOH [13].
Network Diagram for HisCoM-mimi model
miRNA-mRNA structure
$$ {\widehat{X}}_{ijk}={x}_{ijk}-{\gamma}_{jk}{z}_{\mathrm{i}j},j=1,\dots, {G}_j, $$
Equation (1) shows how to obtain mRNA expression before inhibition by miRNA, subscript i means i th individual, x ijk represents the mRNA expression of the kth gene related with j th miRNA, z j the j th miRNA expression, γ jk the inhibition coefficient for the j th miRNA for the k th gene, and G j is the number of inhibited mRNAs by the j th miRNA. By estimating the coefficients γ jk , mRNA expression after removing the inhibition effect of miRNA can be obtained.
miRNA latent structure
$$ {f}_{ij}={\gamma}_{j0}{z}_j+{\sum}_{k=1}^{G_j}{\widehat{X}}_{ij k}{w}_{jk} $$
The miRNA latent variable is defined in Eq. (2). The miRNA latent variable is built by linearly combining miRNA expression values. While γj0 denotes the direct effect of the miRNA on the phenotype. Then, the latent variable f ij represents the global effect of the miRNA's activity through its inhibited mRNAs.
Phenotype-latent structure
$$ logit\left({\pi}_i\right)={\beta}_0+{\sum}_{j=1}^J\left[{\sum}_{k=1}^{G_j}{x}_{ij(k)}^{gene}{w}_{jk}\right]{\beta}_j={\beta}_0+{\sum}_{j=1}^J{f}_{ij}{\beta}_j $$
Let the phenotype variable y i be a binary variable, distinguishing PDAC from normal tissues. Let π i be the probability of y i = 1 (PDAC). logit(π i ) is the logit link function, β j represents the effect of f ij on the phenotype, as interpreted as a log-odds ratio.
Fitting the HisCoM-mimi algorithm
To estimate the parameters for HisCoM-mimi, we adopted our previously developed PHARAOH algorithm [13], which is based on the alternating least squares algorithm for the penalized log-likelihood function, with ridge parameters. Then, the objective function to maximize is given as follows:
$$ logit\left({\pi}_i\right)={\beta}_0+{\sum}_{j=1}^J\left[{\sum}_{k=1}^{G_j}{x}_{ij(k)}^{gene}{w}_{jk}\right]{\beta}_j={\beta}_0+{\sum}_{j=1}^J{f}_{ij}{\beta}_j, $$
$$ {\varphi}_1={\sum}_{i=1}^n\log\ p\left({y}_i;{\beta}_j,\delta \right)-\frac{1}{2}{\lambda}_m{\sum}_{j=1}^J{\sum}_{k=1}^{G_j}{w}_{jk}^2-\frac{1}{2}{\lambda}_{mm}{\sum}_{j=0}^J{\beta}_j^2 $$
where p(y i ; γ i , δ) is the probability distribution for the phenotype of the ith individual. λ m and λ mm are ridge parameters for miRNA-mRNA pairs of interest, representing the integrated latent components.
To maximize the objective function, φ1, the iterative reweighted least squares (IRWLS) algorithm is used. Note that when using IRWLS, maximizing φ1 is equivalent to minimizing the object function φ2.
$$ {\varphi}_2={\sum}_{i=1}^n{\mathrm{v}}_{\mathrm{i}}{\left({z}_i-{\sum}_{j=1}^J{f}_{ij}{\beta}_j\right)}^2-\frac{1}{2}{\lambda}_m{\sum}_{j=1}^J{\sum}_{k=1}^{G_j}{w}_{jk}^2-\frac{1}{2}{\lambda}_{mm}{\sum}_{j=0}^J{\beta}_j^2 $$
Comparative models
To compare the results of HisCoM-mimi with other methods, we considered several alternative regression-based methods.
$$ logit\left({\pi}_i\right)={\beta}_0+{\sum}_{j=1}^J{\theta}_j{z}_{ij}+{\sum}_{k=1}^K{\rho}_k{x}_{ij k},j=1,\dots, J $$
$$ {\varphi}_{LR}\left({\beta}_0,\theta, \rho, \delta; X,Z\right)={\sum}_{i=1}^n\log\ p\left({y}_i;{\beta}_0,\theta, \rho \right)-\delta {P}_{\alpha}\left(\theta, \rho \right),j=1,\dots, J $$
Firstly, we considered the ordinary penalized logistic regression (LR) methods such as lasso or elastic-net (EN) [20, 21]. Equation 7 shows the LR model, where θ j and ρ k represent the effect of the jth miRNA and the kth mRNA, respectively. Equation 8 is the objective function to maximize for finding optimal parameters with the penalty function P α (θ, ρ). When lasso is used, P α (θ, ρ) =∑k ∣ ρ k ∣ + ∑j ∣ θ j ∣.
If EN is used, \( {P}_{\alpha}\left(\theta, \rho \right)=\alpha \left({\sum}_{\mathrm{k}}\left|{\rho}_k\right|+{\sum}_{\mathrm{j}}\left|{\theta}_j\right|\right)+\left(1-\alpha \right)\left({\sum}_{\mathrm{k}}{\rho}_k^2+{\sum}_j{\theta}_j^2\right) \). Lasso or EN can then select the miRNAs and/or mRNAs of interest. However, these methods cannot use group information. Thus, ordinarily penalized LR methods cannot adequately account for the biological structure of miRNA-mRNA.
Secondly, we considered LR with a group lasso penalty (GL) [22], which has the benefit of using group information among the miRNAs and mRNAs of interest. In our analysis, a group can be defined as a set of one miRNA and its corresponding inhibited target mRNAs. GL uses the same LR in (8) with a different penalty function \( P\left(\theta, \rho \right)={\sum}_{j=1}^J\sqrt{\theta_j^2+{\sum}_{k=1}^{G_j}\left|{\rho}_k\right|} \). Via this penalty function, miRNA integration set can be selected together. However, the GL approach does not easily provide p-values for each set of independent variables.
To fit the penalized LR models, we first performed 3-fold cross-validation to find the optimal tuning parameter, δ. after which we fitted the models with all the data sets.
Simulation study
To compare HisCoM-mimi to the other three methods, we performed simulation studies and computed type I errors and power, simulating data from the same miRNA and mRNA data structure in our pancreatic cancer dataset. That is, we selected miRNA and mRNA data from the pancreatic cancer dataset, and then generated phenotype data iteratively from the LR model. We then considered two simulation scenarios. Scenario 1 assumed that a true causal integration set contains two mRNAs, with the same effect size. Scenario 2 assumed that a true causal integration set contains five mRNAs, with the same effect size. For each scenario, we randomly selected one causal miRNA-mRNA subnetwork, and then randomly selected another 9 miRNA-mRNA subnetworks, for which the number of inhibited mRNAs was less than 10. The selected miRNA-mRNA subnetworks for Scenario 1 are summarized in Table 1 and for Scenario 2 are in Table 2.
Table 1 List of used miRNAs and mRNAs for simulation Scenario 1
For Scenario 1, we used miR-217 as a true causal miRNA. To generate phenotypes, we considered the following LR model.
$$ logit\left(\pi \right)={\beta}_{miRNA}{z}_1+{\beta}_1{x}_1+{\beta}_2{x}_2, $$
where π is the probability of observing a disease (Y = 1), z1 represents the true causal miRNA expression, and x1 and x2 represent two causal mRNA expression values. For type I error evaluation, we assumed β miRNA = β1 = β2 = 0. For power comparison, we generated simulation data sets under the assumption that β miRNA = β1 = 0.2, 0.25, 0.3, 0.35. For the given 114 (97 PDAC and 17 normal tissues) values of (z1, x1, x2), from our pancreatic cancer dataset, we simulated 1000 datasets.
For Scenario 2, we assumed that a true causal integration set contains five mRNAs, with the same effect size. In our dataset, miR-381 was the only miRNA having five inhibited target mRNAs. To generate phenotypes, we considered the following LR model:
$$ logit\left(\pi \right)={\beta}_{miRNA}{z}_1+{\beta}_1{x}_1+{\beta}_2{x}_2+{\beta}_3{x}_3+{\beta}_4{x}_4+{\beta}_5{x}_5, $$
where x1, …, x5 represent five causal mRNA expression values. As in Scenario 1, we assumed β miRNA = β1 = β2 = β3 = β4 = β5 = 0, for type I error evaluation, and β miRNA = β1 = β2 = β3 = β4 = β5 = 0.2, 0.25, 0.3, 0.35, for power comparison. For the given 114 values of (z1, x1, x2, x3, x4, x5) from the pancreatic cancer dataset, 1000 simulation datasets were generated. We used the significance level α = 0.05 for HisCoM-mimi, as an false positive rate (FPR) criterion. For lasso, EN, and group-lasso, we selected a threshold T which provides a comparable FPR to the type I error 0.05. T was determined by calculating the FPR for simulation settings such that a miRNA-mRNA subnetwork is selected when β miRNA ≠ 0 and \( K\left(={\sum}_{\mathrm{l}=1}^{\mathrm{L}}I\left({\beta}_l\ne 0\right)\right) \) exceeded the threshold T. Here, L is the number of inhibited mRNAs for true causal miRNA for each scenario: L = 2 for Scenario 1, and L = 5 for Scenario 2.
Simulation results
For our analyses, we first determined the false positive error rates (FPRs) of each method, and chose the threshold values of T to make each penalized method provide (hold) FPRs close to 0.05. In Scenario 1, the type I error rate of HisCoM-mimi was 0.048 when α = 0.05. The FPRs of lasso were 0.054, when T was 1, and that of EN was 0.064, when T was 1. Since type I error rates of lasso and EN were nearly 0.05 when T = 1, we set T = 1 to evaluate power of those two methods. The FPR of GL, when choosing a causal miRNA integration set, 0.064.
For Scenario 2, Table 3 shows the FPRs for lasso and EN, when varying the threshold T. For this result, we found that the type I error of lasso and EN were similar to 0.05, when T = 1 and 2, respectively. The type I error rate of HisCoM-mimi was 0.054. On the other hand, GL did not select a causal miRNA integration set at all, such that the type I error rate was 0. Secondly, we compared the powers of each method for Scenarios 1 and 2. Figure 3 shows bar plots of powers for scenario 1, where the x-axis shows the effect sizes (i.e., beta coefficients), and the y-axis shows the power. HisCoM-mimi showed the highest power, while EN was second, Lasso was third, and GL was last. The same tendency is shown in Fig. 4, for Scenario 2. Figure 5 shows that the differences of power between HisCoM-mimi and the others were much larger than those of Scenario 1. Consequently, GL could not find any significant miRNA-mRNA integration sets under Scenario 1, due to its GL's penalty being too strict for many mRNAs, whose beta values were small.
Table 3 False positive rate when varying the number of selected mRNAs for lasso and EN
Power comparison for scenario 1
Venn Diagram for number of detected miRNAs for each method
Constructing miRNA-mRNA subnetworks
To use human mRNA and miRNA probes, we first filtered out non-annotated mRNA probes and non-human miRNA probes. After filtering, there were 22,077 mRNA probes and 3391 miRNA probes. To construct miRNA-mRNA subnetworks, we checked predicted target mRNAs, for each miRNA, from TargetScan 7.1 (targetscan.org) [19, 23]. Among predicted targets, we only selected mRNAs having significant Pearson correlation coefficients with a specific miRNA. After filtering, there were 55 miRNAs, and 2411 edges connected with mRNAs.
Integration analysis for the PDAC data
Table 4 shows the top significant weights of miRNA-mRNA integrations derived from HisCoM-mimi. To perform multiple comparison, we used false discovery rate (FDR) q-values summarized in the 7th column [24]. We could only find 12 miRNAs having q-values below 0.05. Tables 5 and 6 show the lists of the selected markers by lasso and EN, respectively. Since lasso and EN select markers without any group information, they selected miRNA and mRNA markers independently. There were no miRNAs selected by lasso or EN directly, with lasso yielding only two significant mRNAs, both related to miR-326. Other mRNAs were independently selected from different miRNAs. Consequently, there were only 12 markers selected by lasso. For EN, 58 mRNAs were selected. Similar to the lasso result, there were no selected miRNAs, although four miRNAs (miR-206, miR-3064, miR-222, and miR-326) connected to more than three mRNAs. Figure 5 shows a Venn diagram of the number of miRNAs selected by each method. Each number represents the total number of detected miRNAs and one in the parenthesis does the number of detected miRNAs whose relationship with pancreatic cancer were reported. HisCoM-mimi selected larger number of unique miRNAs and the majority of them were already were reported.
Table 4 Significant miRNAs produced by HisCoM-mimi
Table 5 Selected markers by lasso. Twelve markers (12 mRNAs) were selected. No miRNAs were selected
Table 6 Markers selected by EN
For the lasso group only one miRNA (miR-32) and whose related two mRNA (COL1A2, and BGN) were selected. Although miR-32 is not reported as pancreatic cancer marker, there were some reports that miR-32 is related with other cancers [25, 26].
Table 7 summarizes miRNAs detected by HisCoM-mimi, lasso, EN, or GL. Previously, miR-93, miR-219, miR-141, miR-222, miR-203, miR-132, miR-96, and miR-206 were reported to be pancreatic cancer-related markers [27,28,29,30,31,32,33,34,35]. Although other miRNAs detected by HisCoM-mimi, lasso, EN, or GL have not been reported for pancreatic cancer relation, miR-532, miR-590, miR-133b, miR-326, miR-708, miR-3064, and miR-32 were reported to associate with other cancer types [25, 36,37,38,39,40,41,42].
Table 7 Cancer related miRNAs detected by methods
Table 8 shows the cross-validation (CV) results for comparing prediction performance for marker-sets selected by HisCoM-mimi, Lasso, EN, and Group Lasso. The first column indicates methods used to construct prediction model and the second column does the method to select marker sets. The third column shows the area under the Receiver Operating Characteristic curve (AUC) results performed by leave-one-out cross validation (LOOCV). This setting is from the previous study of Kwon et al. [23]. The fourth column indicates the average AUC values performed by four-fold CV with a hundred iterations. Here, we used four-fold and eight-fold CV to balance the number of samples in CV datasets. The fifth column indicates the average AUC values performed by eight-fold CV with a hundred iterations. For all selected marker-sets, all prediction models built by HisCoM-mimi showed the best performances yielding AUC values higher than 0.9 except the marker-set selected by Group lasso in which the number of markers is less than five and one path coefficient exists.
Table 8 Evaluation of Prediction performance for marker set selected by HisCoM-mimi, Lasso, EN, or Group Lasso in PDAC samples
Discussion and conclusion
In this paper, we proposed and developed a novel method, hierarchical structured component analysis of microRNA-mRNA integration ("HisCoM-mimi"), to construct a component model to identifying significantly integrated miRNA-target-mRNA cognate pairs. Since HisCoM-mimi could use subgroup information, it yelded more results, as related to phenotypes (e.g. cancer, metabolic syndrome, and etc.), than those of other existing methods that lack network information.
In simulation studies, we compared the performances of HisCoM-mimi, lasso, EN, and GL. From that comparison, HisCoM-mimi showed better performance than the other three methods. Controlling type I error, by HisCoM-mimi, was easier for controlling FPRs than other methods, because HisCoM-mimi uses permutation based p-values. In particular, HisCoM-mimi could identify miRNA-mRNA integration sets in a much more flexible way, due to better use of a standard multiple testing framework, as compared to the other methods. In real data analysis, HisCoM-mimi succesfully identified more miRNA-mRNA integration sets for pancreatic ductal adenocarcinoma (PDAC) diagnosis, compared to the other methods. Among 12 miRNAs, whose q-values were below 0.05 by HisCoM-mimi, 7 miRNAs were previously reported to associate with a panreatic cancer [27,28,29,30,31,32,33,34,35]. EN found two miRNAs (miR-222, and miR-206) [30, 34]. Among two miRNAs selected by lasso, only miR-222 was reported to associate with pancreatic cancer.
Although HisCoM-mimi worked well for the PDAC data sets, further biological verification of those results are needed. In future studies, we will perform additional simulation analyses to evaluate the performance of HisCoM-mimi, under numerous conditions. Furthermore, HisCoM-mimi can be extended in many ways, for other types of phenotypes, such as time to event. Second, it can be easily applied to other cancer studies to identify miRNA-mRNA integration sets for early diagnosis and prognosis. Third, it can be extended to combine other types of omics data such as genomics, epignomics, and proteomics data. It is now established that dysregulated miRNAs play substantial roles in a myriad of diseases [43]. We firmly believe that these methods for miRNA identification and their target transcripts could yield effective biomarkers and therapeutic targets, in addition to providing better understanding of disease mechanisms and etiology.
AUC:
Area under the receiver operating characteristic curve
Elastic-net
FPR:
False positive rate
GL:
Group lasso
GSCA:
Generalized structured component analysis
HisCoM-mimi:
Hierarchical structured component analysis of microRNA-mRNA integration
IRWLS:
Iterative reweighted least squares
JJ:
Jin-Young Jang
LR:
PDAC:
Pancreatic ductal adenocarcinoma
PHARAOH:
Pathway-based approach using hierarchical components of collapsed rare variants
Sungkyoung Choi
Sungyoung Lee
SNUH:
Seoul National University Hospital
TP:
Taesung Park
YK:
Yongkang Kim
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This research was supported by a grant of the Korea Health Technology R&D Project through the Korea Health Industry Development Institute (KHIDI), funded by the Ministry of Health & Welfare, Republic of Korea (grant number: HI16C2037010016) and Bio-Synergy Research Project of the Ministry of Science, ICT and Future Planning through the National Research Foundation (grant number: 2013M3A9C4078158). Publication of this article was sponsored by the Bio-Synergy Research Project (grant number: 2013M3A9C4078158).
An implementation of HisCoM-mimi, and normalized intensity microarray data can be downloaded from the website (http://statgen.snu.ac.kr/software/hiscom-mimi).
This article has been published as part of BMC Bioinformatics Volume 19 Supplement 4, 2018: Selected articles from the 16th Asia Pacific Bioinformatics Conference (APBC 2018): bioinformatics. The full contents of the supplement are available online at https://bmcbioinformatics.biomedcentral.com/articles/supplements/volume-19-supplement-4.
Department of Statistics, Seoul National University, Seoul, Korea
Yongkang Kim & Taesung Park
Interdisciplinary program in Bioinformatics, Seoul National University, Seoul, Korea
Sungyoung Lee, Sungkyoung Choi & Taesung Park
Department of Surgery and Cancer Research Institute, Seoul National University College of Medicine, Seoul, Korea
YK performed all analyses and developed the software implementation. YK and TP wrote the manuscript and developed the methodology. SL developed the software implementation. SC helped the analysis. JJ provided clinical interpretation of analysis results. All of the authors have read and approved of the final manuscript.
Correspondence to Taesung Park.
Kim, Y., Lee, S., Choi, S. et al. Hierarchical structural component modeling of microRNA-mRNA integration analysis. BMC Bioinformatics 19, 75 (2018). https://doi.org/10.1186/s12859-018-2070-0
Integration analysis
Generalized Structured Component Analysis (GSCA)
Hierarchical structured component analysis of miRNA-mRNA integration (HisCoM-mimi) | CommonCrawl |
Torsion tensor
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves (or rather the rotation of the Frenet–Serret frame about the tangent vector). In the geometry of surfaces, the geodesic torsion describes how a surface twists about a curve on the surface. The companion notion of curvature measures how moving frames "roll" along a curve "without twisting".
More generally, on a differentiable manifold equipped with an affine connection (that is, a connection in the tangent bundle), torsion and curvature form the two fundamental invariants of the connection. In this context, torsion gives an intrinsic characterization of how tangent spaces twist about a curve when they are parallel transported; whereas curvature describes how the tangent spaces roll along the curve. Torsion may be described concretely as a tensor, or as a vector-valued 2-form on the manifold. If ∇ is an affine connection on a differential manifold, then the torsion tensor is defined, in terms of vector fields X and Y, by
$T(X,Y)=\nabla _{X}Y-\nabla _{Y}X-[X,Y]$
where [X,Y] is the Lie bracket of vector fields.
Torsion is particularly useful in the study of the geometry of geodesics. Given a system of parametrized geodesics, one can specify a class of affine connections having those geodesics, but differing by their torsions. There is a unique connection which absorbs the torsion, generalizing the Levi-Civita connection to other, possibly non-metric situations (such as Finsler geometry). The difference between a connection with torsion, and a corresponding connection without torsion is a tensor, called the contorsion tensor. Absorption of torsion also plays a fundamental role in the study of G-structures and Cartan's equivalence method. Torsion is also useful in the study of unparametrized families of geodesics, via the associated projective connection. In relativity theory, such ideas have been implemented in the form of Einstein–Cartan theory.
The torsion tensor
Let M be a manifold with an affine connection on the tangent bundle (aka covariant derivative) ∇. The torsion tensor (sometimes called the Cartan (torsion) tensor) of ∇ is the vector-valued 2-form defined on vector fields X and Y by
$T(X,Y):=\nabla _{X}Y-\nabla _{Y}X-[X,Y]$
where [X, Y] is the Lie bracket of two vector fields. By the Leibniz rule, T(fX, Y) = T(X, fY) = fT(X, Y) for any smooth function f. So T is tensorial, despite being defined in terms of the connection which is a first order differential operator: it gives a 2-form on tangent vectors, while the covariant derivative is only defined for vector fields.
Components of the torsion tensor
The components of the torsion tensor $T^{c}{}_{ab}$ in terms of a local basis (e1, ..., en) of sections of the tangent bundle can be derived by setting X = ei, Y = ej and by introducing the commutator coefficients γkijek := [ei, ej]. The components of the torsion are then
$T^{k}{}_{ij}:=\Gamma ^{k}{}_{ij}-\Gamma ^{k}{}_{ji}-\gamma ^{k}{}_{ij},\quad i,j,k=1,2,\ldots ,n.$
Here ${\Gamma ^{k}}_{ij}$ are the connection coefficients defining the connection. If the basis is holonomic then the Lie brackets vanish, $\gamma ^{k}{}_{ij}=0$. So $T^{k}{}_{ij}=2\Gamma ^{k}{}_{[ij]}$. In particular (see below), while the geodesic equations determine the symmetric part of the connection, the torsion tensor determines the antisymmetric part.
The torsion form
The torsion form, an alternative characterization of torsion, applies to the frame bundle FM of the manifold M. This principal bundle is equipped with a connection form ω, a gl(n)-valued one-form which maps vertical vectors to the generators of the right action in gl(n) and equivariantly intertwines the right action of GL(n) on the tangent bundle of FM with the adjoint representation on gl(n). The frame bundle also carries a canonical one-form θ, with values in Rn, defined at a frame u ∈ FxM (regarded as a linear function u : Rn → TxM) by
$\theta (X)=u^{-1}(\pi _{*}(X))$
where π : FM → M is the projection mapping for the principal bundle and π∗ is its push-forward. The torsion form is then
$\Theta =d\theta +\omega \wedge \theta .$
Equivalently, Θ = Dθ, where D is the exterior covariant derivative determined by the connection.
The torsion form is a (horizontal) tensorial form with values in Rn, meaning that under the right action of g ∈ GL(n) it transforms equivariantly:
$R_{g}^{*}\Theta =g^{-1}\cdot \Theta $
where g acts on the right-hand side through its adjoint representation on Rn.
Torsion form in a frame
See also: connection form
The torsion form may be expressed in terms of a connection form on the base manifold M, written in a particular frame of the tangent bundle (e1, ..., en). The connection form expresses the exterior covariant derivative of these basic sections:
$D\mathbf {e} _{i}=\mathbf {e} _{j}{\omega ^{j}}_{i}.$
The solder form for the tangent bundle (relative to this frame) is the dual basis θi ∈ T∗M of the ei, so that θi(ej) = δij (the Kronecker delta). Then the torsion 2-form has components
$\Theta ^{k}=d\theta ^{k}+{\omega ^{k}}_{j}\wedge \theta ^{j}={T^{k}}_{ij}\theta ^{i}\wedge \theta ^{j}.$
In the rightmost expression,
${T^{k}}_{ij}=\theta ^{k}\left(\nabla _{\mathbf {e} _{i}}\mathbf {e} _{j}-\nabla _{\mathbf {e} _{j}}\mathbf {e} _{i}-\left[\mathbf {e} _{i},\mathbf {e} _{j}\right]\right)$
are the frame-components of the torsion tensor, as given in the previous definition.
It can be easily shown that Θi transforms tensorially in the sense that if a different frame
${\tilde {\mathbf {e} }}_{i}=\mathbf {e} _{j}{g^{j}}_{i}$
for some invertible matrix-valued function (gji), then
${\tilde {\Theta }}^{i}={\left(g^{-1}\right)^{i}}_{j}\Theta ^{j}.$
In other terms, Θ is a tensor of type (1, 2) (carrying one contravariant and two covariant indices).
Alternatively, the solder form can be characterized in a frame-independent fashion as the TM-valued one-form θ on M corresponding to the identity endomorphism of the tangent bundle under the duality isomorphism End(TM) ≈ TM ⊗ T∗M. Then the torsion 2-form is a section
$\Theta \in {\text{Hom}}\left( \bigwedge }^{2}{\rm {T}}M,{\rm {T}}M\right)$
given by
$\Theta =D\theta ,$
where D is the exterior covariant derivative. (See connection form for further details.)
Irreducible decomposition
The torsion tensor can be decomposed into two irreducible parts: a trace-free part and another part which contains the trace terms. Using the index notation, the trace of T is given by
$a_{i}=T^{k}{}_{ik},$
and the trace-free part is
$B^{i}{}_{jk}=T^{i}{}_{jk}+{\frac {1}{n-1}}\delta ^{i}{}_{j}a_{k}-{\frac {1}{n-1}}\delta ^{i}{}_{k}a_{j},$
where δij is the Kronecker delta.
Intrinsically, one has
$T\in \operatorname {Hom} \left( \bigwedge }^{2}{\rm {T}}M,{\rm {T}}M\right).$
The trace of T, tr T, is an element of T∗M defined as follows. For each vector fixed X ∈ TM, T defines an element T(X) of Hom(TM, TM) via
$T(X):Y\mapsto T(X\wedge Y).$
Then (tr T)(X) is defined as the trace of this endomorphism. That is,
$(\operatorname {tr} \,T)(X){\stackrel {\text{def}}{=}}\operatorname {tr} (T(X)).$
The trace-free part of T is then
$T_{0}=T-{\frac {1}{n-1}}\iota (\operatorname {tr} \,T),$
where ι denotes the interior product.
Curvature and the Bianchi identities
The curvature tensor of ∇ is a mapping TM × TM → End(TM) defined on vector fields X, Y, and Z by
$R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z.$
For vectors at a point, this definition is independent of how the vectors are extended to vector fields away from the point (thus it defines a tensor, much like the torsion).
The Bianchi identities relate the curvature and torsion as follows.[1] Let ${\mathfrak {S}}$ denote the cyclic sum over X, Y, and Z. For instance,
${\mathfrak {S}}\left(R\left(X,Y\right)Z\right):=R(X,Y)Z+R(Y,Z)X+R(Z,X)Y.$
Then the following identities hold
1. Bianchi's first identity:
${\mathfrak {S}}\left(R\left(X,Y\right)Z\right)={\mathfrak {S}}\left(T\left(T(X,Y),Z\right)+\left(\nabla _{X}T\right)\left(Y,Z\right)\right)$
2. Bianchi's second identity:
${\mathfrak {S}}\left(\left(\nabla _{X}R\right)\left(Y,Z\right)+R\left(T\left(X,Y\right),Z\right)\right)=0$
The curvature form and Bianchi identities
The curvature form is the gl(n)-valued 2-form
$\Omega =D\omega =d\omega +\omega \wedge \omega $
where, again, D denotes the exterior covariant derivative. In terms of the curvature form and torsion form, the corresponding Bianchi identities are[2]
1. $D\Theta =\Omega \wedge \theta $
2. $D\Omega =0.$
Moreover, one can recover the curvature and torsion tensors from the curvature and torsion forms as follows. At a point u of FxM, one has[3]
${\begin{aligned}R(X,Y)Z&=u\left(2\Omega \left(\pi ^{-1}(X),\pi ^{-1}(Y)\right)\right)\left(u^{-1}(Z)\right),\\T(X,Y)&=u\left(2\Theta \left(\pi ^{-1}(X),\pi ^{-1}(Y)\right)\right),\end{aligned}}$
where again u : Rn → TxM is the function specifying the frame in the fibre, and the choice of lift of the vectors via π−1 is irrelevant since the curvature and torsion forms are horizontal (they vanish on the ambiguous vertical vectors).
Characterizations and interpretations
Throughout this section, M is assumed to be a differentiable manifold, and ∇ a covariant derivative on the tangent bundle of M unless otherwise noted.
Twisting of reference frames
In the classical differential geometry of curves, the Frenet-Serret formulas describe how a particular moving frame (the Frenet-Serret frame) twists along a curve. In physical terms, the torsion corresponds to the angular momentum of an idealized top pointing along the tangent of the curve.
The case of a manifold with a (metric) connection admits an analogous interpretation. Suppose that an observer is moving along a geodesic for the connection. Such an observer is ordinarily thought of as inertial since they experience no acceleration. Suppose that in addition the observer carries with themselves a system of rigid straight measuring rods (a coordinate system). Each rod is a straight segment; a geodesic. Assume that each rod is parallel transported along the trajectory. The fact that these rods are physically carried along the trajectory means that they are Lie-dragged, or propagated so that the Lie derivative of each rod along the tangent vanishes. They may, however, experience torque (or torsional forces) analogous to the torque felt by the top in the Frenet-Serret frame. This force is measured by the torsion.
More precisely, suppose that the observer moves along a geodesic path γ(t) and carries a measuring rod along it. The rod sweeps out a surface as the observer travels along the path. There are natural coordinates (t, x) along this surface, where t is the parameter time taken by the observer, and x is the position along the measuring rod. The condition that the tangent of the rod should be parallel translated along the curve is
$\left.\nabla _{\frac {\partial }{\partial t}}{\frac {\partial }{\partial x}}\right|_{x=0}=0.$
Consequently, the torsion is given by
$\left.T\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial t}}\right)\right|_{x=0}=\left.\nabla _{\frac {\partial }{\partial x}}{\frac {\partial }{\partial t}}\right|_{x=0}.$
If this is not zero, then the marked points on the rod (the x = constant curves) will trace out helices instead of geodesics. They will tend to rotate around the observer. Note that for this argument it was not essential that $\gamma (t)$ is a geodesic. Any curve would work.
This interpretation of torsion plays a role in the theory of teleparallelism, also known as Einstein–Cartan theory, an alternative formulation of relativity theory.
The torsion of a filament
In materials science, and especially elasticity theory, ideas of torsion also play an important role. One problem models the growth of vines, focusing on the question of how vines manage to twist around objects.[4] The vine itself is modeled as a pair of elastic filaments twisted around one another. In its energy-minimizing state, the vine naturally grows in the shape of a helix. But the vine may also be stretched out to maximize its extent (or length). In this case, the torsion of the vine is related to the torsion of the pair of filaments (or equivalently the surface torsion of the ribbon connecting the filaments), and it reflects the difference between the length-maximizing (geodesic) configuration of the vine and its energy-minimizing configuration.
Torsion and vorticity
In fluid dynamics, torsion is naturally associated to vortex lines.
Geodesics and the absorption of torsion
Suppose that γ(t) is a curve on M. Then γ is an affinely parametrized geodesic provided that
$\nabla _{{\dot {\gamma }}(t)}{\dot {\gamma }}(t)=0$
for all time t in the domain of γ. (Here the dot denotes differentiation with respect to t, which associates with γ the tangent vector pointing along it.) Each geodesic is uniquely determined by its initial tangent vector at time t = 0, ${\dot {\gamma }}(0)$.
One application of the torsion of a connection involves the geodesic spray of the connection: roughly the family of all affinely parametrized geodesics. Torsion is the ambiguity of classifying connections in terms of their geodesic sprays:
• Two connections ∇ and ∇′ which have the same affinely parametrized geodesics (i.e., the same geodesic spray) differ only by torsion.[5]
More precisely, if X and Y are a pair of tangent vectors at p ∈ M, then let
$\Delta (X,Y)=\nabla _{X}{\tilde {Y}}-\nabla '_{X}{\tilde {Y}}$
be the difference of the two connections, calculated in terms of arbitrary extensions of X and Y away from p. By the Leibniz product rule, one sees that Δ does not actually depend on how X and Y{{′}} are extended (so it defines a tensor on M). Let S and A be the symmetric and alternating parts of Δ:
$S(X,Y)={\tfrac {1}{2}}\left(\Delta (X,Y)+\Delta (Y,X)\right)$
$A(X,Y)={\tfrac {1}{2}}\left(\Delta (X,Y)-\Delta (Y,X)\right)$
Then
• $A(X,Y)={\tfrac {1}{2}}\left(T(X,Y)-T'(X,Y)\right)$ is the difference of the torsion tensors.
• ∇ and ∇′ define the same families of affinely parametrized geodesics if and only if S(X, Y) = 0.
In other words, the symmetric part of the difference of two connections determines whether they have the same parametrized geodesics, whereas the skew part of the difference is determined by the relative torsions of the two connections. Another consequence is:
• Given any affine connection ∇, there is a unique torsion-free connection ∇′ with the same family of affinely parametrized geodesics. The difference between these two connections is in fact a tensor, the contorsion tensor.
This is a generalization of the fundamental theorem of Riemannian geometry to general affine (possibly non-metric) connections. Picking out the unique torsion-free connection subordinate to a family of parametrized geodesics is known as absorption of torsion, and it is one of the stages of Cartan's equivalence method.
See also
• Contorsion tensor
• Curtright field
• Curvature tensor
• Levi-Civita connection
• Torsion coefficient
• Torsion of curves
Notes
1. Kobayashi & Nomizu 1963, Volume 1, Proposition III.5.2.
2. Kobayashi & Nomizu 1963, Volume 1, III.2.
3. Kobayashi & Nomizu 1963, Volume 1, III.5.
4. Goriely et al. 2006.
5. See Spivak (1999) Volume II, Addendum 1 to Chapter 6. See also Bishop and Goldberg (1980), section 5.10.
References
• Bishop, R.L.; Goldberg, S.I. (1980), Tensor analysis on manifolds, Dover Publications
• Cartan, É. (1923), "Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie)", Annales Scientifiques de l'École Normale Supérieure, 40: 325–412, doi:10.24033/asens.751
• Cartan, É. (1924), "Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (Suite)", Annales Scientifiques de l'École Normale Supérieure, 41: 1–25, doi:10.24033/asens.753
• Elzanowski, M.; Epstein, M. (1985), "Geometric characterization of hyperelastic uniformity", Archive for Rational Mechanics and Analysis, 88 (4): 347–357, Bibcode:1985ArRMA..88..347E, doi:10.1007/BF00250871, S2CID 120127682
• Goriely, A.; Robertson-Tessi, M.; Tabor, M.; Vandiver, R. (2006), "Elastic growth models" (PDF), BIOMAT-2006, Springer-Verlag, archived from the original (PDF) on 2006-12-29
• Hehl, F.W.; von der Heyde, P.; Kerlick, G.D.; Nester, J.M. (1976), "General relativity with spin and torsion: Foundations and prospects", Rev. Mod. Phys., 48 (3): 393–416, Bibcode:1976RvMP...48..393H, doi:10.1103/revmodphys.48.393, 393.
• Kibble, T.W.B. (1961), "Lorentz invariance and the gravitational field", J. Math. Phys., 2 (2): 212–221, Bibcode:1961JMP.....2..212K, doi:10.1063/1.1703702, 212.
• Kobayashi, S.; Nomizu, K. (1963), Foundations of Differential Geometry, vol. 1 & 2 (New ed.), Wiley-Interscience (published 1996), ISBN 0-471-15733-3
• Poplawski, N.J. (2009), Spacetime and fields, arXiv:0911.0334, Bibcode:2009arXiv0911.0334P
• Schouten, J.A. (1954), Ricci Calculus, Springer-Verlag
• Schrödinger, E. (1950), Space-Time Structure, Cambridge University Press
• Sciama, D.W. (1964), "The physical structure of general relativity", Rev. Mod. Phys., 36 (1): 463, Bibcode:1964RvMP...36..463S, doi:10.1103/RevModPhys.36.463
• Spivak, M. (1999), A comprehensive introduction to differential geometry, Volume II, Houston, Texas: Publish or Perish, ISBN 0-914098-71-3
Various notions of curvature defined in differential geometry
Differential geometry
of curves
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• Affine curvature
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Differential geometry
of surfaces
• Principal curvatures
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Riemannian geometry
• Curvature of Riemannian manifolds
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Tensors
Glossary of tensor theory
Scope
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Notation
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Tensor
definitions
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Related
abstractions
• Affine connection
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Notable tensors
Mathematics
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Mathematicians
• Élie Cartan
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| Wikipedia |
I have a large number of data sets. Each data set has something 200K data points lying in a square times a circle. The square is solid $I\times I$. The circle $S^1$ is hollow (dim 1). By reasoning from the experimental setup that produces the data, I am convinced that this collection of points is a sample from mixed noise plus gaussian, and the gaussian is confined to a small region. (We can assume this at first, but more realistic reasoning would give a mixture of a small number of gaussians, with centres pretty close together. I'll treat it as a single gaussian for the moment.) The problem is that the noise is far from uniform. Moreover, the noise distribution definitely differs from one dataset to the next.
Given one of my datasets D with k points (k varies with D) and underlying noise distribution $Noise(D)$, I have a way of rapidly producing a sample of k points drawn from $Noise(D)$. In other words, the information that comes with D is sufficient to identify $Noise(D)$ in the sense that it is possible to generate a sample of k points from $Noise(D)$.
My guess is that there are something like 100 points explained by the Gaussian, though this could be optimistic.
I'm not a statistician. How could I now estimate my unknown gaussian? It seems that this should be possible, since I "know" $Noise(D)$. Could someone explain to me how I could do something like the EM algorithm in my current situation. Also, could someone please recommend an online explanation of how EM is usually carried out?
Browse other questions tagged mixed-model expectation-maximization or ask your own question.
Should I add noise to my truth data before before training my classifier?
How to calculate likelihood for a mixture model with missing data?
Need intuition - how do they simplify the Q function for gaussian mixture EM? | CommonCrawl |
\begin{document}
\title{Deep factorisation of the stable process II: \
potentials and applications}
\begin{abstract}
Here, we propose a different perspective of the {\it deep factorisation} in \cite{kyprianou2015deep} based on determining potentials. Indeed, we factorise the inverse of the MAP--exponent associated to a stable process via the Lamperti--Kiu transform. Here our factorisation is completely independent from the derivation in \cite{kyprianou2015deep}, moreover there is no clear way to invert the factors in \cite{kyprianou2015deep} to derive our results. Our method gives direct access to the potential densities of the ascending and descending ladder MAP of the Lamperti-stable MAP in closed form.
In the spirit of the interplay between the classical Wiener--Hopf factorisation and the fluctuation theory of the underlying L\'evy process, our analysis will produce a collection of new results for stable processes. We give an identity for the law of the point of closest reach to the origin for a stable process with index $\alpha\in(0,1)$ as well as an identity for the the law of the point of furthest reach before absorption at the origin for a stable process with index $\alpha\in(1,2)$. Moreover, we show how the deep factorisation allows us to compute explicitly the limiting distribution of stable processes multiplicatively reflected in such a way that it remains in the strip $[-1,1]$.
\noindent{\it Subject classification:} 60G18, 60G52, 60G51. \\
\noindent{\it Key words:} Stable processes, self-similar Markov processes, Wiener--Hopf factorisation, radial reflection.
\end{abstract}
\section{Introduction and main results}
Let $(X,\mathbb{P}_x)$ be a one-dimensional L\'evy process started at $x\in\mathbb{R}$. Suppose that, when it exists, we write $\psi$ for its Laplace exponent, that is to say, $\psi(z): = t^{-1}\log \mathbb{E}_0[e^{z X_t}]$ for all $z\in\mathbb{C}$ such that the right-hand side exists. An interesting aspect of the characteristic exponent of L\'evy processes is that they can be written as a product of the so called Wiener--Hopf factors, see for example \cite[Theorem 6.15]{MR3155252}. This means that there exists two Bernstein functions $\kappa$ and $\hat\kappa$ (see \cite{MR2978140} for a definition) such that, up to a multiplicative constant, \begin{equation}\label{eq:normal_WH}
-\psi(i\theta)=\kappa(-i\theta)\hat\kappa(i\theta), \qquad \theta\in\mathbb{R}. \end{equation} There are, of course, many functions $\kappa$ and $\hat\kappa$ which satisfy \eqref{eq:normal_WH}. However imposing the requirements that the functions $\kappa$ and $\hat\kappa$ must be Bernstein functions which analytically extend in $\theta$ to the upper and lower half plane of $\mathbb{C}$, respectively, means that the factorisation in \eqref{eq:normal_WH} is unique up to a multiplicative constant.
The problem of finding such factors has considerable interest since, probablistically speaking, the Bernstein functions $\kappa$ and $\hat\kappa$ are the Laplace exponents of the ascending and descending ladder height processes respectively. The ascending and descending ladder height processes, say $(h_t: t\geq 0)$ and $(\hat{h}_t: t\geq 0)$, are subordinators that correspond to a time change of $\overline{X}_t: = \sup_{s\leq t}X_s$, $t\geq 0$ and $-\underline{X}_t: = -\inf_{s\leq t}X_s$, $t\geq 0$, and therefore have the same range, respectively. Additional information comes from the exponents in that they also provide information about the potential measures associated to their respective ladder height processes. So for example, $U(dx): = \int_0^\infty\mathbb{P}(h_t\in dx)dt$, $x\geq 0$, has Laplace transform given by $1/\kappa$. A similar identity hold for $\widehat{U}$, the potential of $\hat{h}$.
These potential measures appear in a wide variety of fluctuation identities. Perhaps the most classical example concerns the stationary distribution of the process reflected in its maximum, $\overline{X}_t - X_t$, $t\geq 0$ in the case that $\lim_{t\to\infty}X_t = \infty$. In that case, we may take $\lim_{t\to\infty}\mathbb{P}_x(\overline{X}_t -X_t\in dx) = \kappa(0)\widehat{U}(dx)$; c.f. \cite{MR0217858}. The ladder height potential measures also feature heavily in first passage identities that describe the joint law of the triple $(X_{\tau^+_x}, X_{\tau^+_x-}, \overline{X}_{\tau^+_x-})$, where $\tau^+_x = \inf\{t>0: X_t >x\}$ and $x>0$; cf. \cite{DK}. Specifically, one has, for $u>0$, $v\geq y$ and $y\in[0,x]$, \[ \mathbb{P}(X_{\tau^+_x } -
x\in{{d}}u, x - X_{\tau^+_x -} \in {{d}}v , x -
\overline{X}_{\tau^+_x
-}\in{{d}}y)
= U( x-dy)\widehat{U}({{d}}v-y)\Pi({{d}}u + v), \] where $\Pi$ is the L\'{e}vy measure of $X$.
A third example we give here pertains to the much more subtle property of increase times. Specifically, an increase time is a (random) time $t>0$ at which there exists a (random) $\varepsilon>0$ such that $X_{t'}\leq X_t\leq X_{t''}$ for all $t'\in[t-\varepsilon, t]$ and $t''\in[t,t+\varepsilon]$. The existence of increase times occurs with probability 0 or 1. It is known that under mild conditions, increase times exist almost surely if and only if $\int_{0+}\widehat{U}(x)^{-1}U(dx)<0$. See, for example, \cite{Fourati} and references therein.
Within the class of L\'evy processes which experience jumps of both sign, historically there have been very few explicit cases of the Wiener--Hopf factorisation identified within the literature. More recently, however, many new cases have emerged hand-in-hand with new characterisations and distributional identities of path functionals of stable processes; see e.g. the summary in \cite[Section 6.5 and Chapter 13]{MR3155252}. A L\'evy process $(X,\mathbb{P}_x)$ is called a (strictly) $\alpha$-stable process for $\alpha\in(1,2]$ if for every $c>0$, $(cX_{c^{-\alpha}t}:t \geq 0)$ under $\mathbb{P}_x$ has the same law as $(X,\mathbb{P}_{cx})$. The case when $\alpha=2$ corresponds to Brownian motion, which we henceforth exclude from all subsequent discussion. It is known that the Laplace exponent can be parametrised so that \[
\psi(i\theta)=-|\theta|^\alpha (e^{i\pi\alpha(1/2-\rho)}\mathbbm{1}_{\{\theta\geq 0\}} + e^{-i\pi\alpha(1/2-\hat\rho)}\mathbbm{1}_{\{\theta<0\}}), \qquad\theta\in\mathbb{R}, \] where $\rho:=\mathbb{P}(X_1\geq 0)$ is the positivity parameter and $\hat\rho=1-\rho$. In that case, the two factors are easily identified as $ \kappa(\lambda)=\lambda^{\alpha\rho}$ and $\hat\kappa(\lambda)=\lambda^{\alpha\hat\rho}$ for $\lambda \geq 0$, with associated potentials possessing densities proportional to $x^{\alpha\rho-1}$ and $x^{\alpha\hat\rho - 1}$ respectively for $x\geq 0$. In part, this goes a long way to explaining why so many fluctuation identities are explicitly available for stable processes.
In this paper, our objective is to produce a new explicit characterisation of a second type of explicit Wiener--Hopf factorisation embedded in the $\alpha$-stable process, the so-called `deep factorisation' first considered in \cite{kyprianou2015deep}, through its representation as a real-valued self-similar Markov process. In the spirit of the interplay between the classical Wiener--Hopf factorisation and fluctuation theory of the underlying L\'evy process, our analysis will produce a collection of new results for stable processes which are based on identities for potentials derived from the deep factorisation. Before going into detail regarding our results concerning the deep factorisation, let us first present some of the new results we shall obtain {\it en route} for stable processes.
\subsection{Results on fluctuations of stable processes}
The first of our results on stable processes concerns the `point of closest reach' to the origin for stable processes with index $\alpha\in(0,1)$. Recall that for this index range, the stable process does not hit points and, moreover, $\liminf_{t\to\infty}|X_s| = \infty$. Hence, either on the positive or negative side of the origin, the path of the stable process has a minimal radial distance. Moreover, this distance is achieved at the unique time $\underline{m}$ such that $|X_t|\geq |X_{\underline{m}}|$ for all $t\geq 0$. Note, uniqueness follows thanks to regularity of $X$ for both $(0,\infty)$ and $(-\infty,0)$.
\begin{prop}[Point of closest reach]\label{prop:pointofclosestreach} Suppose that $\alpha \in (0,1)$, then for $x>0$ and $|z|\leq x$,
\[
\mathbb{P}_x(X_{\underline m}\in dz) = \frac{\Gamma(1-\alpha\rho) }{\Gamma(1-\alpha)\Gamma(\alpha\hat\rho)}\frac{x+z}{|2z|^\alpha}
\left(x-|z|\right)^{\alpha\hat\rho-1} \left(x+|z|\right)^{\alpha\rho -1}dz.
\] \end{prop}
In the case that $\alpha=1$, the stable process does not hit points and we have that $\limsup_{t\to\infty}|X_t| = \infty$ and $\liminf_{t\to\infty}|X_t| = 0$ and hence it is difficult to produce a result in the spirit of the previous theorem. However, when $\alpha\in (1,2)$, the stable process will hit all points almost surely, in particular $\tau^{\{0\}}: = \inf\{t>0: X_t = 0\}$ is $\mathbb{P}_x$-almost surely finite for all $x\in\mathbb{R}$. This allows us to talk about the `point of furthest' reach until absorption at the origin. To this end, we define $\overline{m}$ to be the unique time such that $|X_t|\leq |X_{\overline{m}}|$ for all $t\leq \tau^{\{0\}}$. Note again, that uniqueness is again guaranteed by regularity of the upper and lower half line for the stable process.
\begin{prop}[Point of furthest reach]\label{prop:pointoffurthestreach}
Suppose that $\alpha \in (1,2)$, then for each $x>0$ and $|z|>x$,
\begin{eqnarray*}
\mathbb{P}_x(X_{\overline m}\in dz) &=&\frac{\alpha-1}{2 |z|^\alpha} \Bigg(|x+z| (|z|-x)^{\alpha\rho-1}(|z|+x)^{\alpha\hat\rho-1}
\\
&&\hspace{2cm}\left.-(\alpha-1)x \int_1^{|z|/x}(t-1)^{\alpha\rho-1}(t+1)^{\alpha\hat\rho-1} dt\right).
\end{eqnarray*} \end{prop}
Finally we are also interested in {\it radially reflected stable processes}: \[
R_t = \frac{X_t}{M_t \vee1} \qquad t \geq 0, \]
where $M_t = \sup_{s \leq t}|X_s|$, $t\geq 0$. It is easy to verify, using the scaling and Markov properties, that for $t,s\geq 0$,
\begin{equation*} M_{t+s}=|X_{t}| \left(\frac{M_{t}}{|X_{t}|}\vee\widetilde{M}_{s|X_{t}|^{-\alpha}}\right),\qquad X_{t+s}=|X_{t}|\widetilde{X}_{s|X_{t}|^{-\alpha}}, \end{equation*} where $(\widetilde{M}, \widetilde{X})$ is such that, for all bounded measurable functions $f$, \[
\mathbb{E}[f(\widetilde{M}_s,\widetilde{X}_s)|\sigma(X_u: u\leq t)] = g({\rm sign}(X_t)), \] where $g(y) = \mathbb{E}_y[f(M_s,X_s)]$.
It follows that, whilst the process $R$ is not Markovian, the pair $(R,M)$ is a strong Markov process. In forthcoming work, in the spirit of \cite{MR2917773}, we shall demonstrate how an excursion theory can be developed for the pair $(R,M)$. In particular, one may build a local time process which is supported on the closure of the times $\{t: |R_t| = 1\}$. The times that are not in this supporting set form a countable union of disjoint intervals during which $R$ executes an excursion {\it into} the interval $(-1,1)$ (i.e. the excursion begins on the boundary $(-1,1)$ and runs until existing this interval). We go no further into the details of this excursion theory here. However, it is worthy of note that one should expect an ergodic limit of the process $R$ which is bounded in $[-1,1]$. The following result demonstrates this prediction in explicit detail.
\begin{figure}
\caption{$\rho=1/2$}
\caption{$\rho=9/10$}
\caption{The limiting distribution of the reflected process for two different values of $\rho$. Note that the concentration of mass at the values $1,-1$ is a consequence of the time that $X$ spends in close proximity to $M$. }
\label{fig:reflected_invariant}
\end{figure}
\begin{theorem}[Stationary distribution of the radially reflected process]\label{thm:reflecting}
Suppose that $\alpha \in (0,1)$. Let $x \in (-1,1)$, then under $\mathbb{P}_x$,
$R$ has a limiting distribution $\mu$, concentrated on $[-1,1],$ given by
\begin{align*}
\frac{d\mu(y)}{dy} = 2^{-\alpha}\frac{\Gamma(\alpha)}{\Gamma(\alpha\rho)\Gamma(\alpha\hat\rho)}
\left[
(1-y)^{\alpha\hat\rho -1}(1+y)^{\alpha\rho} +(1-y)^{\alpha\hat\rho}(1+y)^{\alpha\rho-1}
\right],\qquad y\in[-1,1] \end{align*} (See Figure \ref{fig:reflected_invariant} which has two examples of this density for different values of $\rho$.)
\end{theorem}
\subsection{Results on the deep factorisation of stable processes}
In order to present our results on the deep factorisation, we must first look the Lamperti--Kiu representation of real self-similar Markov processes, and in particular for the case of a stable process.
A Markov process $(X,\mathbb{P}_x)$ is called a real self-similar Markov processes (rssMp) of index $\alpha>0$ if, for every $c >0$, $(cX_{c^{-\alpha}t}:t \geq 0)$ has the same law as $X$. In particular, an $\alpha$-stable L\'evy process is an example of an rssMp, in which case $\alpha\in(0,2)$. Recall, every rssMp can be written in terms of what is referred to as a Markov additive process (MAP) $(\xi,J)$. The details of this can be found in Section \ref{sec:maps}. Essentially $(\xi,J)$ is a certain (possibly killed) Markov process taking values in $\mathbb{R}\times \{1,-1\}$ and is completely characterised by its so-called MAP--exponent $\bm F$ which, when it exists, is a function mapping $\mathbb C$ to $2\times 2$ complex valued matrices\footnote{Here and throughout the paper the matrix entries are arranged by
\[
A=\left(\begin{matrix}
A_{1,1} & A_{1,-1}\\
A_{-1,1} & A_{-1,-1}
\end{matrix}\right).
\]
}, which satisfies,
\[
(e^{\bm F(z) t})_{i,j}=\pmb{\rm E}_{0,i}[e^{z\xi(t)};J_t=j], \qquad i,j = \pm1, t\geq 0.
\]
\begin{figure}
\caption{$X$}
\caption{$|X|$}
\caption{$(\xi,J)$}
\caption{The graphical representation of the Lamperti--Kiu transformation. First figure is the process $X$. In the second figure we take the absolute value of $X$, colouring the positive paths blue and the negative paths red. Then we apply a $\log$ and a time-change to obtain $\xi$, the colours represent the state that $J$ is in.}
\label{fig:lamperti_kiu}
\end{figure}
Next we present the Lamperti--Kiu transfomation which relates a rssMp to a MAP, see also Figure~\ref{fig:lamperti_kiu}.
It follows directly from~\cite[Theorem 6]{MR3160562}
\begin{theorem}[Lamperti--Kiu transformation]
Suppose that $X=(X_t:t\geq 0)$ is a real--valued self--similar Markov process, killed when it hits the origin, then there exists a Markov additive process $(\xi,J)$ on $\mathbb{R}\times \{1,-1\}$ such that started from $X_0=x$,
\begin{equation}\label{eq:lamp_kiu}
X_t =\begin{cases}
|x| e^{\xi_{\varphi(|x|^{-\alpha}t)}} J_{\varphi(|x|^{-\alpha}t)} & \text{if }t <\int_0^\infty e^{\alpha\xi_u}\,{\rm d}u\\
\partial & \text{if }t \geq \int_0^\infty e^{\alpha\xi_u}\,{\rm d}u
\end{cases}
\end{equation}
where $\partial$ is a cemetery state and
\[
\varphi(t):=\inf\left\{s>0:\int_0^s e^{\alpha\xi_u}\,du >t\right\}
\]
with convention that $\inf \emptyset = \infty$ and $e^{\xi_\infty}=0$.
Conversely, every Markov additive process $(\xi,J)$ on $\mathbb{R}\times \{1,-1\}$ defines a real--valued self--similar Markov process, killed when hitting the origin, via \eqref{eq:lamp_kiu}.
\end{theorem}
We will denote the law of $(\xi,J)$ started from $(x,i)$ by $\pmb{\rm P}_{x,i}$.
Note that $\xi$ describes the radial part of $X$ and $J$ the sign, and thus a decrease in $\xi$ when $J=1$, for example, corresponds to an increase in $X$. See again Figure~\ref{fig:lamperti_kiu}.
In the case $X_t\geq 0$ for all $t \geq 0$, we have that $J_t=1$ for all $t \geq 0$. In this special case the transformation is known as the Lamperti transform and the process $\xi$ is a L\'evy process. The Lamperti transformation, introduced in \cite{MR0307358}, has been studied intensively, see for example \cite[Chapter 13]{MR3155252} and references therein. In particular, the law and the Wiener--Hopf factorisation of $\xi$ is known in many cases, for example \cite[Section 13.4]{MR3155252} and \cite{MR2797981}.
Conversely, very little is known about the general case. In this paper, we shall consider the case when $X$ is an $\alpha$--stable process (until first hitting the origin in the case that $\alpha\in(1,2)$) and note that in that case $(\xi,J)$ is not necessarily a L\'evy process. In this case the MAP--exponent is known to be
\begin{equation}\label{eq:MAP_expo}
\bm F({z})=\left(\begin{array}{*2{>{\displaystyle}c}}
-\frac{\Gamma(\alpha-{z})\Gamma(1+{z})}{\Gamma(\alpha\hat\rho-{z})\Gamma(1-\alpha\hat\rho+{z})} & \frac{\Gamma(\alpha-{z})\Gamma(1+{z})}{\Gamma(\alpha\hat\rho)\Gamma(1-\alpha\hat\rho)}\\
&\\
\frac{\Gamma(\alpha-{z})\Gamma(1+{z})}{\Gamma(\alpha\rho)\Gamma(1-\alpha\rho)}& -\frac{\Gamma(\alpha-{z})\Gamma(1+{z})}{\Gamma(\alpha\rho-{z})\Gamma(1-\alpha\rho+{z})}
\end{array}\right),
\end{equation}
for Re$({z}) \in (-1,\alpha)$, and the associated process is called the {\it Lamperti-stable} MAP by analogy to \cite{CC,MR2797981}.
Notice that the rows of $\bm F(0)$ sum to zero which means the MAP is not killed.
Similar to the case of L\'evy processes, we can define $\bm\kappa$ and $\hat{\bm\kappa}$ as the Laplace exponent of the ascending and descending ladder height process for $(\xi,J)$, see Section \ref{sec:maps} for more details.
The analogue of Wiener--Hopf factorisation for MAPs states that, up to pre-multiplying $\bm \kappa$ or $\hat{\bm \kappa}$ (and hence equivalently up to pre-multiplying $\bm F$) by a strictly positive diagonal matrix, we have that
\begin{equation}\label{eq:factorisation}
- \bm F(i\lambda) = {\bm\Delta}^{-1}_{\pi}\hat{\bm \kappa}(i\lambda)^T{\bm\Delta}_{\pi}\bm\kappa(-i\lambda),
\end{equation}
where
\begin{equation}\label{eq:delta_matrix}
{\bm\Delta}_\pi:=\left(\begin{array}{cc}
\sin(\pi\alpha\rho) & 0 \\
0 & \sin(\pi\alpha\hat\rho).
\end{array}\right).
\end{equation}
Note, at later stages, during computations, the reader is reminded that, for example, the term $\sin(\pi\alpha\rho)$ is preferentially represented via the reflection identity $\pi/[\Gamma(\alpha\rho)\Gamma(1-\alpha\rho)]$.
The factorisation in \eqref{eq:factorisation} can be found in \cite{MR650610} and \cite{MR3174223} for example. The exposition in the prequel to this paper, \cite{kyprianou2015deep}, explains in more detail how premultiplication of any of the terms in \eqref{eq:factorisation} by a strictly positive diagonal matrix corresponds to a linear time change in the associated MAP which is modulation dependent. Although this may present some concern to the reader, we note that this is of no consequence to our computations which focus purely on spatial events and therefore the range of the MAPS under question, as opposed to the time-scale on which they are run. Probabilistically speaking, this mirrors a similar situation with the Wiener--Hopf factorisation for L\'evy processes, \eqref{eq:normal_WH}, which can only be determined up to a constant (which corresponds to a linear scaling in time). Taking this into account, our main result identifies the inverse factors $\bm\kappa^{-1}$ and $\hat{\bm\kappa}^{-1}$ explicitly up to post-multiplication by a strictly positive diagonal matrix.
\begin{theorem}\label{thm:factorisation}
Suppose that $X$ is an $\alpha$-stable process then we have that, up to post-multiplication by a strictly positive diagonal matrix, the factors $\bm\kappa^{-1}$ and $\hat{\bm\kappa}^{-1}$ are given as follows. For $a,b,c \in \mathbb{R}$, define
\begin{equation}\label{eq:Psi}
\Psi(a,b,c):=\int_0^1 u^{a}(1-u)^{b}(1+u)^{c} du.
\end{equation}
\underline{For $\alpha\in(0,1)$:}
\begin{align*}
&\bm\kappa^{-1}(\lambda)=
\left(\begin{matrix}
\Psi(\lambda-1,\alpha\rho-1,\alpha\hat\rho) &
\Psi(\lambda-1,\alpha\rho,\alpha\hat\rho-1) \\
&\\
\Psi(\lambda-1,\alpha\hat\rho,\alpha\rho-1) &
\Psi(\lambda-1,\alpha\hat\rho-1,\alpha\rho)
\end{matrix}\right)
\end{align*}
and
\begin{align*}
& \left(\begin{array}{cc} \frac{\Gamma(\alpha\hat\rho)}{\Gamma(1-\alpha\rho) }&0\\
0& \frac{\Gamma(\alpha\rho)}{\Gamma(1-\alpha\hat\rho) }
\end{array}\right)
\hat{\bm\kappa}^{-1}(\lambda) =
\left(\begin{matrix}
\Psi(\lambda-\alpha,\alpha\hat\rho-1,\alpha\rho) & \Psi(\lambda-\alpha,\alpha\hat\rho,\alpha\rho-1) \\
&\\
\Psi(\lambda-\alpha,\alpha\rho,\alpha\hat\rho-1) &\Psi(\lambda-\alpha,\alpha\rho-1,\alpha\hat\rho)
\end{matrix}\right). \end{align*} \underline{For $\alpha=1$:}
\begin{align*}
\bm\kappa^{-1}(\lambda)=\hat{\bm \kappa}^{-1}(\lambda)
&=\left(\begin{matrix}
\Psi(\lambda-1,-1/2,1/2) & \Psi(\lambda-1,1/2,-1/2)\\
&\\
\Psi(\lambda-1,1/2,-1/2)& \Psi(\lambda-1,-1/2,1/2)
\end{matrix}\right).
\end{align*}
\underline{For $\alpha \in (1,2)$:}
\begin{align*}
\bm\kappa^{-1}(\lambda)
&=
\left(\begin{matrix}
\Psi(\lambda-1,\alpha\rho-1,\alpha\hat\rho) & \Psi(\lambda-1,\alpha\rho,\alpha\hat\rho-1)\\
&\\
\Psi(\lambda-1,\alpha\hat\rho,\alpha\rho-1)& \Psi(\lambda-1,\alpha\hat\rho-1,\alpha\rho)
\end{matrix}\right)\\
&\hspace{1cm}- \frac{(\alpha-1)}{(\lambda+\alpha-1)}\left(\begin{matrix}
\Psi(\lambda-1,\alpha\rho-1,\alpha\hat\rho-1) & \Psi(\lambda-1,\alpha\rho-1,\alpha\hat\rho-1)\\
&\\
\Psi(\lambda-1,\alpha\hat\rho-1,\alpha\rho-1) & \Psi(\lambda-1,\alpha\hat\rho-1,\alpha\rho-1)
\end{matrix}\right)
\end{align*}
and
\begin{align*}
\left(\begin{matrix}
\frac{\Gamma(\alpha\hat\rho)}{\Gamma(1-\alpha\rho) }& 0 \\
0 & \frac{\Gamma(\alpha\rho)}{\Gamma(1-\alpha\hat\rho) }
\end{matrix}
\right)
\hat{\bm\kappa}^{-1}(\lambda)&= \left(\begin{matrix}
\Psi(\lambda-\alpha,\alpha\hat\rho-1,\alpha\rho) & \Psi(\lambda-\alpha,\alpha\hat\rho,\alpha\rho-1)\\
&\\
\Psi(\lambda-\alpha,\alpha\rho,\alpha\hat\rho-1)& \Psi(\lambda-\alpha,\alpha\rho-1,\alpha\hat\rho)
\end{matrix}\right) \\
&\hspace{0.3cm}- \frac{(\alpha-1)}{(\lambda+\alpha-1)}\left(\begin{matrix}
\Psi(\lambda-\alpha,\alpha\hat\rho-1,\alpha\rho-1)
&
\Psi(\lambda-\alpha,\alpha\hat\rho-1,\alpha\rho-1)\\
&\\
\Psi(\lambda-\alpha,\alpha\rho-1,\alpha\hat\rho-1)
&
\Psi(\lambda-\alpha,\alpha\rho-1,\alpha\hat\rho-1)
\end{matrix}\right).
\end{align*}
\end{theorem}
Note that the function $\Psi$ can also be written in terms of hypergeometric functions, specifically
\[
\Psi(a,b,c) = \frac{\Gamma(a+1)\Gamma(b+1)}{\Gamma(a+b+2)} \,\setlength\arraycolsep{1pt}
{}_2 F_1\left(-c , a+1 , a+b+2
;-1\right),
\]
where ${}_2 F_1$ is the usual Hypergeometric function.
There are many known identities for such hypergeometric functions, see for example \cite{MR1688958}. The appearance of hypergeometric functions is closely tied in with the fact that we are working with stable processes, for example \cite[Theorem 1]{MR3010227} describes the laws of various conditioned stable processes in terms of what are called hypergeometric L\'evy processes.
The factorisation of $\bm F$ first appeared in Kyprianou \cite{kyprianou2015deep}. Here our factorisation of $\bm F^{-1}$ is completely independent from the derivation in \cite{kyprianou2015deep}, moreover there is no clear way to invert the factors in \cite{kyprianou2015deep} to derive our results. The Bernstein functions that appear in \cite{kyprianou2015deep} have not, to our knowledge, appeared in the literature and are in fact considerably harder to do computations with, whereas the factorisation that appears here is given in terms of well studied hypergeometric functions. Our proof here is much simpler and shorter as it only relies on entrance and exit probabilities of $X$.
Expressing the factorisation in terms of the inverse matrices has a considerable advantage in that the potential measures of the MAP are easily identified. To do this, we let $\bm u$ denote the unique matrix valued function so that, for $\lambda\geq 0$,
\[
\int_0^\infty e^{-\lambda x}\bm u_{i,j}(x) dx = \bm\kappa^{-1}_{i,j}(\lambda) \qquad \text{ for each }i,j=\pm 1.
\]
Similarly, let $\hat{\bm u}$ denote the unique matrix valued function so that, for $\lambda\geq 0$,
\[
\int_0^\infty e^{-\lambda x}\hat{\bm u}_{i,j}(x) dx = \hat{\bm\kappa}_{i,j}^{-1}(\lambda) \qquad \text{ for each }i,j=\pm 1.
\]
The following corollary follows from Theorem \ref{thm:factorisation} by using the substitution $x=-\log u$ in the definition of $\Psi$.
\begin{cor}\label{cor:potentials}
The potential densities are given by the following.
\noindent\underline{For $\alpha\in(0,1)$:}
\begin{align*}
&\bm u(x)=
\left(\begin{matrix}
(1-e^{-x})^{\alpha\rho-1}(1+e^{-x})^{\alpha\hat\rho} &
(1-e^{-x})^{\alpha\rho}(1+e^{-x})^{\alpha\hat\rho-1} \\
&\\
(1-e^{-x})^{\alpha\hat\rho}(1+e^{-x})^{\alpha\rho-1} &
(1-e^{-x})^{\alpha\hat\rho-1}(1+e^{-x})^{\alpha\rho}
\end{matrix}\right)
\end{align*}
and
\begin{align*}
& \left(\begin{array}{cc} \frac{\Gamma(\alpha\hat\rho)}{\Gamma(1-\alpha\rho) }&0\\
0& \frac{\Gamma(\alpha\rho)}{\Gamma(1-\alpha\hat\rho) }
\end{array}\right)\hat{\bm u}(x)=
\left(\begin{matrix}
(e^x-1)^{\alpha\hat\rho-1}(e^x+1)^{\alpha\rho}&(e^x-1)^{\alpha\hat\rho}(e^x+1)^{\alpha\rho-1} \\
&\\
(e^x-1)^{\alpha\rho}(e^x+1)^{\alpha\hat\rho-1}&(e^x-1)^{\alpha\rho-1}(e^x+1)^{\alpha\hat\rho}
\end{matrix}\right).
\end{align*}
\underline{For $\alpha=1$:}
\begin{align*}
\bm u(x)=\hat{\bm u} (x)
&=\left(\begin{matrix}
(1-e^{-x})^{-1/2}(1+e^{-x})^{1/2} & (1-e^{-x})^{1/2}(1+e^{-x})^{-1/2}\\
&\\
(1-e^{-x})^{1/2}(1+e^{-x})^{-1/2}& (1-e^{-x})^{-1/2}(1+e^{-x})^{1/2}
\end{matrix}\right). \end{align*} \underline{For $\alpha\in(1,2)$:}
\begin{align*}
\bm u (x)
&=
\left(\begin{matrix}
(1-e^{-x})^{\alpha\rho-1}(1+e^{-x})^{\alpha\hat\rho} & (1-e^{-x})^{\alpha\rho}(1+e^{-x})^{\alpha\hat\rho-1}\\
&\\
(1-e^{-x})^{\alpha\hat\rho}(1+e^{-x})^{\alpha\rho-1}& (1-e^{-x})^{\alpha\hat\rho-1}(1+e^{-x})^{\alpha\rho}
\end{matrix}\right) \\
&- (\alpha-1) e^{-(\alpha-1)x}\int_0^{e^x}\left(\begin{matrix}
(t-1)^{\alpha\rho-1}(t+1)^{\alpha\hat\rho-1} & (t-1)^{\alpha\rho-1}(t+1)^{\alpha\hat\rho-1}\\
&\\
(t-1)^{\alpha\hat\rho-1}(t+1)^{\alpha\rho-1} & (t-1)^{\alpha\hat\rho-1}(t+1)^{\alpha\rho-1}
\end{matrix}\right)\,dt \end{align*} and \begin{align*}
\left(\begin{matrix}
\frac{\Gamma(\alpha\hat\rho)}{\Gamma(1-\alpha\rho) }& 0 \\
0 & \frac{\Gamma(\alpha\rho)} {\Gamma(1-\alpha\hat\rho) }
\end{matrix}\right)
\hat{\bm u}(x)&=\left(\begin{matrix}
(e^x-1)^{\alpha\hat\rho-1}(e^x+1)^{\alpha\rho}& (e^x-1)^{\alpha\rho}(e^x+1)^{\alpha\rho-1}\\
&\\
(e^x-1)^{\alpha\rho}(e^x+1)^{\alpha\hat\rho-1}& (e^x-1)^{\alpha\rho-1}(e^x+1)^{\alpha\hat\rho}
\end{matrix}\right) \\
&\hspace{1cm}- (\alpha-1)\int_0^{e^x}\left(\begin{matrix}
(t-1)^{\alpha\hat\rho-1}(t+1)^{\alpha\rho-1} &(t-1)^{\alpha\hat\rho-1}(t+1)^{\alpha\rho-1}\\
&\\
(t-1)^{\alpha\rho-1}(t+1)^{\alpha\hat\rho-1} & (t-1)^{\alpha\rho-1}(t+1)^{\alpha\hat\rho-1}
\end{matrix}\right)\,dt,
\end{align*}
where the integral of a matrix is done component-wise.
\end{cor}
Before concluding this section, we also remark that the explicit nature of the factorisation of the Lamperti-stable MAP suggests that other factorisations of MAPs in a larger class of such processes may also exist. Indeed, following the introduction of the Lamperti-stable L\'evy process in \cite{CC}, for which an explicit Wiener--Hopf factorisation are available, it was quickly discovered that many other explicit Wiener--Hopf factorisations could be found by studying related positive self-similar path functionals of stable processes. In part, this stimulated the definition of the class of hypergeometric L\'evy processes for which the Wiener--Hopf factorisation is explicit; see \cite{MR3010227, KPW, KKPvS}. One might therefore also expect a general class of MAPs to exist, analogous to the class of hypergeometric L\'evy processes, for which a matrix factorisation such as the one presented above, is explicitly available. Should that be the case, then the analogue of fluctuation theory for L\'evy processes awaits further development in concrete form, but now for `hypergeometric' MAPs. See for example some of the general fluctuation theory for MAPs that appears in the Appendix of \cite{dereich2015real}.
\subsection{Outline of the paper}
The rest of the paper is structured as follows. In Section \ref{sec:maps} we introduce some technical background material for the paper. Specifically, we introduce Markov additive processes (MAPs) and ladder height processes for MAPs in more detail.
We then prove the results of the paper by separating into three cases.
In Section \ref{sec:proof_a_less} we show Theorem \ref{thm:factorisation} for $\alpha \in (0,1)$, and Proposition \ref{prop:pointofclosestreach}. In Section \ref{sec:proof_a_great} we prove Theorem \ref{thm:factorisation} for $\alpha \in (1,2)$, and Proposition \ref{prop:pointoffurthestreach}. In Section \ref{sec:proof_a_1} we show Theorem \ref{thm:factorisation} for $\alpha=1$. Finally in Section \ref{sec:reflecting} we prove Theorem \ref{thm:reflecting}.
\section{Markov additive processes}\label{sec:maps}
In this section we will work with a (possibly killed) Markov processes $(\xi,J)=((\xi_t,J_t):t \geq 0)$ on $\mathbb{R} \times \{1,-1\}$. For convenience, we will always assume that $J$ is irreducible on $\{1,-1\}$. For such a process $(\xi,J)$
we let $\pmb{\rm P}_{x,i}$ be the law of $(\xi,J)$ started from the state $(x,i)$.
\begin{defn}
A Markov process $(\xi,J)$ is called a Markov additive process (MAP) on $\mathbb{R}\times\{1,-1\}$ if, for any $t \geq 0$ and $j=-1,1$, given $\{J_t=j\}$, the process $((\xi_{s+t}-\xi_t,J_{s+t}):s\geq 0)$ has the same law as $(\xi,J)$ under $\pmb{\rm P}_{0,j}$.
\end{defn}
The topic of MAPs are covered in various parts of the literature. We reference \cite{Cinlar1, Cinlar2, MR889893,MR2766220,MR3160562,MR3174223} to name but a few of the many texts and papers. It turns out that a MAP on $\mathbb{R}\times\{1,-1\}$ requires five characteristic components: two independent and L\'evy processes (possibly killed but not necessarily with the same rates), say $\chi_{1}=(\chi_{1}(t):t \geq 0)$ and $\chi_{-1}=(\chi_{-1}(t):t \geq 0)$, two independent random variables, say $\Delta_{-1,1}$ and $\Delta_{1,-1}$ on $\mathbb{R}$ and a $2\times 2$ intensity matrix, say $\bm Q=(q_{i,j})_{i,j=\pm 1}$. We call the quintuple $(\chi_{1},\chi_{-1},\Delta_{-1,1},\Delta_{1,-1},\bm Q)$ the driving factors of the MAP.
\begin{defn}
A Markov additive processes on $\mathbb{R}\times\{1,-1\}$ with driving factors $(\chi_{1},\chi_{-1},\Delta_{-1,1},\Delta_{1,-1},\bm Q)$ is defined as follows. Let $J=(J(t):t\geq 0)$ be a continuous time Markov process on $\{1,-1\}$ with intensity matrix $\bm Q$. Let $\sigma_1,\sigma_2,\dots$ denote the jump times of $J$. Set $\sigma_0=0$ and $\xi_0=x$, then for $n \geq 0$ iteratively define
\[
\xi(t)=\mathbbm{1}_{n>0}(\xi(\sigma_n-) + U^{(n)}_{J(\sigma_n-) , J(\sigma_n)} ) + \chi^{(n)}_{J(\sigma_n)}(t-\sigma_n) \qquad t \in [\sigma_n, \sigma_{n+1}),
\]
where $(U^{(n)}_{i,j})_{n \geq 0}$ and $\chi^{(n)}_{i}$ are i.i.d. with distributions $\Delta_{i,j}$ and $\chi_{i}$ respectively.
\end{defn}
It is not hard to see that the construction above results in a MAP. Conversely we have that every MAP arises in this manner, we refer to \cite[XI.2a]{MR889893} for a proof.
\begin{prop}
A Markov process $(\xi,J)$ is a Markov additive process on $\mathbb{R}\times\{1,-1\}$ if and only if there exists a quintuple of driving factors $(\chi_{1},\chi_{-1},\Delta_{-1,1},\Delta_{1,-1},\bm Q)$. Consequently, every Markov additive process on $\mathbb{R}\times\{1,-1\}$ can be identified uniquely by a quintuple $(\chi_{1},\chi_{-1},\Delta_{-1,1},\Delta_{1,-1},\bm Q)$ and every quintuple defines a unique Markov additive process.
\end{prop}
Let $\psi_{-1}$ and $\psi_1$ be the Laplace exponent of $\chi_{-1}$ and $\chi_1$ respectively (when they exist). For ${z} \in \mathbb C$, let $\bm G({z})$ denote the matrix whose entries are given by $\bm G_{i,j}({z})=\pmb{\rm E}[e^{{z} \Delta_{i,j}}]$ (when they exists), for $i \neq j$ and $\bm G_{i,i}({z})=1$. For ${z} \in \mathbb C$, when it exists, define
\begin{equation}\label{eq:F_def}
\bm F({z}):= \text{diag}(\psi_{1}({z}),\psi_{-1}({z})) - \bm Q \circ \bm G({z}),
\end{equation}
where diag$(\psi_{1}({z}),\psi_{-1}({z}))$ is the diagonal matrix with entries $\psi_{1}({z})$ and $\psi_{-1}({z})$, and $\circ$ denotes element--wise multiplication.
It is not hard to check that $\bm F$ is unique for each quintuple $(\chi_{1},\chi_{-1},\Delta_{-1,1},\Delta_{1,-1},\bm Q)$ and furthermore, see for example \cite[XI, Proposition 2.2]{MR1978607}, for each $i,j = \pm 1$ and $t \geq 0$,
\[
\pmb{\rm E}_{0,i}[e^{{z} \xi_t}; J_t=j] = (e^{t\bm F({z})})_{i,j}
\]
where $e^{t\bm F({z})}$ is the exponential matrix of $t\bm F({z})$. For this reason we refer to $\bm F$ as a MAP-exponent.
\subsection{Ladder height process}\label{reflected}
Here we will introduce the notion of the ladder height processes for MAPs and introduce the matrix Wiener--Hopf factorisation. It may be useful for the reader to compare this to the treatment of these topics for L\'evy processes in \cite[Chapter 6]{MR3155252}.
Let $(\xi,J)$ be a MAP and define the process $\bar \xi=(\bar \xi_t:t \geq 0)$ by setting $\bar \xi_t=\sup_{s \leq t}\xi_s$. Then it can be shown (see \cite[Theorem 3.10]{MR650610} or \cite[Chapter IV]{MR1406564}) that there exists two non-constant increasing processes $\bar L^{(-1)}=(\bar L^{(-1)}_t:t \geq 0)$ and $\bar L^{(1)}=(\bar L^{(1)}_t:t \geq 0)$ such that $\bar L^{(-1)}$ increases on the closure of the set $\{t: (\xi_t,J_t)=(\bar\xi_t,-1)\}$ and $\bar L^{(1)}$ increases on the closure of the set $\{t: (\xi_t,J_t)=(\bar\xi_t,1)\}$. Moreover $\bar L^{(-1)}$ and $\bar L^{(1)}$ are unique up to a constant multiples. We call $\bar L=\bar L^{(-1)}+\bar L^{(1)}$ the local time at the maximum. It may be the case that $\bar L_\infty<\infty$, for example if $\xi$ drifts to $- \infty$. In such a case, both $\bar L^{(-1)}_\infty$ and $\bar L^{(1)}_\infty$ are distributed exponentially. Since the processes $\bar L^{(-1)}$ and $\bar L^{(1)}$ are unique up to constants, we henceforth assume that whenever $\bar L_\infty<\infty$, the normalisation has been chosen so that
\begin{equation}\label{eq:killing_ass}
\text{ both } \bar L^{(-1)}_\infty \text{ and } \bar L^{(1)}_\infty \text{ are distributed as exponentials with rate } 1 .
\end{equation}
The ascending ladder height processes $H^+=(H^+_t:t \geq 0)$ is defined as
\[
(H^+_t, J^+_t):= (\bar \xi_{\bar L^{-1}_t}, J_{\bar L^{-1}_t}) \qquad t \in [0,\bar L_\infty),
\]
where the inverse in the above equation is taken to be right continuous. At time $\bar L_\infty$ we send $(H^+, J^+)$ to a cemetery state and declare the process killed. It is not hard to see that $(H^+,J^+)$ is itself a MAP. \[ \text{ We denote by $ \bm \kappa$ the Laplace exponent of } (H^+,J^+), \] that is to say \[ ({\rm e}^{-\bm\kappa(\lambda)t})_{i,j} :=\mathbf{E}_{0,i}[{\rm e}^{-\lambda H^+_t}; J^+_t =j], \qquad \lambda \geq0. \]
Similarly, we define $(H^-,J^-)$, called the descending ladder height, by using $- \xi$ in place of $\xi$. We denote by $\underline{\bm\kappa}$ the MAP--exponent of $(H^-,J^-)$.
Recalling that $\bm\kappa$ can only be identified up to pre-multiplication by a strictly positive diagonal matrix, the choice of normalisation in the local times \eqref{eq:killing_ass} is equivalent to choosing a normalisation $\bm\kappa$.
For a L\'evy process, its dual is simply given by its negative. The dual of a MAP is a little bit more involved. Firstly, since $J$ is assumed to be irreducible on $\{1,-1\}$, it follows that it is reversible with respect to a unique stationary distribution $\pi=(\pi_{1},\pi_{-1})$. We denote by $\hat{\bm Q}=(\hat q_{i,j})_{i,j=\pm 1}$ the matrix whose entries are given by
\[
\hat q_{i,j} = \frac{\pi_j}{\pi_i}q_{j,i}.
\]
The MAP--exponent, $\hat{\bm F}$, of the dual $(\hat \xi, \hat J)$ is given by
\begin{equation}\label{eq:F_hat}
\hat{\bm F}({z})= \text{diag}(\psi_{1}(-{z}),\psi_{-1}(-{z})) - \hat{\bm Q} \circ \bm G(-{z}),
\end{equation}
whenever the right-hand side exists.
The duality in this case corresponds to time-reversing $(\xi,J)$, indeed, as observed in \cite[Lemma 21]{dereich2015real}, for any $t\geq 0$,
\[
(((\xi_{(t-s)-}-\xi_t,J_{(t-s)-}):0\leq s\leq t),\pmb{\rm P}_{0,\pi}) \overset{d}{=} (((\hat \xi_s,\hat J_s): 0\leq s\leq t),\hat\pmb{\rm P}_{0,\pi})
\]
where we define $(\xi_{0-},J_{0-})=(\xi_0,J_0)$.
Next define
\[
{\bm\Delta}_\pi= \left(\begin{array}{*2{>{\displaystyle}c}}
\pi_{1} & 0\\
0 & \pi_{-1}
\end{array}\right).
\]
Then the following lemma follows immediately from \eqref{eq:F_hat}.
\begin{lemma}\label{lemma:F_hat_transform}
For each ${z} \in \mathbb C$,
\[
\hat{\bm F}({z}) = {\bm\Delta}_\pi^{-1}\bm F(-{z})^T{\bm\Delta}_\pi,
\]
where $\bm F(-{z})^T$ denotes the transpose of $\bm F(-{z})$.
\end{lemma}
\begin{rem}
Notice that
\[
\bm F(0)=\left(\begin{matrix}
q_{1,1} & -q_{1,1} \\
-q_{-1,-1} & q_{-1,-1}
\end{matrix}\right) = -\bm Q.
\]
Hence the matrix ${\bm\Delta}_\pi$ can be computed leading to the form in \eqref{eq:delta_matrix}. Also note that it is sufficient to use a constant multiple of the matrix ${\bm\Delta}_\pi$.
\end{rem}
Similarly to how we obtained $(H^+,J^+)$, we denote by $(\hat H^+,\hat J^+)$ the ascending ladder height process of the dual MAP $(\hat \xi, \hat J)$.
\[
\text{We denote by } \hat{\bm \kappa} \text{ the Laplace exponent of } (\hat H^+,\hat J^+).
\]
\begin{lemma}\label{lemma:hat_from_des} Let $ \underline{\bm\kappa}$ be the matrix exponent of the ascending ladder height processes of the MAPs $(-\xi,J)$. Then we have, up to post-multiplication by a strictly diagonal matrix,
\begin{equation*}
\underline{\bm\kappa}(\lambda)=\hat{\bm \kappa}(\lambda)\qquad \lambda \geq 0.
\end{equation*}
\end{lemma}
\begin{proof}
The MAP--exponent $\hat{\bm F}$ of $(\hat\xi,\hat J)$ is given explicitly in~\cite[Section 7]{kyprianou2015deep} and it is not hard to check that $\bm F(-z)=\hat{\bm F}(z)$. As a consequence, the MAP $(-\xi,J)$ is equal in law to $(\hat{\xi}, \hat{J})$.
Since $\underline{\bm\kappa}$ and $\hat{\bm\kappa}$ are the matrix Laplace exponent of the ascending ladder height processes of the MAPs $(-\xi,J)$ and $(\hat{\xi}, \hat{J})$, respectively, it follows that $\underline{\bm\kappa}(\lambda) = \hat{\bm\kappa}(\lambda)$ as required.
\end{proof}
We complete this section by remarking that if $X$ is an rssMp with Lamperti--Kiu exponent $(\xi,J)$, then $\xi$ encodes the radial distance of $X$ and $J$ encodes the sign of $X$. Consequently if $(H^+,J^+)$ is the ascending ladder height process of $(\xi, J)$, then $H^+$ encodes the supremum of $|X|$ and $J^+$ encodes the sign of where the supremum is reached. Similarly if $(H^-,J^-)$ is the descending ladder height process of $(\xi, J)$, then $H^-$ encodes the infimum of $|X|$ and $J^-$ encodes the sign of where the infimum is reached.
\begin{figure}
\caption{$X$}
\caption{$|X|$}
\caption{$(\xi,J)$}
\caption{A visualisation of the ladder height process $(H^+,J^+)$. The colours represent the state of $J$. This figure is the reverse of the process described in Figure~\ref{fig:reflected_invariant}.}
\label{fig:ladder_height}
\end{figure}
Although it is not so obvious, one can obtain $\hat{\bm\kappa}$ from $\bm\kappa$ as given by the following lemma which is a consequence of particular properties of the stable process. The proof can be found in \cite[Section 7]{kyprianou2015deep}.
\begin{lemma}\label{lemma:K_hat_transform}
For each $\lambda \geq 0$,
\[
\hat{\bm \kappa}(\lambda)={\bm\Delta}^{-1}_\pi \bm\kappa(\lambda +1 -\alpha)|_{\rho \leftrightarrow \hat\rho} {\bm\Delta}_\pi,
\]
where $|_{\rho\leftrightarrow\hat\rho}$ indicates exchanging the roles of $\rho$ with $\hat\rho$.
\end{lemma}
\section{Results for \texorpdfstring{$\alpha \in(0,1)$}{a in (0,1)}}\label{sec:proof_a_less}
In this section we will prove Theorem \ref{thm:factorisation} for $\alpha \in (0,1)$ and Proposition \ref{prop:pointofclosestreach}.
Suppose that $(X,\mathbb{P}_x)$ is a $\alpha$-stable process started at $x \neq 0$ with $\alpha < 1$ and let $(\xi,J)$ be the MAP in the Lamperti--Kiu transformation of $X$. Let $(H^-,J^-)$ be the descending ladder height process of $(\xi,J)$ and define $\bm U^-$ by
\[
\bm U^-_{i,j}(dx) = \int_0^\infty \pmb{\rm P}_{0,i}(H^-_t \in dx; J^-_t = j; t< \underline L_\infty)d t, \qquad x\geq 0.
\]
Note that we set $\pmb{\rm P}_{0,i}(H^-_t \in dx; J^-_t = j)=0$ if $(H^-,J^-)$ is killed prior to time $t$. The measure $\bm U^-_{i,j}$ is related to the exponent $ \underline{\bm\kappa}$ by the relation
\[
\int_0^\infty e^{-\lambda x} \bm U_{i,j}^-(dx) = \int_0^\infty\pmb{\rm E}_{0,i}[e^{-\lambda H^-_t}; J^-_t = j]dt = \underline{\bm\kappa}^{-1}_{i,j}(\lambda), \qquad \lambda\geq0.
\]
\noindent We present an auxiliary result.
\begin{lemma}\label{lemma:u_minus}
For an $\alpha$-stable process with $\alpha\in(0,1)$ we have that the measure $\bm U^-_{i,j}$ has a density, say $\bm u^-$, such that
\begin{align*}
&\bm u^-(x)= \left(\begin{array}{cc} \frac{\Gamma(1-\alpha\rho) }{\Gamma(\alpha\hat\rho)}&0\\
0& \frac{\Gamma(1-\alpha\hat\rho) }{\Gamma(\alpha\rho)}
\end{array}\right)
\left(\begin{array}{*2{>{\displaystyle}c}}
(e^x-1)^{\alpha\hat\rho-1}(e^x+1)^{\alpha\rho} & (e^x-1)^{\alpha\hat\rho}(e^x+1)^{\alpha\rho-1} \\
&\\
(e^x-1)^{\alpha\rho}(e^x+1)^{\alpha\hat\rho-1} & (e^x-1)^{\alpha\rho-1}(e^x+1)^{\alpha\hat\rho}
\end{array}\right), \qquad x\geq 0. \end{align*}
\end{lemma}
Now we show that Theorem \ref{thm:factorisation} follows from Lemma \ref{lemma:u_minus}.
\begin{proof}[Proof of Theorem \ref{thm:factorisation} for $\alpha \in(0, 1)$]
Thus from Lemma \ref{lemma:u_minus}, we can take Laplace transforms to obtain e.g., for $i,j=-1$ and $\lambda\geq0$,
\begin{align*}
\underline{\bm\kappa}^{-1}_{-1,-1}(\lambda) &=\frac{\Gamma(1-\alpha\hat\rho) }{\Gamma(\alpha\rho)} \int_0^\infty e^{-\lambda x}(e^x-1)^{\alpha\rho-1}(e^x+1)^{\alpha\hat\rho}dx \\
&= \frac{\Gamma(1-\alpha\hat\rho) }{\Gamma(\alpha\rho)} \int_0^1 u^{\lambda - \alpha}(1-u)^{\alpha\rho-1}(1+u)^{\alpha\hat\rho}du\\
&=\frac{\Gamma(1-\alpha\hat\rho) }{\Gamma(\alpha\rho)} \Psi(\lambda-\alpha, \alpha\rho-1,\alpha\hat\rho),
\end{align*}
where we have used the substitution $u=e^{-x}$. {\color{black}Once the remaining components of $\underline{\bm\kappa}^{-1}$ have been obtained similarly to above, we use Lemma~\ref{lemma:hat_from_des} to get $\hat{\bm\kappa}^{-1}$ and then apply Lemma \ref{lemma:K_hat_transform} to get $\bm\kappa^{-1}$. The reader will note that a direct application of the aforesaid Lemma will not give the representation of $\bm\kappa^{-1}$ stated in Theorem \ref{thm:factorisation} but rather the given representation post-multiplied by the diagonal matrix
\[
\left(\begin{array}{cc} \frac{\Gamma(1-\alpha\rho) }{\Gamma(\alpha\hat\rho)}&0 \\
0& \frac{\Gamma(1-\alpha\hat\rho) }{\Gamma(\alpha\rho)}
\end{array}\right),
\]
and this is a because of the normalisation of local time chosen in \eqref{eq:killing_ass}. Note that this is not important for the statement of Theorem \ref{thm:factorisation} as no specific normalisation is claimed there.
The details of the computation are left out.}
\end{proof}
We are left to prove Lemma \ref{lemma:u_minus}. We will do so by first considering the process $X$ started at $x>0$. The case when $x<0$ will follow by considering the dual $\hat X=-X$.
Recall that
$\underline m$ is the unique time such that
\[
|X_t| \geq |X_{\underline m}| \qquad \text{ for all } t \geq 0.
\]
Our proof relies on the analysis of the random variable $X_{\underline m}$. Notice that when $X_0=x$, $X_{\underline m}$ may be positive or negative and takes values in $[-x,x]$.
Before we derive the law of $X_{\underline m}$, we first quote the following lemma which appears in \cite[Corollary 1.2]{MR3161489}.
\begin{lemma}\label{lemma:avoid_strip}
Let $\tau^{(-1,1)}:=\inf\{t \geq 0: |X_t|<1\}$. We have that, for $x>1$,
\[
\mathbb{P}_x(\tau^{(-1,1)}=\infty) = \Phi(x),
\]
where
\[
\Phi(x) =\frac{\Gamma(1-\alpha\rho)}{\Gamma(\alpha\hat\rho)\Gamma(1-\alpha)} \int_0^{(x-1)/(x+1)} t^{\alpha\hat\rho-1}(1-t)^{-\alpha} \, dt.
\]
\end{lemma}
Lemma \ref{lemma:avoid_strip} immediately gives that the law of $|X_{\underline m}|$ as
\[
\mathbb{P}_x(|X_{\underline m}| > z) = \Phi(x/z), \qquad \text{ for } z\in [0,x].
\]
Indeed, the event $\{|X_{\underline m}| > z\}$ occurs if and only if $\tau^{(-z,z)}=\infty$. From the scaling property of $X$ we get that $\mathbb{P}_x(\tau^{(-z,z)}=\infty)=\mathbb{P}_{x/z}(\tau^{(-1,1)}=\infty)=\Phi(x/z)$.
We first begin to derive the law of $X_{\underline m}$ which shows Proposition \ref{prop:pointofclosestreach}.
\begin{proof}[Proof of Proposition \ref{prop:pointofclosestreach}]
Fix $x>0$. Similarly to the definition of $\underline m$, we define $\underline m^+$ and $\underline m^-$ as follows: Let $\underline m^+$ be the unique time such that $X_{\underline m^+}>0$ and
\[
X_t \geq X_{\underline m^+} \qquad \text{ for all } t \geq 0 \text{ such that } X_t>0.
\]
Similarly let $\underline m^-$ be the unique time such that $X_{\underline m^-}<0$ and
\[
X_t \leq X_{\underline m^-} \qquad \text{ for all } t \geq 0 \text{ such that } X_t < 0.
\]
In words, $\underline m^+$ and $\underline m^-$ are the times when $X$ is at the closest point to the origin on the positive and negative side of the origin, respectively. Consequently, we have that $X_{\underline m}>0$ if and only if $X_{\underline m^+} < | X_{\underline m^-}|$.
We now have that
\[
\mathbb{P}(| X_{\underline m^-}|> u; X_{\underline m^+} > v )=\mathbb{P}_x(\tau^{(-u,v)}=\infty) =\Phi\left(\frac{2x+u-v}{u+v}\right),
\]
where $\Phi$ is defined in Lemma \ref{lemma:avoid_strip} and in the final equality we have scaled space and used the self-similarity of $X$.
Next we have that for $z \geq 0$,
\begin{align}\label{eq:minreach_phi}
\frac{\mathbb{P}_x(X_{\underline m} \in \,d z)}{\,d z} = -\frac{\partial}{\partial v}\mathbb{P}_x(| X_{\underline m^-}|> z; X_{\underline m^+} > v )|_{v=z}&= -\frac{\partial}{\partial v} \Phi\left(\frac{2x+z-v}{z+v}\right)|_{v=z}\nonumber\\
&= \frac{x+z}{2z^2}\Phi'\left(\frac{x}{z}\right).
\end{align}
The proposition for $z>0$ now follows from an easy computation. The result for $z<0$ follows similarly.
\end{proof}
Now we will use \eqref{eq:minreach_phi} to show Lemma \ref{lemma:u_minus}. We will need the following simple lemma which appears in the Appendix of \cite{dereich2015real}.
\begin{lemma}\label{lemma:crossing_U}
Let $T_0^-:=\inf\{t \geq 0: \xi_t <0\}$. Under the normalisation \eqref{eq:killing_ass}, for $i,j=-1,1$ and $y>0$,
\[
\pmb{\rm P}_{y,i}(T_0^-=\infty; J_{\varphi(\underline m)} = j)=\bm U^-_{i,j}(y):= \bm U^-_{i,j}[0,y].
\]
\end{lemma}
The basic intuition behind this lemma can be described in terms of the descending ladder MAP subordinator $(H^-, J^-)$. The event $\{T_0^-=\infty; J_{\varphi(\underline m)} = j\}$ under $ \pmb{\rm P}_{y,i}$ corresponds the terminal height of $H^-$ immediately prior to being killed being of type $j$ and not reaching the height $y$. This is expressed precisely by the quantity $\bm U^-_{i,j}(y)$. It is also important to note here and at other places in the text that $\xi$ is regular for both $(0,\infty)$ and $(-\infty,0)$. Rather subtly, this allows us to conclude that the value of $J_{\varphi(\underline{m})} = \lim_{s\uparrow \varphi(\underline{m})}J_s$, or, said another way, the process $\xi$ does not jump away from its infimum as a result of a change in modulation (see~\cite{ivanovs2015splitting} for a discussion about this).
\begin{proof}[Proof of Lemma \ref{lemma:u_minus}]
Let us now describe the event $\{T_0^-=\infty; J_{\varphi(\underline m)} = j\}$ in terms of the underlying process $X$. The event $\{T_0^-=\infty; J_{\varphi(\underline m)} = 1\}$ occurs if and only if $\tau^{(-1,1)}=\infty$ and furthermore the point at which $X$ is closest to the origin is positive, i.e. $X_{\underline m} > 0$. Thus $\{T_0^-=\infty; J_{\varphi(\underline m)} = 1\}$ occurs if and only if $X_{\underline m} > 1$. Using Lemma \ref{lemma:crossing_U} and \eqref{eq:minreach_phi} we have that
\begin{align*}
\bm U^-_{1,1}(x)&= \pmb{\rm P}_{x,1}(T_0^-=\infty; J_{\varphi(\underline m)} = 1)\\
&=\mathbb{P}_{e^x}(X_{\underline m} > 1)\\
&= \frac{1}{2}\int_{1}^{e^x} (e^x+z) \Phi'\left(\frac{e^x}{z}\right) z^{-2}dz\,\\
&= \frac{1}{2} \int_1^{e^x} (1+1/u)\Phi'(u)du,
\end{align*}
where in the final equality we have used the substitution $u=e^{x}/z$.
Differentiating the above equation we get that
\begin{align}\label{eq:u_phi_prime}
\bm u^-_{1,1}(x)&= \frac{1}{2}(e^{x} + 1)\Phi'(e^x)=\frac{\Gamma(1-\alpha\rho) }{2^\alpha\Gamma(1-\alpha) \Gamma(\alpha\hat\rho)} (e^x-1)^{\alpha\hat\rho-1}(e^x+1)^{\alpha\rho}.
\end{align}
Similarly considering the event $\{T_0^+=\infty; J_{\varphi(\underline m)} = -1\}$ we get that
\[
\bm u^-_{1,-1}(x) = \frac{1}{2}(e^{x} - 1)\Phi'(e^x) = \frac{\Gamma(1-\alpha\rho) }{2^\alpha\Gamma(1-\alpha) \Gamma(\alpha\hat\rho)}(e^x-1)^{\alpha\hat\rho}(e^x+1)^{\alpha\rho-1}.
\]
Notice now that $\bm u^-_{1,j}$ only depends on $\alpha$ and $\rho$. Consider now the dual process $\hat X =(-X_t:t \geq 0)$. This process is the same as $X$ albeit $\rho\leftrightarrow \hat\rho$. To derive the row $\bm u^-_{-1,j}$ we can use $\hat X$ in the computations above and this implies that $\bm u^-_{-1,j}$ is the same as $\bm u^-_{1,-j}$ but exchanging the roles of $\rho$ with $\hat\rho$. This concludes the proof of Lemma \ref{lemma:u_minus}.
\end{proof}
\section{Proof of Theorem \ref{thm:factorisation} for \texorpdfstring{$\alpha \in (1,2)$}{a in (1,2)}}\label{sec:proof_a_great}
In this section we will prove Theorem \ref{thm:factorisation} for $\alpha \in (1,2)$. Let $X$ be an $\alpha$-stable process with $\alpha \in (1,2)$ and let $(\xi,J)$ be the MAP associated to $X$ via the Lamperti--Kiu transformation. The notation and proof given here are very similar to that of the case when $\alpha<1$, thus we skip some of the details.
Since $\alpha \in (1,2)$ we have that $\tau^{\{0\}}:=\inf\{t \geq 0: X_t=0\}< \infty$ and $X_{\tau^{\{0\}}-} = 0$ almost surely. Hence it is the case that $\xi$ drifts to $-\infty$. Recall that $\overline m$ as the unique time for which $\overline m < \tau^{\{0\}}$ and
\[
|X_{\overline m}| \geq |X_t| \qquad \text{ for all } t < \tau^{\{0\}}
\]
where the existence of such a time follows from the fact that $X$ is a stable process and so $0$ is regular for $(0,\infty)$ and $(-\infty,0)$.
The quantity we are interested in is $X_{\overline m}$. We begin with the following lemma, which is lifted from \cite[Corollary 1]{profeta2015harmonic} and also can be derived from the potential given in \cite[Theorem 1]{kyprianou2014potentials}.
\begin{lemma}\label{lemma:upper_crossing}
For every $x \in (0,1)$ and $y \in (x,1)$,
\[
\mathbb{P}_x(\tau^{\{y\}}<\tau^{(-1,1)^c}) = (\alpha-1)\left(\frac{x-y}{1-y^2}\right)^{\alpha-1}\bar\Phi\left(\left|\frac{1-xy}{x-y}\right|\right),
\]
where
\[
\bar\Phi(z) = \int_1^{z}(t-1)^{\alpha\rho-1}(t+1)^{\alpha\hat\rho-1} dt.
\]
\end{lemma}
Next we prove Proposition \ref{prop:pointoffurthestreach} by expressing exit probabilities in terms of $\bar\Phi$. In the spirit of the proof of Proposition \ref{prop:pointofclosestreach}, we apply a linear spatial transformation to the probability $\mathbb{P}_x(\tau^{(-u,v)^c}<\tau^{\{0\}})$ and write it in terms of $\bar\Phi$.
\begin{proof}[Proof of Proposition \ref{prop:pointoffurthestreach}]
Similar to the derivation of \eqref{eq:minreach_phi} in the proof of Proposition \ref{prop:pointofclosestreach}, for each $x>0$ and $|z|>x$,
\begin{equation}\label{eq:maxpoint_phi}
\frac{\mathbb{P}_x(X_{\overline m}\in dz)}{dz} = \frac{\alpha-1}{2 x^{2-\alpha} |z|^\alpha} \left(|x+z| \bar\Phi'\left(\frac{|z|}{x}\right)-(\alpha-1)x \bar\Phi\left(\frac{|z|}{x}\right)\right).
\end{equation}
The result now follows from straight forward computations.
\end{proof}
Again we introduce the following lemma from the Appendix of \cite{dereich2015real} (and again, the subtle issue of regularity of $\xi$ for the positive and negative half-lines is being used).
\begin{lemma}\label{lemma:crossing_a_big}
Let $T_0^+:=\inf\{t \geq 0: \xi_t >0\}$, then, with the normalisation given in \eqref{eq:killing_ass}, for $i,j=-1,1$ and $y>0$,
\[
\pmb{\rm P}_{-y,i}(T_0^+=\infty; J_{\varphi(\overline m)} = j)={\bm U}_{i,j}(y).
\]
\end{lemma}
Similar to the derivation in \eqref{eq:u_phi_prime}, we use \eqref{eq:maxpoint_phi} and Lemma \ref{lemma:crossing_a_big} to get that
\begin{align}\label{eq:u_phi_a_big}
\bm u_{1,1}(x) &= \frac{\,d}{dx} \mathbb{P}_{e^{-x}}(X_{\overline m} \in (e^{-x},1)) \nonumber\\
&= \frac{\alpha-1}{2}\frac{\,d}{dx} \int_{e^{-x}}^1 dz\frac{1}{z^\alpha} e^{(2-\alpha)x}\left((e^{-x}+z) \bar\Phi'\left(\frac{z}{e^{-x}}\right)-(\alpha-1)e^{-x} \bar\Phi\left(\frac{z}{e^{-x}}\right)\right)\nonumber \\
&= \frac{\alpha-1}{2}\frac{\,d}{dx} \int_1^{e^x} du\, \frac{1}{u^\alpha} \left((1+u) \bar\Phi'\left(u\right)-(\alpha-1) \bar\Phi\left(u\right)\right) \nonumber\\
&= \frac{\alpha-1}{2} e^{-(\alpha-1)x}\left((1+e^x) \bar\Phi'\left(e^x\right)-(\alpha-1) \bar\Phi\left(e^x\right)\right)\nonumber\\
&= \frac{\alpha-1}{2} e^{-(\alpha-1)x} \left((e^x-1)^{\alpha\rho-1}(e^x+1)^{\alpha\hat\rho}-(\alpha-1) \bar\Phi\left(e^x\right)\right),
\end{align}
where in the third equality we have used the substitution $u=z/e^{-x}$. Now we will take the Laplace transform of $\bm u_{1,1}$. The transform of the $\bar\Phi$ term is dealt with in the following lemma. The proof follows from integration by parts which we leave out.
\begin{lemma}\label{lemma:integrate}
Suppose that $\gamma> \alpha-1$, then
\[
\int_0^\infty e^{-\gamma x}\bar\Phi(e^x)dx = \frac{1}{\gamma} \Psi(\gamma-\alpha,\alpha\rho-1,\alpha\hat\rho-1).
\]
\end{lemma}
Next we have
\[
\int_0^\infty e^{-(\lambda+\alpha-1)x} (e^x-1)^{\alpha\rho-1}(e^x+1)^{\alpha\hat\rho}dx =\int_0^1 u^{\lambda-1}(1-u)^{\alpha\rho-1}(1+u)^{\alpha\hat\rho}du = \Psi(\lambda-1,\alpha\rho-1,\alpha\hat\rho),
\]
where we have used the substitution $u=e^{-x}$. Integrating \eqref{eq:u_phi_a_big} and using the above equation together with Lemma \ref{lemma:integrate} we get that
\[
\bm\kappa^{-1}_{1,1}(\lambda)=\frac{\alpha-1}{2}\Psi(\lambda-1,\alpha\rho-1,\alpha\hat\rho) - \frac{(\alpha-1)^2}{2(\lambda+\alpha-1)}\Psi(\lambda-1,\alpha\rho-1,\alpha\hat\rho-1).
\]
Similar proofs give $\bm\kappa_{i,j}^{-1}$ for the remaining $i,j$. The given formula for $\hat{\bm\kappa}^{-1}$ follows from Lemma \ref{lemma:K_hat_transform} as well as the straightforward the matrix algebra
\begin{equation}
\boldsymbol{\Delta}_{\boldsymbol \pi}^{-1} {\bm M} \boldsymbol{\Delta}_{\boldsymbol \pi}\left[ \begin{matrix}
\frac{\Gamma(1-\alpha\rho) }{\Gamma(\alpha\hat\rho)}& 0 \\
0 & \frac{\Gamma(1-\alpha\hat\rho) }{\Gamma(\alpha\rho)}
\end{matrix}
\right] = \left[ \begin{matrix}
\frac{\Gamma(1-\alpha\rho) }{\Gamma(\alpha\hat\rho)}& 0 \\
0 & \frac{\Gamma(1-\alpha\hat\rho) }{\Gamma(\alpha\rho)}
\end{matrix}
\right]{\bm M},
\label{matrixalgebra}
\end{equation}
where ${\bm M}$ is any $2\times 2$ matrix.
\section{Proof of Theorem \ref{thm:factorisation} for \texorpdfstring{$\alpha=1$}{a=1}}\label{sec:proof_a_1}
In the case when $\alpha=1$, the process $X$ is a Cauchy process, which has the property that $\limsup_{t\to\infty}|X_t| = \infty$ and $\liminf_{t\to\infty}|X_t|=0$. This means that the MAP $(\xi,J)$ oscillates and hence the global minimum and maximum both do not exist so that the previous methods cannot be used. Instead we focus on a two sided exit problem as an alternative approach. (Note, the method we are about to describe also works for the other cases of $\alpha$, however it is lengthy and we do not obtain the new identities {\it en route} in a straightforward manner as we did in Proposition \ref{prop:pointofclosestreach} and Proposition \ref{prop:pointoffurthestreach}.)
The following result follows from the compensation formula and the proof of it is identical to the case for L\'evy processes, see \cite[Chapter III Proposition 2]{MR1406564} and \cite[Theorem 5.8]{MR3155252}.
\begin{lemma}\label{lemma:cauchy_exit_potential}
Let $(H^+,J^+)$ be the height process of $(\xi,J)$. For any $x >0$ define $T_x:=\inf\{t>0: H^+_t>x\}$, then for any $x>0$ and $i=\pm 1$,
\[
\pmb{\rm P}_{0,i}( x-H^+_{T_x-}\in du; J^+_{T_x-}=1; J^+_{T_x}=1 ) = {\bm u}_{i,1}(x-u) \Lambda[u,\infty)du,
\]
where $\Lambda$ is some $\sigma$--finite measure on $[0,\infty)$.
\end{lemma}
Next we will calculate the over and under shoots in Lemma \ref{lemma:cauchy_exit_potential} by using the underlying process $X$. This is done in the following lemma.
\begin{lemma}\label{lemma:cauchy_exit_prob}
Let $\tau^+_1:=\inf\{t \geq 0: X_t >1\}$ and $\tau^-_{-1}:=\inf\{t \geq 0: X_t <-1\}$. Then for $x \in (-1,1)$, $u\in[0,(1-x)\vee 1)$ and $y \geq 0$,
\begin{align*}
\mathbb{P}_x(1 -&\bar X_{\tau^+_1-} \in du; X_{\tau^+_1}-1 \in dy; X_{\tau^+_1-}>0;\tau^+_1<\tau^-_{-1})= \frac{(1-u+x)^{1/2}}{(1-u-x)^{1/2}} \frac{(u+y)^{3/2}}{(2-u+y)^{1/2}}dudy.
\end{align*}
\end{lemma}
\begin{proof}
First \cite[Corollary 3]{kyprianou2014potentials} gives that for $z \in (0,1)$, $u\in[0,1-z)$ and $v \in (u,1]$,
\begin{align*}
\mathbb{P}_z&(1 -\bar X_{\tau^+_1-} \in du; 1-X_{\tau^+_1-}\in dv;X_{\tau^+_1}-1 \in dy; \tau^+_1<\tau^-_{0}) \\
&= \frac{1}{\pi} \frac{z^{1/2}(1-v)^{1/2}}{(1-u-z)^{1/2}(v-u)^{1/2} (1-u)(v+y)^2}dudy,
\end{align*}
where $\tau^-_{0}:=\inf\{t \geq 0: X_t <0\}$. We wish to integrate $v$ out of the above equation. To do this, we make the otherwise subtle observation that
\begin{align*}
\int_u^1dv \, (1-v)^{1/2} (v-u)^{-1/2} (v+y)^{-2}&= (u+y)^{-2}(1-u)\int_0^1dz \, (1-z)^{1/2} z^{-1/2} \left(1+z\frac{1-u}{u+y}\right)^{-2} \\
&= (u+y)^{-2}(1-u) \frac{\pi}{2}\, \,_2F_1\left(2,1/2,2; -\frac{1-u}{u+y}\right)\\
&=\frac{\pi}{2} (u+y)^{-3/2}(1-u) (1+y)^{-1/2},
\end{align*}
where in the first equality we have used the substitution $z=(v-u)/(1-u)$. In the second equality we have used \cite[Theorem 2.2.1]{MR1688958} and the final equality follows from the Euler--transformation \cite[Theorem 2.2.5]{MR1688958}.
Hence, for $z\in (0,1)$, $u\in[0,1-z)$ and $y \geq 0$,
\begin{align}\label{eq:double_exit_unshifted}
\mathbb{P}_z&(1 -\bar X_{\tau^+_1-} \in du; X_{\tau^+_1}-1 \in dy; \tau^+_1<\tau^-_{0}) \nonumber\\
&= \frac{1}{2} \frac{z^{1/2}}{(1-u-z)^{1/2}(1+y)^{1/2} (u+y)^{3/2}} dudy.
\end{align}
Next we have that for $x\in (-1,1)$, $u \in [0,(1-x)\vee 1)$ and $y \geq 0$,
\begin{align*}
&\frac{\mathbb{P}_x(1 -\bar X_{\tau^+_1-} \in du; X_{\tau^+_1}-1 \in dy; \bar X_{\tau^+_1-}>-\underline X_{\tau^+_1-};\tau^+_1<\tau^-_{-1})}{du\, dy}\\
&= \frac{\partial}{\partial v}\frac{\partial}{\partial y}\mathbb{P}_x(1 -\bar X_{\tau^+_1-} \leq v; X_{\tau^+_1}-1 \leq y; \tau^+_1<\tau^-_{u-1})|_{v=u}\\
&= \frac{\partial}{\partial v}\frac{\partial}{\partial y}\mathbb{P}_{\frac{x+1-u}{2-u}}\left(1-\bar X_{\tau^+_1-} \leq \frac{v}{2-u}; X_{\tau^+_1}-1 \leq \frac{y}{2-u}; \tau^+_1<\tau^-_{0}\right)|_{v=u}\\
& = \frac{1}{2}(2-u)^{-2} \frac{\left(\frac{x+1-u}{2-u}\right)^{1/2}}{\left(1-\frac{u}{2-u}-\frac{x+1-u}{2-u}\right)^{1/2}\left(1+\frac{y}{2-u}\right)^{1/2} \left(\frac{u+y}{2-u}\right)^{3/2}}\\
&= \frac{(1-u+x)^{1/2}}{(1-u-x)^{1/2}} \frac{1}{(2-u+y)^{1/2}(u+y)^{3/2}},
\end{align*}
where in the first equality we have used that the event $\{\bar X_{\tau^+_1-}>-\underline X_{\tau^+_1-}, 1 -\bar X_{\tau^+_1-} \in du\}$ constrains $\underline{X}$ and thus is equivalent to $\{ \tau^+_1<\tau^-_{u-1}, 1 -\bar X_{\tau^+_1-} \in du\}$. In the second equality we have used the scaling property of $X$ and in the third equality we have used \eqref{eq:double_exit_unshifted}.
\end{proof}
Notice now that for each $x \geq 0$ and $j = \pm 1$,
\begin{align}\label{eq:Lambda_find}
&\frac{\partial}{\partial u}\frac{\partial}{\partial y}\pmb{\rm P}_{0,j}(x-H^+_{T_x-}\leq u; H^+_{T_x}-x\leq y; J^+_{T_x-}=1; J^+_{T_x}=1 ) \nonumber\\
&=\frac{\partial}{\partial u}\frac{\partial}{\partial y}\mathbb{P}_{j}(\bar X_{\tau^+_{e^x}-} \geq e^{x-u};X_{\tau^+_{e^x}} \leq e^{y+x}; X_{\tau^+_{e^x}-}>0;\tau^+_{e^{x}}<\tau^-_{-e^{x}})\nonumber\\
&=\frac{\partial}{\partial u}\frac{\partial}{\partial y}\mathbb{P}_{je^{-x}}(\bar X_{\tau^+_{1}-} \geq e^{-u};X_{\tau^+_{1}} \leq e^y; X_{\tau^+_{1}-}>0;\tau^+_{1}<\tau^-_{-1})\nonumber\\
&=e^{y-u}\frac{(e^{-u}+je^{-x})^{1/2}}{(e^{-u}-je^{-x})^{1/2}} \frac{1}{(e^y+e^{-u})^{1/2}(e^y-e^{-u})^{3/2}},
\end{align}
where in the second equality we have used the scaling property of $X$ and in the final equality we applied Lemma \ref{lemma:cauchy_exit_prob}. The above equation together with Lemma \ref{lemma:cauchy_exit_potential} gives that for $x \geq 0$,
\begin{equation}\label{eq:cauchy_sim_eq1}
\frac{\bm{u}_{1,1}(x-u)}{\bm{u}_{-1,1}(x-u)}= \frac{\pmb{\rm P}_{0,1}( x-H^+_{T_x-}\in du; J^+_{T_x-}=1; J^+_{T_x}=1)/du}{\pmb{\rm P}_{0,-1}( x-H^+_{T_x-}\in du; J^+_{T_x-}=1; J^+_{T_x}=1 )/du} = \frac{1+e^{-(x-u)}}{1-e^{-(x-u)}}.
\end{equation}
Next we claim that for any $x\geq 0$,
\begin{equation}\label{eq:cauchy_pot_sum}
\sum_{i=\pm 1}\bm{u}_{1,i}(x) = (1-e^{-x})^{-1/2}(1+e^{-x})^{1/2} + (1-e^{-x})^{1/2}(1+e^{-x})^{-1/2},
\end{equation}
which also fixes the normalisation of local time (not necessarily as in \eqref{eq:killing_ass}). Again we remark that this is not a concern on account of the fact that Theorem \ref{thm:factorisation} is stated up to post-multiplication by a strictly positive diagonal matrix.
This follows from existing literature on the Lamperti transform of the Cauchy process and we briefly describe how to verify it. It is known (thanks to scaling of $X$ and symmetry) that $(|X_t|:t \geq 0)$ is a positive self-similar Markov process with index $\alpha$. As such, it can can be expressed in the form
$
|X_t| = \exp\{\chi_{\beta_t}\},
$
for $ t\leq \tau^{\{0\}}$,
where $\beta_t = \inf\{s>0 : \int_0^s \exp\{\alpha \chi_u\}du >t\}$, see for example \cite[Chapter 13]{MR3155252}. The sum on the left-hand side of \eqref{eq:cauchy_pot_sum} is precisely the potential of the ascending ladder height process of the L\'evy process $\chi$. We can verify that the potential of the ascending ladder height process of $\chi$ has the form given by the right-hand side of \eqref{eq:cauchy_pot_sum} as follows. Laplace exponent of the ascending ladder height process of $\chi$ is given in \cite[Remark 2]{MR2797981}. Specifically, it takes the form
$\kappa_\chi(\lambda): = \Gamma((\lambda +1 )/2)/\Gamma(\lambda/2)$, $\lambda \geq 0$. Then the identity in \eqref{eq:cauchy_pot_sum} can be verified by checking that, up to a multiplicative constant, its Laplace transform agrees with $1/\kappa_\chi(\lambda)$, $\lambda\geq 0$.
Now we can finish the proof. Notice first that the Cauchy process is symmetric, thus $\bm{u}_{i,j}=\bm{u}_{-i,-j}$ for each $i,j\in\{1,-1\}$. Thus from \eqref{eq:cauchy_pot_sum} we get
\begin{equation}\label{eq:cauchy_sim_eq2}
\sum_{i=\pm 1}\bm{u}_{i,1}(x) = (1-e^{-x})^{-1/2}(1+e^{-x})^{1/2} + (1-e^{-x})^{1/2}(1+e^{-x})^{-1/2}.
\end{equation}
Solving the simultaneous equations \eqref{eq:cauchy_sim_eq1} and \eqref{eq:cauchy_sim_eq2} together with the fact $\bm{u}_{i,j}=\bm{u}_{-i,-j}$ gives the result for $\bm{u}$. To obtain $\hat{\bm{u}}$ we note that the reciprocal process $\widetilde{X}: = 1/X_{\theta_t}$, $t \geq 0$ has the law of a Cauchy process, where $\theta_t = \inf\{s>0: \int_0^{s}|X_u|^{-2}du >t\}$ (see \cite[Theorem 1]{MR2256481}). Theorem 4 in \cite{kyprianou2015deep} also shows that $\widetilde{X}$ has an underlying MAP which is the dual of the MAP underlying $X$. It therefore follows that
$\hat{\bm{u}}=\bm u$. This finishes the proof.
\begin{rem}
Using the form of $\bm u$ and \eqref{eq:Lambda_find}, we also get the jump measure $\Lambda$ appearing in Lemma \ref{lemma:cauchy_exit_potential} as
\[
\Lambda(d y)= \frac{e^{y}}{(e^y+1)^{1/2}(e^y-1)^{3/2}}\,d y.
\]
Up to a multiplicative constant, this can also be seen in \cite[equation (14)]{kyprianou2015deep}.
\end{rem}
\section{Proof of Theorem \ref{thm:reflecting}}\label{sec:reflecting}
Recall that $(R,M)$ is a Markov process. Since $R$ takes values on $[-1,1]$ and is recurrent, it must have a limiting distribution which does not depend on its initial position. For $x\in[-1,1]$ and $j = \pm1$, when it exists, define
\[
\mu_j(dy): = \lim_{t\to\infty}\mathbb{P}_x(|R_t|\in dy; \, {\rm sgn}(R_t) = j)\qquad y \in [0,1].
\]
Notice that the stationary distribution $\mu$ is given by $\mu(A) = \mu_1(A\cap[0,1])+ \mu_{-1}(-A\cap[0,1])$ (here we are pre-emptively assuming that each of the two measures on the right-hand side are absolutely continuous with respect to Lebesgue measure and so there is no `double counting' at zero) and hence it suffices to establish an identify for $\mu_j$.
For $i,j=\pm1$,
\[
\pmb{\rm E}_{0,i}\left[ e^{-\beta (\overline{\xi}_{\mathbbm{e}_q} - \xi_{\mathbbm{e}_q})} ; J_{\mathbbm{e}_q}=j\right]= \sum_{k= \pm1}\pmb{\rm E}_{0,i}\left[ e^{-\beta (\overline{\xi}_{\mathbbm{e}_q} - \xi_{\mathbbm{e}_q})} ; J_{\overline{m}_{\mathbbm{e}_q}}=k, \,J_{\mathbbm{e}_q}=j\right],
\]
where $\mathbbm{e}_q$ is an independent and exponentially distributed random variable with rate $q$ and $\overline{m}_{\mathbbm{e}_q} $ is the unique time at which $\xi$ obtains its maximum on the time interval $[0,\mathbbm{e}_q]$. Appealing to the computations in the Appendix of \cite{dereich2015real}, specifically equation (22) and Theorem 23, we can develop the right-hand side above using duality, so that
\begin{eqnarray*}
\int_0^1 y^{\beta}\tilde\mu_j(d y)&:=&\lim_{q\downarrow0}\pmb{\rm E}_{0,i}\left[ e^{-\beta (\overline{\xi}_{\mathbbm{e}_q} - \xi_{\mathbbm{e}_q})} ; J_{\mathbbm{e}_q}=j\right]\\
&&=\lim_{q\downarrow0}\sum_{k=\pm1}\pmb{\rm E}_{0,i}\left[ e^{-\beta (\overline{\xi}_{\mathbbm{e}_q} - \xi_{\mathbbm{e}_q})} ;
J_{\overline{m}_{\mathbbm{e}_q}}=k, \,
J_{\mathbbm{e}_q}=j\right]\\
&&=\lim_{q\downarrow0} \sum_{k= \pm1}\pmb{\rm P}_{0,i}\left( J_{\overline{m}_{\mathbbm{e}_q}}=k\right)\hat{\pmb{\rm E}}_{0,j}\left[ e^{-\beta \overline{\xi}_{\mathbbm{e}_q} } ; J_{\overline{m}_{\mathbbm{e}_q}}=k\right]\frac{\pi_j}{\pi_k}\\
&&={\pi_j}\sum_{k=\pm1}[\hat{\bm\kappa}(\beta)^{-1}]_{j,k}c_k,
\end{eqnarray*}
for some strictly positive constants $c_{\pm1}$,
where in the first equality we have used the Lamperti--Kiu transform and in the third equality, we have split the process at the maximum and used that, on the event $\{J_{\overline{m}_{\mathbbm{e}_q}} = j, J_{\mathbbm{e}_q} = k\}$, the pair $( \overline{\xi}_{\mathbbm{e}_q}-\xi_{\mathbbm{e}_q} , \mathbbm{e}_q - \overline{m}_{\mathbbm{e}_q} )$ is equal in law to the pair
$( \overline{\hat{\xi}}_{\mathbbm{e}_q}, {\overline{\hat{m}}}_{\mathbbm{e}_q} )$ on $\{\hat{J}_0 = k, \hat{J}_{\overline{\hat{m}}_{\mathbbm{e}_q}} =j\}$, where $\{(\hat\xi_s, \hat{J}_s): s\leq t\}: = \{ (\xi_{(t-s)-} -\xi_t, J_{(t-s)-}): s\leq t\}$, $t\geq 0$, is equal in law to the dual of $\xi$, $\overline{\hat{\xi}}_t = \sup_{s\leq t}\hat{\xi}_s$ and $\overline{\hat{m}} = \sup\{s\leq t: \overline{\hat\xi}_s = \hat{\xi}_t\}$.
Note, we have also used the fact that, $\overline{m}_{\mathbbm{e}_q}$ converges to $+\infty$ almost surely as $q\to\infty$ on account of the fact that $\limsup_{t\to\infty}|X_t|=\infty$.
Since $[\hat{\bm\kappa}(\lambda)^{-1}]_{j,k}$ is the Laplace transform of $\hat{\bm u}_{j,k}$, it now follows that, \[
\left.\frac{\,d\tilde\mu_j(y)}{{d y}}\right|_{y = {\rm e}^{-x}}={\pi_j} \sum_{k=\pm1}\hat{\bm u}_{j,k}(x)c_k, \qquad x\geq 0. \] Said another way, \[ \tilde\mu_j({d y}) = \frac{\pi_j}{y}\sum_{k=\pm1}\hat{\bm u}_{j,k}(-\log y)c_k\,d y, \qquad y\in[0,1]. \] The constants $c_k$, $k=\pm1$, can be found by noting that, for $j = \pm1$, $\mu_j([0,1]) = \pi_j$ and hence, for $j =\pm1$, \begin{equation} c_1\left(\int_0^\infty \hat{\bm u}_{j,1}(x) \,d x\right) + c_{-1}\left(\int_0^\infty \hat{\bm u}_{j,-1}(x) \,d x\right)= 1. \label{c1c-1} \end{equation} Using~\cite{hyper} and Theorem \ref{cor:potentials} (i), \begin{align*} &\int_0^\infty [\hat{\bm u}_{1,1}(x) - \hat{\bm u}_{-1,1}(x)]\,d x \\
&= \frac{\Gamma(1-\alpha\rho)}{\Gamma(\alpha\hat\rho)} \int_0^1 u^{-\alpha} (1-u)^{\alpha\hat\rho -1}(1+u)^{\alpha\rho}
{\,d u}\\ &\hspace{1cm}- \frac{\Gamma(1-\alpha\hat\rho)}{\Gamma(\alpha\rho)}\int_0^1 u^{-\alpha} (1-u)^{\alpha\rho}(1+u)^{\alpha\hat\rho-1}du \\ &= \frac{\Gamma(1-\alpha\rho)}{\Gamma(\alpha\hat\rho)}B(1-\alpha, \alpha\hat\rho) \,\,_{2}F_1(-\alpha\rho, 1-\alpha, 1-\alpha\rho; -1)\\ &\hspace{1cm}- \frac{\Gamma(1-\alpha\hat\rho)}{\Gamma(\alpha\rho)}B(1-\alpha, \alpha\rho + 1) \,\, _{2}F_1(1-\alpha\hat\rho, 1-\alpha, 2-\alpha\hat\rho; -1)\\ &=\Gamma(1-\alpha\rho)\Gamma(1-\alpha\hat\rho). \end{align*} Now subtracting \eqref{c1c-1} in the case $j = -1$ from the case $j =1$, it appears that \[ \Gamma(1-\alpha\rho)\Gamma(1-\alpha\hat\rho)(c_1 - c_{-1}) = 0, \] which is to say, $c_1 = c_{-1}.$
In order to evaluate either of these constants, we appeal to the definition of the Beta function to compute \begin{align*} &\int_0^\infty [\hat{\bm u}_{1,1}(x) +\hat{\bm u}_{1,-1}(x)]\,d x \\ &=\frac{\Gamma(1-\alpha\rho)}{\Gamma(\alpha\hat\rho)} \int_0^1 u^{-\alpha}(1-u)^{\alpha\hat\rho -1}(1+u)^{\alpha\rho} + u^{-\alpha}(1-u)^{\alpha\hat\rho}(1+u)^{\alpha\rho -1} \,d u\\ &=2\frac{\Gamma(1-\alpha\rho)}{\Gamma(\alpha\hat\rho)} \int_0^1 u^{-\alpha}(1-u)^{\alpha\hat\rho -1}(1+u)^{\alpha\rho-1} \,d u\\
&=2^{\alpha}\frac{\Gamma(1-\alpha\rho)}{\Gamma(\alpha\hat\rho)} \int_0^1 v^{\alpha\hat\rho-1}(1-v)^{-\alpha}\,d v \\ &=2^{\alpha}\Gamma(1-\alpha), \end{align*} where in the third equality, we have made the substitution $v = (1-u)/(1+u)$. It now follows from \eqref{c1c-1} that \[ c_1= c_{-1} = \frac{1}{2^{\alpha}\Gamma(1-\alpha)} \] and hence e.g. on $y\in[0,1]$, \begin{align*} \tilde{\mu}(dy)& =\frac{\sin(\pi\alpha\rho)\Gamma(1-\alpha\rho)}{2^{\alpha}\Gamma(\alpha\hat\rho)\Gamma(1-\alpha)[\sin(\pi\alpha\rho) + \sin(\pi\alpha\hat\rho)]} \left\{y^{-\alpha} (1-y)^{\alpha\hat\rho -1}(1+y)^{\alpha\rho} +y^{-\alpha}(1-y)^{\alpha\hat\rho}(1+y)^{\alpha\rho-1} \right\}\\ &=\frac{2^{-\alpha}\pi}{\Gamma(\alpha\rho)\Gamma(\alpha\hat\rho)\Gamma(1-\alpha)[\sin(\pi\alpha\rho) + \sin(\pi\alpha\hat\rho)]} \left\{y^{-\alpha} (1-y)^{\alpha\hat\rho -1}(1+y)^{\alpha\rho} +y^{-\alpha}(1-y)^{\alpha\hat\rho}(1+y)^{\alpha\rho-1} \right\} \end{align*} The proof is completed by taking account of the the time change in the representation (2) in the limit (see for example the discussion at the bottom of p240 of \cite{Walsh} and references therein) and noting that, up to normalisation by a constant, $K$, \[ \mu_j(dy) = Ky^\alpha \tilde\mu_j(dy), \] for $j = \pm1$. Note that \begin{align*} 1 &=\int_{-1}^1\mu(dy) \\&= K\frac{2^{1-\alpha}\pi}{\Gamma(\alpha\rho)\Gamma(\alpha\hat\rho)\Gamma(1-\alpha)[\sin(\pi\alpha\rho) + \sin(\pi\alpha\hat\rho)]}\\ &\times \left\{ \int_0^1 (1-y)^{\alpha\hat\rho -1}(1+y)^{\alpha\rho-1} dy +\int_0^1 (1-y)^{\alpha\rho -1}(1+y)^{\alpha\hat\rho-1} dy \right\}. \end{align*} Appealing to the second hypergeometric identity in \cite{hyper}, the curly brackets is equal to \begin{align*} &\frac{1}{\alpha\hat\rho}\, {_{2}F_1}(1-\alpha\rho, 1, 1+\alpha\hat\rho;-1) + \frac{1}{\alpha\rho}\, {_{2}F_1}(1-\alpha\hat\rho,1,1+\alpha\rho;-1)\\ &=\frac{1}{\alpha\hat\rho} \frac{1}{\alpha\rho}\left( \alpha\rho\,{_{2}F_1}(1-\alpha\rho, 1, 1+\alpha\hat\rho;-1) + \alpha\hat\rho\,{_{2}F_1}(1-\alpha\hat\rho,1,1+\alpha\rho;-1)\right)\\ &=2^{\alpha-1} \frac{\Gamma(\alpha\rho)\Gamma(\alpha\hat\rho)}{\Gamma(\alpha)} \end{align*} and hence \[ K = \frac{[\sin(\pi\alpha\rho) + \sin(\pi\alpha\hat\rho)]}{\sin(\alpha\pi)}. \] In conclusion, we have that \begin{align*}
\frac{d\mu(y)}{dy} =2^{-\alpha} \frac{\Gamma(\alpha)}{\Gamma(\alpha\rho)\Gamma(\alpha\hat\rho)}\begin{cases}
(1-y)^{\alpha\hat\rho -1}(1+y)^{\alpha\rho} +(1-y)^{\alpha\hat\rho}(1+y)^{\alpha\rho-1} & \text{ if }y \in [0,1]\\
&\\
(1-|y|)^{\alpha\rho}(1+|y|)^{\alpha\hat\rho -1} +(1-|y|)^{\alpha\rho -1}(1+|y|)^{\alpha\hat\rho}
& \text{ if } y \in [-1,0) .
\end{cases} \end{align*}
as required.
$\square$
\end{document} | arXiv |
\begin{document}
\begin{titlepage} \newcommand{\HRule}{\rule{\linewidth}{0.5mm}}
\includegraphics[width = 4cm]{./figures/imperial}\\[0.5cm]
\center
\textsc{\Large Imperial College London}\\[0.5cm] \textsc{\large Department of Computing}\\[0.5cm]
\HRule \\[0.4cm] { \Large \bfseries An FPGA Accelerated Method for Training Feed-forward Neural Networks Using Alternating Direction Method of Multipliers and LSMR \\[0.3cm]} \HRule \\[1.5cm]
\begin{minipage}{0.4\textwidth} \begin{flushleft} \large \emph{Author:}\\ Seyedeh Niusha Alavi Foumani \end{flushleft} \end{minipage} ~ \begin{minipage}{0.4\textwidth} \begin{flushright} \large \emph{Supervisor:} \\ Professor Wayne Luk \end{flushright} \end{minipage}\\[4cm]
Submitted in partial fulfilment of the requirements for the MSc degree in Advanced Computing~of Imperial College London\\[0.5cm]
\makeatletter \@date \makeatother
\end{titlepage}
\pagenumbering{roman}
{\pagestyle{empty}\cleardoublepage} \setcounter{page}{1} \pagestyle{fancy} \renewcommand{1.25}{1.25}\normalsize
\begin{abstract} In this project, we have successfully designed, implemented, deployed and tested a novel FPGA accelerated algorithm for neural network training. The algorithm itself was developed in an independent study option. This training method is based on Alternating Direction Method of Multipliers algorithm, which has strong parallel characteristics and avoids procedures such as matrix inversion that are problematic in hardware designs by employing LSMR. As an intermediate stage, we fully implemented the ADMM-LSMR method in C language for feed-forward neural networks with a flexible number of layers and hidden size. We demonstrated that the method can operate with fixed-point arithmetic without compromising the accuracy. Next, we devised an FPGA accelerated version of the algorithm using Intel FPGA SDK for OpenCL and performed extensive optimisation stages followed by successful deployment of the program on an Intel Arria 10 GX FPGA. The FPGA accelerated program showed up to 6 times speed up comparing to equivalent CPU implementation while achieving promising accuracy.
\\
\textbf{Keywords:} Feed-forward Neural Network, Alternating Direction Method of Multipliers (ADMM), LSMR, FPGA, OpenCL, Fixed-point
\end{abstract}
\cleardoublepage
\begin{itemize}[label=$\sqbullet$]
\item My supervisor, Prof Wayne Luk for all his valuable help and advice and for giving me the opportunity to get involved in an exciting area of research.
\item Dr Ce Guo for all of his guidance and constructive feedback. He introduced the fascinating world of hardware for neural networks to me and patiently taught me a lot throughout the course of this project.
\item My family and my fiancé for their unconditional love and support.
\end{itemize}
{\pagestyle{empty}\cleardoublepage}
\fancyhead[RE,LO]{\sffamily {Table of Contents}} \doublespacing \tableofcontents
\renewcommand{1.25}{1.25}\normalsize
{\pagestyle{empty}\cleardoublepage} \pagenumbering{arabic} \setcounter{page}{1} \newcommand{\changefont}{
\fontsize{7}{7}\selectfont} \fancyhead[LE,RO]{\changefont\slshape \rightmark \tiny} \fancyhead[LO,RE]{\changefont\slshape \leftmark \small}
\chapter{Introduction} \label{intro}
In this project, we have designed and implemented a hardware-accelerated neural network training algorithm. This project is, in fact, a continuation of an independent study option, in which a hardware-friendly approach for training neural networks using ADMM and LSMR was introduced \cite{iso}. In this work, we present an implementation of the ADMM-LSMR algorithm, which is accelerated with FPGA using OpenCL. To the best of our knowledge, this is the first hardware-accelerated implementation of an ADMM-based training method that uses LSMR to avoid matrix inversion. This implementation takes advantage of parallel characteristics of ADMM and LSMR and uses fixed-point arithmetic to suit hardware design restrictions.
\section{Motivation}
Machine learning and in particular neural networks have shown promising performance in many domains both in academia and industry. The models are getting more and more complex, and the available amount of data is rapidly growing. As a result, more sophisticated challenges are emerging in this field, and many techniques have been employed to make training algorithms more efficient and keep up with the data volume and complexity. One way to address some of these challenges is to fortify the training platforms by employing hardware acceleration or even designing custom hardware.
\\
Several approaches can be taken in order to use hardware acceleration in neural network training algorithms. Gradient-based methods, which are a commonly used category of algorithms for training neural networks, have many characteristics that complicate the use of hardware acceleration. In \cite{iso}, the ADMM-LSMR method is described as an alternative to gradient-based methods alongside a high-level Python implementation as proof of concept. This method is, in fact, orthogonal to common approaches as it is focused on the algorithm itself to be suitable for hardware design rather than implementing a hardware-accelerated variant of a conventional training algorithm. However, a method being theoretically suitable for hardware design and a Python implementation is far away from a practical and deployable hardware design.
\section{Objectives}
In this project, we improved the proposed method in \cite{iso} by applying some common techniques and made an even more hardware friendly variant of the algorithm. Then we took multiple steps to optimise the hardware design and finally, a fully operational deployment on an FPGA card using OpenCL was achieved.
\\
The key components of this project and the achievements can be enumerated as follow:
\begin{itemize}[label=$\sqbullet$]
\item Full low-level C implementation of the ADMM-LSMR method.
\item Implementation of fixed-point arithmetic with four different rounding methods.
\item Implementation of 16-bit and 32-bit fixed-point variants of LSMR method to be more hardware efficient.
\item Primary design and implementation of both CPU (host) side and device (FPGA) side of an OpenCL program and applying detailed amendments to emulate the program successfully.
\item Deployment of the design on Intel Arria® 10 GX FPGA on Intel DevCloud stack which required another set of design amendments to achieve a successful deployment.
\item Applying multiple stages of optimisation to increase speed and maximise FPGA board utilisation.
\item Conducting experiments to assess the accuracy and efficiency of the final hardware-accelerated program.
\end{itemize}
\section{Outline}
In the background chapter of this report, first, we briefly described neural networks, ADMM optimisation method and LSMR algorithm. Next, we explored common approaches in hardware acceleration of neural networks followed by a description of FPGAs and their structures. Then, usage of FPGAs in neural network training acceleration is explored. Finally, an overview of the employed technologies, including OpenCL and Intel FPGA development stack is provided.
\\
In chapter \ref{ch3}, software and algorithmic aspects of the implementation are described. First, the ADMM-LSMR algorithm \cite{iso} is explained, and it is demonstrated how and why the method is a perfect candidate for hardware acceleration. Later, details of the low-level implementation of the method, fixed-point arithmetic and their combination are described.
\\
In chapter \ref{ch4}, the route taken to convert the C implementation to an OpenCL accelerated program and finally a fully working FPGA deployment is explained. A separate section is dedicated to applied optimisation techniques and their outcome.
\\
Chapter \ref{ch5} contains the results of experiments conducted to assess the accuracy and time efficiency of the implemented method.
\\
Finally, the report is closed on chapter \ref{ch6}, providing the conclusion and potential areas to be improved.
\chapter{Background} \label{ch2} \section{Artificial Neural Networks} \label{Background-ANN}
The main goal of the neural networks is to find a function $f$ that best approximates some target function $f^*$. This goal is achieved through a process called \textit{training}, where the network learns a set of parameters from the input data. Using the learned parameters to predict the output of a new data is called \textit{inference}.
\\ Two key components of a neural network are neurons and layers. Layers are a collection of neurons, and the network is composed by connecting these layers. Different approaches for connecting layers together leads to different types of neural networks like feed-forward neural networks, convolutional neural networks and recurrent neural networks. Each of these networks is suitable for a specific set of applications \cite{Deep}. \begin{gather}
\label{NNopt}
\min_{W} \ell(f(x_0,W), y) \end{gather} The optimisation problem of a neural network can be written as equation \ref{NNopt}, where $\ell$ is a loss function, and $W$ is the learnable parameter. As we mentioned, the goal is to learn $W$ such that we can minimise the difference between the output of $f$ given the input $x_0$ and the actual output $y$.
\subsection{Feed-forward Neural Networks} \label{FNN} \begin{figure}
\caption{A simple feed-forward neural network }
\label{fig:fnn}
\end{figure} In a feed-forward neural network, all of the neurons of a layer are connected to all of the neurons of the next layer as it is shown in figure \ref{fig:fnn}. This type of neural network is called feed-forward as the information always flows forward in them. It has been proposed that a feed-forward neural network with one hidden layer can approximate any continuous function, but the required hidden size may be significantly large and that would make the process of learning impossible \cite{Deep}. Feed-forward neural networks are suitable for unstructured data for example when the data is not time-dependent or sequential.
\\ Following statements hold for a three-layer feed-forward neural network: \begin{gather*}
\text{Input data }\hspace{5pt} x_{0} \in {\rm I\!R}^{D*N} \\
W_1 \in {\rm I\!R}^{HS*D} \\ z_1= W_1x_0 ,\hspace{5pt} z_1\in {\rm I\!R}^{HS*N} \\
\text{Input of hidden layer }\hspace{5pt} x_1 = h_1(z_1) \in {\rm I\!R}^{HS*N} \\
W_2 \in {\rm I\!R}^{HS*HS}
\\
z_2= W_2x_1 ,\hspace{5pt} z_2\in {\rm I\!R}^{HS*N} \\
\text{Input of last layer}\hspace{5pt} x_2 = h_2(z_2) \in {\rm I\!R}^{HS*N} \\ W_3 \in {\rm I\!R}^{OS*HS}
\\
\text{Output}\hspace{5pt} z_3= W_3x_2 ,\hspace{5pt} z_3\in {\rm I\!R}^{OS*N} \end{gather*} Where we have $N$ training samples, $D$ features, $HS$ number of neurons in the hidden layers and $OS$ is the dimensions of output. $h_l$ is the activation function of layer l. This notation is used in this report.
\subsection{Gradient-Based Methods} \label{Background:gradientbased} Gradient-based methods \cite{ruder2016overview} are the most common optimisers which are used alongside back-propagation \cite{rumelhart1986learning} to solve the optimisation problem of neural networks. These iterative algorithms use the first-order derivative of the objective function to move towards the optimal solution.
\\ Many variants of gradient-based methods have been proposed. Such as SGD \cite{bottou1991stochastic} \cite{bottou2012stochastic}, AdaDelta \cite{zeiler2012adadelta}, AdaGrad \cite{duchi2011adaptive}, Nadam and Adam \cite{kingma2014adam} which is the most popular and most commonly used. In general, it is observed that these methods suffer from several limitations, such as:
\subsubsection{Vanishing and Exploding Gradient}
This problem which occurs as a result of repeated matrix multiplication, is known as one of the fundamental limitations of the gradient-based methods.
Multiplying small values of gradient several times results in a very small value that can slow down or even stop the training process. This defect is called vanishing gradient.
On the other hand, exploding gradient happens as a consequence of multiplying big values of gradient multiple times. This can make the learning process extremely unstable. In Recurrent Neural Networks, this becomes even more crucial \cite{bengio1994learning}. Proposed methods to reduce the effect of this problem include changing the architecture to Long Short Term Memory networks \cite{hochreiter1997long} and using Rectified Linear Units \cite{nair2010rectified} for activation function which helps with vanishing gradient and also clipping gradients to mitigate exploding gradients.
\subsubsection{Sequential Dependency}
Gradient-based methods are sequential by nature. In these methods the gradient computation of a batch can only start after the computation of the previous batch has been completed and weight updates take place when gradients of a batch are available. This characteristic makes the gradient-based methods not a suitable candidate for parallel implementation. This is even a more critical issue in FPGA implementation of neural networks when the parallelisation involves hardware pipelines and an algorithm with fewer dependencies and the ability to be pipelined is desired.
\subsubsection{Converging to Local Minima or Saddle Points}
Generally, the optimisation problem of neural networks is non-convex. Therefore, converging to local minima or saddle points is another concern. Also, it has been discussed that in higher dimensions, saddle points can cause more crucial issues \cite{dauphin2014identifying}.
\subsubsection{Sensitivity to Ill-conditioning}
Another common issue in training neural networks using gradient-based methods is ill-conditioning. When the Hessian of the objective function is ill-conditioned, it can drastically affect the convergence rate of gradient-based methods and slow down the training process \cite{chong1996chong}.
\section{Alternating Direction Method of Multipliers} \label{Background-ADMM}
Recently, Alternating Direction Method of Multipliers (ADMM) \cite{gabay1976dual} has been used as an optimisation method in a wide variety of applications \cite{lin2013design} including machine learning and neural networks \cite{kiaee2016alternating}. This powerful iterative optimisation method breaks down the objective function into smaller pieces that can be solved easier \cite{boyd2011distributed}.
\\ This method can be applied in parallel, and this characteristic makes it a suitable replacement for gradient-based methods in the optimisation problem of neural networks. The inherent parallelism of ADMM also makes it a good option for hardware implementation. However, as we discussed in \cite{iso} this method includes matrix inversion, which is not a hardware friendly operation. ADMM is a combination of dual decomposition and method of multipliers. We have explored the mathematical details of these methods in \cite{iso}.
\begin{comment} ADMM is a combination of dual decomposition and method of multipliers. To elaborate ADMM, first we briefly discuss these two algorithms in the following sections.
\subsection{Dual Decomposition} Consider the following optimisation problem: \begin{gather} \label{DDopt}
\min f(x)
\\
\text{subject to } Ax=b \nonumber \end{gather} Where $x \in {\rm I\!R}^{n}$, $A \in {\rm I\!R}^{m*n}$, $b \in {\rm I\!R}^{m}$ and $f$ from ${\rm I\!R}^{n}$ to $ {\rm I\!R}$ is a convex function.\\ The Lagrangian function associated with the optimisation problem \ref{DDopt} is: \begin{gather}
L(x,\lambda) = f(x) + \lambda^T(Ax-b) \end{gather} Where $\lambda \in {\rm I\!R}^{m}$. In order to perform dual ascent method on \ref{DDopt}, we use gradient ascent to solve the dual problem \ref{dual}. \begin{gather} \label{dual} \max g(\lambda) \\ g(\lambda) = inf_x L(x,\lambda) \end{gather} Where $g(\lambda)$ is the dual function.
\\ Assuming that $L(x,\lambda^*)$ has only one minimiser ($f$ is strictly convex), the primal optimal point $x^*$ is calculated using the dual optimal point $\lambda^*$: \begin{gather} x^* = \argmin_x L(x,\lambda^*) \end{gather} To apply gradient ascent on the dual problem we have to iteratively update $\lambda$. Assuming that $g$ is differentiable we can write: \begin{gather}
\lambda^{k+1} = \lambda^k + \alpha^k \nabla g(\lambda^k) \end{gather} Where $\alpha^k > 0 $ is the step size at iteration $k$. \\ We can compute $\nabla g(\lambda)$ from the following: \begin{gather}
\nabla g(\lambda^k) = A\hat{x} - b \\
\hat{x} = \argmin_x L(x,\lambda^k) \end{gather} In summary, the dual ascent method is two iterating updates: \begin{gather}
\label{xmin}
x^{k+1} = \argmin_x L(x,\lambda^k)
\\
\lambda^{k+1} = \lambda^k + \alpha^k(Ax^{k+1}-b) \end{gather} Now assume that the objective function $f$ is separable: \begin{gather}
f(x) = \sum_{i=1}^N f_i(x_i) \\
x = (x_1,...,x_N) \end{gather} Then the Lagrangian function is also separable in $x$: \begin{gather}
L(x,\lambda) = \sum_{i=1}^N L_i(x_i,\lambda) \\
L_i(x_i,\lambda) = f_i(x_i) + \lambda^TA_ix_i - (1/N)\lambda^Tb \\
A = [A_1,...,A_N] \end{gather} In this case, we can split \ref{xmin} into $N$ minimisations that are independent and can be computed in parallel. This algorithm is called dual decomposition \cite{everett1963generalized}: \begin{gather}
x_i^{k+1} = \argmin_{x_i} L_i(x_i,\lambda^k), \hspace{5} i = 1,...N
\\
\lambda^{k+1} = \lambda^k + \alpha^k(Ax^{k+1}-b) \end{gather} This algorithm can be used to solve large problems with numerous strong assumptions. \subsection{Method of Multipliers} Applying the dual ascent method on the augmented Lagrangian of an optimisation problem is called method of multipliers \cite{hestenes1969multiplier} . Augmented Lagrangian methods are used to make the dual ascent algorithm converge under milder assumptions (to be specific we can eliminate the assumption of convexity of $f$ ).
\\ The augmented Lagrangian associated with the optimisation problem \ref{DDopt} is: \begin{gather}
L_p(x,\lambda) = f(x) + \lambda^T(Ax-b) + (p/2)||Ax-b||_2^2 \end{gather} Where $p>0$ is the penalty term. The augmented Lagrangian of \ref{DDopt} can be seen as the standard Lagrangian for the equivalent problem \ref{DDequi}: \begin{gather} \label{DDequi}
\min f(x) +(p/2)||Ax-b||_2^2
\\
\text{subject to } Ax=b \nonumber \end{gather} We can write the dual function: \begin{gather}
\label{pdual}
g_p(\lambda) = inf_x L_p(x,\lambda) \end{gather} \ref{pdual} can be shown to be differentiable under milder assumptions compare to the original problem. By applying the dual ascent method with step size equal to $p$ we have: \begin{gather}
\label{xminp}
x^{k+1} = \argmin_x L_p(x,\lambda^k)
\\
\lambda^{k+1} = \lambda^k + p(Ax^{k+1}-b) \end{gather}
It worth mentioning that when $f$ is separable we can not conclude that augmented Lagrangian is also separable. Therefore we can not break the minimisation step of the algorithm \ref{xminp} into subproblems that can be solved in parallel. \subsection{ADMM Algorithm} In ADMM minimisation over different variables is separated.Consider the following optimisation problem: \begin{gather} \label{admmopt}
\min f(x) + g(z)
\\
\text{subject to } Ax + Bz = c \nonumber \end{gather} Where $x \in {\rm I\!R}^{n}$, $z \in {\rm I\!R}^{m}$, $A \in {\rm I\!R}^{p*n}$, $B \in {\rm I\!R}^{p*m}$, $c \in {\rm I\!R}^{p}$ and $f$ and $g$ are convex functions. The augmented Lagrangian associated with \ref{admmopt} is: \begin{gather}
L_p(x,z,\lambda) = f(x) + g(z)+ \lambda^T(Ax + Bz - c) + (p/2)||Ax + Bz - c||_2^2 \end{gather} The method of multipliers for \ref{admmopt} can be written as: \begin{gather}
\label{jointly}
(x^{k+1} , z^{k+1} ) = \argmin_{x,z} L_p(x,z,\lambda^k)
\\
\lambda^{k+1} = \lambda^k + p(Ax^{k+1} + Bz^{k+1} - c ) \end{gather} In \ref{jointly} we minimise over $x$ and $z$ jointly. In case of ADMM algorithm the minimisation over $x$ and $z$ is separated (minimise over $x$ while holding $z$ fixed and vice versa). Each iteration of the ADMM algorithm to solve the problem \ref{admmopt} includes three updates: \begin{gather}
\label{admm}
x^{k+1} = \argmin_x L_p(x,z^k,\lambda^k)
\\
z^{k+1} = \argmin_z L_p(x^k+1,z,\lambda^k)
\\
\lambda^{k+1} = \lambda^k + p(Ax^{k+1} + Bz^{k+1} - c ) \end{gather} \end{comment}
\section{LSMR} \label{Background-LSMR}
LSMR \cite{fong2011lsmr} is an iterative method that solves linear systems and least-square problems like \ref{linearsystem} where $A \in {\rm I\!R}^{m * n}$, $b \in {\rm I\!R}^{m}$ and $x \in {\rm I\!R}^{n}$. This method is based on the Golub-Kahan bidiagonalization process \cite{golub1965calculating}, which is shown in pseudo-code \ref{alg:GKB}. Other iterative least-square solvers also exist which are based on Golub-Kahan bidiagonalization process \cite{paige1982lsqr} \cite{estrin2019lslq}. The difference between these methods is usually their early stopping criterion. These algorithms are sequential in principle, but it is worth mentioning that they can be parallelized for solving problems where $X$ and $B$ are matrices as each column can be processed separately. This characteristic is desirable for hardware implementation since it allows pipelining and parallelisation. \begin{gather} \label{linearsystem}
Ax = b \\ \nonumber
\min_x ||Ax -b ||_2 \end{gather}
\begin{algorithm} \SetAlgoLined \KwIn{$A \in {\rm I\!R}^{m * n} , b \in {\rm I\!R}^{m}$ }
$\beta_1 \leftarrow ||b||_2 $\\
$u_1 \leftarrow b/\beta_1$ \\
$\alpha_1 \leftarrow ||A^T u_1||_2$ \\
$v_1 \leftarrow A^T u_1/\alpha_1 $\\
\For{$k =1,2,...$}
{
$\beta_{k+1} \leftarrow ||A v_k - \alpha_k u_k||_2$ \\
$u_{k+1} \leftarrow (A v_k - \alpha_k u_k )/ \beta_{k+1}$\\
$\alpha_{k+1} \leftarrow ||A^T u_{k+1} - \beta_{k+1} v_k||_2$ \\
$v_{k+1} \leftarrow (A^T u_{k+1} - \beta_{k+1} v_k )/ \alpha_{k+1}$
}
\caption{Golub-Kahan Bidiagonalization Process \cite{estrin2019lslq}} \label{alg:GKB} \end{algorithm}
\section{ADMM for Neural Networks } \label{Background-ADMMNN}
The implemented ADMM-LSMR method in \cite{iso} and this work, is based one an ADMM-based training method proposed in \textit{"Training Neural Networks Without Gradients: A Scalable ADMM Approach"} \cite{taylor2016training}. Here, their work is briefly discussed using the notation provided in section \ref{FNN}.
\\ To utilise ADMM for solving the optimisation problem of neural networks, the key idea proposed in \cite{taylor2016training} is to use a variable called pre-activation $z_l$ for each layer $l$. This will enable us to decouple the weights from the activation function and rewrite the optimisation problem of an $L$ layer neural network to the following:
\begin{gather}
\label{admmNN}
\min_{W_l , x_l,z_l} \ell(z_L, y)
\\
\text{subject to }\hspace{5pt} z_l = W_lx_{l-1}, \text{ for } l = 1,2,...L \nonumber
\\ x_l = h_l(z_l), \text{ for }l = 1,2,...L-1\nonumber \end{gather} The augmented Lagrangian of \ref{admmNN} can be written as: \begin{gather}
\ell(z_L, y) + \beta_L||z_L - W_Lx_{L-1}||_2^2 \\ \nonumber + \sum_{l=1}^{L-1}[\gamma_l||x_l - h_l(z_l)||_2^2 + \beta_l||z_l - W_lx_{l-1}||_2^2] \\ \nonumber +
\sum_{l=1}^{L-1} \lambda_l^T(z_l - W_lx_{l-1}) + \delta_l^T(x_l - h_l(z_l)) \\ \nonumber +
\lambda_L^T(z_L - W_Lx_{L-1}) + \delta_L^T(x_L - h_L(z_L)) \end{gather} Where $\lambda_l$ and $\delta_l$ are vectors of Lagrangian multipliers and $\gamma_l$ and $\beta_l$ are penalty parameters. The proposed method in \cite{taylor2016training} uses just one Lagrangian multiplier vector since they observed that by applying the classic ADMM where each constant has its own Lagrangian vector, the method would be unstable. This results in \ref{ADMML} where the only Lagrangian multiplier vector is $\lambda$
\begin{gather}
\label{ADMML}
\ell(z_L, y) + \beta_L||z_L - W_Lx_{L-1}||_2^2 \\ \nonumber + \sum_{l=1}^{L-1}[\gamma_l||x_l - h_l(z_l)||_2^2 + \beta_l||z_l - W_lx_{l-1}||_2^2] \\ \nonumber +
\lambda^T(z_L - W_Lx_{L-1}) \end{gather}
Pseudo-code of their proposed method is provided in algorithm \ref{alg:ADMM-NN}. In this method, variables are updated one at a time while the others are fixed. Minimisation steps of this algorithm are explained in the following sections.
\begin{algorithm} \SetAlgoLined
\While{not converged}{
\For { $l =1,2,... L-1$} {
$W_l \leftarrow z_lx_{l-1}^\dagger$ \\
$x_l \leftarrow ( \gamma_l I +\beta_{l+1}W_{l+1}^T W_{l+1}) ^{-1}(\gamma_lh_l(z_l) + \beta_{l+1}W_{l+1}^T z_{l+1}) $
\\
$z_l \leftarrow \argmin_z {\gamma_l||x_l - h_l(z_l)||_2^2 + \beta_l||z_l - W_lx_{l-1}||_2^2 }$
}
$W_L \leftarrow z_Lx_{L-1}^\dagger$ \\
$z_L \leftarrow \argmin_z { \ell(z_L, y) + \beta_L||z_L - W_Lx_{L-1}||_2^2 + \lambda^T(z_L - W_Lx_{L-1})}$\\
$\lambda \leftarrow \lambda + \beta_L(z_L - W_Lx_{L-1})$
} \caption{ADMM for Neural Networks \cite{taylor2016training}} \label{alg:ADMM-NN} \end{algorithm} \subsubsection{Weight Update} Solution of minimising \ref{ADMML} with respect to $W_l$ can be written as \ref{weightupdate} where $x_{l-1}^\dagger$ is the pseudo-inverse of the matrix $x_{l-1}$. \begin{gather} \label{weightupdate} W_l \leftarrow z_lx_{l-1}^\dagger \end{gather} \subsubsection{Activation Update} $x_l$ is updated using the equation \ref{activationupdate} in each in each step. Details of how this equation is derived of are discussed in \cite{iso}.
\begin{gather}
\label{activationupdate}
x_l \leftarrow ( \gamma_l +\beta_{l+1}W_{l+1}^T W_{l+1}) ^{-1}(\gamma_lh_l(z_l) + \beta_{l+1}W_{l+1}^T z_{l+1}) \end{gather}
\subsubsection{Output Update}
The new value of $z_L$ is calculated using the optimisation problem \ref{outputupdate}. This optimisation problem is non-convex and non-quadratic because of the activation function $h$. However, it can be solved easily in closed form when $h$ is piece-wise linear since the activation function works element-wise on its inputs.
\begin{gather} \label{outputupdate}
\argmin_z {\gamma_l||x_l - h_l(z_l)||_2^2 + \beta_l||z_l - W_lx_{l-1}||_2^2 } \end{gather} \subsubsection{Lagrangian Multiplier Update} The Lagrangian multiplier is updated using the following equation: \begin{gather}
\lambda \leftarrow \lambda + \beta_L(z_L - W_Lx_{L-1}) \end{gather}
\section{Hardware for Neural Networks} \label{Background-HWNN}
As the amount of available data and also the complexity of neural networks are increasing, computation and storage cost of these models are growing rapidly. In some cases, these requirements have made the use of large neural networks impossible, especially in applications where low power consumption or small latency is critical. As a result, choosing and designing efficient computing platforms for neural network applications is becoming more critical than ever \cite{guo2017survey} \cite{sze2017hardware}.
\\ Training or inference of neural networks on general-purpose CPUs with von Neumann architecture is inefficient since a significant amount of MAC operations are involved in these processes. CPUs neither have high performance in this area nor low power consumption and are not suitable for either cloud or mobile applications of neural networks. Also, breakdown of Dennard scaling, failure to increase the clock frequency, and the low rate of data transfer between CPU and memory, known as von Neumann bottleneck, have made the use of custom hardware architectures for neural networks more interesting.
\\ GPUs have a higher arithmetic density compared to CPUs. As a result, nowadays, neural networks are usually trained on GPUs which have very high power consumption. The need for low-power and efficient platform for training neural networks has led to a significant increase in research in using custom hardware architecture for neural networks in the last decades \cite{farabet2011large} \cite{chen2014dadiannao} \cite{esser2015backpropagation}.
\subsection{Motivation} One of the main reasons for interest in the use of custom hardware is exploiting the inherent parallelism of neural networks. Also, as we mentioned, von Neumann bottleneck and breakdown of Dennard scaling result in limitations for CPUs and highlight the need for more efficient hardware platforms. Computations of neural network models usually take place in the cloud, but, there are some applications where local embedded processing is preferred because of privacy. In these cases, small footprint and low power consumption become more important. These two factors are also critical in wearable or implantable medical devices. Other important motivations are increasing the speed of computation and decreasing the latency, which is always desirable and also is critical in applications like autonomous vehicles and robotics. It has been observed that with custom chips, we can achieve much faster neural networks compared to von Neumann architectures \cite{sze2017hardware} \cite{schuman2017survey}.
\\ Considering the above, the main motivations behind the growing interest in hardware implementations of neural networks can be summarised as: \begin{itemize}[label=$\sqbullet$]
\item Parallelism
\item CPU limitations
\item Low power consumption
\item Small footprint
\item High Speed \end{itemize} \subsection{Approaches} Key factors that should be considered when implementing hardware-based neural networks are the following \cite{sze2017hardware}: \begin{itemize}[label=$\sqbullet$]
\item Accuracy: A common measure that demonstrates how performant a neural network is.
Accuracy of neural networks should not be compromised for the hardware implementation.
\item Power consumption: The energy consumed by the platform. Data movements usually cost more energy than computation.
\item Throughput/latency: How much data can be processed at a time and how fast can the network respond to queries. Latency is more important in inference.
\item Cost: Is determined by many factors. If a small number of chips are needed, ASIC design costs much more than FPGA. Also, the complexity of the circuit and the amount of memory required on the chip have a direct impact on the cost. \end{itemize}
Various techniques have been developed to maintain a trade-off between these factors while making neural networks more hardware-compatible. These techniques usually aim to reduce data movement, computation and required storage on the chip, while maintaining the accuracy.
Hardware implementation of neural networks can be split into three major categories \cite{schuman2017survey} \cite{girau2006fpna}: \begin{itemize}[label=$\sqbullet$]
\item Analog: ASIC and FPAA (Field Programmable Analog Arrays).
\item Digital: ASIC and FPGA.
\item Mixed Analog/Digital systems. \end{itemize} Analog ASIC designs are fast and dense with low power consumption, but, they are expensive and lack flexibility. In general analog designs are noisier than digital designs, and they may suffer from problems such as not being precise and robust along with data storage problems. Another problem with FPAAs is that currently there are very few FPAA manufacturers and their on-chip resources, which are critical in neural network implementations, are much less than FPGAs. Digital ASIC designs provide more accuracy and robustness compared to analog ASIC, but again they are very expensive and not flexible with a very time-consuming and challenging development process. There is also ongoing research to implement mixed analog/digital circuits for networks. These systems have the overhead of ADC and DAC conversion \cite{sze2017hardware} \cite{girau2006fpna}.
\\ FPGAs usually are less performant comparing to ASIC designs in term of area, power and speed. On the other hand, they have a faster design process and are less expensive. These reconfigurable platforms also benefit from increased processing density (greater performance per unit of silicon area) compared to general-purpose processors, and they can have better cost:performance ratio compared to both ASIC and general purpose-processors. In addition, FPGAs have the advantage of being reconfigurable, which means that they are flexible and can be programmed to be used on different neural networks on-demand \cite{omondi2006fpga} \cite{liu2009survey} \cite{moussa2006arithmetic}.
\section{Field-programmable Gate Arrays} \label{Background-FPGA}
\begin{figure}
\caption{FPGA Architecture \cite{bestpracticeguide}}
\label{fig:FPGAarchitecture}
\end{figure}
\begin{figure}
\caption{FPGA design flow}
\label{fig:FPGAflow}
\end{figure}
A field-programmable gate array or FPGA is a semiconductor integrated circuit (IC) which is made of small computation units, usually called logic blocks, connecting together with programmable interconnections. FPGAs can execute an infinite variety of functions as they can be configured over and over again. Configuring FPGAs is actually programming their logic blocks in a way that their output(s) becomes a specific function of their inputs. Because of this capability, we are able to build custom data-paths and program the dataflow directly into the hardware.
\\ A simple view of an FPGA architecture is shown in figure \ref{fig:FPGAarchitecture}. Interconnections have the task of connecting the logic blocks and making the flow of signals inside the chip possible. A logic block usually consists of lookup tables (LUT), flip flops (FF) and multiplexers. The primary structure of FPGA is a two-dimensional array of logic blocks, interconnections and I/O blocks. These days FPGAs usually include on-die processors, RAM blocks, digital signal processors (DSP) and embedded multipliers.
\\ The main advantages of FPGAs can be enumerated as following:
\begin{itemize}[label=$\sqbullet$]
\item They are programmable, and their functionality can be changed by downloading a new configuration file into the device. FPGAs can be considered as platforms that can implement a custom instruction set for a target application. This is while multiple instructions must be combined to perform the same operations in CPUs, DSPs or GPUs.
\item One of the main advantages of FPGAs is their support for pipeline implementations. In FPGAs, the parallelism is not necessarily achieved by replication of compute units. The pipelining approach allows parallelism while maximising hardware utilisation \cite{bestpracticeguide}.
\item Besides being re-programmable, FPGA design process is considerably faster and easier than ASIC design. The development cost is also significantly lower compared to ASIC. \end{itemize} These features have made FPGAs very popular over the past decades and they are employed in a wide range of applications like speech recognition, image processing, video compression, ASIC prototyping and medical applications.
\\ The FPGA design flow is shown in figure \ref{fig:FPGAflow}. Each step is explored in the following paragraphs.
\\ Design entry is performed by using schematic or a hardware description language (HDL). By using schematic, the designer has to design the low-level hardware. As a result, this technique can be used only in small projects while HDLs such as Verilog and VHDL can be used for more complex systems and make the design process faster. Recently, this step can also be done using higher-level programming languages like C and let the C-to-FPGA compilers translate the C code into HDL. Such translation is performed when developing FPGA accelerated programmes using OpenCL. By using higher-level languages, the designer has less control over the FPGA resources and may not be able to utilise all of the available hardware resources compared to HDL designs, but, the design process will be less time-consuming.
\\ In the synthesis step, the design is translated into a circuit using a netlist. A netlist lists the required logic elements and interconnections. First, a syntax check is applied and then an optimisation process is performed in order to eliminate redundant logic and reduce the size of the design. Next, the details of the design are planned, and the timing properties are estimated.
\\ The layout of the design is determined in the implementation step. In this stage, the design is mapped into logic blocks of the FPGA, and then the IO blocks and logic blocks are connected.
\\ In the last step, the mapped and routed design is loaded into the FPGA using a generated bitstream file.
\\ In order to test the design, at the end of each step, a simulation can be performed. Behavioural simulation is performed before synthesis to check the functionality of the design. Functional simulation or post-synthesis simulation is a netlist level simulation which ignores the timing. In timing simulation, wiring and delays are also taken into account. This simulation usually is more time consuming but is the most accurate one \cite{xfpga} \cite{ifpga} \cite{hfpga}.
\section{FPGA for Neural Networks} \label{Background-FPGANN} FPGAs have a parallel architecture which makes them suitable for massive convolution, MAC, and other essential matrix operations in training neural networks \cite{hao2017general}. They also benefit from flexibility in design and short development process like software, while having the performance closer to ASIC designs \cite{moussa2006arithmetic}.
\\ An FPGA-based neural network system consists of two parts: CPU part (host) and FPGA part (device). These two parts are usually connected with PCIe connections. FPGAs generally have on-chip SRAM (Static Random Access Memory) which are usually not enough for storing neural network parameters, and we have to use off-chip memory. The performance of the system is usually bounded by bandwidth and power consumption of this external memory. An abstract structure of a typical FPGA implementation of a neural network is illustrated in figure \ref{fig:FPGANN}. The role of the host is to monitor the FPGA and issue commands to it. The FPGA usually has a controller which is responsible for communicating with the host and also issuing signals for other modules in FPGA. This controller can be a finite state machine or a decoder \cite{guo2017survey}.
\\ In general, two main factors limit the performance of an FPGA-based neural network: \begin{itemize}[label=$\sqbullet$]
\item On-chip resources.
\item Off-chip memory bandwidth. \end{itemize} \begin{figure}
\caption{A typical FPGA-based Neural Network \cite{guo2017survey}}
\label{fig:FPGANN}
\end{figure} The main proposed ideas in order to make neural-networks more suitable for FPGA-implementation fall into three categories \cite{guo2017survey} \cite{sze2017hardware}: \begin{itemize}[label=$\sqbullet$]
\item Reduce precision.
\item Sparsity
\item Compression \end{itemize} In the following sections, first precision reduction techniques are explained in details as this approach is employed in our implementation. Next, an overview of the other two approaches is given. There are also other techniques that do not fall into these categories. The common challenge in all the existing approaches is to make an optimal trade-off between accuracy and hardware speed and energy. For example, by reducing the size of each computation unit, we would be able to place multiple replicas of the units on the FPGA and increase the throughput. One possible way to reduce the compute unite size could be using fixed-point arithmetic which results in sacrificing the precision. It is also worth mentioning that some implementation details like data access pattern can affect the efficiency of hardware utilisation.
\subsection{Precision Reduction}
In order to meet the computation requirements of training neural networks, domain-specific accelerators which have densely packed floating-point arithmetic units are being utilised. One of the factors that limit the speed of training neural networks is the arithmetic density of hardware platforms. As a result, there is a fair amount of ongoing research to replace the floating-point arithmetic with fixed-point or even using less number of bits which increase the performance density \cite{drumond2018training}.
\\ Techniques of narrowing arithmetic are widely used in neural network customised hardware and are not specific to FPGA implementations. For example, NVIDIA TESLA V100 GPU \cite{nvidia2017v100} takes advantage of 16bit-32bit mixed-precision arithmetic and google TPU v2 and v3 \cite{jouppi2017datacenter} use Bfloat 16 which is a 16-bit floating-point representation that has been tailored for training neural networks and has a better performance compared to IEEE half-precision representation in neural network applications.
\\ One of the main issues of implementing neural networks on FPGA is selecting the best numerical precision. Single and double-precision floating-point representations decrease the quantisation error because of their high precision at the cost of a significant amount of FPGA resources. For example, in FP32 (Single precision floating-point) 24 bits are dedicated for mantissa, which results in a very high precision that is not needed for our purpose. In \cite{drumond2018training} they trained ResNet 20 \cite{he2016deep} on CIFAR-10 using floating-point representations, and they altered mantissa and exponent width to observe the validation error. Their observations can be summarised as following: 1.Convergence without precision loss, using 8-bit mantissa. 2.Convergence with a small precision loss, using 4-bit mantissa. 3.Divergence using 2-bit mantissa. They also mentioned that the exponent width could not be narrowed as it has a significant impact on the representable range. FP16 (half-precision floating-point) is denser than FP32 but still needs more hardware than fixed-point. In this representation, 11 bits are assigned to mantissa, and the remaining 5 are for the exponent. FP16 suffers from the issue of narrow representable range. On the other hand, fixed-point numbers which have less precision and narrow range, increase the quantisation error while requiring less amount of FPGA resources \cite{drumond2018training}.
\\ In \cite{micikevicius2017mixed}, it has been discussed that training with fixed-point or half-precision floating-point has mixed results because of the limited representable range. This fact makes a vital trade-off between hardware resources and precision. The precision affects the neural network accuracy and also the speed of its convergence. However, higher precision is associated with more hardware requirement. The challenge is to find an optimal point and a balance between the required precision and hardware resources.
\\ Since there is a limited amount of resources available on FPGAs, and in order to make efficient use of them, we aim to find the minimum viable precision and minimum viable range. This is equivalent to finding the maximum amount of quantisation error that can be tolerated without affecting the accuracy drastically. By using fewer bits for neural network computations, we can also reduce the bandwidth requirement, which is one of the main issues of the FPGA-based implementation of neural networks.
\\ To summarise, using fewer bits and simpler representation have the following advantages which make the FPGA implementation of neural networks more feasible: \begin{itemize}[label=$\sqbullet$]
\item Less memory requirement
\item Less computation cost
\item Less hardware requirement
\item Less bandwidth requirement \end{itemize}
Considering these advantages, using standard floating-point numbers is not the best choice and usually, more area-efficient numeric representations like 16 or 32 bit fixed-point are used. In order to use low-precision computation, we have to quantise the weights and activations of the neural network \cite{gupta2015deep}.
One of the simplest techniques is to use the nearest fixed-point number representation of each parameter. This method suffers from overflow and underflow because the range of floating-point is highly dynamic and easily exceeds the representable range with fixed-point. It has been found that the range of parameters of a neural network (weights and activation) is limited in a single layer, but, this range differs when comparing different layers \cite{qiu2016going}. It is also possible to use more bits for the first and last layer and utilise ternary or binary representations for hidden layers \cite{wang2018design}.
\\
Low-precision computation is widely used in the inference part of neural networks to make the run-time faster, and they can usually reach 32-bit floating-point accuracy \cite{umuroglu2017finn} \cite{chen2016eyeriss} \cite{ghasemzadeh2018rebnet} \cite{han2015deep} \cite{krishnamoorthi2018quantizing} \cite{choi2018pact} \cite{zhou2017incremental} \cite{anwar2015fixed}. Even binary and ternary representations have been used for inference \cite{hubara2017quantized} \cite{li2016ternary} \cite{zhu2016trained}.
\\
On the contrary, using low-precision computation in training neural networks usually has an evident negative effect on accuracy. This is mainly due to the nature of back-propagation and gradient-based methods \cite{fox2019training} \cite{siddhartha2018simultaneous} \cite{liu2017fpga} \cite{courbariaux2016binarized} \cite{wang2018training} \cite{han2017ese} \cite{wu2018training} \cite{courbariaux2014training} \cite{banner2018scalable}. For example, in stochastic gradient descent, which is a common optimiser used in neural networks, many small noisy steps take place for each parameter update. It is obvious that to keep track of these small steps, and for SGD to work at all, high precision is required \cite{courbariaux2015binaryconnect} \cite{hubara2017quantized}. One of the widely used methods to tackle this problem is to use high precision for gradient accumulation and then use lower-precision for other parts of the learning \cite{wang2018training} \cite{wu2018training} \cite{courbariaux2015binaryconnect} \cite{rastegari2016xnor}. Gradient accumulators are frequently updated during training, and the fact that storing them in low-precision adversely affects the accuracy is not desirable \cite{yang2019swalp}. Training neural networks with end-to-end low precision has been done in \cite{zhang2017zipml}, \cite{zhou2016dorefa} and \cite{koster2017flexpoint}. They have used different methods to restrict the range of activations and selecting quantisation points. It is also worth mentioning that in \cite{zhou2016dorefa}, parameters of first and last layers of networks are not quantised.
\\
In general, we can conclude that while many benefits are observed in researches on low-precision training, reduction in the accuracy is also reported\cite{de2018high}.
\subsection{Sparsification} It is possible to reduce the number of MAC operations by removing some weights of the network. There are different approaches to achieve this, including removing the weights with small absolute value \cite{han2015deep} \cite{nakahara2019fpga}, or values with minimal impact on the output. Another approach is to set the value of some weights to zero in order to remove them. This techniques are beneficial for both computation costs and required storage \cite{guo2017survey} \cite{sze2017hardware}. \subsection{Compression}
As we mentioned before, data movement and storage are two critical issues in implementing FPGA-based neural networks. Using compression techniques helps with both of these issues. In order to compress parameters of a neural network, both lossless and lossy compression can be utilised. In some cases, codes are assigned to values and a translation table should be used \cite{han2015deep} \cite{han2016eie} \cite{chen2015compressing} \cite{guo2017survey} \cite{sze2017hardware}. Using low-precision for inference of neural networks can also be considered as a type of compression. In \cite{samragh2017customizing}, they proposed a greedy algorithm to encode the parameters of networks considering the platform's memory and the accuracy required for the given task. Also, in \cite{chen2015compressing}, they used a hash function to compress and reduce the size of a trained model.
\section{Fixed-point Arithmetic} \label{Background-fixedpoint} As previously stated, fixed-point data types are widely used in FPGA implementation of neural networks. In general, fixed-point numbers can be utilised whenever performance is more critical than precision.
\\ Two parameters are associated with a fixed-point data type definition: \begin{itemize}[label=$\sqbullet$]
\item Bit width of representation. We refer to this as word length ($WL$).
\item Number of fractional bits which determines the position of the binary point. We refer to this as fraction length ($FL$). \end{itemize} We will use the notation $fixed \langle WL, FL\rangle$ in this report. We can also calculate the integer length ($IL$), the representable range ($RR$) and the smallest representable positive number ($\epsilon$) as the following: \begin{gather}
IL = WL - FL.\\
RR = [ -2^{IL-1} , 2^{IL-1} - 2^{-FL}]\\
\epsilon = 2^{-FL} \end{gather}
The $\epsilon$ is a crucial parameter in fixed-point data type and is used frequently in fixed-point arithmetic. It is important to highlight the fact that in computer a bit pattern can represent different values in different number systems. When working with fixed-point numbers we should consider both the \textbf{\textit{representation}} and the \textbf{\textit{value}} of a number. In a given $fixed\langle WL, FL\rangle$ we have $FL$ number of fraction bits in \textbf{\textit{representation}} and to interpret the \textbf{\textit{value}} of a given bit pattern we have to do the following: \begin{enumerate}
\item Calculate the value of the bit pattern in two's complement format.
\item Divide the calculated value by $2^{FL}$ (or multiply with $\epsilon$). \end{enumerate} Consider the following examples with $fixed\langle 16, 10\rangle$ data type: \begin{gather}
\epsilon \text{ \textbf{\textit{representation}}: } 0000 0000 0000 0001 \\
\epsilon \text{ \textbf{\textit{value}} : } 2 ^ {-10} \end{gather} \begin{gather}
\text{ \textbf{\textit{representation}}: } 0101 1100 1000 1001 \\ \nonumber
\text{ \textbf{\textit{value}} : } 23689 * 2^{-10} = 23.1337890625 \end{gather} \begin{gather}
\text{ \textbf{\textit{representation}}: } 1001 0001 1010 0010 \\ \nonumber
\text{ \textbf{\textit{value}} : } -28254 * 2^{-10} = -27.591796875 \end{gather}
It is also worth mentioning that the two's complement representation can be considered a special case of fixed-point representation with $FL = 0$.
\\ As we mentioned earlier, although using fixed-point data types can save a considerable amount of hardware resources and computation, the precision and representable range of a floating-point data type with equivalent word length is significantly higher. \begin{table}[H] \def1.25{1.5} \caption{Comparing representable range and smallest positive representable number in floating-point and fixed-point data types} \begin{center} \label{tab:RRepsilon}
\begin{tabular}{|l|l|l|} \hline \textbf{Data Type} & \textbf{Representable Range} & \textbf{\begin{tabular}[c]{@{}l@{}}Smallest Positive \\ Representable Value\end{tabular}} \\ \hline \textit{ \begin{tabular}[c]{@{}l@{}}32bit Floating-point\\ IEEE 754 Single Precision\end{tabular} } & $[-3.4 * 10^{38} , +3.4 * 10^{38}]$ & $1.18 * 10^{-38}$ \\ \hline $fixed\langle32,18\rangle$ & $[-2 ^{13} , 2 ^{13} - 2 ^{-18}]$ & $2 ^{-18}$ \\ \hline \end{tabular} \end{center} \end{table}
\subsection{Rounding Methods} As the precision of floating-point data types is less than fixed-point numbers, a rounding procedure is required when converting from floating-point to fixed-point. Different rounding methods can be used for this conversion. The implemented methods in this project are the following: \begin{itemize}[label=$\sqbullet$]
\item Downward rounding: Rounds to the nearest representable number which is smaller than the input.
\item Upward rounding: Rounds to the nearest representable number which is bigger than the input.
\item Nearest rounding: Rounds to the nearest representable number to the input.
\item Stochastic rounding: Rounds to the nearest representable number greater or less than the input based on a probability calculated proportionally to the distance. \end{itemize}
Given a number $x$ and a fixed-point representation $fixed \langle WL,FL \rangle$ we define $\lfloor x \rfloor$ as the largest integer multiple of $\epsilon =2^{-FL}$ less than or equal to $x$. The above methods can be mathematically defined as: \begin{gather} downwardRound(x) = \lfloor x \rfloor \end{gather}
\begin{gather} upwardRound(x) = \lfloor x \rfloor + \epsilon \end{gather}
\begin{gather}
nearestRound(x) =
\begin{cases}
\lfloor x \rfloor & \text{if $ \frac{x - \lfloor x \rfloor}{\epsilon} < 0.5$ }\\
\lfloor x \rfloor + \epsilon & \text{otherwise}\\
\end{cases} \end{gather}
\begin{gather}
stochasticRound(x) =
\begin{cases}
\lfloor x \rfloor & \text{with probability $1 - \frac{x - \lfloor x \rfloor}{\epsilon}$ }\\
\lfloor x \rfloor + \epsilon & \text{with probability $ \frac{x - \lfloor x \rfloor}{\epsilon}$ }\\
\end{cases}
\label{SR} \end{gather}
Fixed-point quantisation with stochastic rounding has shown promising results in training neural networks with gradient-based methods. However, it has the overhead of pseudo-random number generator compared to nearest rounding \cite{gupta2015deep} \cite{li2017training}. \\
\section{Development Stack}
\subsection{OpenCL} \label{Background-opencl} OpenCL (Open Computing Language) is a free standard for parallel-programming across heterogeneous processing platforms including CPUs, GPUs, DSPs, FPGAs and other processors or hardware accelerators. This framework is used to write portable yet efficient programmes. \\ The OpenCL specification \cite{khronos2012opencl} uses four models to describe OpenCL concepts: \begin{itemize}[label=$\sqbullet$]
\item Platform Model
\item Memory Model
\item Execution Model
\item Programming Model \end{itemize}
\subsubsection{Platform Model} The platform model contains a host connected to one or more devices, as shown in figure \ref{fig:platform}. Each OpenCL device consist of one or more compute units. To perform computations on the compute units of a device, relevant commands should be submitted from the host. \begin{figure}
\caption{OpenCL platform model}
\label{fig:platform}
\end{figure}
\subsubsection{Execution Model} OpenCL programmes execute in two parts: \begin{enumerate}
\item Kernels that execute on devices (CPUs, GPUs, FPGAs and DSPs)
\item Host program that executes on the host (Usually a general-purpose CPU) \end{enumerate} Unit of concurrent execution in OpenCL standard is work-item. Each work-item executes the kernel body. A single iteration of a loop is usually mapped to a work-item. Work-items are divided into work-groups.
The host program manages the execution of kernels by defining and controlling a context. The context includes the following: \begin{itemize}[label=$\sqbullet$]
\item Devices: A list of available devices.
\item Kernels: Functions that run on devices.
\item Program Objects: Kernels executable
\item Memory Objects: Memory objects that are visible to host and devices \end{itemize} The host creates one or more command-queues for each device and enqueues different commands to them to manage the execution of kernels. Different queues run independently and concurrently. Commands can be kernel execution commands, memory commands or synchronisation commands. A command queue can accept all the command types and schedules them. Commands in a command queue can execute in-order or out-of-order relative to each other. Whenever a kernel execution or a memory command is submitted to a queue, an event is created. These events can be used by the host program and other commands. Using these events, execution of different commands and their dependencies on each other can be orchestrated and synchronisation points of host program and kernels are managed.
\subsubsection{Memory Model}
Four different memory regions are defined in OpenCL: \begin{enumerate}
\item Global Memory: All work-items in all work-groups have read/write access to this memory. Host can access global memory.
\item Local Memory: This memory is local to a work-group and all work-items within a work-group have read/write access to this memory.
\item Constant Memory: A subset of global memory that does not change during the execution of a kernel.
\item Private Memory: This memory is private to a work-item and is not visible to other work-items. \end{enumerate}
\subsubsection{Programming Model} OpenCL supports data-parallel and task-parallel programming models. By using data-parallelism, we are able to apply a sequence of instruction on different elements of memory. In task parallelism, we execute a kernel using one work-item. We can then enqueue multiple tasks to achieve parallelism.
\subsection{Intel® FPGA SDK for OpenCL} \label{Background-intelopencl}
Intel FPGA SDK for OpenCL \cite{programmingguide} provides a compiler and a set of tools for building and running OpenCL programmes on Intel FPGA products. Two main components of applications implemented using Intel FPGA SDK for OpenCL are: \begin{itemize}[label=$\sqbullet$]
\item Bitstream for programming FPGA
\item Host program for managing the application flow and FPGA \end{itemize} This software development kit includes two compilers. A C++ compiler and an AOC compile. AOC is a specialised offline compiler that compiles C code written for the FPGA (OpenCL kernel) to generate emulator executable or a hardware programming image. The regular C++ compiler generates the executable that runs on the host.
\\ As it is shown in figure \ref{fig:CLmodel}, first, the offline compiler compiles OpenCL kernels to an FPGA image file with \textbf{.aocx} extension. This image file is then is used by the host to program the FPGA. The C++ compiler in the host side compiles the host program and links it to the run time libraries of Intel FPGA SDK for OpenCL. The host application, which has the task of programming and executing the hardware image onto the FPGA, is then run by the host.
\\ \begin{figure}
\caption{Diagram of the Intel FPGA SDK for OpenCL programming model \cite{programmingguide}}
\label{fig:CLmodel}
\end{figure} As we explained, in order to program an Intel FPGA the following components should work jointly: \begin{itemize}[label=$\sqbullet$]
\item The host compiler
\item The host application
\item The offline compiler
\item The OpenCL kernel(s)
\item The custom platform \end{itemize}
When a kernel is compiled, a custom dataflow circuit is generated, and acompute unit is made of different pre-optimised components including load/store units, arithmetic units and flow control units. These components are connected together depending on the dataflow that is implied by the kernel(s). One of the main advantages of Intel FPGA SDK for OpenCL is that it enables the use of FPGAs without requiring programs written in their specific programming languages such as VHDL or Verilog. Another important feature of Intel FPGA SDK for OpenCL is the detailed report generation in \textit{html} format which contains plenty of information about the resource and area usage alongside performance bottlenecks. We will discuss the information provided in this report more in section \ref{Background-AOCreport}.
\subsection{Intel AOC Compiler Report} \label{Background-AOCreport} Intel recommends using single-work item kernels which are called task instead of NDRange kernels. When a kernel is written as a task, the Intel FPGA SDK FOR OpenCL compiler is able to heavily apply pipeline parallelism to iterations of the loops to achieve high-throughput.
\\ The kernel's report.html file gives us kernel analytical data including memory and area usage and also kernel pipeline information. This report includes a summary section alongside three categories of information that can be enumerated as: \begin{enumerate}
\item Summary
\item Throughput Analysis
\item Area Analysis
\item System Viewers \end{enumerate} We discuss each these sections in more details in the following.
\subsubsection{Summary Report} Provides an overview of the design, including compile information such as target FPGA family, device and board and AOC version, basic information about kernels including the number of their compute units and their resource usage and achieved fmax of the design.
\subsubsection{Throughput Report}
The information in this section includes the fmax, bottleneck summary, loop analysis and latency estimation which aids the developer with optimising the kernel. This section is divided into two parts:
\begin{itemize}[label=$\sqbullet$]
\item \textbf{Loop Analysis:} Provides useful information for all of the loops, including whether the loops are pipelined or not, and their II (initiation interval). These information helps the developer to maximise the throughput of the designed kernel.
\\
Pipelined loops make efficient use of hardware by keeping more resources occupied and making the process of several data chunks concurrently possible.
\\
\textbf{II or initiation interval} is an important parameter in loop pipelining. When a loop is pipelined, the next iteration will begin before the previous one is finished. \textbf{II} determines the number of clock cycles that are required for launching a new iteration. This is actually the number of clock cycles that is needed to resolve dependencies between iterations of the loop.
It is obvious that smaller values of II are desirable. In the best case, II value is equal to one, which means that one loop iteration is launched every clock cycle.
\item \textbf{Fmax Report:}
The scheduled fmax of all the blocks is provided in this section. The maximum frequency at which the output of registers is updated is called the \textbf{fmax}.
The duration of the clock cycle is limited by the physical propagation delay of signals between two successive registers which is a function of the complexity of logic of the path. The path with the highest delay limits the speed of the entire circuit and is called the critical path. The \textbf{fmax} is calculated as the inverse of the critical path delay.
A high value of fmax means higher performance when there is no other bottleneck and hence is desirable. The AOC compiler tries to optimise the design to achieve the highest possible fmax. When the value of desired fmax and II are not specified in the design, the compiler uses a heuristic to achieve the best fmax/II trade-off.
\end{itemize}
\subsubsection{Area Report} This report provides details about the resource utilisation of the kernels, which helps with optimising the kernel to be more area efficient. The resource usage information is available in three levels of hierarchy: \begin{itemize}[label=$\sqbullet$]
\item System area: Resources that are utilised by all kernels, including global interconnects and board interface.
\item Kernel area: Resources that are utilised by each of the kernels of the design, including kernel dispatch logic.
\item Block area: Resources that are utilised by each of the blocks inside the kernels. Each Block is usually a branch-free section of the code like a loop body. \end{itemize}
\subsubsection{System Viewers} This section presents a graphical representation of the generated hardware. This section has three parts: \begin{itemize}[label=$\sqbullet$]
\item Graph viewer: Provides graphical report that includes information about sizes of loads and stores, latency and stalls.
\item Kernel memory viewer: Demonstrates the AOC compiler interpretation of the data movements of the kernels.
\item Schedule viewer: Illustrates the scheduled cycle and latency of a group of instructions in the design. \end{itemize}
\subsection{Intel® FPGA Devcloud} \label{Background-inteldevcloud} Intel FPGA Devcloud \cite{inteldevcloud} is an Intel hosted cloud service which provides Intel XEON processors and FPGA acceleration cards for the developers to devise and test their designs. This cloud infrastructure allows users to experiment with their designs' functionality on high-end FPGA accelerator cards. The access to this service will be granted upon request. You will benefit from remote access to Intel servers which are equipped with: \begin{itemize}[label=$\sqbullet$]
\item Latest Intel FPGA programmable acceleration cards like Intel Stratix 10 and Intel Arria 10 devices
\item Intel Core processors 6th to 8th generation
\item Intel optimised frameworks and libraries
\item Software tools needed for FPGA design, development and workload testing, including Intel FPGA SDK for OpenCL.
\end{itemize}
\chapter{Software Design} \label{ch3}
As we mentioned in the previous sections, there are some limitations in gradient-based methods that encouraged us to look for a better solution to solve the optimisation problem of neural networks. The proposed method in \cite{iso} which is based on \cite{taylor2016training}, applies ADMM \cite{boyd2011distributed} for training feed-forward neural networks in order to make this process more feasible for hardware implementation. This method implements a simple version of LSMR \cite{fong2011lsmr} as an iterative least-squares method to avoid performing matrix inversion. As it has been discussed in \cite{iso}, the key characteristics of this method are the following: \begin{itemize}[label=$\sqbullet$]
\item Since the method does not use the gradient-based optimisers, their sequential dependency is avoided, and the method is parallel by nature. To be more specific, line 2 and 3 in algorithm \ref{alg:ADMM-LSMR} (\textbf{\textit{weight\_update}} and \textbf{\textit{activation\_update}} procedures) can run in parallel since they don't have any dependencies.
\item The proposed method had the potential of being combined with fixed-point arithmetic, and since back-propagation is not used in this method, it was expected that the accuracy would not be severely affected.
\item By using an iterative least-squares method, there is no need to perform matrix inversion, which is the only obstacle for hardware-implementation of ADMM-based training method. In this implementation, LSMR is used to avoid matrix inversion and pseudo-inversion.
\item LSMR is perfectly suitable for pipeline parallelism as it is an iterative method. This can be observed in algorithm \ref{alg:LSMR}. It also allows independent computation of each column of the result. \end{itemize} The pseudo-code of ADMM-LSMR method for training feed-forward neural networks and the implemented LSMR can be seen in the algorithms \ref{alg:ADMM-LSMR} and \ref{alg:LSMR} respectively.
\begin{algorithm}
\SetAlgoLined \SetKwProg{Fn}{Function}{ } {end}
\While{not converged}{
\For { $l =1,2,... L-1$} {
$
W_l \leftarrow \textbf{ weight\_update}(z_l, x_{l-1})$\\
$x_l \leftarrow \textbf{ activation\_update}(W_{l+1}, z_{l+1}, z_l, \beta, \gamma )$\\
$ z_l \leftarrow \argmin_z {\gamma_l||x_l - h_l(z_l)||_2^2 + \beta_l||z_l - W_lx_{l-1}||_2^2 }$
\\
}
$W_L \leftarrow \textbf{weight\_update}( z_L, x_{L-1})$\\
$z_L \leftarrow \argmin_z { \ell(z_L, y) + \beta_L||z_L - W_Lx_{L-1}||_2^2 + \lambda^T(z_L - W_Lx_{L-1})}$\\
$ \lambda \leftarrow \lambda + \beta_L(z_L - W_Lx_{L-1})$
}
\Fn{weight\_update} { \KwIn{$ z_l \in {\rm I\!R}^{m * n} , x_{l-1} \in {\rm I\!R}^{p * n} $} \KwOut{$W_l \in {\rm I\!R}^{m * p}$}
\For{$i =1, 2, ..., m $}
{
$ W_l^T[:,i] \leftarrow \textbf{LSMR}( x_{l-1}^T, z_l^T[:,i])$\\
} }
\Fn{activation\_update} { \KwIn{$W_{l+1} \in {\rm I\!R}^{m * n}, z_{l+1} \in {\rm I\!R}^{m * p}, z_l \in {\rm I\!R}^{p * n}, \beta, \gamma$} \KwOut{$x_l \in {\rm I\!R}^{n * p}$}
$ part1 \leftarrow \gamma_l I +\beta_{l+1}W_{l+1}^T W_{l+1} $\\
$part2 \leftarrow \gamma_lh_l(z_l) + \beta_{l+1}W_{l+1}^T z_{l+1}$ \\
\For{$i =1, 2, ..., m $}
{
$ x_l[:,i] \leftarrow \textbf{LSMR}(part1 , part2[:,i])$\\
} }
\caption{ADMM-LSMR for Neural Networks \cite{iso}} \label{alg:ADMM-LSMR} \end{algorithm}
\begin{algorithm} \SetAlgoLined \SetKwProg{Fn}{Function}{ } {end} \Fn{LSMR}{ \KwIn{$\mat{A} \in {\rm I\!R}^{m * n} , \vec{b} \in {\rm I\!R}^{m}$ } \KwOut{$\vec{x} \in {\rm I\!R}^{n}$}
$\beta_1 \leftarrow ||\vec{b}||_2 , \hspace{50pt} \vec{u_1} \leftarrow \vec{b}/\beta_1$\\
$\alpha_1 \leftarrow ||\mat{A^T} \vec{u_1}||_2 , \hspace{25pt} \vec{v_1} \leftarrow \mat{A^T} \vec{u_1}/\alpha_1$ \\
$ \overline{\zeta_1} \leftarrow \alpha_1 * \beta_1 ,\hspace{42pt} \overline{\alpha_1} \leftarrow \alpha_1$ \\
$\rho_1 \leftarrow \overline{\rho_1} \leftarrow \overline{c_1} \leftarrow 1,\hspace{17pt} \overline{s_1} \leftarrow \zeta_1 \leftarrow 0 $\\
$\vec{h_1} \leftarrow \vec{v_1}$\\
$\vec{x} \leftarrow \vec{\overline{h_1}} \leftarrow \vec{0}$\\
\For{$k =1, 2, ..., min(m,n) $}
{
$\beta_{k+1} \leftarrow ||\mat{A} \vec{v_k} - \alpha_k \vec{u_k}||_2$ \\
$\vec{u_{k+1}} \leftarrow (\mat{A} \vec{v_k} - \alpha_k \vec{u_k} )/ \beta_{k+1}$\\
$\alpha_{k+1} \leftarrow ||\mat{A^T} \vec{u_{k+1}} - \beta_{k+1} \vec{v_k}||_2$ \\
$\vec{v_{k+1}} \leftarrow (\mat{A^T} \vec{u_{k+1}} - \beta_{k+1} \vec{v_k} )/ \alpha_{k+1}$\\
$ c_{k+1}, s_{k+1}, r_{k+1} \leftarrow \textbf{\textit{sym}} (\overline{\alpha_k} , \beta_{k+1})$ \\
$\overline{\alpha_{k+1}} \leftarrow c_{k+1} * \alpha_{k+1}$\\
$\overline{c_{k+1}}, \overline{s_{k+1}}, \overline{\rho_{k+1}} \leftarrow \textbf{\textit{sym}}(\overline{c_k} * \rho_k, s_{k+1} * \alpha_{k+1}) $\\
$\zeta_{k+1} \leftarrow \overline{c_{k+1}} * \overline{\zeta_k} ,\hspace{14pt} \overline{\zeta_{k+1}} \leftarrow - \overline{s_{k+1}} * \overline{\zeta_k}$\\
$\vec{\overline{h_{k+1}}} \leftarrow - (\overline{s_k} * \rho_{k+1} * \rho_{k+1}) / ( \rho_k * \overline{\rho_k} ) \vec{\overline{h_k}} + \vec{h_k}$\\
$\vec{x_{k+1}} \leftarrow (\zeta_{k+1} / (\rho_{k+1} * \overline{\rho_{k+1}}) \vec{\overline{h_{k+1}}} + \vec{x_k} $ \\
$ \vec{h_{k+1}} \leftarrow - ((s_{k+1} * \alpha_{k+1}) / \rho_{k+1}) \vec{h_k} + \vec{v}$
} }
\Fn{sym}{ \KwIn{ a, b } \KwOut{c, s, r} \If {abs(b) $>$ abs(a)} { $\tau \leftarrow a / b$\\
$s \leftarrow sign(b) / sqrt(1 + \tau^2)$\\
$c \leftarrow s * \tau,\hspace{15pt} r \leftarrow b / s$\\
} \Else {
$\tau \leftarrow b / a$\\
$c \leftarrow sign(a) / sqrt(1 + \tau^2)$\\
$s \leftarrow c * \tau , \hspace{15pt} r \leftarrow a / c $ \\
} }
\caption{LSMR } \label{alg:LSMR} \end{algorithm}
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\section{C Implementation} \label{achievements-c} As the first step towards a hardware implementation, a low-level C implementation was required. Each of the sub-procedures of the method including LSMR, \textbf{\textit{weight\_update}}, \textbf{\textit{activation\_update}} ,\textbf{\textit{output\_update}} and \textbf{\textit{lagrangian\_update}}) have been implemented and tested individually and a likewise comparison with Python modules implemented in \cite{iso} was performed for a sanity check. \subsection{Motivation} A Python version of the proposed method had been implemented in \cite{iso}. This implementation was heavily using the NumPy library for matrix operations, and the underlying details were out of control of the developer. First of all, C implementation is required for an OpenCL accelerated program as the host program can only be written in C or C++. Secondly, the parts which were aimed to be accelerated also should have been implemented in C because a high-level Python implementation can not be used as a reference to measure the speed up gain of the accelerated version. On the other hand, the device kernel programming language is a derivation of C language and a C implementation which is easier to be tested and modified, can be converted to an OpenCL kernel with minimal effort. \subsection{Code Structure} \label{ccode} This program takes the number of hidden layers in the network and the number of neurons in each layer as input.
\\ In order to implement algorithms \ref{alg:ADMM-LSMR} and \ref{alg:LSMR} in C, we implemented a data structure for storing matrices and also functions to perform primary matrix operations. In this implementation, we used double data type to store the data of matrices. We defined the matrix data type as a struct: \begin{lstlisting}[style=CStyle] typedef struct {
int rows;
int cols;
double * data; } matrix; \end{lstlisting} In this struct, we store the number of rows and columns of the matrix and a pointer to where elements of the matrix are stored row-wise in the memory. Numerous matrix operations were required to be implemented for this matrix datatype. There was no particular challenge associated with this implementation apart from dealing with low-level concepts of C language and avoiding memory leaks. \\
\subsection{Bottleneck Analysis} \label{achievements-ctime} Using the C implementation, we measured the execution time of each procedure in a single iteration of training of a 4 layer neural network with the hidden size of 28 on a subset of HIGGS data set \cite{HIGGS} and a 3 layer neural network with the hidden size of 8 on IRIS data set \cite{IRIS}. These measurements were performed for 3000 iterations. The percentage of execution time associated with each of the procedures is reported in tables \ref{tab:timeHIGGS} and \ref{tab:timeIRIS} and figure \ref{fig:piechart}. As it is evident from the results, the most time-consuming sub-procedures are \textit{\textbf{activation\_update}} and \textit{\textbf{weight\_update}}. Considering algorithm \ref{alg:ADMM-LSMR}, we concluded that these procedures are considerably more time-consuming because of several \textit{\textbf{LSMR}} calls. Therefore, we chose the LSMR function as the primary target for hardware acceleration.
\begin{table}[H] \def1.25{1.5} \caption{Percentage of the execution time of different procedures in one iteration on HIGGS} \begin{center} \label{tab:timeHIGGS}
\begin{tabular}{|l|l|l|l|} \hline \textit{\textbf{activation\_update}} & \textit{\textbf{weight\_update}} & \textit{\textbf{output\_update}} & \textit{\textbf{lagrangian update}} \\ \hline
54.35 \% & 39.46 \% & 6.12 \% & 0.05 \% \\ \hline \end{tabular} \end{center} \end{table}
\begin{table}[H] \def1.25{1.5} \caption{Percentage of execution time of different procedures in one iteration on IRIS} \begin{center} \label{tab:timeIRIS}
\begin{tabular}{|l|l|l|l|} \hline \textit{\textbf{activation\_update}} & \textit{\textbf{weight\_update}} & \textit{\textbf{output\_update}} & \textit{\textbf{lagrangian update}} \\ \hline
68.06 \% & 29.98 \% & 4.69\% & 0.21\% \\ \hline \end{tabular} \end{center} \end{table}
\begin{figure}
\caption{Piechart of execution time of different procedures in one iteration}
\label{fig:piechart}
\end{figure}
\section{Implementation of Fixed-point Arithmetic} \label{achievements-fixedpoint} Fixed-point arithmetic is widely used in FPGA implementation of neural networks, for both inference and training and is considerably faster and more efficient compared to floating-point. In the following sections, we discuss the motivations, challenges and the code structure of our implementation.
\subsection{Motivation} Since fix-point arithmetic is composed of simpler data types and operations compared to floating-point, it is widely used for general speed up optimisations and especially for hardware designs because it requires less silicon area. Nevertheless, depending on the algorithm, the disadvantage of narrower range and lower precision may adversely affect precision efficiency.
\\
In stochastic gradient descent which is a primary version of gradient-based methods, problem space of parameters is explored using small and noisy steps. Such exploration demands relatively high precision during the updates in SGD algorithm. By observing the implemented low-precision algorithms \cite{de2017understanding} and also considering the theoretical upper bound on the performance of low-precision SGD \cite{de2015taming} we can conclude that precision-accuracy trade-off has limited the performance of current training algorithms.
\\
As the ADMM-LSMR does not involve gradient calculation and by being parallel avoids heavily sequential processes which cause the required precision being accumulated, we conclude that our method would be much less vulnerable to fixed-point errors comparing to SGD.
\\ Also it has been proposed that noise in neural networks is a form of regularisation and can help the model to generalise better. This concept has been assessed in previous works like Dropout \cite{srivastava2013improving} \cite{srivastava2014dropout}, DropConnect \cite{wan2013regularization} and Binary Connect \cite{courbariaux2015binaryconnect}.
\\ Considering the above and the fact that fixed-point arithmetic requires less hardware resources and is faster, we deduced that employing this technique will probably result in an overall improvement in hardware-accelerated ADMM-LSMR algorithm. We implemented both 16bit and 32bit fixed-point with four different rounding methods.
\subsection{Challenges} The challenges in the process of developing the fixed-point arithmetic and embedding it in matrix operations can be summarised as: \begin{itemize}[label=$\sqbullet$]
\item Numerous edge cases to be considered
\item Keeping track of fraction bits in chain operations
\item Boundary checks
\item Temporary containers overflow check and precision
\item Challenging bit-wise operations and generalisation to support flexible change of fractional bits
\item Difficult testing procedure \end{itemize}
\subsection{Fixed-point Arithmetic Details } First, we defined our fixed-point data type and associated arithmetic functions. Also, we implemented a set of conversion functions for each rounding method. Next, we implemented a data structure for storing matrices with our fixed-point data type. Finally, we implemented the fixed-point version of all the relevant matrix functions. In the following sections, details of selected functions are explained. We will only demonstrate the detail for stochastic rounding method as it was the most complicated version. We also only discuss the implementation of 32bit fixed-point. 16bit implementation is similar to the 32bit version with minor differences.
\subsubsection{Fixed-point Data Type} Arithmetic on fixed-point numbers is almost identical to integers with some adjustment. We used types int32\_t and int16\_t to define 32bit and 16bit fixed-point data types.
\begin{lstlisting}[style=CStyle]
#define WL 32 // Word Length
#define Fixed int32_t
#define FL 18 // Fraction Length
#define IL WL - FL // Integer Length
#define Epsilon pow(2, - FL)
#define Ubound_value (float) (pow(2, IL - 1) - pow(2, -FL))
#define Lbound_value (float) (- (1 << (IL - 1)))
#define Ubound (Fixed) ((1LL << (WL - 1)) - 1)
#define Lbound (Fixed) (- (1 << (WL - 1)))
#define ONE_F (Fixed)(1 << FL)
#define MINUS_ONE_F ((Fixed)((1 << FL) - 1)) ^ 0xFFFFFFFF \end{lstlisting}
We also defined important constants like $WL$ , $FL$ ,$IL$. These constants define the fixed-point data type $fixed\langle WL, FL\rangle$. Another important constant is $\epsilon$. As we mentioned in section \ref{Background-fixedpoint}, $\epsilon$ is the smallest positive number that can be represented given a fixed-point data type.
\\ As previously stated in section \ref{Background-fixedpoint}, we should consider both the \textbf{\textit{representation}} and the \textbf{\textit{value}} when working with fixed-point numbers. \textit{Ubound} and \textit{Lbound} define the \textbf{\textit{representation}} of boundaries in our data type. Since our representation is the same as two's complement, \textit{Ubound} is represented by setting all bits except the leftmost bit to \textit{1}, and the \textit{Lbound} is represented by setting the left most bit to \textit{1} and the others to zero. \textit{Ubound\_value} and \textit{Lbound\_value} store the \textbf{\textit{value}} of boundaries in float. \textbf{\textit{Representation}} of $1$ and $-1$ are also stored in ONE\_F and MINUS\_ONE\_F.
\\ Considering $fixed\langle 32, 18\rangle$, which is defined above, these constant can be written as below: \begin{gather}
\text{Ubound }= 0111 1111 1111 1111 1111 1111 1111 1111 \\
\text{Lbound }= 1000 0000 0000 0000 0000 0000 0000 0000 \\
\text{ONE\_F }= 0000 0000 0000 0100 0000 0000 0000 0000 \\
\text{MINUS\_ONE\_F }= 1111 1111 1111 1100 0000 0000 0000 0000 \end{gather}
To convert a given float number to fixed-point, first, a boundary check is required. This boundary check takes place in \textit{convert} function and either saturation to boundaries is applied, or a specified rounding function is called.
\begin{gather}
\text{convert(x)} =
\begin{cases}
\text{Ubound \hspace{10pt} if x $\geq$ Ubound\_value} \\
\text{Lbound \hspace{10pt} if x $\leq$ Lbound\_value} \\
\text{round\_f(x) \hspace{10pt} otherwise }
\end{cases} \end{gather}
Two essential functions were implemented to cast 64bit fixed-point with $FL$ bits of the fraction and $2*FL$ bits of fraction to our defined data type. These two functions are frequently used in the fixed-point arithmetic. The pseudo-code of these functions can be seen in algorithm \ref{alg:cast32}.
\\ \begin{algorithm} \SetAlgoLined \SetKwProg{Fn}{Function}{ }{end} \Fn{cast\_f64\_simple} { \KwIn{int64\_t $x$ } \KwOut{Fixed out} \If{$ x \leq (\text{int64\_t})$ Lbound } {
out $\leftarrow$ Lbound \\ } \If{$x \geq (\text{int64\_t})$ Ubound } {
out $\leftarrow$ Ubound \\ } \Else {
out $\leftarrow$ (Fixed) $x$ \\
} }
\Fn{cast\_f64 } { \KwIn{int64\_t $x$ } \KwOut{Fixed out}
\If{ $x \leq ((\text{int64\_t})$ Lbound$) \ll FL$ }
{
out $\leftarrow$ Lbound \\
}
\If{ $x \geq ((\text{int64\_t})$ Ubound$) \ll FL$}
{
out $\leftarrow$ Ubound\\
}
\Else
{
diff $\leftarrow (x \& (\text{int64\_t})((1 \ll (FL)) -1 )) $\\
prob $\leftarrow 1 - \text{diff} * \epsilon $\\
\If { \text{random} $\leq \text{prob} $}
{
out $\leftarrow (\text{Fixed}) (x \gg FL)$\\
}
\Else
{
out $\leftarrow ( (\text{Fixed}) (x\gg FL) + 1 )$ \\
}
} }
\caption{Casting 64bit fixed-point to 32bit fixed-point } \label{alg:cast32} \end{algorithm} The function \textit{cast\_f64\_simple} preforms an overflow check, and then the input is either saturated to boundaries or its leftmost 32bits are discarded.
\\ The other function, \textit{cast\_f64}, is more complicated. As it can be seen in algorithm \ref{alg:cast32}, first a boundary check is performed. To perform this check, we first cast \textit{Lbound} and \textit{Ubound} to int64\_t, so they become 64 bits and then in order to align them with the input we shift them to left $FL$ times. After the boundary check, if the input is in range, we change it to fit in 32 bit. Since the input number has $2*FL$ bits of fraction, the rightmost $FL$ bits can not be presented in our target data type. We mask these bits using SHIFT and AND operations and store them in diff. We can write: \begin{gather}
x - \lfloor x \rfloor = \text{diff} * \epsilon^2 \end{gather} Where $\lfloor x \rfloor$ is defined as the largest integer multiple of $\epsilon$ less than or equal to $x$. We use $\text{diff} * \epsilon = \frac{ x - \lfloor x \rfloor}{\epsilon} $ to compute the rounded version of input using \ref{SR} formula.
\\ Three functions were also implemented to perform primary operations on our defined fixed-point data type. The pseudo-code of these functions are provided in algorithm \ref{alg:32fixprimary}.
\\ \begin{algorithm} \SetAlgoLined \SetKwProg{Fn}{Function}{ } {end} \Fn{add\_f } { \KwIn{Fixed $a$, Fixed $b$ } \KwOut{Fixed out}
\text{int64\_t }temp $\leftarrow (\text{int64\_t}) a + (\text{int64\_t})b$\\
cast\_f64\_simple(temp, out) }
\Fn{multiply\_f } { \KwIn{Fixed $a$, Fixed$ b$ } \KwOut{Fixed out}
\text{int64\_t} temp $\leftarrow (\text{int64\_t}) a * (\text{int64\_t}) b$ \\
cast\_f64(temp, out) \\ }
\Fn{divide\_f } { \KwIn{Fixed $a$, Fixed $b$ } \KwOut{Fixed out}
\text{int64\_t }temp $\leftarrow (((\text{int64\_t}) a) \ll FL) / ((\text{int64\_t}) b)$ \\
cast\_f64(temp, out)\\ }
\caption{Primary operations on fixed-point data type} \label{alg:32fixprimary} \end{algorithm} For adding two 32 bit fixed-point numbers we simply cast each of them to 64bits then add them to avoid overflow. In the end, \textit{cast\-f\_simple} function is called to fit the result in 32 bits.
\\
By multiplying two fixed-point numbers, $a$ and $b$, with $FL$ bits of fraction, we have: \begin{gather}
\text{ \textbf{\textit{representation}}: } a * b = temp \\ \nonumber
\text{ \textbf{\textit{value}} : } a * \epsilon * b * \epsilon = temp * \epsilon^2 \end{gather} So the result has $2*FL$ bits of fractions and we call \textit{cast\_f} function to perform fraction adjustment.
\\ In fixed-point by fixed-point division, assuming both numbers having $FL$ bits of fraction, the dividend should be shifted to left $FL$ times. Consider the division $a / b = result $. The value of the result should be: \begin{gather}
\text{ result \textbf{\textit{value}}: } \frac{a}{b} \end{gather}
Since the \textbf{\textit{value}} of a representation in our data type is calculated as $\textbf{\textit{representation}} * \epsilon $, the representation of the result in our data type should be: \begin{gather}
\text{result \textbf{ \textit{representation}}: } \frac{a}{b * \epsilon} = \frac{a * \epsilon^{-1}}{b} \end{gather}
To achieve this we divide $a$ (dividend) by $\epsilon$ using a SHIFT operation.
\subsubsection{Fixed-point Matrix} In order to use our implemented fixed-point data type in matrix operations, we defined a new struct: \begin{lstlisting}[style=CStyle] typedef struct {
int rows;
int cols;
Fixed * data; } fmatrix; \end{lstlisting} This struct is the same as the other matrix struct that we defined in section \ref{achievements-c}, except that the type of stored data is \textit{Fixed}. All the associated matrix operations were also implemented for fmatrix. Most of these functions are quite identical to their equivalent version with double data type matrices, with primary operations (add, multiply, divide) being replaced with fixed-point variants.
\\ On the other hand, some functions including \textit{matMul}, \textit{dot} and \textit{norm} required more modification. The pseudo-code of these functions are provided in algorithms \ref{alg:32fixmatrixmatmul}, \ref{alg:32fixmatrixdot} and \ref{alg:32fixmatrixnorm} respectively.
\\
\begin{algorithm} \SetAlgoLined \SetKwProg{Fn}{Function}{ } {end} \Fn{matMul\_f } { \KwIn{$\text{fmatrix } \mat{mat1} \in {\rm I\!R}^{m * n} \text{ , fmatrix } \mat{mat2} \in {\rm I\!R}^{n * p}$} \KwOut{fmatrix prod} \For{$ col =1, 2, ..., p $} { \For{$ row =1, 2, ..., m$}
{
$(\text{int64\_t}) \text{ sum} \leftarrow 0$ \\
\For{$ k = 1 ,2 ,... , n $ }
{
$(\text{int64\_t}) \text{ temp} \leftarrow (\text{int64\_t}) \mat{mat1}[row][k] * (\text{int64\_t}) \mat{mat2}[k][col] $\\
$ \text{ sum} \leftarrow \text{ temp} + \text{ sum} $ \\
\textit{ overflow check and saturation of} sum \\
}
Fixed result\\
cast\_f64(sum , result)\\
prod[row][col] $\leftarrow$ result\\
} }
}
\caption{Matrix multiplication of fmatrix} \label{alg:32fixmatrixmatmul} \end{algorithm}
In all of these functions MAC (Multiply and Accumulate) operation is performed. To apply this critical operation in fixed-point with minimal error, the following approached was used: \begin{itemize}[label=$\sqbullet$]
\item We store the result of each multiplication which is a fixed-point number with $2*FL$ bits of fraction and $2 *IL$ bits of integer in an int64\_t. The conversion of these results to our \textit{Fixed} data type is avoided and is delayed to the next step, but, after each addition, overflow check is applied.
\item We convert the sum of all the results to \textit{Fixed} data type using \textit{cast\_f} function since it is a 64bit fixed-point with $2*FL$ bits of fraction. \end{itemize}
\begin{algorithm} \SetAlgoLined \SetKwProg{Fn}{Function}{ } {end} \Fn{dot\_f } { \KwIn{$\text{fmatrix } \vec{v1} \in {\rm I\!R}^{1 * m} \text{ , fmatrix } \vec{v2} \in {\rm I\!R}^{m * 1}$} \KwOut{Fixed prod}
$(\text{int64\_t}) \text{ sum} \leftarrow 0$ \\ \For{$ k =1, 2, ..., m $} {
$(\text{int64\_t}) \text{ temp} \leftarrow (\text{int64\_t}) \mat{v1}[0][k] * (\text{int64\_t}) \mat{v2}[k][0] $\\
$ \text{ sum} \leftarrow \text{ temp} + \text{ sum} $ \\
\textit{ overflow check and saturation of} sum \\
cast\_f64(sum , prod)\\
}
}
\caption{Dot product of fmatrix} \label{alg:32fixmatrixdot} \end{algorithm}
\begin{algorithm} \SetAlgoLined \SetKwProg{Fn}{Function}{ } {end} \Fn{norm\_f } { \KwIn{$\text{fmatrix } \vec{v} \in {\rm I\!R}^{m * 1} $} \KwOut{Fixed n} $(\text{int64\_t}) \text{ sum} \leftarrow 0$ \\
\For{$ row =1, 2, ..., m$}
{
$(\text{int64\_t}) \text{ temp} \leftarrow (\text{int64\_t}) \vec{v}[row][0] * (\text{int64\_t}) \vec{v}[row][0] $\\
$ \text{ sum} \leftarrow \text{ temp} + \text{ sum} $ \\
\textit{ overflow check and saturation of} sum \\
}
n $\leftarrow$ integer sqrt of sum
} \caption{L2-norm of fmatrix} \label{alg:32fixmatrixnorm} \end{algorithm} A function to perform integer square root was needed in norm function. We used the proposed algorithm in \cite{intsqrt} with a little modification for this purpose.
\section{Using Fixed-point LSMR in ADMM} \label{achievement-fixedpointlsmr} As we discussed in section \ref{achievements-ctime}, LSMR function in \textbf{\textit{weight\_update}} and \textbf{\textit{activation\_update}} was the most time consuming part of the C implementation. In order to speed up the training process, we aimed to make the LSMR function faster by running it on hardware and taking advantage of both task and pipeline parallelism.
\\ As previously stated, fixed-point arithmetic is considerably more efficient comparing to floating-point. Therefore, in order to make the hardware implementation more feasible, fixed-point arithmetic was used in the LSMR module. The fixed-point version of LSMR works with fixed-point matrices and uses their relevant functions to perform matrix operations which were explained in section \ref{achievements-fixedpoint}.
\\ Also, some experiments were performed to check if the ADMM-LSMR algorithm works with low precision using different rounding methods. Results of these experiments can be found in section \ref{exp-floatvsfixed}.
\\ In summary, we observed that: \begin{itemize}[label=$\sqbullet$]
\item We were able to achieve near float accuracy using 32bit fixed-point implementation of LSMR.
\item The proposed ADMM-LSMR method failed to converge using 16bit fixed-point implementation of LSMR. \end{itemize}
\chapter{Hardware Design} \label{ch4} \section{Hardware-accelerated ADMM-LSMR} \label{achievements-HWADMMLSMR}
To achieve our final goal, which was hardware implementation of the proposed method, we used Intel FPGA SDK for OpenCL.
A Programmable Acceleration Card with Intel Arria® 10 GX FPGA was used for our design.
\\
In this implementation we were able to run \textbf{\textit{weight\_update}} and \textbf{\textit{activation\_update}} procedures in parallel and speed up or training process. To the best of our knowledge, this is the first hardware implementation of ADMM, which also uses LSMR for training neural networks. \subsection{Motivation} The proposed ADMM-LSMR method in \cite{iso} is a hardware-friendly approach for training neural networks. From the early stages of this work, our goal was to use hardware acceleration to take advantage of the inherent parallelism of this method. After implementing the fixed-point version of the method and observing its promising results, our next step was to develop an OpenCL program to perform FPGA emulation and finally run on our target FPGA card.
\\ In the following sections, the latest version of the implementation, which is a product of extensive optimisation is explained. The details of these optimisation stages are discussed in section \ref{achievements-optimisation}.
\subsection{Challenges} \begin{itemize}[label=$\sqbullet$]
\item Extreme system requirements of the development tools. Development had to be performed fully remote on department workstations or Intel Devcloud
\item Unstable emulator
\item Lack of informative error/crash reports
\item Long hardware compilation time.
\item Getting access to Intel FPGA boards and technical difficulties in working with Intel servers
\item Minor but undocumented differences between the emulator and physical FPGA \end{itemize} \subsection{OpenCL for FPGA Implementation}
As we mentioned in section \ref{Background-opencl}, an OpenCL accelerated program has two parts: \begin{enumerate}
\item A C++ program to run on the host. This program is compiled using g++.
\item An OpenCL program including kernels to run on the device which is complied using Intel AOC compiler. \end{enumerate} In the following sections, an overview of the program flow and more details of the host and device sections are explained.
\begin{figure}
\caption{Gantt chart of execution time of one iteration of ADMM-LSMR on a hidden layer with hidden size of 28 on HIGGS data set}
\label{fig:gantt}
\end{figure}
\begin{figure}
\caption{Activity diagram of one iteration of ADMM-LSMR on a single layer }
\label{fig:sequence}
\end{figure} \subsubsection{Program Flow} A high-level overview of the program can be described as: \begin{enumerate} \item Initialising OpenCL run-time and resources \item Preparing the inputs and setup network architecture \item Performing ADMM-LSMR until converged as: \begin{enumerate} \item Apply the following for each layer: \begin{enumerate} \item Performing \textbf{\textit{weight\_update}} and \textbf{\textit{activation\_update}} with LSMR call commands being sent to the FPGA. \item Wait for stage \textit{i} results and perform \textbf{\textit{output\_update}} if not in last layer \end{enumerate} \item Apply \textbf{\textit{last\_output\_update}} for last layer and \textbf{\textit{lagrangian\_update}} \end{enumerate} \item Save the results and clean up the resources \end{enumerate}
The hardware-accelerated part takes place on stage \textit{3.a}. An activity diagram of this process is shown in figure \ref{fig:sequence}. Also, a Gantt chart illustrating this stage on a single hidden layer of size 28 on HIGGS data set is provided in figure \ref{fig:gantt}.
\subsubsection{Device Program}
As it is shown in pseudo-code \ref{alg:ADMM-LSMR}, in both \textbf{\textit{activation\_update}} and \textbf{\textit{weight\_update}} procedures, the LSMR function is called inside a for loop. This is because pf the fact that original LSMR method solves least-square problems like \ref{linearsystem} and its output has the following form: \begin{gather}
x =A^{-1} b \nonumber \\
A \in {\rm I\!R}^{m * n} , b \in {\rm I\!R}^{m * 1} , x \in {\rm I\!R}^{n * 1} \end{gather} Therefore, in order to solve a problem like \ref{lsmrmatrix}, the LSMR function should be called $p$ times. These $p$ different LSMR calls are independent in relation to each other which makes them an ideal candidate to be implemented in hardware and be parallelised. \begin{gather} \label{lsmrmatrix}
X =A^{-1}B \nonumber \\
A \in {\rm I\!R}^{m * n} , B \in {\rm I\!R}^{m * p} , X \in {\rm I\!R}^{n * p} \end{gather}
Our implemented LSMR kernel takes 3 matrices as input to solve a problem of a form \ref{lsmrmatrix}. Intel AOC compiler applies pipeline parallelism when it translates the kernel to bitstream for FPGA. \begin{lstlisting}[style=CStyle] __kernel void lsmr(const int m, const int n, const int p, __global const Fixed * restrict base_input , __global Fixed * restrict base_At, __global Fixed * restrict base_output, __local Fixed * restrict base_u , __local Fixed * restrict base_v, __local Fixed * restrict base_h ,__local Fixed * restrict base_hbar, const int offset ) \end{lstlisting}
As dynamic allocation is not allowed inside the kernel, all of the allocations take place in the host and OpenCL memory objects are passed to the kernel. This memory objects can be seen as a pointer inside the kernel. These memory objects which appear in the header of the kernel are:
\begin{itemize}[label=$\sqbullet$]
\item \textit{\_\_global const Fixed * restrict base\_input}: Pointer to start of a global memory that stores inputs $A$ and $B$. We refer to this as input buffer in the host side.
\item \textit{\_\_global Fixed * restrict base\_At}: Pointer to starts of a global memory for storing $A^T$ which is a matrix that is needed in internal computing of the kernel. We refer to this as internal buffer in the host side.
\item \textit{\_\_global Fixed * restrict base\_output}: Pointer to starts of a global memory for storing the output $X$. We refer to this as output buffer in the host side.
\item \textit{\_\_local Fixed * restrict base\_u ,\_\_local Fixed * restrict base\_v, \_\_local Fixed * restrict base\_h ,\_\_local Fixed * restrict base\_hbar}: Pointers to start of local memories for storing internal matrices needed for kernel computations.
\end{itemize}
In addition to these memory objects, the kernel takes four integers. $m$, $n$ and $p$ determine dimensions of the input and output matrices.
\\ In order to utilise most of the available FPGA resources, after the extensive optimisations, we were able to fit \textbf{\textit{four compute units}} of the LSMR kernel in the target FPGA. In every iteration of training, two of these compute units are used for activation update LSMR, and the other two perform weight update LSMR. As we mentioned earlier, computing the columns of the output of LSMR are independent in relation to each other. Therefore in order to split the workload between the compute units, we made each compute unit responsible for computing half of the columns of the output. We used an offset (a const int) to tell the compute units which part of the input they should use and in which part of the output they should write.
\\ Another important matter to mention is that this implementation uses round to nearest method for fixed-point numbers. All of the inputs are converted to fixed-point in the host using the specified rounding method in that program. But, the used rounding method inside the kernel is nearest rounding.
\\ Since the inputs of LSMR were memory objects, we had to define another version of fmatrix in the kernel program. All of the matrix operation functions had to be modified consequently. Most of these functions are not used explicitly in the final version since we had to make the matrix operations inline to optimise the kernel performance and resource utilisation. The final implementation is quite complex, tangled and hard to read as a result of hardware optimisations. \subsubsection{Host Program} The host program has three main tasks: \begin{enumerate}
\item Configure OpenCL runtime and initialising OpenCL objects.
\item Perform the skeleton of the algorithm.
\item Orchestrate the acceleration and device commands. \end{enumerate}
\subsubsection{\textit{Configuration and Initialisation}} We defined a manager for OpenCL to perform all of the initial configurations of the OpenCL and also to keep track of important objects including platform, device, context, program and command queues. As we mentioned earlier, we have four compute units that we want to work in parallel. Therefore it was required to define four different command queues. \begin{lstlisting}[style=CStyle] typedef struct {
cl_platform_id platform;
scoped_array<cl_device_id> device;
cl_context context;
cl_command_queue queue0;
cl_command_queue queue1;
cl_command_queue queue2;
cl_command_queue queue3;
cl_program program;
} opencl_manager; \end{lstlisting}
We also defined a function to perform initialisation, \textit{init\_opencl}, and another one to release allocated objects, \textit{cleanup\_opencl}. These two functions are invoked at the start and end of the host program respectively.
\\ In the \textit{init\_opencl} function we perform the following: \begin{enumerate}
\item Get the OpenCL platform.
\item Query the available OpenCL devices and pick the first one.
\item Create the context.
\item Create the program and build it.
\item Create the command queues. \end{enumerate}
Each LSMR kernel invocation requires a set of parameters and OpenCL objects. Therefore the following struct was defined. This struct includes dimensions of the matrices, all the input, output and other buffers that the kernel requires, and event objects used to define dependencies of commands and synchronisation with the host. \\ \begin{lstlisting}[style=CStyle] typedef struct {
int m;
int n;
int p;
cl_mem input_buf;
cl_mem internal_buf1;
cl_mem internal_buf2;
cl_mem output_buf;
cl_kernel kernel1;
cl_kernel kernel2;
cl_event kernel_event[2];
cl_event finish_event[2];
} lsmr_module; \end{lstlisting} In the host program we defined two dynamic arrays of \textit{lsmr\_module}. \textit{weight\_update\_lsmr} and \textit{activation\_update\_lsmr}. We constructed one \textit{lsmr\_module} for each call of \textit{weight\_update} and \textit{activation\_update} in one training iteration. As the dimensions of inputs and output of the function in different iterations are the same, these modules are reused in iterations of the training. Therefore \textit{weight\_update\_lsmr} and \textit{activation\_update\_lsmr} arrays are consist of \textit{layers} and \textit{ layers -1} number of \textit{lsmr\_modules} respectively.
\\ We also defined a function in order to initialise \textit{lsmr\_modules} with the appropriate parameters. The important parameters are m, n and p which determine the dimensions of the inputs and output of the LSMR.
\\ This function performs the following: \begin{enumerate}
\item Create input buffer, output buffer and internal buffer with the appropriate size.
\item Create the kernel and set its arguments. \end{enumerate}
\subsubsection{\textit{Device Invocation}}
The host program is responsible for managing the device kernel calls. This process includes setting kernel arguments, uploading the inputs, invoking the kernel and downloading the results. It also manages dependencies and synchronisation of upload, invocation, download and in general host-device synchronisation.
\\ In our implementation, a function called \textit{run\_lsmr\_opencl} is responsible to perform OpenCL commands given an initialised \textit{lsmr\_module} object. This function gets called from the \textbf{\textit{weight\_update}} and \textbf{\textit{activation\_update}} functions and performs the following actions: \begin{enumerate}
\item \textit{clEnqueueWriteBuffer} to upload input matrix $A$.
\item \textit{clEnqueueWriteBuffer} to upload input matrix $B$, we upload this matrix column wise.
\item \textit{clEnqueueWriteBuffer} to fill the output buffer with zero.
\item \textit{clEnqueueTask} to invoke the kernel for computing first half of the result.
\item \textit{clEnqueueTask} to invoke the kernel for computing second half of the result.
\item \textit{clEnqueueReadBuffer} to download the output $X$ from the output buffer. \end{enumerate} It is worth mentioning that for each \textit{clEnqueueTask}, we had to use a different queue so they would be able to run in parallel. Also, OpenCL events are used to ensure the kernels start after the upload is finished and likewise, the download is started when the kernels are completed.
\\ In the main training loop, after calling the \textbf{\textit{weight\_update}} and \textbf{\textit{activation\_update}} a function called post process is invoked where we wait for the results of the two enqueued kernels and perform the further required operations.
\\ As previously stated, we use four LSMR compute units in parallel and each of them is pipelined internally. As a result, the two most consuming parts of the training process execute in parallel and pipelined fashion which leads to a noticeable speed up.
\section{Optimisations of FPGA Implementation} \label{achievements-optimisation}
\
In this section, we describe the applied optimisation steps and their results on timing and resource usage. These optimisations were critical to maximise the performance and utilisation of the available resources. Our goal was to achieve the maximum frequency of our target device (240 MHz), and II equal to one for most parts of the design.
\\ In each step, we have provided two tables for showing some of the non-optimised blocks of code and critical issues based on II and fmax values. A separate table is also provided to show the resource usage of each design.
\subsubsection{Version 1} Our first OpenCL implementation of the LSMR kernel was similar to its C implementation with minor modification. This version took as input a matrix $A \in {\rm I\!R}^{m * n}$ and a vector $b \in {\rm I\!R}^{m * 1}$ and produced an output of the form $x \in {\rm I\!R}^{n * 1}$. Some critical deficiencies of this design is shown in tables \ref{tab:v1f} and \ref{tab:v1II}. One of the main issues of this implementation was the memory dependency between the load and store operations which caused a high value of II in different sections of the code. \begin{table}[H] \def1.25{1.25} \caption{Non-optimised blocks of design based on II. Version 1. } \begin{center} \label{tab:v1II}
\begin{tabular}{|l|l|l|} \hline \textbf{Location in source code} & \textbf{II} & \textbf{Details} \\ \hline \textit{Computing norm} & $\sim$172 & \begin{tabular}[c]{@{}l@{}}Data dependency. \\ Load from global memory.\end{tabular} \\ \hline \textit{Summation of matrices} & $\sim$258 & \begin{tabular}[c]{@{}l@{}}Memory dependency.\\ Load and then store to global memory.\end{tabular} \\ \hline \textit{Transposing matrix} & $\sim$257 & \begin{tabular}[c]{@{}l@{}}Memory dependency.\\ Load and then store to global memory.\end{tabular} \\ \hline \end{tabular} \end{center} \end{table}
\begin{table}[H] \def1.25{1.25} \caption{Non-optimised blocks of design based on fmax. Version 1. } \begin{center} \label{tab:v1f}
\begin{tabular}{|l|l|l|} \hline \textbf{Location in source code} & \textbf{Scheduled fmax} & \textbf{Details} \\ \hline \textit{Computing Integer SQRT } & 98.3 & Loop feedback \\ \hline \textit{ Performing Matrix Multiplication} & 175.0 & Loop feedback \\ \hline \textit{Computing norm } & 135.0 & Loop feedback \\ \hline \end{tabular}
\end{center} \end{table}
\begin{table}[H] \def1.25{1.25} \caption{Estimated resource of system. Version 1. } \begin{center} \label{tab:v1area}
\begin{tabular}{l|l|l|l|l|l|} \cline{2-6} & \textbf{ALUTs} & \textbf{FFs} & \textbf{RAMs} & \textbf{MLABs} & \textbf{DSPs} \\ \hline
\multicolumn{1}{|l|}{\textbf{Board Interface}} & 8\% & 8\% &6\% & 0\% & 0\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Kernel System}} & 28\% & 18\% & 36\% & 3\%& 33\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Total}} &36\% & 26\%& 42\% & 3\% & 33\% \\ \hline \end{tabular} \end{center} \end{table}
\subsubsection{Version 2} In this version, we changed the LSMR kernel to work on inputs of the form $A \in {\rm I\!R}^{m * n}$ and vector $B \in {\rm I\!R}^{m * p}$ and produce an output of the form $B \in {\rm I\!R}^{n * p}$. We also used local memory for two of internal vectors, which were accessed more frequently than others, to reduce the size of internal buffer and to reduce the number of load and store from the global memory. This modification was done to address one of the issues of the former version. As a general rule, if possible, it is better to copy parts of memory that are accessed more than once into local memory.
\begin{table}[H] \def1.25{1.25} \caption{Non-optimised blocks of design based on II. Version 2. } \begin{center} \label{tab:v2II}
\begin{tabular}{|l|l|l|} \hline \textbf{Location in source code} & \textbf{II} & \textbf{Details} \\ \hline \textit{\begin{tabular}[c]{@{}l@{}}Computing norm of a vector \\ in local memory\end{tabular}} & $\sim$41 & \begin{tabular}[c]{@{}l@{}}Data dependency. \\ Load from local memory.\end{tabular} \\ \hline \textit{\begin{tabular}[c]{@{}l@{}}Computing norm of a vector \\ in global memory\end{tabular}} & $\sim$156 & \begin{tabular}[c]{@{}l@{}}Data dependency. \\ Load from global memory.\end{tabular} \\ \hline \textit{\begin{tabular}[c]{@{}l@{}}Summation of two vectors \\ in global memory\end{tabular}} & $\sim$214 & \begin{tabular}[c]{@{}l@{}}Memory dependency.\\ Load and then store to global memory.\end{tabular} \\ \hline \end{tabular}
\end{center} \end{table}
\begin{table}[H] \def1.25{1.25} \caption{Estimated resource usage of system. Version 2. } \begin{center} \label{tab:v2area}
\begin{tabular}{l|l|l|l|l|l|} \cline{2-6}
& \textbf{ALUTs} & \textbf{FFs} & \textbf{RAMs} & \textbf{MLABs} & \textbf{DSPs} \\ \hline
\multicolumn{1}{|l|}{\textbf{Board Interface}} & 8\% & 8\% & 6\% & 0\% & 0\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Kernel System}} & 32\% & 20\% & 68\% & 5\% & 33\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Total}} & 40\% & 28\% & 74\% & 5\% & 33\% \\ \hline \end{tabular} \end{center} \end{table}
The scheduled fmax value did not change significantly in this step.
\\ By comparing tables \ref{tab:v1area} and \ref{tab:v2area} we can observe that the only major difference in resource usage between these two versions is the RAM usage, even though the version 2 design can perform the same operation as the version 1 but on multiple columns of input using a for loop. This comparison demonstrates that the \textbf{\textit{AOC compiler does not replicate the hardware for each column, and tries to apply pipeline parallelism.}}
\\ The high amount of RAM usage in version 2 is resolved in the next versions. \subsubsection{Version 3} In previous versions, in most of the nested loops, the outer loop was not pipelined. This was because the compiler was not able to recognise that number of iterations of the inner loop are the same for different iterations of the outer loop. In this version, we guided the compiler to pipeline these outer loops by using constant type copies of the variables for specifying the number of loop iterations.
\\ Also in previous versions, there was a compile warning about not using \textit{"restrict"} keyword for pointers to memories in kernel signature. This keyword was used in this version which helps the compiler with some cache optimisations and also restricts the effect of pointer aliasing.
\\ Another technique used in this step was coalescing nested loops manually wherever possible. In coalescing a nested loop, we transform it into a single loop without changing its functionality. This technique reduces the loop overhead and also latency, and as a result, reduces the kernel resource usage.
\\ Fusing adjacent loops is also applied manually in this step. This technique also reduces the loop overhead and therefore the area usage. The main effect of this technique is running the adjacent loops concurrently as they are considered a single loop and this increases the performance. In this version, we managed to use one nested loop instead of three, for calculating \ref{av} as well as calculating \ref{au} which correspond to lines 10 and 12 of algorithm \ref{alg:LSMR}. This was achieved by fusing the following loops: matrix-vector multiplication nested loop, vector-scalar multiplication loop and subtracting vectors loop for both calculations. This also helped with precision and reduced the amount of internal buffer needed by eliminating some internal vectors.
\begin{gather} \label{av}
\vec{u_{k+1}} \leftarrow (\mat{A} \vec{v_k} - \alpha_k \vec{u_k} ) \\ \label{au}
\vec{v_{k+1}} \leftarrow (\mat{A^T} \vec{u_{k+1}} - \beta_{k+1} \vec{v_k} ) \end{gather} Again the scheduled fmax did not change significantly in this step.
\\ The II value changed a little for computing norm of a vector in global memory and summation of vectors.
\\ There was a considerable change in the amount of resource usage specially RAM usage. This reduction in resource usage is because of eliminating the overhead of some loops. \begin{table}[H] \def1.25{1.25} \caption{Non-optimised blocks of design based on II. Version 3. } \begin{center} \label{tab:v3II}
\begin{tabular}{|l|l|l|} \hline \textbf{Location in source code} & \textbf{II} & \textbf{Details} \\ \hline \textit{\begin{tabular}[c]{@{}l@{}}Computing norm of a vector \\ in local memory\end{tabular}} & $\sim$41 & \begin{tabular}[c]{@{}l@{}}Data dependency. \\ Load from local memory.\end{tabular} \\ \hline \textit{\begin{tabular}[c]{@{}l@{}}Computing norm of a vector \\ in global memory\end{tabular}} & $\sim$150 & \begin{tabular}[c]{@{}l@{}}Data dependency. \\ Load from global memory.\end{tabular} \\ \hline \textit{\begin{tabular}[c]{@{}l@{}}Summation of two vectors \\ in global memory\end{tabular}} & $\sim$196 & \begin{tabular}[c]{@{}l@{}}Memory dependency.\\ Load and then store to global memory.\end{tabular} \\ \hline \end{tabular} \end{center} \end{table}
\begin{table}[H] \def1.25{1.25} \caption{Estimated resource usage of system. Version 3. } \begin{center} \label{tab:v3area}
\begin{tabular}{l|l|l|l|l|l|} \cline{2-6}
& \textbf{ALUTs} & \textbf{FFs} & \textbf{RAMs} & \textbf{MLABs} & \textbf{DSPs} \\ \hline
\multicolumn{1}{|l|}{\textbf{Board Interface}} & 8\% & 8\% & 6\% & 0\% & 0\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Kernel System}} & 27\% & 17\% & 50\% & 5\% & 28\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Total}} & 35 \% & 25\% & 56 \% & 5\% & 28\% \\ \hline \end{tabular} \end{center} \end{table}
\subsubsection{Version 4} In this step, more loop fusing was performed. We fused the loop for computing the norm of vector to its adjacent loop, which itself was the result of a loop fusion in the previous step. By this modification, we were able to perform \ref{avn} in one nested loop and \ref{aun} in another nested loop. \ref{avn} and \ref{aun} correspond to lines 10 and 12 of algorithm \ref{alg:LSMR}.
\begin{gather} \label{avn}
\vec{u_{k+1}} \leftarrow (\mat{A} \vec{v_k} - \alpha_k \vec{u_k} )
\\
\beta_{k+1} \leftarrow ||\mat{A} \vec{v_k} - \alpha_k \vec{u_k}||_2 \end{gather} \begin{gather} \label{aun}
\vec{v_{k+1}} \leftarrow (\mat{A^T} \vec{u_{k+1}} - \beta_{k+1} \vec{v_k} ) \\
\alpha_{k+1} \leftarrow ||\mat{A^T} \vec{u_{k+1}} - \beta_{k+1} \vec{v_k}||_2 \end{gather} Also we fused the loops for computing \ref{hbark}, \ref{xk} and \ref{hk} (corresponding to lines 17, 18 and 19 of the algorithm \ref{alg:LSMR}) and we used just one loop to perform all of them. \begin{gather}
\label{hbark}
\vec{\overline{h_{k+1}}} \leftarrow - (\overline{s_k} * \rho_{k+1} * \rho_{k+1}) / ( \rho_k * \overline{\rho_k} ) \vec{\overline{h_k}} + \vec{h_k} \\
\label{xk}
\vec{x_{k+1}} \leftarrow (\zeta_{k+1} / (\rho_{k+1} * \overline{\rho_{k+1}}) \vec{\overline{h_{k+1}}} + \vec{x_k} \\\label{hk}
\vec{h_{k+1}} \leftarrow - ((s_{k+1} * \alpha_{k+1}) / \rho_{k+1}) \vec{h_k} + \vec{v} \end{gather} These loop fusions again helped with reducing resource usage. The only factor that changed considerably in this step of optimisation was resource usage: \begin{table}[H] \def1.25{1.25} \caption{Estimated resource usage of system. Version 4. } \begin{center} \label{tab:v4area}
\begin{tabular}{l|l|l|l|l|l|} \cline{2-6}
& \textbf{ALUTs} & \textbf{FFs} & \textbf{RAMs} & \textbf{MLABs} & \textbf{DSPs} \\ \hline
\multicolumn{1}{|l|}{\textbf{Board Interface}} & 8\% & 8\% & 6\% & 0\% & 0\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Kernel System}} & 25\% & 15\% & 37\% & 4\% & 28\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Total}} & 33\% & 22\% & 44\% & 4\% & 28\% \\ \hline \end{tabular} \end{center} \end{table} \subsubsection{Version 5} In this version, we used one of the OpenCL built-in integer functions, \textit{add\_sat}. Using this function, eliminated a great number of conditional statements inside the main loop. which were related to overflow check logic. This change, led to a significant reduction in resource usage.
\\ We also reduced the value of II significantly and as it can be seen in table \ref{tab:v5II}, the only block of code with value of II bigger than 1 was the square root block. The value of II for integer square root was 4 in all the other versions but not mentioned in the tables as there were more critical blocks in previous versions. Again, the scheduled fmax did not change significantly.
\begin{table}[H] \def1.25{1.25} \caption{Non-optimised blocks of design based on II. Version 5. } \begin{center} \label{tab:v5II}
\begin{tabular}{|l|l|l|} \hline \textbf{Location in source code} & \textbf{II} & \textbf{Details} \\ \hline \textit{Integer SQRT} & 4 & Data dependency. \\ \hline \end{tabular} \end{center} \end{table}
\begin{table}[H] \def1.25{1.25} \caption{Estimated resource of system. Version 5. } \begin{center} \label{tab:v5area}
\begin{tabular}{l|l|l|l|l|l|} \cline{2-6}
& \textbf{ALUTs} & \textbf{FFs} & \textbf{RAMs} & \textbf{MLABs} & \textbf{DSPs} \\ \hline
\multicolumn{1}{|l|}{\textbf{Board Interface}} & 8\% & 8\% & 6\% & 0\% & 0\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Kernel System}} & 11\% & 7\% & 20\% & 3\% & 18\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Total}} & 19\% & 15\% & 26\% & 3\% & 18\% \\ \hline \end{tabular} \end{center} \end{table}
\subsubsection{Version 6} To solve to problem of II value of integer square root block, we used the OpenCL built-in function for computing the float square root. In order to use this function, we had to convert the input from fixed-point to float and convert the output back to fixed-point. By this change, we were able to reduce the II value to 1 as the Intel implementation of the built-in square root function is highly optimised. However, we sacrificed a small amount of RAM usage (Less than one percent) that is negligible. This also solved the problem of low fmax in integer sqrt and we were able to increase the fmax value from 98 (as it is shown in table \ref{tab:v1f}) to 240 for this block
\\ In this version all of the blocks of code shown the II value of (1 , \~ 1 , $\geq$ 1) based on the report. We also used \textit{prefetch\_load} in this version for reading from the global memory. \begin{table}[H] \def1.25{1.25} \caption{Estimated resource of system. Version 6. } \begin{center} \label{tab:v6area}
\begin{tabular}{l|l|l|l|l|l|} \cline{2-6}
& \textbf{ALUTs} & \textbf{FFs} & \textbf{RAMs} & \textbf{MLABs} & \textbf{DSPs} \\ \hline
\multicolumn{1}{|l|}{\textbf{Board Interface}} & 8\% & 8\% & 6\% & 0\% & 0\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Kernel System}} & 11\% & 7\% & 20\% & 3\% & 18\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Total}} & 19\% & 15\% & 26\% & 3\% & 18\% \\ \hline \end{tabular} \end{center} \end{table} \subsubsection{Version 7} In this version we replaced the \textit{add\_sat} with add and we observed that this alteration did not lead to any loss in accuracy. Instead, we were able to solve the problem of low fmax values. In this version, all of the code blocks were showing the scheduled fmax equal to \textit{240 MHz}, which is the maximum achievable frequency on our target board. The area usage did not change significantly.
\\ In this version, optimisation was completed. In the next versions, we focused on achieving maximum hardware utilisation. \subsubsection{Version 8} At this stage, we used two LSMR compute units to run the LSMR in the \textbf{\textit{activation\_update}} and \textbf{\textit{weight\_update}} in parallel. It can be observed in table \ref{tab:v8area} that there were still unused hardware resources. \begin{table}[H] \def1.25{1.25} \caption{Estimated resource of system. Version 8. } \begin{center} \label{tab:v8area}
\begin{tabular}{l|l|l|l|l|l|} \cline{2-6}
& \textbf{ALUTs} & \textbf{FFs} & \textbf{RAMs} & \textbf{MLABs} & \textbf{DSPs} \\ \hline
\multicolumn{1}{|l|}{\textbf{Board Interface}} & 8\% & 8\% & 6\% & 0\% & 0\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Kernel System}} & 23\% & 14\% & 40\% & 6\% & 37\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Total}} & 31\% & 22\% & 46\% & 6\% & 37\% \\ \hline \end{tabular} \end{center} \end{table} \subsubsection{Version 9} To take advantage of the remaining available hardware resources, we increased the number of compute units of LSMR kernel to 4 on our FPGA. Consequently, the host code was modified to split the workload of LSMR in each of the \textbf{\textit{activation\_update}} and \textbf{\textit{weight\_update}} between two compute units. Also, the LSMR kernel was modified to only work on a segment of the problem to fit into this purpose.
\\ This modification led to about 2X speed up compared to the previous version as we doubled the task parallelism.
\\ It is also worth mentioning that the board interface resource usage, which was the same in all of the steps, slightly increased in this step.
\begin{table}[H] \def1.25{1.25} \caption{Estimated resource of system. Version 9. } \begin{center} \label{tab:v9area}
\begin{tabular}{l|l|l|l|l|l|} \cline{2-6}
& \textbf{ALUTs} & \textbf{FFs} & \textbf{RAMs} & \textbf{MLABs} & \textbf{DSPs} \\ \hline
\multicolumn{1}{|l|}{\textbf{Board Interface}} & 16\% & 10\% & 15\% & 0\% & 0\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Kernel System}} & 46\% & 27\% & 81\% & 11\% & 75\% \\ \hline
\multicolumn{1}{|l|}{\textbf{Total}} & 62\% & 37\% & 96\% & 11\% & 75\% \\ \hline \end{tabular} \end{center} \end{table}
\chapter{Experimental Results} \label{ch5} In this project, two data sets were used for experiments. IRIS \cite{IRIS} which is a small data set and a subset of HIGGS which is a bigger and more complex data set. The achieved test accuracy of the ADMM-LSMR method was assessed against two state-of-the-art gradient-based methods in \cite{iso}. It was observed that the ADMM-LSMR algorithm is able to achieve better accuracy compared to SGD and Adam on HIGGS and IRIS data sets. In this work, first, we compared the achieved test accuracy of the implemented method when using fixed-point arithmetic in LSMR against the original floating-point implementation. Second, we compared the execution time of each training iteration of C implementation and hardware-accelerated FPGA implementation. Finally, we studied the impact of increasing hidden size on the execution time of each iteration in both CPU and FPGA accelerated implementations.
\\ The key observation can be summarised as the following: \begin{itemize}[label=$\sqbullet$]
\item We were able to achieve near floating point accuracy, with less than one percent penalty using fixed point LSMR with nearest rounding method.
\item The nearest rounding method was the best choice when using fixed-point version of LSMR in ADMM. This was an important observation as it is reported that using the stochastic rounding in gradient-based methods has the best accuracy compared to other rounding methods \cite{gupta2015deep}.
\item We were able to achieve up to 6X speed up when using the hardware-accelerated FPGA implementation compared to the C implementation.
\item The speed up gain of using hardware-accelerated implementation grows by increasing the hidden size of the network. \end{itemize}
The experiments were conducted on two different machines. For CPU runs, one of the custom computing lab workstations (cccad5) was used and for FPGA accelerated runs Intel Devcloud nodes were employed. The specification of these platforms were the following: \begin{description} \item \textbf{cccad5 workstation}
CPU model name: Intel(R) Xeon(R) Gold 6154 CPU @ 3.00GHz\\ CPU cache size: 25344 KB\\ CPU cores: 18\\ Memory: 768GB
\item \textbf{Intel Devcloud node}
CPU model name: Intel(R) Xeon(R) Gold 6230 CPU @ 2.10GHz\\ CPU cache size: 28160 KB\\ CPU cores: 20\\ Memory: 18 GB\\ FPGA Board : Intel® Programmable Acceleration Card with Intel Arria® 10 GX FPGA\\ \end{description}
\section{Comparing Floating-point vs Fixed-point} \label{exp-floatvsfixed} In this section, we compared the accuracy of training algorithm using fixed-point and floating-point arithmetic. The results were derived from 500 runs of each training algorithm. \subsection{IRIS} \subsubsection{3 layer network with hidden size equal to 8} In this section, a 3 layer network with hidden size equal to 8 is used to train on IRIS data set using the ADMM-LSMR algorithm. We trained this model using fixed-point LSMR with four different rounding methods and floating-point LSMR. \\ \begin{table}[ht] \def1.25{1.25} \caption{Comparing accuracy of using floating-point vs fixed-point with different rounding methods on IRIS } \begin{center} \label{tab:IRISfixedvsfloat}
\begin{tabular}{l|l|l|} \cline{2-3} \textbf{} & \textbf{Mean} & \textbf{STDV} \\ \hline
\multicolumn{1}{|l|}{\textit{Floating-point}} & 83.7 \% & 1.6\% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with nearest rounding}} & 82.8 \%& 2.0\% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with stochastic rounding}} & 82.3 \% & 3.1\% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with upward rounding}} & 81.1\% & 5.3\% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with downward rounding}} & 80.8\% & 5.2\% \\ \hline \end{tabular} \end{center} \end{table}
We observed that the average of achieved accuracy using nearest rounding is better than the other rounding methods. Also, the achieved accuracy of this method has less variance compared to other rounding methods.
\\ It is worth mentioning that as opposed to the conventional training methods such as SGD which perform better using the stochastic rounding method with fixed-point, we did not observe significant difference by using this rounding method.
\\ We also observed less than 1\% penalty in accuracy when using fixed-point with nearest rounding method compared to floating-point.
\subsection{HIGGS} In this section, we trained three neural networks with hidden sizes of 8, 14 and 28 on a subset of HIGGS data set using ADMM-LSMR algorithm. We trained these models using fixed-point LSMR with four different rounding methods and also using floating-point LSMR. \subsubsection{3 layer network with hidden size equal to 8}
\begin{table}[H] \def1.25{1.25} \caption{Comparing accuracy of using floating-point vs fixed-point with different rounding methods on HIGGS } \begin{center} \label{tab:HIGGS188fixedvsfloat}
\begin{tabular}{l|l|l|} \cline{2-3} \textbf{} & \textbf{Mean} & \textbf{STDV} \\ \hline
\multicolumn{1}{|l|}{\textit{Floating-point}} & 63.6 \% & 0.1\% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with nearest rounding}} & 62.8 \% & 0.2\% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with stochastic rounding}} & 62.6 \% & 0.2\% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with upward rounding}} & 62.6\% & 0.1\% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with downward rounding}} & 62.6\% & 0.2\% \\ \hline \end{tabular} \end{center} \end{table}
\subsubsection{3 layer network with hidden size equal to 14}
\begin{table}[H] \def1.25{1.25} \caption{Comparing accuracy of using floating-point vs fixed-point with different rounding methods on HIGGS } \begin{center} \label{tab:HIGGS11414fixedvsfloat}
\begin{tabular}{l|l|l|} \cline{2-3} \textbf{} & \textbf{Mean} & \textbf{STDV} \\ \hline
\multicolumn{1}{|l|}{\textit{Floating-point}} & 63.6 \% & 0.1\% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with nearest rounding}} & 62.7 \% & 0.2 \% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with stochastic rounding}} & 62.7 \% & 0.1\% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with upward rounding}} & 62.6\% & 0.1\% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with downward rounding}} & 62.7\% & 0.2\% \\ \hline \end{tabular} \end{center} \end{table}
\subsubsection{3 layer network with hidden size equal to 28}
\begin{table}[H] \def1.25{1.25} \caption{Comparing accuracy of using floating-point vs fixed-point with different rounding methods on HIGGS } \begin{center} \label{tab:HIGGS12828fixedvsfloat}
\begin{tabular}{l|l|l|} \cline{2-3} \textbf{} & \textbf{Mean} & \textbf{STDV} \\ \hline
\multicolumn{1}{|l|}{\textit{Floating-point}} & 61.3 \% & 0.1\% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with nearest rounding}} & 62.8 \% & 0.2 \% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with stochastic rounding}} & 62.8 \% & 0.1\% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with upward rounding}} & 62.7\% & 0.2\% \\ \hline
\multicolumn{1}{|l|}{\textit{Fixed-point with downward rounding}} & 62.7\% & 0.2\% \\ \hline \end{tabular} \end{center} \end{table}
We observed that the average of achieved accuracy using nearest rounding is better than the other rounding methods.
\\ It is worth mentioning that as opposed to the conventional training methods such as SGD which perform better using the stochastic rounding method with fixed-point, we did not observe significant difference by using this rounding method.
\\ We also observed less than 1\% penalty in accuracy when using fixed-point with nearest rounding method compared to floating-point.
\\ Additionally, it is evident that by increasing the hidden size of the network, the floating-point is subject to minor loss in accuracy while such behaviour is not observed in fixed-point LSMR implementation. This can be due to the fact that using fixed-point injects noise to the neural network, which delays the overfitting and helps the network to generalise better. \section{Comparing CPU Implementation and FPGA Implementation: Accuracy} \label{exp-cpuvsfpga-accuracy} In this section, we compared the accuracy of FPGA implementation with CPU implementation. The results are from running each training algorithm 500 times.
\\ As expected, the FPGA implementation was able to achieve the same accuracy as the C implementation. This set of experiments were done to assess the correctness of the FPGA implementation. \subsection{IRIS} \subsubsection{3 layer network with hidden size equal to 8} \begin{table}[H] \def1.25{1.25} \caption{Comparing accuracy of C implementation vs FPGA implementation } \begin{center} \label{tab:IRISCPUvsFPGA}
\begin{tabular}{l|l|l|} \cline{2-3}
\textbf{} & \textbf{Mean} & \textbf{STDV} \\ \hline
\multicolumn{1}{|l|}{\textit{CPU implementation}} & 82.8 \% & 2.0\% \\ \hline
\multicolumn{1}{|l|}{\textit{FPGA implementation}} & 82.7 \%& 2.4\% \\ \hline
\end{tabular} \end{center} \end{table}
\subsection{HIGGS} \subsubsection{3 layer network with hidden size equal to 8}
\begin{table}[H] \def1.25{1.25} \caption{Comparing accuracy of C implementation vs FPGA implementation } \begin{center} \label{tab:HIGGSCPUvsFPGA188}
\begin{tabular}{l|l|l|} \cline{2-3}
\textbf{} & \textbf{Mean} & \textbf{STDV} \\ \hline
\multicolumn{1}{|l|}{\textit{CPU implementation}} & 62.8 \% & 0.2\% \\ \hline
\multicolumn{1}{|l|}{\textit{FPGA implementation}} & 62.6 \%& 0.2\% \\ \hline
\end{tabular} \end{center} \end{table}
\subsubsection{3 layer network with hidden size equal to 14}
\begin{table}[H] \def1.25{1.25} \caption{Comparing accuracy of C implementation vs FPGA implementation } \begin{center} \label{tab:HIGGSCPUvsFPGA11414}
\begin{tabular}{l|l|l|} \cline{2-3}
\textbf{} & \textbf{Mean} & \textbf{STDV} \\ \hline
\multicolumn{1}{|l|}{\textit{CPU implementation}} & 62.7 \% & 0.2\% \\ \hline
\multicolumn{1}{|l|}{\textit{FPGA implementation}} & 62.7 \%& 0.1\% \\ \hline
\end{tabular} \end{center} \end{table}
\subsubsection{3 layer network with hidden size equal to 28}
\begin{table}[H] \def1.25{1.25} \caption{Comparing accuracy of C implementation vs FPGA implementation } \begin{center} \label{tab:HIGGSCPUvsFPGA12828}
\begin{tabular}{l|l|l|} \cline{2-3} \textbf{} & \textbf{Mean} & \textbf{STDV} \\ \hline
\multicolumn{1}{|l|}{\textit{CPU implementation}} & 62.8 \% & 0.2\% \\ \hline
\multicolumn{1}{|l|}{\textit{FPGA implementation}} & 62.8 \%& 0.2\% \\ \hline
\end{tabular} \end{center} \end{table} \section{Comparing CPU Implementation and FPGA Implementation: Time} \label{exp-cpuvsfpga-time} In this section, we compare the execution time of each loop iteration of implemented ADMM-LSMR algorithm in CPU and FPGA. The execution time of 2500 iterations of each algorithm has been measured to produce these results. A subset of HIGGS data set was used for these experiments.
\\ As it is evident, we were able to achieve up to 6 times speed up depending on the architecture of the network. \subsection{HIGGS} \subsubsection{3 layer network with hidden size equal to 8}
\begin{table}[H] \def1.25{1.25} \caption{Comparing execution time of C implementation vs FPGA implementation } \begin{center} \label{tab:HIGGSCPUvsFPGATIME188}
\begin{tabular}{l|l|l|c|} \cline{2-4} \textbf{} & \textbf{Mean} & \textbf{STDV} & \textbf{Speed up} \\ \hline
\multicolumn{1}{|l|}{\textit{CPU Implementation}} & 589.8 $ms$ & 13.3 $ms$ & \multirow{2}{*}{\textbf{4.1}} \\ \cline{1-3}
\multicolumn{1}{|l|}{\textit{FPGA Implementation}} & 143.7 $ms$ & 0.4 $ms$ & \\ \hline \end{tabular}
\end{center} \end{table}
\subsubsection{3 layer network with hidden size equal to 14}
\begin{table}[H] \def1.25{1.25} \caption{Comparing execution time of C implementation vs FPGA implementation } \begin{center} \label{tab:HIGGSCPUvsFPGATIME11414}
\begin{tabular}{l|l|l|c|} \cline{2-4} \textbf{} & \textbf{Mean} & \textbf{STDV} & \textbf{Speed up} \\ \hline
\multicolumn{1}{|l|}{\textit{CPU Implementation}} & 1391.4 $ms$ & 24.0 $ms$ & \multirow{2}{*}{\textbf{5}} \\ \cline{1-3}
\multicolumn{1}{|l|}{\textit{FPGA Implementation}} & 277.7 $ms$ & 0.9 $ms$ & \\ \hline \end{tabular}
\end{center} \end{table}
\subsubsection{3 layer network with hidden size equal to 28}
\begin{table}[H] \def1.25{1.25} \caption{Comparing execution time of C implementation vs FPGA implementation } \begin{center} \label{tab:HIGGSCPUvsFPGATIME1282828}
\begin{tabular}{l|l|l|c|} \cline{2-4} \textbf{} & \textbf{Mean} & \textbf{STDV} & \textbf{Speed up} \\ \hline
\multicolumn{1}{|l|}{\textit{CPU Implementation}} & 5523.7 $ms$ &93.1 $ms$ & \multirow{2}{*}{\textbf{5.9}} \\ \cline{1-3}
\multicolumn{1}{|l|}{\textit{FPGA Implementation}} & 931.1 $ms$ & 2.3 $ms$ & \\ \hline \end{tabular} \end{center} \end{table}
\subsubsection{4 layer network with hidden size equal to 28} \begin{table}[H] \def1.25{1.25} \caption{Comparing execution time of C implementation vs FPGA implementation } \begin{center} \label{tab:HIGGSCPUvsFPGATIME228282828}
\begin{tabular}{l|l|l|c|} \cline{2-4} \textbf{} & \textbf{Mean} & \textbf{STDV} & \textbf{Speed up} \\ \hline
\multicolumn{1}{|l|}{\textit{CPU Implementation}} & 8437.2 $ms$ & 156.2 $ms$ & \multirow{2}{*}{\textbf{6}} \\ \cline{1-3}
\multicolumn{1}{|l|}{\textit{FPGA Implementation}} & 1387.2 $ms$ & 3.5$ms$ 22 & \\ \hline \end{tabular}
\end{center} \end{table}
\section{Run-time Relation to Network Complexity} \label{exp-relation} \subsection{HIGGS} In this section, we investigated the relation of the execution time of each training iteration with hidden size in a 3 layer neural network on a subset of HIGGS.
\begin{figure}
\caption{Correlation of execution time to hidden of network}
\label{fig:HIGGSRelationFPGAHS}
\end{figure}
It is observed that while the FPGA implementation constantly performs faster, the run time of both implementations grows in a non-linear fashion when the hidden size is increased. Also, it is evident that the CPU implementation is more sensitive to hidden size changes and the gap between the execution time of implementations also grows with hidden size.
\chapter{Conclusion and Future Work} \label{ch6}
The ADMM-LSMR method was introduced for the first time in the \cite{iso} alongside a Python implementation. It is also known that floating-point operations will cause performance issues on hardware designs. Hence we had to replace most of the arithmetic with custom implemented fixed-point arithmetic. Therefore, the feasibility of a hardware design that maintains accuracy while being deploy-able on a reasonable FPGA board was not clear at the beginning. Not only we were able to achieve such design, but also several stages of optimisation were applied to improve the initial algorithm and maximise the parallelism and hardware utilisation and achieve noticeable speed up comparing to equivalent CPU implementation.
\\ Based on the experimental results, the FPGA accelerated program was able to achieve the same accuracy as the original implementation with \textbf{\textit{less than 1\% loss using the fixed-point LSMR with nearest rounding}}. Additionally, the implementation has shown up to \textbf{\textit{6 times speed up}} depending on the network size, and architecture and it is evident that the acceleration is more effective on larger networks regarding the hidden size.
\section{Technical Achievements } The achievements of this project can be summarised as the following:
\begin{itemize}[label=$\sqbullet$]
\item \textbf{C implementation}:
ADMM-LSMR training method fully implemented in C for the first time. This was a necessity for OpenCL implementation and enabled us to perform a bottleneck analysis and identify LSMR as the target for acceleration.
\item \textbf{Fixed-point implementation}: As a known technique for hardware implementations, fixed-point arithmetic with various rounding methods were implemented. Also, 16-bit and 32-bit were used with flexibility of setting the precision. These variants were employed in the LSMR module.
\item \textbf{OpenCL implementation}: Conversion of the C implementation to an OpenCL accelerated program composed of CPU(host) program and device(FPGA) kernels. This was achieved by several structural changes both to adopt OpenCL models and Intel OpenCL SDK for FPGA guidelines and led to successful training using the emulated FPGA.
\item \textbf{FPGA deployment}: By getting access to Intel DevCloud environment, we were able to deploy and test the program on an actual FPGA board. This step demanded more design changes and primary optimisations as the emulation is not exactly equivalent to real hardware and hardware capacity has been added to the constrains.
\item \textbf{Optimisations}: There have been multiple stages of optimisation applied to the primary design. We were able to both speed up the design and reduce the utilised hardware resource in each iteration and finally fit multiple duplicates of the design on the target board to maximise utilisation and consequently the speed up.
\end{itemize}
\section{Results and Observations}
The key observations of this work can be summarised as following:
\begin{itemize}[label=$\sqbullet$]
\item
Accuracy of ADMM-LSMR method was assessed in \cite{iso}, and it was observed that this method is able to achieve higher accuracy compared to SGD and Adam, which are two commonly used gradient-based optimisers. In this work, the accuracy of the implementation has been constantly assessed during the development both for checking the implementation correctness and more importantly, verifying the feasibility of applied techniques like variants of the fixed-point arithmetic. We were able to maintain the accuracy of the original ADMM-LSMR method with less than 1\% penalty while using nearest rounding method on HIGGS and IRIS data sets.
\item
After reaching an acceptable design regarding the hardware utilisation and performance reports, the performance of the program was measured in several ways. In general, we were able to demonstrate 6 times speed up comparing to CPU implementation. We also assessed the impact of the architecture on acceleration by increasing the hidden size and observed more effectiveness on larger networks.
\item We observed that the nearest rounding method is more effective in the ADMM-LSMR method. This observation was unexpected as it is reported that stochastic rounding is more efficient when fixed-point arithmetic is used with gradient-based methods. The nearest rounding is simpler than stochastic rounding as it does not have the overhead of pseudo-random number generator and requires less resources. Considering this fact and our observation, the nearest rounding method was used in the final implementation. \end{itemize}
\section{Future Work} Some areas can be improved and many ideas can be employed to extend this work, such as: \begin{itemize}[label=$\sqbullet$]
\item Design and utilisation of a 16-bit iterative least-square solver.
\item Using other iterative least-square solvers like LSLQ
\item Perform more hardware optimisations and potentially increase the speed up.
\item Using more than one devices.
\item Employ full or partial HDL implementation to maximise hardware utilisation and efficiency.
\item Assess the method on other architectures of neural networks. \end{itemize}
\end{document} | arXiv |
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Latin American Economic Review
Existing literature recovering WTP
The experiments
Empirical strategy and results
External validity and heterogeneity analysis
Time goes by so slowly (for those who wait): a field experiment in health care
Sofía Garrido1 and
Emilio Gutiérrez2Email author
Latin American Economic Review201928:1
We exploit a unique field experiment to recover the willingness to pay (WTP) for shorter waiting times at a cataract detection clinic in Mexico City, and compare the results with those obtained through a hypothetical dichotomous choice questionnaire. The WTP to avoid a minute of wait obtained from the field experiment ranges from 0.59 to 0.82 Mexican pesos (1 USD = 12.5 Mexican pesos at the time of the survey), while that from the hypothetical choice experiment ranges from 0.33 to 0.48 Mexican pesos. WTP to avoid the wait is lower for lower income individuals, and it is larger the more accurately the announced expected waiting time matches the true values. Finally, we find evidence that the marginal disutility of waiting is not constant.
Obtaining reliable measures of the cost implied by waiting times in the Mexican health-care sector seems particularly relevant given recent changes in the health-care system. For instance, after a law requiring a prescription for antibiotics was passed in 2010, pharmacy-adjacent doctors' offices (PADOs) expanded rapidly across the county (being almost inexistent in the mid-2000s). Their success1 is attributed, among other things, to a very large difference in waiting times at doctors' offices between PADOs and public clinics (Pérez-Cuevas et al. 2012).2
Economists understand the willingness to pay (WTP) for a specific good or a good's attribute as a monetary measure of the value that consumers assign to it. Ideally, to recover such measures from data, econometricians can rely on variation in prices and goods' attributes readily available on the market and relate them to consumers' choices. Nonetheless, a consistent estimation of the WTP from observed choices requires sufficient variation in the goods' attributes and prices that is uncorrelated with other factors that could influence WTP. When this variation is unavailable from real-world data, WTP measures obtained from hypothetical choice experiments are very commonly used, and growingly so in the field of health economics. However, they have been widely criticized, as they are likely to deliver biased estimates for a variety of reasons.
This paper exploits a unique field experiment in which individuals were allowed to pay a randomly assigned price to avoid the waiting time to be seen by a physician at a cataract detection clinic in Mexico City to recover the WTP for shorter waiting times. In addition, it compares the estimates obtained from these real choices in the field with those obtained through a hypothetical dichotomous choice questionnaire administered to patients from the same clinic (throughout the text, we refer to this second as a contingent valuation (CV) exercise).
Our findings according to the field experiment indicate that the clinic's patients' WTP to avoid a minute of wait ranges from 0.59 to 0.82 Mexican pesos.3 Participants in the hypothetical choice experiment are significantly less responsive to variations in price and waiting times, and the point estimates for the WTP to avoid a minute of wait in this case ranges from 0.33 to 0.48 Mexican pesos. While our experiment may suffer from lack of external validity, we present a series of heterogeneity tests to explore how informative our results may be for the Mexican context. For instance, while our sample of patients is drawn from the lower tail of the income distribution in the city, we find suggestive evidence that, for lower income individuals in our sample, the WTP to avoid the wait is lower. The WTP to avoid a minute of wait is larger the more accurately the announced expected waiting time matches the true parameters. Finally, we find evidence that the marginal disutility of waiting is not constant, casting doubt on the appropriateness of indirect measures of the cost of waiting times, such as forgone wages.
The rest of the paper is presented as follows. The next section motivates the need for consistent measures of WTP to shorten waiting times in health care. Section III discusses the existing methods to recover such measures, and the potential advantages and caveats associated with doing so in a dichotomous choice setting, both through hypothetical choice questionnaires and actual choices. Section IV describes the context for the field and hypothetical choice experiments conducted for this paper. Section V presents the empirical strategy and results. Section VI discusses concerns of external validity and performs heterogeneity analyses. The last section concludes the paper.
Scholars have devoted their attention to understanding how different policies may have an impact on waiting times in health care (see for example Propper et al. 2002; Hurst and Siciliani 2003; Siciliani 2007; Siciliani et al. 2009; Brekke, et al. 2008). However, while measuring the direct impact of these policy interventions on waiting times may be useful per se, a full cost–benefit analysis requires quantifying welfare gains from changes in waiting times for the relevant population (Cullis et al. 2000).
Recovering consistent estimates of WTP is a challenging task. Ideally, to recover such measures from data, econometricians can rely on variation in prices and goods' attributes readily available on the market and relate them to consumers' choices. Nonetheless, a consistent estimation of the WTP from observed choices requires sufficient variation in the goods' attributes and prices that is uncorrelated with other factors that could influence WTP. When this variation is unavailable from real-world data, WTP measures obtained from hypothetical choice experiments are very commonly used, and growingly so in the field of health economics. In light of this, an assessment of whether WTP measures recovered from hypothetical choices accurately reflect true valuations may contribute to the academic debate on the reliability of contingent valuation techniques.
3 Existing literature recovering WTP
Many theorists have relied on measures of the cost of waiting times in health care using concrete measures of the opportunity cost that they entail, such as forgone wages. However, while useful from a theoretical perspective, such approximations fail to take into account that the specific conditions under which time is spent (or lost) may have an influence on the cost that waiting represents for patients. The disutility from sitting in a doctor's office waiting to be seen may differ substantially from the utility of time spent in another context. Moreover, the marginal disutility of waiting in a doctor's office may not be constant (an extra minute of wait after having waited for an hour may not cause the same disutility than that of the first minute in the waiting room). The cost of waiting may then be context specific and very different from forgone wages.4
Some researchers have exploited real-world data to recover estimates of the cost of waiting times. Deacon and Sonstelie (1985) exploit a mandated price decrease specifically for Chevron gas stations in California in the 1980s (which implied a large increase in the waiting times at these stations, and not at others where there was no price decrease) to estimate the value of time spent waiting for gasoline purchases. Apart from the fact that their study is performed in a very different context than a doctor's office, in their data the variation in waiting times and prices is only present across gas stations. Aguiar and Hurst (2007) exploit scanner data to structurally estimate the cost of time spent grocery shopping. Besley et al. (1999) show that longer waiting times in the British National Health Service (NHS) are associated with larger purchases of private health insurance, and calculate patients' willingness to pay for shorter waiting times. Their results assume that the characteristics of private health insurance do not vary with price, and they rely on the assumption that private and public care only differ in terms of the waiting times.
Since the WTP for some goods or goods' attributes (such as waiting times for health care) is hard to recover from real data, over the last decades and increasingly so in health economics, researchers have tried to recover such measures from hypothetical surveys, which use a wide array of techniques to ask individuals about their "reservation price" for a specific good or good's attributes. While widely used, an ongoing debate regarding the elicitation of such questions has been taking place for decades. Open-ended questions are widely believed to deliver biased estimates. However, since the National Oceanic and Atmospheric Administration (NOAA) panel (Arrow et al. 1993) performed a critical review of the existing methods to recover WTP through hypothetical surveys, dichotomous choice (DC) questionnaires are perhaps seen as the most reliable alternative, both because of their simplicity and reduced incentives for strategic behavior (Hoehn and Randall 1987; Carson et al. 1999).
DC questionnaires have been widely used by health economists to recover measures of WTP for a variety of health-care attributes. For example, Propper (1990) estimates WTP for shorter waiting times in the NHS waiting list, and Bishai and Lang (2000) estimate differences in WTP for shorter waiting times for cataract surgery in Canada, Denmark and Spain. Johannesson et al. (1991) also estimate the WTP for shorter waiting times in the Swedish health-care system through an experiment of this kind.5
However, the literature that questions the validity of contingent valuation methods to recover WTP is also widespread (Portney 1994, Cummings et al. 1995; Klose 1999; Ryan et al. 2004; Donaldson and Shackley 2002; Smith 2005; Harrison and Rutström 2008; Hausman 2012).
Three main concerns arise when recovering WTP from hypothetical surveys: "hypothetical bias", which simply refers to the fact that individuals' responses to hypothetical choices may not fully correspond to their behavior in real life (for example, subjects may choose to please the interviewer or may infer that their answers could have a policy impact); the consistent difference in estimates obtained from "willingness to pay" and "willingness to accept" questionnaires, which can be generalized as evidence of subjects' sensitivity to the framing of the hypothetical choice questions; and the difficulty to correctly isolate in the questions' wording the good's attribute for which the WTP wants to be recovered (Hausman 2012).
While some theory-based techniques to assess the consistency of WTP estimates obtained through hypothetical surveys have been proposed (Diamond and Hausman 1994), to determine whether a hypothetical survey delivers biased estimates of WTP, one would ideally know the true values for respondents. The existing literature circumvents the problem by conducting laboratory experiments, in which respondents' true value is recovered through experiments where the choice is real, and then compared to those obtained through hypothetical surveys. But the debate on whether the results obtained from laboratory experiments can be extended to the real world is large and growing. In particular, while within a laboratory the researcher has full control over the environment under which choices are made, this is never the case in the real world. A bias in hypothetical surveys may arise in the field regardless of its absence in a laboratory. Smith and Mansfield (1998) do compare estimates from a hypothetical choice questionnaire and real choices, recovering the willingness to accept spending time answering to a phone interview, finding no significant differences between hypothetical and real choices.
This paper contributes then to this literature by effectively randomizing the price faced by individuals when making a decision in the field about whether to wait or not to be seen by a physician, and comparing the estimates obtained with those from a hypothetical choice experiment. Both exercises use patients of the same cataract detection clinic in Mexico City as subjects, and they both keep all characteristics of the service provided constant, except for the waiting time and the price for not having to wait.
4 The experiments
4.1 The field experiment
The field experiment was conducted during the last 3 weeks in October 2014, from Friday to Saturday, at a health-care facility in Mexico City, specializing in cataract detection and surgery. Patients of this clinic arrive at the reception desk and are announced the expected waiting time to be seen by a doctor. This waiting time is calculated by the clinic's personnel, based on the number of patients in the waiting room and the number of doctors at the clinic. Patients generally stay in a waiting room until their name is called. From the clinic's records, no patients chose to leave the clinic after this waiting time was announced, even before the option to pay to avoid the wait was offered.
In January 2013, this clinic introduced a new product, which consisted in the possibility of paying $300 Mexican pesos (approximately, 25 US dollars) to be seen by one of the doctors without having to wait. The field experiment consisted in randomizing the price at which this product was offered.
When patients arrived at the reception desk,6 apart from being announced the expected waiting time to be seen by the physician, they were also informed that the clinic was offering a "no waiting time consultation" at promotional prices. This offer consisted in a lottery that assigned to each patient, randomly, a different price for this product. The prices offered were $200, $250 and $300 (the baseline price) Mexican pesos ($1 USD = 12.5 Mexican pesos). Because all patients could potentially interact in the waiting room, the receptionist informed them that the price offered was a "promotion aimed at improving their experience at the clinic" and that a different promotional price was offered, randomly, to each patient.7 All patients were explicitly informed that the quality of the service, apart from the difference in waiting times, would be identical from the one offered to the rest of the patients.
All of the patients at the clinic, paying and non-paying, were asked by the physician to fill in a questionnaire that captures some basic socioeconomic characteristics. The sample obtained through the field experiment consists of 279 patients that arrived at the clinic individually. Table 1 presents the descriptive statistics.
Price offered for the non-waiting time consult (Mexican pesos)
0.24***
Waiting time (in minutes)
[6.40]
Gender (male = 1)
Household head
Was accompanied
Responsible for own health decisions
Dirt floor
Standard errors of means in brackets
* Significant at 10%
** Significant at 5%
*** Significant at 1% for the test of that category against the rest
Not surprisingly, patients were sensitive to the price of the non-waiting offer: 6, 12 and 24 percent of those offered this product for $300, $250 and $200 Mexican pesos, respectively, chose to pay for it. As expected, due to the random assignment of prices to patients, individuals did not seem to differ in any other observable characteristic, including the waiting time announced at the arrival to the clinic.
4.2 The contingent valuation exercise
Two weeks after the field experiment took place, a hypothetical contingent valuation (CV) questionnaire was administered to a comparable sample of 251 patients (all arriving to the clinic seeking an appointment with an ophthalmologist). The implementation of this survey also lasted three full weeks, from Friday to Saturday. Subjects participating in the hypothetical choice survey did so before being informed that the non-waiting time consultation was available. Surveyors approached them before their arrival to the reception desk and stated that the clinic had an interest in improving the experience of future patients and asked if they were willing to answer a brief questionnaire. All subjects agreed8 to participate in the short survey, which consisted in asking each patient the following:
"If the expected waiting time to be seen by the doctor today was T hours, would you pay P pesos to not have to wait and be seen by the doctor right away, or choose to wait T hours at no cost?"
Prices (P) and waiting times (T) were randomly assigned to questionnaires in this hypothetical exercise. The point values for the price of the hypothetical non-waiting consultation were the same as those offered in the field experiment: 89, 84 and 82 subjects were assigned a 200, 250 and 300 pesos price, respectively. The waiting times randomly assigned to questionnaires were chosen to lie on the same range as those announced during the field experiment. In particular, 42, 45, 35, 30, 45, and 56 questionnaires stated a waiting time of 90, 120, 150, 180, 240, and 300 min, respectively. Because patients were not randomly assigned to the field experiment or the contingent valuation exercise, Table 2 shows the descriptive statistics of the socioeconomic variables listed in Table 1, this time comparing patients participating in each of the experiments.
Field experiment
CV exercise
0.14**
116.16 [3.58]***
61.28 [1.03]
0.40 [0.03]**
3.21 [0.14]
*** Significant at 1% for the difference in means test
The two samples differ significantly on the fraction of individuals that chose to pay to avoid waiting, although they also differ in the average waiting time announced, which is higher for the CV sample9 that chose to pay more frequently. Apart from that, small differences are observed in the rest of the variables, and only the fraction of interviewed individuals that declared to be the head of their household differs significantly between samples.
5 Empirical strategy and results
5.1 Random utility model (RUM) framework
In the context analyzed, both for the field and hypothetical choice experiments, the basic formulation of consumers' utility that can allow for recovering their WTP to avoid waiting at the doctor's office can be embodied in a random utility model (RUM) of the following form:
$$U_{ia} \left( {t_{ia} ,p_{ia} ,X_{i} } \right) = V_{ia} \left( {t_{ia} ,p_{ia} ,X_{i} } \right) + \varepsilon_{ia} ,$$
where \(U_{ia}\) is the utility that individual i derives from choosing alternative a, at a price \(p_{ia}\), with a waiting time of \(t_{ia}\) minutes, \(X_{i}\) is a vector of observable characteristics, and \(\varepsilon_{ia}\) is an error term, which captures unobserved heterogeneity in individuals' preferences.
For empirical purposes, it is common practice to assume that time and money are linearly separable in the individuals' utility function, and that the marginal utility of both is constant across alternatives and individuals. For the specific context analyzed, we can incorporate these assumptions by describing the utility from each alternative as:
$$\begin{gathered} U_{{i1}} \left( {t_{{i1}} ,p_{{i1}} ,X_{i} } \right) = \alpha _{1} + \beta _{1} t_{{i1}} + \beta _{2} p_{{i1}} + F_{1} \left( {X_{i} } \right) + \varepsilon _{{i1}} \; \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad {\text{and}} \hfill \\ \;U_{{i2}} \left( {t_{{i2}} ,p_{{i2}} ,X_{i} } \right) = \alpha _{2} + \beta _{1} t_{{i2}} + \beta _{2} p_{{i2}} + F_{2} \left( {X_{i} } \right) + \varepsilon _{{i2}} . \hfill \\ \end{gathered}$$
Given this setup, \(\beta_{1}\) and \(\beta_{2}\) are the marginal utilities of waiting times and price, respectively, and the WTP to avoid a unit of wait can be easily computed as:
$${\text{WTP}} = - {\text{MRS}}_{t,p} = \frac{{\beta_{1} }}{{\beta_{2} }},$$
which is interpreted in consumer choice theory as the units of currency that the individual is willing to pay to avoid a unit of wait.
In our setting, an individual will choose to pay to avoid waiting (alternative 1) over not paying and waiting to be seen by the physician (alternative 2) when:
$$\begin{gathered} U_{{i1}} \left( {t_{{i1}} ,p_{{i1}} ,X_{i} } \right) \ge U_{{i2}} \left( {t_{{i2}} ,p_{{i2}} ,X_{i} } \right) \hfill \\ \quad \quad \quad \quad \quad \quad {\text{or}} \hfill \\ \varepsilon _{{i2}} - \varepsilon _{{i1}} \le V_{{i1}} \left( {t_{{i1}} ,p_{{i1}} ,X_{i} } \right) - V_{{i2}} \left( {t_{{i2}} ,p_{{i2}} ,X_{i} } \right). \hfill \\ \end{gathered}$$
Given the parametrization of the individuals' utility function above, and given that \(t_{i1}\), the expected waiting time for the non-waiting time consultation, and \(p_{i2}\), the price individuals pay for the consultation if they choose to wait, are both equal to zero; individuals will choose to pay for the non-waiting time consultation if:
$$\varepsilon_{i2} - \varepsilon_{i1} \le (\alpha_{1} - \alpha_{2} ) - \beta_{1} t_{i2} + \beta_{2} p_{i1} + F_{1} \left( {X_{i} } \right) - F_{2} \left( {X_{i} } \right).$$
Assuming a specific distribution for \((\varepsilon_{i2} - \varepsilon_{i1} )\) and a functional form for \(\left( {F_{1} \left( {X_{i} } \right) - F_{2} \left( {X_{i} } \right)} \right)\), it is then possible to estimate the marginal disutility of time spent waiting (\(\beta_{1}\)) and the marginal disutility of the price paid for the consultation (\(\beta_{2}\)), and thus compute the WTP to avoid waiting: \(\left( {\frac{{\beta_{1} }}{{\beta_{2} }}} \right)\).
It is common practice to assume (as we do in our empirical analysis) that the \(\varepsilon_{i}\) are independently and identically extreme-value distributed, so that the difference \((\varepsilon_{i2} - \varepsilon_{i1} )\) is distributed logistically. Then, defining \(P_{1}\) as the probability that an individual will choose alternative 1 (the non-waiting time consultation, in this case) implies that:
$$P_{1} = \frac{1}{{1 + {\text{e}}^{{ - \left( {(\alpha_{1} - \alpha_{2} ) - \beta_{1} t_{i2} + \beta_{2} p_{i1} + F_{1} \left( {X_{i} } \right) - F_{2} \left( {X_{i} } \right)} \right)}} }}$$
and all the relevant parameters can be estimated through a logit regression.10
5.2 Empirical specification
In this particular case, to allow preferences to vary with respect to individual characteristics, and to directly test for potential differences in WTP in the field and in the hypothetical surveys, we assume that the difference in utility between the non-waiting and waiting consultation alternatives can be parametrized as:
$$\begin{aligned} \Delta U_{i} &= \alpha_{1} + \alpha_{2} {\text{CV}}_{i} + \beta_{1} {\text{Time}}_{i} + \beta_{2} {\text{Price}}_{i} + \beta_{3} {\text{Time}}_{i} *{\text{CV}}_{i} \\ & \quad + \beta_{4} {\text{Price}}_{i} *{\text{CV}}_{i} + \mathop \sum \limits_{n} \delta_{n} {\text{Control}}_{ni} + e_{i} ,\end{aligned}$$
where \({\text{Time}}\) measures the announced waiting time (in minutes) for individuals participating in the field experiment, and the hypothetical waiting time listed in the hypothetical choice setting for the participants in the CV exercise. \({\text{Price}}\) indicates the price (in Mexican pesos) randomly assigned to each patient for the non-waiting time consultation in the field experiment, and the hypothetical price of the non-waiting time consultation for those participating in the CV exercise. \({\text{CV}}\) is a dummy variable taking a value of one if the individual corresponds to the CV sample. \(Control_{n}\) are the control variables listed in Table 1, and \(e_{i}\) is an error term.
As outlined above, this then implies that the probability of choosing the non-waiting time consultation can be written as:
$$P_{1} = { \Pr }(\Delta U_{i} < 0) = \frac{1}{{1 + e^{{ - \left( {\alpha_{1} + \alpha_{2} CV_{i} + \beta_{1} Time_{i} + \beta_{2} Price_{i} + \beta_{3} Time_{i} *CV_{i} + \beta_{4} Price_{i} *CV_{i} + \mathop \sum \nolimits_{n} \delta_{n} Control_{ni} } \right)}} }},$$
where the functional form follows from the common assumption on the distribution of the difference in the error terms. We can then estimate this equation via a logit regression, using the dummy variable indicating if individuals chose the non-waiting time consultation as our dependent variable.
Under the assumptions listed so far, we can recover estimates of the individuals' utility function arguments: the marginal disutility of time and money in the field experiment (\(- \beta_{1}\) and \(\beta_{2}\), respectively), the marginal disutility of hypothetical time and money in the CV exercise (\(- \beta_{1} - \beta_{3}\) and \(\beta_{2} + \beta_{4}\), respectively), and the implied WTP to avoid a minute of wait implied by the field experiment \(\left( {\frac{{ - \beta_{1} }}{{\beta_{2} }}} \right)\) and by the CV exercise \(\left( {\frac{{ - \beta_{1} - \beta_{3} }}{{\beta_{2} + \beta_{4} }}} \right)\).
5.3 Main results
Table 3 shows the results of the logit regression.11 Column 1 includes no controls. Column 2 includes all control variables listed in Table 1, and column 3 additionally includes date fixed effects. As can be seen, throughout specifications, the coefficients on price and announced waiting time have the expected signs. Participants are less likely to pay for the non-waiting time consult when its (randomly assigned) price is higher, and more likely to do so when the announced expected waiting time is higher. The coefficients for the interactions between price and waiting time with the dummy variable indicating if the individuals' responses correspond to those in the CV exercise roughly suggest that individuals are less responsive to both price and waiting time in the hypothetical scenario. Throughout specifications, the coefficient associated with the assigned price and the interaction with the CV indicator is positive, and the coefficient associated with the interaction between the CV dummy and the announced waiting time is negative. When including socioeconomic variables as controls, both of these coefficients are significantly different from zero at a 10% confidence level. When we additionally include date fixed effects, although the coefficient for the interaction between price and the CV dummy loses significance, its sign and magnitude remain relatively stable.
Logit regression results
Dummy = 1 if paid
− 0.014638 [0.003328]***
0.009147 [0.002274]***
Price*contingent valuation
0.003635 [0.002286]
0.004291 [0.002305]*
Waiting time*contingent valuation
− 0.004812 [0.003025]
− 0.005167 [0.003025]*
− 0.010074 [0.005038]**
WTP-field experiment
Chi squared for test WTP = 0)
12.39***
WTP-CV exercise
Difference in WTP between field and CV
Chi squared for test of difference in WTP = 0
Socioeconomic controls
Date fixed effects
Robust standard errors in brackets
For participants in the field experiment, the dependent variables are dummy, taking value of one if individuals paid for the non-waiting consult. For participants in the CV exercise, the dependent variables are dummy taking value of one if individuals declared they would have paid for the non-waiting consult, given the hypothetical price and waiting time announced
*** Significant at 1%
As shown in Table 3, the implied WTP to avoid a minute of wait ranges between 0.59 and 0.86 Mexican pesos, while for the CV sample, it ranges from 0.33 to 0.48 Mexican pesos. The WTP recovered from the hypothetical survey is considerably lower than the one recovered from true choices. While we cannot reject the hypothesis that the WTP recovered from the field differs significantly from that recovered from the CV exercise, taking the results from column 2, we can reject the hypotheses that individuals respond similarly to price and announced waiting time in a hypothetical setting and when facing true choices. The results then cast doubt on the validity of WTP measures recovered from hypothetical choices.
To better put into perspective the WTP estimates recovered from the field experiment, a simple back-of-the-envelope calculation may be useful. Approximately, ten million patients are seen monthly only by physicians in The Mexican Social Security Institute12 (IMSS, for its acronym in Spanish), which provides health care to 39.2 percent of the Mexican population.13 According to the 2016 Mexican Health and Nutrition Survey (ENSANUT), the average wait to be seen by a physician in the IMSS system is approximately 70 min. Given our estimate of the cost per minute of wait (0.59 Mexican pesos), the monthly cost associated to waiting times in the IMSS system could roughly account to 413 million pesos. Given that in Mexico there were approximately 225,000 practicing physicians in 2014, and that the average monthly salary of a doctor in Mexico is 12,722 pesos,14 the cost associated with waiting times only in the IMSS system is then approximately equivalent to 15 percent of the sum of all physicians' salaries in the country.
6 External validity and heterogeneity analysis
In this section, we further explore the data recovered from the field experiment to investigate to what extent our results may be informative to public policy. All the estimates presented thus far are based on the assumption that the marginal disutility of time and money is constant for individuals with different observable characteristics. Testing for this assumption is particularly relevant in the context analyzed, as the experiment exploited in this paper was performed in a very particular setting: a private clinic where typical patients lack any kind of health insurance. The clinic's promotion strategies around the time we conducted the experiment were to visit remote, low income neighborhoods in the city to inform its citizens of the availability of the no-cost consultations. It is then perhaps not surprising that, according to Table 1, socioeconomic characteristics of participating subjects differ considerably from those of Mexico City's population. For example, more than 45 percent of participating subjects' dwellings have a dirt floor, while this number for Mexico City's population is less than 2 percent, according to the 2010 census. The extent to which the WTP estimates recovered through the experiment are informative for the Mexican health-care system crucially depends on how the estimated WTP differs by subjects' characteristics.
The model's assumptions can be easily relaxed to allow for heterogeneity in the WTP based on observables. In particular, we can then allow the marginal utility of time and money to vary with respect to observable characteristics by assuming that the differences in utility from alternative 1 versus alternative 2 can be expressed as:
$$\Delta U_{i} = \alpha_{1} + \beta_{1} {\text{Time}}_{i} + \beta_{2} {\text{Price}}_{i} + \beta_{3} {\text{Time}}_{i} *X_{i} + \beta_{4} {\text{Price}}_{i} *X_{i} + \mathop \sum \limits_{n} \delta_{n} {\text{Control}}_{ni} + e_{i} ,$$
where all variables are defined as above and \(X\) is any observable characteristic of interest. The coefficients associated with the interaction between this variable and \({\text{Time}}\) or \({\text{Price}}\) (\(\beta_{3}\) and \(\beta_{4}\), respectively) will indicate if the marginal utility of time or money differs for individuals for which \(X\) takes different values. We explore if this is the case for men vs. women, and for individuals whose dwelling has a dirt floor. The latter is particularly relevant to explore if the WTP varies with the closest proxy for income available in our dataset. In addition, we explore if the WTP varies with respect to the uncertainty about whether the announced waiting time at the time of arrival to the clinic reflects the true waiting times that patients face. In particular, we compute the standard deviation in the announced waiting times within each of the days the experiment was conducted. Our assumption is that the variation we observe in the announced waiting times within a day can proxy for how much waiting times varied within each day and, thus, how close to the true waiting time the announced wait was. For ease of interpretation, we compute a dummy variable taking value of one if the standard deviation of waiting times within each day is above or below the sample median.
The results of the logistic regression restricting the sample to participants in the field experiment are presented in Table 4.15 Column 1 does not have interaction of the price and time variables with any other. Column 2 presents the results by gender (interacting with an indicator for males). Column 3 shows estimates by type of floor in the individual's dwelling, and column 4 shows interaction of the price and time variables with an indicator of whether the standard deviation in the announced waiting time during the day of the patient's visit was higher than the median in our sample. Results in column 2 suggest that women have a lower WTP for shorter waiting times than men, although the difference between the estimates is not significantly different from zero. Column 3 shows that lower income individuals are more responsive to the price, and less responsive to waiting times (although, again, the coefficients associated with these variables and the interaction between the dummy variable for dirt floor are not significantly different from zero). These estimates also imply that their willingness to pay for shorter waiting times is considerably lower than for the rest of the sample. Finally, column 4 shows that the WTP for shorter waiting times is significantly higher when the standard deviation in waiting times announced during the patients' day of visit is larger than the median. We interpret this last result as evidence that, if the uncertainty in the actual waiting times faced given the announcement can bias our estimates of WTP, this bias is likely to be reflecting a lower bound of the WTP in the context analyzed.
Male = 1
Dirt floor = 1
SD (time) > median
0.008095 [0.003407]**
0.02406 [0.009977]**
Price*X = 1
Waiting time*X = 1
WTP − X = 0
Chi squared for test WTP = 0
7.1***
Difference in WTP between X = 0 and X = 1
− 0.58
The dependent variables are dummy taking value of one if individuals paid for the non-waiting consult
Column 2 presents the results by gender (interacting with an indicator for males). Column 3 shows estimates by type of floor in the individual's dwelling and column 4 shows interactions of the price and time variables with an indicator of whether the standard deviation in the announced waiting time during the day of the patient's visit was higher than the median in our sample
In addition, it is perhaps worth testing the assumption that the marginal disutility of waiting is constant, as rejecting this hypothesis may be informative of the appropriateness of using indirect measures for the cost of waiting, such as forgone wages. For this purpose, we rewrite the difference in utility from choosing alternative 1 vs 2 as:
$$\Delta U_{i} = \alpha_{1} + \beta_{1} {\text{Time}}_{i} + \beta_{2} {\text{Price}}_{i} + \beta_{3} {\text{Time}}_{i}^{2} + \mathop \sum \limits_{n} \delta_{n} {\text{Control}}_{ni} + \varepsilon_{i} .$$
We test the null hypothesis that \(\beta_{3} = 0\). In this particular context, rejection of the null indicates that the marginal disutility of waiting may not be constant.
Results are presented in Table 5.16 Column 1 includes no controls, column 2 controls for all observable characteristics listed in Table 1, and column 3 additionally includes date fixed effects. Across specifications, we reject the hypothesis that the marginal disutility of waiting is constant, as the coefficient associated with the square of the waiting time is always negative and statistically different from zero. We interpret this last result as strong evidence against using indirect measures for the cost of time, such as forgone wages, for the welfare analysis of policies that may have an impact on waiting times.
Waiting time squared
− 0.00009 [0.000035]***
WTP-field experiment (at T = 150)
WTP is calculated as the ratio of marginal utilities between time and money given the coefficients presented. The marginal utility of time is computed at T = 150
7 Conclusions
In this paper, we recover estimates of the WTP for shorter waiting times at a cataract detection clinic in Mexico City through a field experiment and a contingent valuation exercise. Results from the field experiment indicate that the clinic's patients' WTP to avoid a minute of wait ranges from 0.59 to 0.82 Mexican pesos, and further heterogeneity analysis suggests that this estimate is likely a lower bound of the WTP of Mexico City's population in similar contexts. A simple back-of-the-envelope calculation suggests that reducing the average waiting time to be seen by a doctor at IMSS (Mexico's Social Security System) by 70 min, on average, would imply welfare gains for patients equivalent to 15 percent of the sum of all physicians' salaries in the country.
The estimates obtained cast doubt on the appropriateness of indirect measures of the cost of time, such as forgone wages. For instance, the average daily salary of full-time formal workers in Mexico City in 2014 was 366 pesos,17 or 0.76 pesos per minute. Given that our sample is drawn from the lower tail of Mexico City's income distribution and that WTP is lower for relatively lower income participants in our experiment, indirect estimates of the cost of time obtained from forgone wages may heavily underestimate them. Moreover, our results suggest that the marginal disutility of waiting is not constant.
In addition, estimates recovered from the field experiment differ considerably from those obtained through the contingent valuation exercise. Participants in the hypothetical choice experiment are significantly less responsive to variations in price and waiting times, and the WTP to avoid a minute of wait according to the hypothetical choice experiment ranges from 0.33 to 0.48 Mexican pesos.
While our results then also cast doubt on the appropriateness of contingent valuation techniques to consistently recover WTP for goods or goods' attributes unavailable on the market, they should be taken with caution. Field experiments to recover WTP as the one exploited in this paper are generally very hard to implement, and even if implementable, they can be prohibitively costly. Rather than completely dismissing the appropriateness of CV questionnaires to recover WTP, we hope that in cases where the cost of large-scale field experiments is too costly, small-scale experiments as the one analyzed in this paper can be exploited to identify the best correction techniques to approximate WTP from data generated from CV questionnaires (Harrison 2006).
According to the industry's estimates, roughly the same number of patients visit PADOs every day in Mexico as the main social security system's outpatient clinics (about 300,000 daily visits).
In the USA, some of the arguments regarding the potential cost of the health-care reform proposed and passed by the Obama administration revolved precisely around the potential increase in waiting times to receive health-care services.
1 USD = 12.5 Mexican pesos at the time of the survey.
For instance, Aguiar and Hurst (2007) document that the least educated consume more leisure, which they find at odds with standard predictions from income and substitution effects, suggesting that the conditions under which leisure is spent may affect its marginal utility. Ramey and Francis (2009) also characterize the evolution of leisure in the USA during the past 100 years. Lee et al. (2012) show that, for the Japanese context, the marginal rate of substitution between work hours and different kinds of non-work activities (i.e., leisure and in-home production) differs.
See Olsen and Smith (2001) and Diener et al. (1998) for a review of this literature.
Patients waited in line, outside the clinic, approximately for 10 min before being seen by the receptionist. All announced waiting times only considered the wait after patients registered at the reception desk.
This feature of the experimental setting implies that our results should be interpreted with caution. Part of the patients' response to the price offered may be driven by its "promotional" nature.
The 100% compliance rate may seem surprising. Nonetheless, it is a result of the fact that participating individuals were waiting in line, outside the clinic, before being seen by the receptionist.
This is the result of the fact that higher and lower waiting times were assigned to the CV questionnaires with similar probabilities, while, in the field experiment, longer waiting times are relatively less common.
For a thorough discussion of random utility models and the use of logit to estimate them, see Train (2009).
The analogous results estimating a linear probability model through OLS are presented in Appendix Table 6.
http://www.imss.gob.mx/conoce-al-imss/memoria-estadistica-2017.
http://www.beta.inegi.org.mx/temas/derechohabiencia/.
http://www.beta.inegi.org.mx/contenidos/saladeprensa/aproposito/2014/medico0.pdf.
http://archivo.eluniversal.com.mx/finanzas-cartera/2014/impreso/cinco-estados-concentran-los-salarios-mas-elevados-107715.html.
Both authors participated in the experimental design. SG supervised the field work and collected the data. Both authors worked in the data analysis. EG wrote the final version of the paper. Both authors read and approved the final manuscript.
We are grateful to Javier Okhuysen and Carlos Orellana for the opportunity to conduct the field experiment. Sofia Cerrilla provided outstanding assistance throughout the experiment's implementation. All errors are ours.
The title of this paper makes a reference to the lyrics of Madonna's song "Hung Up".
SG was employed at the clinic in which the experiment described in this paper was carried out. She received no compensation for conducting it. EG declares no competing interests.
All data generated or analyzed during this study are included in this article.
Emilio Gutierrez is grateful for support from the Asociacion Mexicana de Cultura.
See Tables 6, 7 and 8.
OLS regression results
F-statistic for test WTP = 0
F-statistic for test of difference in WTP = 0
For participants in the field experiment, the dependent variables are dummy taking value of one if individuals paid for the non-waiting consult. For participants in the CV exercise, the dependent variables are dummy taking value of one if individuals declared they would have p aid for the non-waiting consult, given the hypothetical price and waiting time announced
Table 3 in the main text is the analog of this table. However, in this case we assume a linear probability model and estimate the regression through ordinary least squares (tables in the main text are estimated through a logit regression)
WTP − X
4.2**
The dependent variables are dummy taking value of one if individuals paid for the non-waiting consult Column 2 presents the results by gender (interacting with an indicator for males). Column 3 shows estimates by type of floor in the individual's dwelling and column 4 shows interactions of the price and time variables with an indicator of whether the standard deviation in the announced waiting time during the day of the patient's visit was higher than the median in our sample
WTP is calculated as the ratio of marginal utilities between time and money given the coefficients presented
The marginal utility of time is computed at T = 150
General Atlantic, Mexico City, Mexico
Department of Economics, ITAM, Camino a Santa Teresa 930, Héroes de Padierna, CDMX, CP 10700, Mexico
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Municipal Debt
Title: Municipal Debt
Bond valuation
High-yield debt
Registered share
Stock certificate
Credit derivative
Futures exchange
Hybrid security
Reinsurance market
Practical trading
Financial market participants
Finance series
A municipal bond is a bond issued by a local government, or their agencies. Potential issuers of municipal bonds include states, cities, counties, redevelopment agencies, special-purpose districts, school districts, public utility districts, publicly owned airports and seaports, and any other governmental entity (or group of governments) at or below the state level. Municipal bonds may be general obligations of the issuer or secured by specified revenues.
In the United States, interest income received by holders of municipal bonds is often exempt from the federal income tax, and may be exempt from state income tax, although municipal bonds issued for certain purposes may not be tax exempt.[1]
Unlike new issue stocks that are brought to market with price restrictions until the deal is sold, municipal bonds are free to trade at any time once they are purchased by the investor. Professional traders regularly trade and retrade the same bonds several times a week.
2 Types of tax-exempt bonds
3 Purpose of municipal bonds
3.1 Municipal bond issuers
3.2 Municipal bond holders
3.3 Bond measure
4 Characteristics of municipal bonds
4.1 Taxability
5 Risk
6 Disclosures to investors
7 Comparison to corporate bonds
8 Subprime mortgage crisis
9 Default rates
10 Build America Bonds
11 Statutory history
Historically municipal debt predates corporate debt by several centuries: the early Renaissance Italian city-states borrowed money from major banking families. Borrowing by American cities dates to the nineteenth century; records of U.S. municipal bonds indicate use around the early 1800s. Officially the first recorded municipal bond, a general obligation bond was issued by the City of New York for a canal in 1812.[2] During the 1840s, many U.S. cities were in debt; roughly by 1843 cities had about 25 million in outstanding debt. In the pursuing decades rapid urban development demonstrated a correspondingly explosive growth in municipal debt. The debt was used to finance both urban improvements and a growing system of free public education.
Years after the civil war, significant local debt was issued to build railroads. Railroads were private corporations and these bonds were very similar to today's industrial revenue bonds. Construction costs in 1873 for one of the largest transcontinental railroads, the northern pacific, closed down access to new capital.[3]
The largest bank of the country, which was owned by the same investor as by Northern Pacific, collapsed. Smaller firms followed suit as well as the stock market. The 1873 panic and years of depression that followed put an abrupt but temporary halt to the rapid growth of municipal debt.[4]
Responding to widespread defaults that jolted the municipal bond market of the day, new state statutes were passed that restricted the issuance of local debt. Several states wrote these restrictions into their constitutions. Railroad bonds and their legality were widely challenged; this gave rise to the market-wide demand that an opinion of qualified bond counsel accompany each new issue.
The U.S. economy began to move forward once again, municipal debt continued its momentum, this maintained well into the early part of the twentieth century. The great depression of the 1930s halted growth, although defaults were not as severe as in the 1870s.[5] Outstanding municipal debt then fell during World War II. Many American resources were devoted to the military. Prewar municipal debt burst into a new period of rapid growth for an ever-increasing variety of uses. After World War II, state and local debt was $145 per capita. In 1998, according to the U.S. census, state debt was $1,791 per capita. Local per capita debt in 1996, the most current Census data available, was $2,704. Federal debt was $20,374 per capita at the end of 1998.
In addition to the 50 states and their local governments (including cities, counties, villages and school districts), the District of Columbia and U.S. territories and possessions (American Samoa, the commonwealth of Puerto Rico, Guam, the Northern Mariana Islands, and the U.S. virgin Islands) can and do issue municipal bonds. Another important category of municipal bond issuers includes authorities and special districts, which have grown in number and variety in recent years.
The two most prominent early authorities were the Port of New York Authority, formed in 1921 and renamed Port Authority of New York and New Jersey in 1972, and the Triborough Bridge Authority (now the Triborough Bridge and Tunnel Authority), formed in 1933. The debt issues of these two authorities are exempt from federal, state and local governments taxes.[6]
Types of tax-exempt bonds
Municipal bonds provide tax exemption from federal taxes and many state and local taxes, depending on the laws of each state. Municipal securities consist of both short-term issues (often called notes, which typically mature in one year or less) and long-term issues (commonly known as bonds, which mature in more than one year). Short-term notes are used by an issuer to raise money for a variety of reasons: in anticipation of future revenues such as taxes, state or federal aid payments, and future bond issuances; to cover irregular cash flows; meet unanticipated deficits; and raise immediate capital for projects until long-term financing can be arranged. Bonds are usually sold to finance capital projects over the longer term.
The two basic types of municipal bonds are:
General obligation bonds: Principal and interest are secured by the full faith and credit of the issuer and usually supported by either the issuer's unlimited or limited taxing power. In many cases, general obligation bonds are voter-approved.[7]
Revenue bonds: Principal and interest are secured by revenues derived from tolls, charges or rents from the facility built with the proceeds of the bond issue. Public projects financed by revenue bonds include toll roads, bridges, airports, water and sewage treatment facilities, hospitals and subsidized housing. Many of these bonds are issued by special authorities created for that particular purpose.[7]
Most municipal notes and bonds are issued in minimum denominations of $5,000 or multiples of $5,000.
Purpose of municipal bonds
Municipal bonds are securities that are issued for the purpose of financing the infrastructure needs of the issuing municipality. These needs vary greatly but can include schools, streets and highways, bridges, hospitals, public housing, sewer and water systems, power utilities, and various public projects.
Municipal bond issuers
Municipal bonds are issued by states, cities, and counties, (the municipal issuer) to raise funds. The methods and traces of issuing debt are governed by an extensive system of laws and regulations, which vary by state. Bonds bear interest at either a fixed or variable rate of interest, which can be subject to a cap known as the maximum legal limit. If a bond measure is proposed in a local county election, a Tax Rate Statement may be provided to voters, detailing best estimates of the tax rate required to levy and fund the bond.
The issuer of a municipal bond receives a cash payment at the time of issuance in exchange for a promise to repay the investors who provide the cash payment (the bond holder) over time. Repayment periods can be as short as a few months (although this is rare) to 20, 30, or 40 years, or even longer.
The issuer typically uses proceeds from a bond sale to pay for capital projects or for other purposes it cannot or does not desire to pay for immediately with funds on hand. Tax regulations governing municipal bonds generally require all money raised by a bond sale to be spent on one-time capital projects within three to five years of issuance.[8] Certain exceptions permit the issuance of bonds to fund other items, including ongoing operations and maintenance expenses, the purchase of single-family and multi-family mortgages, and the funding of student loans, among many other things.
Because of the special tax-exempt status of most municipal bonds, investors usually accept lower interest payments than on other types of borrowing (assuming comparable risk). This makes the issuance of bonds an attractive source of financing to many municipal entities, as the borrowing rate available in the open market is frequently lower than what is available through other borrowing channels.
Municipal bonds are one of several ways states, cities and counties can issue debt. Other mechanisms include certificates of participation and lease-buyback agreements. While these methods of borrowing differ in legal structure, they are similar to the municipal bonds described in this article.
Municipal bond holders
Municipal bond holders may purchase bonds either directly from the issuer at the time of issuance (on the primary market), or from other bond holders at some time after issuance (on the secondary market). In exchange for an upfront investment of capital, the bond holder receives payments over time composed of interest on the invested principal, and a return of the invested principal itself (see bond).
Repayment schedules differ with the type of bond issued. Municipal bonds typically pay interest semi-annually. Shorter term bonds generally pay interest only until maturity; longer term bonds generally are amortized through annual principal payments. Longer and shorter term bonds are often combined together in a single issue that requires the issuer to make approximately level annual payments of interest and principal. Certain bonds, known as zero coupon or capital appreciation bonds, accrue interest until maturity at which time both interest and principal become due.
A bond measure is an initiative to sell bonds for the purpose of acquiring funds for various public works projects, such as research, transportation infrastructure improvements, and others. These measures are put up for a vote in general elections and must be approved by a plurality or majority of voters, depending on the specific project in question.
Such measures are very often used in the United States when other revenue sources, such as taxes, are limited or non-existent.
Characteristics of municipal bonds
Taxability
One of the primary reasons municipal bonds are considered separately from other types of bonds is their special ability to provide tax-exempt income. Interest paid by the issuer to bond holders is often exempt from all federal taxes, as well as state or local taxes depending on the state in which the issuer is located, subject to certain restrictions. Bonds issued for certain purposes are subject to the alternative minimum tax.
The type of project or projects that are funded by a bond affects the taxability of income received on the bonds held by bond holders. Interest earnings on bonds that fund projects that are constructed for the public good are generally exempt from federal income tax, while interest earnings on bonds issued to fund projects partly or wholly benefiting only private parties, sometimes referred to as private activity bonds, may be subject to federal income tax. However, qualified private activity bonds, whether issued by a governmental unit or private entity, are exempt from federal taxes because the bonds are financing services or facilities that, while meeting the private activity tests, are needed by a government.
Purchasers of municipal bonds should be aware that not all municipal bonds are tax-exempt, and not all tax-exempt bonds are exempt from all federal and state taxes. The laws governing the taxability of municipal bond income are complex. At the federal level they are contained in the IRS Code, (Sections 103, 141-150), and rules promulgated thereunder. Each state will have its own laws governing what bonds, if any, are exempt from state taxes. For publicly offered bonds and most private placements, at the time of issuance a legal opinion will be provided indicating that the bonds are tax-exempt. Offering documents, such as an official statement or placement memorandum, will contain further information regarding tax treatment of interest on the bonds. Investors should be aware that there are also post-issuance compliance requirements that must be met to ensure that the bonds remain tax-exempt. The IRS has a specific section of their website, www.irs.gov, devoted to tax exempt bonds and compliance with federal requirements.
Main article: Credit risk
The risk ("security") of a municipal bond is a measure of how likely the issuer is to make all payments, on time and in full, as promised in the agreement between the issuer and bond holder (the "bond documents"). Different types of bonds are secured by various types of repayment sources, based on the promises made in the bond documents:
General obligation bonds promise to repay based on the full faith and credit of the issuer; these bonds are typically considered the most secure type of municipal bond, and therefore carry the lowest interest rate.
Revenue bonds promise repayment from a specified stream of future income, such as income generated by a water utility from payments by customers.
Assessment bonds promise repayment based on property tax assessments of properties located within the issuer's boundaries.
In addition, there are several other types of municipal bonds with different promises of security.
The probability of repayment as promised is often determined by an independent reviewer, or "rating agency". The three main rating agencies for municipal bonds in the United States are Standard & Poor's, Moody's, and Fitch. These agencies can be hired by the issuer to assign a bond rating, which is valuable information to potential bond holders that helps sell bonds on the primary market.
Municipal bonds have traditionally had very low rates of default as they are backed either by revenue from public utilities (revenue bonds), or state and local government power to tax (general obligation bonds). However, sharp drops in property valuations resulting from the 2009 mortgage crisis have led to strained state and local finances, potentially leading to municipal defaults. For example, Harrisburg, PA, when faced with falling revenues, skipped several bond payments on a municipal waste to energy incinerator and did not budget more than $68m for obligations related to this public utility. The prospect of Chapter 9 municipal bankruptcy was raised by the Controller of Harrisburg, although it was opposed by Harrisburg's mayor.[9]
Disclosures to investors
Key information about new issues of municipal bonds (including, among other things, the security pledged for repayment of the bonds, the terms of payment of interest and principal of the bonds, the tax-exempt status of the bonds, and material financial and operating information about the issuer of the bonds) typically is found in the issuer's official statement. Official statements generally are available at no charge from the the EMMA continuing disclosure service.
Comparison to corporate bonds
Because municipal bonds are most often tax-exempt, comparing the coupon rates of municipal bonds to corporate or other taxable bonds can be misleading. Taxes reduce the net income on taxable bonds, meaning that a tax-exempt municipal bond has a higher after-tax yield than a corporate bond with the same coupon rate.
This relationship can be represented mathematically, as follows:
r_m = r_c ( 1 - t ) \,
rm = interest rate of municipal bond
rc = interest rate of comparable corporate bond
t = tax rate
For example if rc = 10% and t = 38%, then
r_m = (10%)(100% - 38%) = 6.2% \,
A municipal bond that pays 6.2% therefore generates equal interest income after taxes as a corporate bond that pays 10% (assuming all else is equal).
The marginal tax rate t at which an investor is indifferent between holding a corporate bond yielding rc and a municipal bond yielding rm is:
t = 1- \frac{r_m}{r_c}.
All investors facing a marginal rate greater than t are better off investing in the municipal bond than in the corporate bond.
Alternatively, one can calculate the taxable equivalent yield of a municipal bond and compare it to the yield of a corporate bond as follows:
r_c = \frac{r_m}{( 1 - t )}
Because longer maturity municipal bonds tend to offer significantly higher after-tax yields than corporate bonds with the same credit rating and maturity, investors in higher tax brackets may be motivated to arbitrage municipal bonds against corporate bonds using a strategy called municipal bond arbitrage.
Some municipal bonds are insured by monoline insurers that take on the credit risk of these bonds for a small fee.
The municipal bond market was affected by the subprime mortgage crisis. During the crisis, monoline insurers that insured municipal bonds incurred heavy losses on the collateralized debt obligations (CDOs) and other structured financial products that they also insured. Consequently, the credit ratings of these monoline insurers were called into question, and the prices of municipal bonds fell.
The historical default rate for municipal bonds is lower than that of corporate bonds. The Municipal Bond Fairness Act (HR 6308),[10] introduced September 9, 2008, included the following table giving bond default rates up to 2007 for municipal versus corporate bonds by rating and rating agency.
Build America Bonds
Main article: Build America Bonds
Build America Bonds are a taxable municipal bond created under the American Recovery and Reinvestment Act of 2009 that carry special tax credits and federal subsidies for either the bond holder or the bond issuer. Many issuers have taken advantage of the Build America Bond provision to secure financing at a lower cost than issuing traditional tax-exempt bonds. The Build America Bond provision is open to governmental agencies issuing capital expenditure bonds before January 1, 2011.[11][12][13]
Statutory history
The U.S. Supreme Court held in 1895 that the federal government had no power under the U.S. Constitution to tax interest on municipal bonds.[14] But, in 1988, the Supreme Court stated the Congress could tax interest income on municipal bonds if it so desired on the basis that tax exemption of municipal bonds is not protected by the Constitution.[15] In this case, the Supreme Court stated that the contrary decision of the Court 1895 in the case of Pollock v. Farmers' Loan & Trust Co. had been "effectively overruled by subsequent case law."
The Revenue Act of 1913 first codified exemption of interest on municipal bonds from federal income tax.[16]
The Tax Reform Act of 1986 greatly reduced private activities that may be financed with tax-exempt bond proceeds.[17]
IRC 103(a) is the statutory provision that excludes interest on municipal bonds from federal income tax.[18] As of 2004[update], other rules, however, such as those pertaining to private activity bonds, are found in sections 141–150, 1394, 1400, 7871.
http://www.citymayors.com/finance/bonds.html Municipal bonds have been issued by US local governments since 1812
MSRB's EMMA Education Center
The Bond Buyer, website, conferences and newspaper focusing on the municipal bond industry.
Securities Industry and Financial Markets Association, the industry trade group.
Municipal Finance Journal, the only peer-reviewed journal devoted to municipal securities and state & local public finance.
About Municipal Bonds
Types of bonds by issuer
Agency bond
Corporate bond (Senior debt, Subordinated debt)
Types of bonds by payout
Accrual bond
Auction rate security
Callable bond
Convertible bond
Exchangeable bond
Extendible bond
Fixed rate bond
Floating rate note
Inflation-indexed bond
Inverse floating rate note
Perpetual bond
Puttable bond
Reverse convertible
Zero-coupon bond
Clean price
Convexity
Current yield
Dirty price
I-spread
Mortgage yield
Nominal yield
Yield to maturity
Z-spread
Collateralized mortgage obligation
Commercial mortgage-backed security
Yield-curve spread
Bond options
Embedded option
Option-adjusted spread
Commercial Mortgage Securities Association (CMSA)
International Capital Market Association (ICMA)
Securities Industry and Financial Markets Association (SIFMA)
Debt management plan
Debt-snowball method
DIP financing
Debt collection and evasion
Debt compliance
Debtors' prison
Tax refund interception
Deposit account
Debt buyer
Debt in economics
Consumer leverage ratio
Debt levels and flows | CommonCrawl |
List of complexity classes
This is a list of complexity classes in computational complexity theory. For other computational and complexity subjects, see list of computability and complexity topics.
Many of these classes have a 'co' partner which consists of the complements of all languages in the original class. For example, if a language L is in NP then the complement of L is in co-NP. (This does not mean that the complement of NP is co-NP—there are languages which are known to be in both, and other languages which are known to be in neither.)
"The hardest problems" of a class refer to problems which belong to the class such that every other problem of that class can be reduced to it. Furthermore, the reduction is also a problem of the given class, or its subset.
#PCount solutions to an NP problem
#P-completeThe hardest problems in #P
2-EXPTIMESolvable in doubly exponential time
AC0A circuit complexity class of bounded depth
ACC0A circuit complexity class of bounded depth and counting gates
ACA circuit complexity class
AHThe arithmetic hierarchy
APThe class of problems alternating Turing machines can solve in polynomial time.[1]
APXOptimization problems that have approximation algorithms with constant approximation ratio[1]
AMSolvable in polynomial time by an Arthur–Merlin protocol[1]
BPPSolvable in polynomial time by randomized algorithms (answer is probably right)
BQPSolvable in polynomial time on a quantum computer (answer is probably right)
co-NP"NO" answers checkable in polynomial time by a non-deterministic machine
co-NP-completeThe hardest problems in co-NP
DSPACE(f(n))Solvable by a deterministic machine with space O(f(n)).
DTIME(f(n))Solvable by a deterministic machine in time O(f(n)).
ESolvable in exponential time with linear exponent
ELEMENTARYThe union of the classes in the exponential hierarchy
ESPACESolvable with exponential space with linear exponent
EXPSame as EXPTIME
EXPSPACESolvable with exponential space
EXPTIMESolvable in exponential time
FNPThe analogue of NP for function problems
FPThe analogue of P for function problems
FPNPThe analogue of PNP for function problems; the home of the traveling salesman problem
FPTFixed-parameter tractable
GapLLogspace-reducible to computing the integer determinant of a matrix
IPSolvable in polynomial time by an interactive proof system
LSolvable with logarithmic (small) space
LOGCFLLogspace-reducible to a context-free language
MASolvable in polynomial time by a Merlin–Arthur protocol
NCSolvable efficiently (in polylogarithmic time) on parallel computers
NESolvable by a non-deterministic machine in exponential time with linear exponent
NESPACESolvable by a non-deterministic machine with exponential space with linear exponent
NEXPSame as NEXPTIME
NEXPSPACESolvable by a non-deterministic machine with exponential space
NEXPTIMESolvable by a non-deterministic machine in exponential time
NL"YES" answers checkable with logarithmic space
NONELEMENTARYComplement of ELEMENTARY.
NP"YES" answers checkable in polynomial time (see complexity classes P and NP)
NP-completeThe hardest or most expressive problems in NP
NP-easyAnalogue to PNP for function problems; another name for FPNP
NP-equivalentThe hardest problems in FPNP
NP-hardAt least as hard as every problem in NP but not known to be in the same complexity class
NSPACE(f(n))Solvable by a non-deterministic machine with space O(f(n)).
NTIME(f(n))Solvable by a non-deterministic machine in time O(f(n)).
PSolvable in polynomial time
P-completeThe hardest problems in P to solve on parallel computers
P/polySolvable in polynomial time given an "advice string" depending only on the input size
PCPProbabilistically Checkable Proof
PHThe union of the classes in the polynomial hierarchy
PNPSolvable in polynomial time with an oracle for a problem in NP; also known as Δ2P
PPProbabilistically Polynomial (answer is right with probability slightly more than ½)
PPADPolynomial Parity Arguments on Directed graphs
PRSolvable by recursively building up arithmetic functions.
PSPACESolvable with polynomial space.
PSPACE-completeThe hardest problems in PSPACE.
PTASPolynomial-time approximation scheme (a subclass of APX).
QIPSolvable in polynomial time by a quantum interactive proof system.
QMAQuantum analog of NP.
RSolvable in a finite amount of time.
REProblems to which we can answer "YES" in a finite amount of time, but a "NO" answer might never come.
RLSolvable with logarithmic space by randomized algorithms (NO answer is probably right, YES is certainly right)
RPSolvable in polynomial time by randomized algorithms (NO answer is probably right, YES is certainly right)
SLProblems log-space reducible to determining if a path exist between given vertices in an undirected graph. In October 2004 it was discovered that this class is in fact equal to L.
S2Pone round games with simultaneous moves refereed deterministically in polynomial time[2]
TFNPTotal function problems solvable in non-deterministic polynomial time. A problem in this class has the property that every input has an output whose validity may be checked efficiently, and the computational challenge is to find a valid output.
UPUnambiguous Non-Deterministic Polytime functions.
ZPLSolvable by randomized algorithms (answer is always right, average space usage is logarithmic)
ZPPSolvable by randomized algorithms (answer is always right, average running time is polynomial)
References
1. Sanjeev Arora, Boaz Barak (2009), Computational Complexity: A Modern Approach, Cambridge University Press; 1 edition, ISBN 978-0-521-42426-4
2. "S2P: Second Level of the Symmetric Hierarchy". Stanford University Complexity Zoo. Archived from the original on 2012-10-14. Retrieved 2011-10-27.
External links
• Complexity Zoo - list of over 500 complexity classes and their properties
| Wikipedia |
1 FFT(x, T, Freq)
2 The Discrete Fourier Transform
2.1 Units and spacing
2.2 Nyquist frequency, aliasing, mirroring
2.3 Amplitude and phase
4.1 Seasonality detection and adjustment
4.2 Fast convolution
4.3 Deconvolution
4.4 Low pass or high pass filtering
4.5 Characteristic functions of probability distributions
FFT(x, T, Freq)
The Fast Fourier Transform (FFT) converts a time series of equally spaced values, «x», from the discrete time (or spatial) domain to the discrete frequency domain. «T» is the time index and «Freq» is the Frequency index, and these two indexes should have the same length. The time series, «x», is indexed by «T» and may be real or complex, and the result is indexed by «Freq» and contains complex numbers in general. The units of «Freq» are cycles per «units of T».
The FFTInv function does the inverse transformation from the frequency domain back to the time (or spatial) domain.
The Discrete Fourier Transform
The FFT computes the discrete Fourier transform (DFT) in an efficient manner. The DFT is defined given by
$ H_k = \sum_{i=0}^{n-1} x_i e^{2j\pi i k/n} $
where $ j $ is the imaginary number $ \sqrt{-1} $, and n is the number of points in «T» and «Freq». This is a discrete approximation to the continuous Fourier Transform given by
$ H(f) = \int_{-\infty}^\infty x(t) e^{2j \pi f t} dt $
Units and spacing
Let $ \Delta t $ denote the spacing between points in «T» and $ n $ be the number of points. Then the spacing $ \Delta F $ of the points in «Freq» is $ 1/n\Delta t $. The quantity $ 1/\Delta t $ is called the sampling frequency, which should be at least twice the smallest frequency that is present in the underlying continuous signal.
For example, suppose your time series is sampled at 1000Hz for just over 10 seconds, so that $ \Delta t=1 millisecond $ and $ n=10K $. Then $ \Delta F=0.1 Hz $. It would make sense to define «Freq» as (0..10K-1)*0.1 so that the numeric values of «Freq» are in Hertz.
Nyquist frequency, aliasing, mirroring
When all frequencies that are present in the underlying continuous signal are below the Nyquist frequency, $ 1/2\Delta t $, then the discretely sampled time series contains all of the information in the original continuous signal. This is known as the sampling theorem and is a remarkable fact. Even though the continuous signal appears to contain infinitely more information -- all the values between the discrete points in time where the signal is sampled -- the discrete series uniquely determines the full continuous series.
When a frequency is present that exceeds the Nyquist frequency, the power from that frequency is still transferred to the FFT result, but it gets mapped to a bin in the result as if it had wrapped around. This phenomena is termed aliasing. So you can think of the resulting spectrum as being a circular buffer, representing signal at each frequency modulo the maximum frequency of $ 1/\Delta t $. A frequency of $ 1/2 \Delta t $ would map to the middle bin, but so would a frequency of $ 3/2\Delta t $ as well as a frequency of $ 5/2\Delta t $.
The same wrap-around occurs for negative frequencies. When the real-valued time series contains a component sine wave with a frequency of 100 Hz, it implicitly also contains a frequency of -100Hz. This -100Hz component also appears in the result of the FFT, but instead of mapping to a negative bin, it wraps around and appears in the second half the the spectrum. Hence, the result of FFT can be divided into a first half and a second half. For a band-limited signal with all frequencies below the Nyquist frequency, the first half of the spectrum corresponds to positive frequencies, the second half of the spectrum is the negative frequencies.
For example, here is the magnitude of an FFT for a 200 Hz sine wave, Sin(200*T*360) where T is in seconds
In the example, $ \Delta t $ is 1 ms, so the maximum frequency is 1KHz. Notice that the 200Hz signal shows up both at 200Hz and at 800Hz -- the 800 Hz actually capturing the -200Hz component.
In many applications, an FFT is taken, the signal is multiplied by a transfer function and the inverse FFT is then applied. In these cases, it is usually most convenient to keep the frequency spectrum in this wrapped-around representation. But when displaying a spectrum, you might want to either extract only the first half the array, thus showing only the positive frequencies, or remap it to a frequency index that has negative values. In either case, you'll need a second frequency index.
Suppose your F index is (0..10K-1)*0.1. To extract the positive part:
Index Freq_pos := (0..5K-1) * 0.1 Do fft_result[Freq=Freq_pos]
To map to an index with a negative part:
Index Frequency := (-5K + 1..5K)*0.1 Do fft_result[@Freq = Mod(@Frequency + 5K - 1, 10K) + 1]
The use of positional indexing here avoids potential complications due to numeric round-off.
Amplitude and phase
Each number in the result of FFT is a complex number. You can think of this as an encoding of both the amplitude and phase of each frequency component. For example, if a 200 Hz component is present, the magnitude of the result at 200Hz (given by the Abs function) gives the power density at that frequency. But if you are to recover the original signal, the phase of the component is also relevant. Even though the power density at 200Hz is the same for a Sin(200*T*360), a Cos(200*T*360), or a Sin(200*T*360 + 30), the phase is different in each of these cases. The ComplexDegrees or ComplexRadians function can be used to extract the relative phase of each component.
I always find the absolute height of an FFT to be somewhat confusing. What is usually important is the relative height across different frequencies -- e.g., there 1000 more power at 200Hz than at 100Hz. The height is a reflection of power density, so if you double the sampling frequency, and hence half the width of each frequency bin, you'll double the amplitude of the FFT result.
The FFT is a fast implementation of the Discrete Fourier Transform (DFT), which is most efficient when the number of elements in «T» and «Freq» is an even power of 2. Analytica's implementation is still considerably more efficient than a straight DFT as long as the number of elements in «T» is the product of several small factors. So, for example, it would still be very efficient when «T» has 531,441 items, since $ 531441=3^{12} $. It is also pretty good on even powers of 10, since even powers of 10 have factors of 2 and 5, both of which are small numbers. The worst efficiency would occur when «T» has a prime number of elements, in which case the FFT would be equivalent in efficiency to a DFT and would have $ O(n^2) $ time complexity.
A straight DFT requires $ O(n^2) $ steps. If you were to apply this to a time series having n=1M points, it would require 1 trillion steps to compute, which is likely to be infeasible. The complexity of an FFT when the number of points is an even power of 2 is computed in $ O(n \log(n)) $ steps. For n=1M this comes out to 6 million steps, about 150,000 times faster than a straight DFT.
I haven't worked out the complexity for the case when n is not a power of 2, but is a product of small factors, but I think it should be something like $ O(k n \log(n)) $, where k is the largest factor and k « n.
Seasonality detection and adjustment
One application of the FFT is to detect (and remove) cyclic or seasonal components in data before applying regression techniques to fit forecasting models to the data. You plot the absolute value of the FFT for your data and see if any dominant frequencies stand out -- these would correspond to seasonalities. These are usually spotted by human eye, and then removed.
To demonstrate, here is a graph of average weekly gasoline price at the pump in the US over the previous 1024 weeks (from U.S. Energy Administration Information):
Each of the 1024 data points is spaced 7 days apart, so we define the «Freq» index to be (0..1023) / (1024*7)*365.25. The last 365.25 factor is to put the units in cycles per year, so Freq = 1 would be the annual cycle, and Freq = 12 would be a monthly seasonality.
Abs(FFT(Price_of_gasoline, Date1, Freq) →
Surprisingly, this is an example where no seasonal component is evident. Pure white noise has a flat spectrum. Even though this graph is shown on a log scale, it is quite flat.
In another example, we look at the frequency spectrum for hourly electrical demand in New England from 1-Jan-2011 to 31-Oct-2011. Here the time-series data is on the Hour index, and we define the Freq index as (0..size(hour)-1)*1/(size(hour))*24. The 24 here is to convert frequency units to cycles per day instead of cycles per hour. «Freq» ranges from 0 to 24, with the following graph manually scaled to the section showing frequencies under 3 cycles per day.
In this case two cyclic components are evident a daily cycle (at Freq = 1) and a 12 hour cycle (at Freq = 2). Before applying regression techniques, these two peaks could be removed. Instead of setting the peaks to zero, I'm going to reduce the peaks to be similar in amplitude to the neighboring data points. By looking at the table view, we find the peaks to be about 3 cells wide (although the middle cells is dominant) so I will adjust a few neighboring cells. The adjusted_spectrum is:
Var h := FFT(Electricity_demand, Hour, Freq);
Var adjust := If 0.995 < Freq < 1.005 Then 550K/Abs(h)
Else If 1.995 < Freq < 2.005 Then 80K/Abs(h)
Else 1;
h*adjust
and the time series with this seasonality removed is given by
Abs(FFTInv(adjusted_spectrum, Freq, Hour))
We estimated the 550K and 80K target levels for Freq = 1 and Freq = 2 by looking at the neighboring cells. In this example, the seasonally adjusted time series is nearly indistinguishable from the original (since the seasonality is so small), but the steps at least demonstrate the technique.
Fast convolution
Another application of the FFT is fast convolution. Convolution itself has several uses. It can be used to average neighboring values together to smooth out noise, or conversely to enhance change points. It is also used to a transfer function describing how one component acts on a signal. When computed in the time domain, convolution requires $ O(n^2) $ steps, whereas it becomes a simple multiplication in the frequency domain. Hence, it is actually more efficient to use the FFT to transform a time series and a filter shape or transfer function into the frequency domain, multiply their spectra together, and then use FFTInv to transform the result back to the time domain.
Deconvolution
Low pass or high pass filtering
Characteristic functions of probability distributions
The FFT can be useful for computing a full probability density curve for the distribution of a sum of independent random variables. Suppose x1, x2, ... xn are each random variables with distributions p1(x1), p2(x2),..., pn(xn), and let p(x) be the distribution of x=x1+x2+...+xn. Then to within a constant, k,
$ FFT(p) = k \sum_{i=1}^n FFT(p_i) $
This means that in many cases, the full distribution curve for a sum of random variables can be quickly computed by way of the FFT.
The FFT is closely related to the characteristic function of a probability distribution p(x), which is defined as:
$ CF(\omega) = E_p[e^{i \omega x}] = \int_{-\infty}^\infty p(x) e^{i \omega x} dx $
Comparing $ CF(\omega) $ to the Fourier transform, H(p), you can see that the CF is simply the FFT with a change of units -- radians instead of cycles.
The application of the FFT to computing the probability curve involves a trade-off between the number of points to use and what range of values to span. When your distributions have long tails, as you cover more of the tail, you get sparser coverage of the area around the mode.
Introduced in Analytica 4.5.
FFTInv
Retrieved from "https://wiki.analytica.com/index.php?title=FFT&oldid=52743" | CommonCrawl |
TSME: a trust-based security scheme for message exchange in vehicular Ad hoc networks
Ryma Abassi ORCID: orcid.org/0000-0003-2148-79651,
Aida Ben Chehida Douss1 &
Damien Sauveron2
Human-centric Computing and Information Sciences volume 10, Article number: 43 (2020) Cite this article
A Vehicular Ad hoc NETwork (VANET) is a self-organized network formed by connected vehicles, which allows the exchange of useful traffic information in a timely manner. In such a context, evaluating the reliability of transmissions is vital. Trust can be used to promote such healthy collaboration. In fact, trust enables collaborating vehicles to counter uncertainty and suspicion by establishing trustworthy relationships. The main contribution of this paper is the proposition of a trust-based security scheme for message exchange in a VANET called TSME. Because of VANET characteristics, including dynamicity and high speed, we first proposed a VANET Grouping Algorithm (VGA); a suitable clustering algorithm organizing the network into groups with elected Group-Heads. Second, built on the VGA, we defined our trust management scheme dealing with vehicles' reputations. Finally, we proposed a formal specification of the scheme using an inference system, and conducted a formal validation to assess its completeness and soundness rather than conducting simulations where some potentially rare conflicting or malfunctioning situations might not be detected. Soundness was proven by showing that there were no conflicts in our scheme, and completeness was established by assessing that all potential situations could be handled. The results obtained showed that our scheme for evaluating the veracity of exchanged messages is formally sound and complete.
A Vehicular Ad hoc NETwork (VANET) is a special case of a Mobile Ad hoc NETwork (MANET), where the nodes are vehicles equipped with On-Board Units (OBUs) [1]. These vehicles can directly inter-communicate, or communicate through routers called Road Side Units (RSUs). The first case is called Vehicle-to-Vehicle communication (V2V), while the second case is Vehicle-to-Infrastructure communication (V2I). In both cases, Trusted Authorities (TA) control the whole network. VANETs are mainly used to improve traffic security (such as traffic services, alarms and warning messaging) and efficiency. In this context, a security problem can have disastrous consequences since an attacker may have the ability to broadcast false alerts and/or messages for its own benefit.
Trust can be used to promote healthy collaboration by enabling collaborating vehicles to counter uncertainty and suspicion by establishing trustworthy relationships [2, 3]. Because of the importance of the challenge, trust is associated with an abstract system that helps decision making, referred to as Trust Management (TM) [4]. Hence our main proposal is a trust-based security scheme for message exchanges in VANETs. Because of VANET characteristics such as dynamicity and high speed, this scheme is built upon a new grouping algorithm called the VANET Grouping Algorithm (VGA), which organizes vehicles into groups characterized by a Group-Head (GH) and member nodes. When grouping vehicles into multiple groups, the system becomes scalable by having message relay done between GHs instead of between two neighboring peers. Grouping is generally deployed using two phases: setup and maintenance [5]. In the first phase, some nodes are chosen to act as coordinators (GHs) and each GH is associated with a number of member nodes, the whole making one group. Because the network topology changes over time, mainly due to displacement, failure, arrival, or departure of a node, a maintenance phase is required to update the group's organization. The next goal in the definition of a security architecture for a VANET is its validation. To obtain a comprehensive assessment, we decided to conduct a formal validation rather than simulations, where some potentially rare conflicting or malfunctioning situations might not be detected. Hence, we proposed an inference system for handling the VGA maintenance phase. Next, we formally validated the proposed inference system according to two main properties: (1) soundness, which ensures that the proposed model reacts correctly; and (2) completeness, which determines that the model is complete—i.e. no other situations can be found.
This paper proposes TSME, a dedicated trust-based scheme to secure message exchange in vehicular ad hoc networks, which is formally verified.
The salient contributions of this paper are as follows:
Proposing a new grouping algorithm known as VGA to organize the VANET into scalable groups and deal with specific vehicular network characteristics, including dynamicity (i.e. vehicle arrival and departure), high speed and other salient characteristics.
Proposing a trust management scheme, built upon VGA, to handle message exchange into the VANET and deal with vehicles' reputations.
Formally validating the completeness and soundness of TSME; i.e., the whole proposal including VGA and the trust management scheme, with regard to the defined specification using an inference system.
Structure of the paper
Section 2 reviews some existing studies dealing with trust in VANETs. Section 3 introduces our clustering algorithm for VANETs, called the VANET Grouping Algorithm. Section 4 provides a description of the proposed trust management scheme built upon the VGA. Section 5 details the formal specification of the scheme using an inference system and elaborates the formal validation procedure for assessing soundness and completeness. Section 6 concludes this paper.
Several reputation systems have been proposed for peer-to-peer networks [6, 7], ad hoc networks [8,9,10], wireless sensor networks [11, 12] and Internet of Vehicle [13,14,15]. However, these systems cannot be applied to VANETs in their existing forms they do not consider the main VANET characteristics: dynamicity and high speed.
In [16], the authors proposed a beacon-based trust management system, called BTM, which aims to prevent internal attackers from sending false messages in privacy-enhanced VANETs. A vehicle can use not only direct or indirect event messages, but also beacon messages to construct trust relationships in order to distinguish trustworthy event messages.
Al Falasi and Mohamed [17] proposed a "similarity-based trust management system for detecting fake safety messages in VANETs". Their scheme uses similarity-based trust relationships to detect false safety event messages from abnormal vehicles in VANETs. Moreover, it reacts to safety event claims made by a vehicle and predicts that the source vehicle will react to a truthful safety event report.
Zhang et al. [18] proposed a "trust-modeling framework for message propagation and evaluation in VANETs". In their model, a vehicle can decide whether to trust a message or not by evaluating others' opinions. However, such decentralized trust systems, relying on interactions with neighbors, are not practical in the highly dynamic environment of a VANET.
In [19], a trust-extended authentication mechanism (TEAM) was proposed. It is a decentralized, lightweight authentication scheme for highly dynamic VANETs, ensuring integrity and non-repudiation and thus increasing the vehicles' confidence in communications. However, TEAM does not deal with the reliability of the message data itself.
In [20], the authors proposed a trust-based relay selection scheme, called PTRS, which, based on Dirichlet distribution, differentiates the trust levels of the vehicles, while preserving robustness. The PTRS scheme is robust against some attacks, such as packets analysis attacks, reputation link attacks, packets dropping attacks [21], and fake reputation attacks.
Recently, Das et al. [22] proposed schemes for finding the trusted location of a vehicle. Firstly, the trust percentage of the information is computed using the responses received from vehicles. Based on this, the trust percentage of the information is calculated on the basis of the number of requests and the number of positive responses. Each vehicle giving a positive response about the information is rewarded with points for providing true information, thus enabling calculation of the trustworthiness of each node present in the network. In the second case, when the trust is below 50%, instead of going to an RSU or a TA, the vehicle will check the trustworthiness of each node in the network and accept the response of the most trustworthy node.
Mahmood et al. [23] proposed a hybrid trust management scheme to identify malicious vehicles and to prevent them from being elected as the GH. Their scheme encompasses a composite metric (i.e., trust values assigned to the vehicles coupled with their resource availability) for GH and proxy GH selection via intermittent elections. This approach helps to form trustworthy and resource-efficient vehicular networks.
In [24], Sugumar et al. proposed a trust-based authentication scheme for cluster-based VANETs. The vehicles are clustered, and the trust degree of each node is estimated. The trust degree is a combination of direct trust degree and indirect trust degree. Based on this estimated trust degree, cluster heads are selected. Then, each vehicle is monitored by a set of verifiers, and the messages are digitally signed by the sender's private key and encrypted using a public key (keys are distributed by a trusted authority and decrypted at the destination). This verifies the identity of the sender as well as the receiver, thus providing authentication to the scheme.
Hasrouny et al. [25] proposed a Trust Model for VANETs. It is a combination of centralized and distributed cooperation between vehicles and infrastructure to achieve the selection of the trustiest node as the GH. This proposed model is based on different metrics to analyze the behavior of the vehicles in the group while preserving the privacy of the participants and maintaining low network overheads.
Hao et al. [26] proposed the concepts of local trust and global trust to indicate the local and global trust relationships between vehicles. They adopted the PageRank algorithm [27], used to rank web pages to calculate the global trust of vehicles.
To summarize, the trust management models discussed above are not dynamic enough to cope with VANETs' characteristics. Some of them [16, 17] were proposed to deal with a specific type of message while others [3, 6, 19] used decentralized trust systems, relying on interactions with neighbors, which is not practical in the highly dynamic environment of a VANET. Several works [4, 10, 15, 23] tried to cope with a specific type of attack, which can be a limitation whereas several others were designed to handle a specific challenge such as authentication [24], privacy [20, 25] or localization [18, 22]. Therefore, unlike other solutions, we propose an adaptive trust-based security scheme for message exchange in VANETs based on VGA, our new clustering algorithm designed to handle the dynamicity such networks. It is described in Section 3.
VGA: a VANET grouping algorithm
Our overall proposal, TSME, a trust-based security scheme for message exchange in VANETs, depicted in Fig. 1, comprises two parts: (a) our VANET Grouping Algorithm (VGA), which is described below, to handle VANET characteristics, including dynamicity and high speed; and (b) a trust management scheme described in Section 4 built upon VGA.
The proposed Trust-based Security Scheme for Message Exchanges in VANETs
VGA is composed of three phases: pre-processing, setup and maintenance. In the pre-processing phase, initial reputations are set, whereas in the setup, groups are formed. The maintenance phase is used to handle the mobility of the vehicles and update the formed groups.
The trust management scheme uses the formed groups as well as the veracity of exchanged event messages to increase or decrease vehicles' reputations via the reputation update module. The veracity of the event messages is computed based on three indicators: \(L_{c}\) the location closeness, \(F_{c}\) the number of forwarders and \(T_{c}\) the time closeness. More details about these indicators are provided later in this section.
According to [28], the general procedural flow of a clustering algorithm is described in five steps: neighborhood discovery; cluster head selection; affiliation; announcement; and maintenance. Hence, VGA is based on similar steps with some additional ones needed to fit our specific need: trust management.
Our algorithm is based on the following assumptions:
OBUs periodically broadcast single-hop beacon messages containing (at least) position, velocity and direction.
Each vehicle generates its keys, sends a certificate signing request to the TA and retrieves a properly signed certificate as proof.
All the vehicles' clocks are synchronized.
Notations used are depicted in Table 1.
Table 1 Used notations
Pre-processing phase
During this phase, initial reputations are assigned to vehicles based on some exchanged information (called credentials). A credential measures the level of trust that we can attribute to the node during the set-up phase and when the node is entering a new cluster or moving from one cluster to another. Hence, a classification of these credentials into three levels of sensitivity is presented and a set of negotiation policies to be used during the negotiation process is defined. Our negotiation approach allows the nodes to provide more information about themselves in order to increase their degree of trust. Here, we highlight that we use a classification similar to [2], which is used to define a XeNa negotiation framework [29]. When the node is able to provide more sensitive information about itself, its trust level is enhanced. Hence, the following resources will be considered during the negotiation process: driver identity, direction, identifier, photo, position and manufacturer. These parameters are classified as follows:
Level 1 (most sensitive): driver identity, direction.
Level 2 (normally sensitive): identifier, photo, velocity.
Level 3 (less sensitive): position, manufacturer.
Two different negotiation strategies are defined: (1) vehicles send to the GH the set of all their information in order to get the highest level of trust; (2) vehicles prefer to protect their private information and preserve their privacy preferences by sending only less sensitive information.
Table 2 presents the negotiation policies corresponding to the different resources. Each vehicle builds trust in other vehicles gradually by gathering information related to the other vehicles. For less sensitive resources no negotiation policies are defined. For other resources, a negotiation policy is considered based on the received information. For instance, to be able to get the driver identity of vehicle \(v_{i}\), vehicle \(v_{j}\) must give its identifier \({Id_{v_{j}}}\), direction \({d_{v_{j}}}\) and driver identity \({dId_{v_{j}}}\).
Table 2 Negotiation policies
Once the negotiation procedure between vehicles is completed, initial reputations are affected as depicted by Algorithm 1 as follows: if the trust level is 1, the reputation value is initialized to 3, if it is 2, reputation is initialized to 2 and when the trust level is 3, reputation is initialized to 1.
Setting up phase
Once the pre-processing phase is completed, the setting up phase is triggered in order to discover the neighborhood, select a GH, and form the groups. It is based on six steps: (1) the exchange of signed beacon messages; (2) the reception of these messages; (3) the sending of acknowledge messages; (4) the reception of acknowledgement messages; (5) the election of GHs; and (6) constitution of the group. These steps are detailed in the following and the whole process is described by Algorithm 2. Exchanged messages during this phase are depicted in Table 3.
Sending of beacon messages
The beacon message, \(M_{b}\), is structured as follows:
$$\begin{aligned} M_{b}\equiv <Id_{v_{i}},\,(x_{i},y_{i}),\,vel_{v_{i}},\,d_{v_{i}},\,ts> \end{aligned}$$
where \(Id_{v_{i}}\) is the vehicle identifier sending the message, \((x_{i},y_{i})\) designates the location of vehicle \(v_{i}\), \(vel_{v_{i}}\) corresponds to the velocity of the vehicle, \(d_{v_{i}}\) is the vehicle direction and ts is the time stamp associated with the message.
Initially, each vehicle \(v_{i}\) uses its private key \(Pr_{v_{i}}\) to generate \(sign_{M_{b}}\) a signature for \(M_{b}\), the beacon message, as follows: \(sign_{M_{b}}=E(Pr_{v_{i}},M_{b})\).
Then the vehicle \(v_{i}\) broadcasts \((M_{b}||sign_{M_{b}})\).
Reception of beacon messages
Each vehicle \(v_{j}\) receiving the signed message from \(v_{i}\), verifies it using the public key \(Pu_{v_{i}}\), which is verified using \(MP_{TA}\), the public key of the TA, as well as the vehicle's identifier \(Id_{v_{i}}\). If the verification is successful, then a second verification is made: the ability of the vehicle to belong to the same group. Hence, vehicle \(v_{j}\) verifies the direction \(d_{v_{i}}\) of the sending vehicle \(v_{i}\), then its proximity (\(L_{c}\)), as formalized by Eq. 1.
$$\begin{aligned} L_{c}= \left\{\begin{array}{ll} \frac{1}{x_{i}+y_{i}} &{} if\,(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}<\varDelta ^{2}\\ \\ 0 &{} otherwise \end{array}\right. \end{aligned}$$
\(v_{j}\) is used as the origin (\(x_{j}=0\), \(y_{j}=0\)). Any vehicle located at (x, y) around the event position within a radius of \(\varDelta \) can be trusted with a level of confidence that decreases with increases in \((x_{i}+y_{i})\). The value of \(\varDelta \) can be fixed initially.
If these checks are successful, vehicle \(v_{j}\) computes its reputation score \(s_{v_{j}}\) as shown in Eq. 2 and sends an acknowledgement message \((M_{ack}\Vert sign_{M_{ack}})\) to \(v_{i}\). Otherwise, the score is set to a value of 0.
$$\begin{aligned} s_{v_{j}}=\left\{\begin{array}{ll} vel_{v_{j}}\times rep_{v_{j}} &{} if\,L_{c}\ne 0\,and\,same\,direction\\ \\ 0 &{} otherwise \end{array}\right. \end{aligned}$$
This score depends on the reputation of the vehicle as well as its velocity. Due to the dynamicity of the network, a trustworthy vehicle must have a good reputation and a similar velocity to other vehicles in order to communicate with them. Otherwise the vehicle will be left behind or will overtake other vehicles and communication with it is not of interest.
Sending of acknowledgement messages
An acknowledgement message \(M_{ack}\) is structured as follows:
$$\begin{aligned} M_{ack}\equiv <Id_{v_{j}},\,s_{v_{j}},\,ts> \end{aligned}$$
where \(Id_{v_{j}}\) is the identifier of the vehicle sending the message, \(s_{v_{j}}\) is its computed reputation score and ts is the time stamp associated with the message.
Reception of acknowledgement messages
Each vehicle \(v_{i}\) receiving an acknowledgement message (from \(v_{j}\) for instance), verifies the vehicle's existence by comparing its identifier with the one on the certificate, to avoid flooding attacks based on random identifiers. If confirmed, the message is added to its neighbors table \(Tab_{N}\). This table is maintained by each node and contains, for each vehicle, its initial reputation \(rep_{v_{j}}\) as negotiated in the pre-processing phase; its reputation score \(s_{v_{j}}\); its membership, which is a Boolean indicating whether the vehicle belongs to a current group; and a state flag \(s-flag\) indicating whether the vehicle is Benevolent or Hostile.
Group-head election
The next step in this phase is the election of the Group-Head (GH). Each vehicle compares its reputation score with the received scores and the vehicle having the greatest score sends a GH Message \(M_{GH}\) to all its members to inform them of the creation of the cluster and the election of the GH as follows:
$$\begin{aligned} M_{GH}\equiv <GH_{id},\,G_{id},\,Members,\,ts> \end{aligned}$$
where \(GH_{id}\) is the identifier of the GH, \(G_{id}\) is the identifier of the group given by the elected GH, Members corresponds to the set of vehicles belonging to this group and ts is the time stamp associated with the message.
Group constitution
Each member that receives the GH message from a GH vehicle replies with a \(M_{join}\) message to confirm its role as a member node.
Each vehicle then creates a group list \(GL=(G_{id},\,GH_{id},\,Members)\) containing the elected GH as well as the member vehicles belonging to the group identified by \(G_{id}\).
$$\begin{aligned} M_{join}\equiv <Id_{v_{i}},\,GH_{id},\,ts> \end{aligned}$$
Table 3 Exchanged messages
Maintenance phase
The maintenance phase reacts to all topology changes that may occur in the VANET, such as the departure of a vehicle or the arrival of a new vehicle. In this section, the procedures relative to these two kinds of topology change are presented.
Vehicle departure
Two cases are possible: (1) A member vehicle quits the group; and (2) The GH quits the group.
When the GH detects a member vehicle departure, it removes the vehicle from its \(Tab_{N}\) and sends a new \(M_{GH}\) informing the rest of the members of this change.
When the closest vehicle to the GH detects the departure of the GH, it informs other members, and the member vehicle having the highest reputation score s sends a new \(M_{GH}\) informing other members that a new GH has been elected.
Vehicle arrival
When the GH receives a beacon message from a new vehicle \(v_{new}\), it simply:
Verifies its signature,
Negotiates its initial reputation \(rep_{v_{new}}\),
Adds \(v_{new}\) to the \(Tab_{N}\),
Sends a \(M_{GH}\) to all the group members in order to inform them that a new vehicle belongs to their group.
Trust management scheme
Our main objective of this section is to propose a trust management scheme for VANETs, as depicted in Fig. 1. Hence, the first stage was the design of a trust-suitable grouping algorithm for VANETs while the second stage is dedicated to the trust management scheme.
Our proposal is based on the following assumptions:
Vehicles are exchanging event messages.
Each vehicle has a reputation.
Reputations are between − 3 and 3. A negative reputation is a synonym for a malicious vehicle.
Reputation is maintained through direct observations, as well as reputation messages exchanged with other vehicles.
Only GHs maintain reputation tables.
Active GHs are safe; i.e. cannot behave maliciously.
When a vehicle needs to communicate with another vehicle or a group of vehicles in order to declare an incident and/or request road liberation, a V2V warning propagation is used. A vehicle observing an event sends an event message \(M_{event}\) to its GH as follows:
$$\begin{aligned} M_{event}\equiv <Id_{v_{i}},\,type,\,(x,y),\,l,\,t_{r}> \end{aligned}$$
The GH, on receiving such an alert, verifies it by calculating a veracity score VS, on the basis of which the message can be forwarded or stopped. The veracity score, VS, is defined as follows:
$$\begin{aligned} VS=(Lc+Tc)*Fc \end{aligned}$$
where Lc is the location closeness, as defined by Eq. 1, Tc is the time closeness representing the freshness of the reported event and formalized in Eq. 4, and Fc is the forwarding chain closeness, estimating the number of vehicles that have forwarded the reported event, and formalized in Eq. 5.
$$\begin{aligned} T_{c}=\left\{\begin{array}{ll} 1-\frac{1}{|t_{r}-t_{e}|} &{} if\,|t_{r}-t_{e}|<\delta _{t}\\ \\ 0 &{} otherwise \end{array}\right. \end{aligned}$$
Where the time of occurrence of the event \(t_{e}\) is used as the origin, \(t_{r}\) corresponds to the reporting time and the value of \(\delta _{t}\) is fixed initially.
$$\begin{aligned} F_{c}=\left\{\begin{array}{ll} \frac{1}{n} &{} if\,n<\delta _{n}\\ \\ 0 &{} otherwise \end{array}\right. \end{aligned}$$
Where n is the number of forwarders and the value of \(\delta _{n}\) is fixed initially.
More precisely, this score satisfies the following hypotheses:
The closer the sender is to the event location, the higher is the veracity score.
As time closeness decreases, the veracity score decreases.
As the number of senders increases, the \(F_{c}\) decreases and consequently, the score decreases. In fact, the greater the number of vehicles that have forwarded the reported event, the higher is the probability of a modification of the event or the loss of it, e.g. due to a malevolent node.
If \(VS>0\), the message is considered trustworthy, the reputation score is increased by +0.2, is added to the message, and is forwarded. Otherwise, it is seen as untrustworthy and is simply stopped.
Whenever the score VS is zero, the node reputation score is decreased by 1. It is worth noting that values +0.2 and -1 are derived from our previous work [30]. Once it reaches zero, the vehicle is blacklisted and the GH sends a blacklist message \(M_{Blacklist}\) to the RSU as follows:
$$\begin{aligned} M_{Blacklist}\equiv <GH_{id},\,Id_{v_{i}},\,ts> \end{aligned}$$
The RSU then informs other RSUs and the TA about this misbehaving vehicle. Each RSU receiving this message informs the GHs under its control by sending them a warning message as follows:
$$\begin{aligned} M_{warning}\equiv <Id_{v_{i}},\,s-flag=`H',\,ts> \end{aligned}$$
Each GH receiving this message notifies it in its neighbors table and each time a message is sent from this vehicle, it is simply ignored.
We also propose a rehabilitation mechanism enabling malicious nodes to change their behavior so they can rejoin the system. Each GH monitors the behavior of any member nodes detected as malicious. If the malicious node subsequently behaves well, its reputation score is incremented by \(+0.1\) using the reputation module, until it reaches the neutral value of 0. Once reached, the rehabilitated node is removed from the blacklist.
Formal specification and validation
The VGA can malfunction due to conflicts between exchanges or lack of necessary functionality in messages or in the scheme phases presented in Section 3. Hence, it is necessary to validate it prior to implementation. The rest of this section is divided into two parts. First, the VGA maintenance phase is specified using a formal and automated method referred to as an inference system, based on the use of logical rules; i.e. a function that takes premises, analyses their applicability and returns a conclusion. Second, a validation task is performed using the proposed inference system. According to [31], validating a model can be done by showing that this model is free of conflict or lack of functionality in the proposed message exchange. Specifically, two main properties have to be considered as proposed in [31, 32]: (1) soundness, by checking that a topology change does not have any influence on clusters; and (2) completeness, by assessing whether the proposed inference system handles all possible situations. In the next subsection, we describe the inference systems for the proposed VGA maintenance algorithms.
The proposed inference system is based on the following assumptions:
The VGA set-up phase is already complete. The VANET is organized into groups with members and elected GHs. Each vehicle has a unique role (member or GH) and belongs to only one group.
Vehicles perform the pre-processing phase periodically; i.e. each vehicle exchanges credentials with its neighbors and initial reputations are established for each one of them.
The proposed inference system is triggered in three cases: (1) When a novel vehicle arrives in the VANET; (2) when a vehicle moves from its group to another group; and (3) when a member or a GH fails.
The inference system stops when all new, moving or failing vehicles have been handled.
Formal specification
VGA's inference system
In this section, the proposed inference system handling the changes that could occur in a VANET topology is presented in Fig. 2. Table 4 summarizes the notations used.
The rules of the system, known as inference rules, apply to couples (N, GL) whose first component N is a set of couples (a, b), where a denotes a new, moving or failing vehicle in the VANET and b denotes the vehicle detecting the arrival or the failure of node x. The second component, GL, represents the initial set of groups generated after the VGA setting up phase. Groups belonging to the GL set are composed by a couple \((\{V_{GH}\},Members)\), where \(V_{GH}\) is the elected vehicle head and Members is the set of vehicles belonging to the same group as \(V_{GH}\). Three inference rules are proposed. \(VM_{Failure}\) and \(VGH_{Failure}\) are concerned with existing members or GH failures. \(V_{Arrival}\) addresses the case of the arrival of new vehicles in the VANET or the displacement of existing members or GHs. The inference system stops when all vehicles (new, moving or failing vehicle) are dealt with.
Each of the proposed inference rules is detailed below.
\(VM_{Failure}\) inference rule. \(VM_{Failure}\) is triggered when a GH or a member vehicle detects the failure of a member \((Failure(\{V_{m}\}\equiv True))\). In this case, the \(VM_{Failure}\) inference rule is applied to remove the member vehicle \(V_{m}\) for the members set Members in an existing group of GL.
\(VGH_{Failure}\) inference rule. The \(VGH_{Failure}\) inference rule is applied when a member vehicle detects the failure of its \(GH(Failure(\{V_{GH}\}\equiv True))\). In this case, the \(VGH_{Failure}\) inference rule is triggered in order to elect another vehicle member \(V_{m}\), having the highest reputation score, to take over the old GH's role.
\(V_{Arrival}\) inference rule. \(V_{arrival}\) is triggered when a vehicle \(V_{i}\) (a GH or a member) detects a new vehicle \(V_{j}\) by receiving a Beacon message (\(V_{j}\) is detached from its group or has joined the VANET for the first time). In this case \(V_{i}\) verifies the direction and the closeness of \(V_{j}\). If these checks are successful, \(V_{i}\) computes its reputation score and sends a \(M_{ack}\) message to \(V_{j}\) in order to integrate it into its group as member.
In this section, verification of the soundness and completeness of the proposed inference system is achieved. Soundness is proved by showing that groups remain safe even after a VANET's topology changes (due to the arrival of a new vehicle, or the displacement or failure of an existing vehicle). Completeness is proved by checking that all expected potential scenarios are handled by the proposed inference system. The groups' safety proof is built upon three formal properties: (1) independence: each vehicle belongs to only one group; (2) single role: each vehicle has a unique role i.e. GH or member; and (3) stability: each group has a unique GH.
Soundness Verification
In this section, we prove that the proposed inference system is sound by showing that groups remain safe even after a VANET topology change. Hence, three properties have to be considered: independence, single role and stability.
In the following, these properties are first defined and then proved using appropriate theorems.
(Independence) "Two groups \(G_{i}\) and \(G_{j}\) are independent \(iff\,G_{i}\bigcap G_{j}=/\!\!\!{O}\)".
Theorem 1
(Groups Independence) Initially, all groups in a VANET are independent (by assumption). If \((N,GL)\,|-^*\,stop\) then the independence property is preserved.
\(If\,(N,GL)\,|-^*\,stop\) then only one inference rule among \(VM_{Failure}\), \(VGH_{Failure}\) or \(V_{Arrival}\) can be applied for each element in N. Hence, we must verify whether the application of each inference rule locally maintains this property.
When a new vehicle \(V_{n}\) arrives in the VANET network, only the \(V_{Arrival}\) inference rule is triggered in order to integrate \(V_{n}\) in a GL's group as a member node. Therefore, groups remain separate and the independence property is preserved.
When the set N includes failing members, the \(VM_{Failure}\) inference rule is applied to remove the failing \(V_{n}\) member from the Members set.
Otherwise (if a GH vehicle fails), the \(VGH_{Failure}\) inference rule is applied to remove the GH and to elect another member as GH.
\(\square \)
In these three cases, modifications occur in a single group without altering the others. Hence, the independence property is preserved.
(Single vehicle role) "A vehicle has a unique role (GH or member): Given a group \(GL(\{GH\},\,Members),\,\{GH\}\bigcap Members=/\!\!\!{O}\)".
(Vehicles' single role) Assuming that initially, all vehicles in the VANET have a single role, if \((N,GL)\,|-^*\,stop\) then the single role property is preserved.
After the VGA setup phase, vehicles have a single role (GH or member). Hence, we have to check whether the application of each rule of the proposed inference system locally maintains this property. If \((N,GL)\,|-^*\,stop\) then only one inference rule of \(VM_{Failure}\), \(VGH_{Failure}\) or \(V_{Arrival}\) applies for each element in N.
When a new vehicle \(V_{n}\) arrives in the VANET, \(V_{Arrival}\) the inference rule is applied by including \(V_{n}\) in the Members' set in GL's group. Therefore, \(\{GH\}\) and Members remain disjoint.
For a failing member vehicle \(V_{n}\), only the \(VM_{Failure}\) inference rule is triggered by removing \(V_{n}\) from the members set Members in a GL's group and all the other roles are maintained. Therefore, \(\{GH\}\) and Members remain disjoint.
For the case of GH failure, only the \(VGH_{Failure}\) inference rule is applied by removing the failing GH and by electing another member vehicle from the Members set as the new GH. Its role as a member disappears. Hence, the single role property is preserved.
(Stability). " A group GL is stable if it has a unique GH".
(Groups' single GH) Assuming that initially, all vehicles in a VANET have a single role, if \((N,GL)\,|-^*\,stop\) then the stability property is preserved.
By assuming that, after the VGA setup phase, groups are stable, we must check whether the application of each inference rule locally maintains this property. If \((N,GL)\,|-^*\,stop\) then only one inference rule among \(VM_{Failure}\), \(VGH_{Failure}\) or \(V_{Arrival}\) applies for each element in N.
When a new vehicle \(V_{n}\) arrives in the network, the \(V_{Arrival}\) inference rule is applied to integrate \(V_{n}\) into the detected GL's group as a member. Hence, the unique \(\{GH\}\) vehicle in GL is preserved.
For a failing member \(V_{m}\) in a GL group, only the \(VM_{Failure}\) inference rule is applied to remove it from its group in the Members' set. In this case, the stability property is preserved because the \(\{GH\}\) vehicle in GL remains unique.
\(VGH_{Failure}\) is applied when a GH fails, by removing it, and electing another member \(V_{m}\) as the GH. The GH vehicle remains unique.
Corollary 1
(Safety) "A group is safe if it is independent from any other groups, all its vehicles have a unique role, and it is stable".
(Soundness) Assuming that initially, the VANET is safe, if \((N,GL)\,|-^*\,stop\) then the safety property is preserved.
Using Theorems 1, 2 and 3, if \((N,GL)\,|-^* stop\), the independence, single role and stability properties are preserved. Hence, the VANET remains safe. \(\square \)
Completeness Verification
Once the soundness of the proposed inference system has been established, we can proceed to the verification of its completeness. This is achieved by determining whether all potential situations are handled by the inference system.
(Completeness) If the VANET remains safe after the arrival, displacement or failure of vehicles then \((N,GL)\,|-^*\,stop\).
Assume that a VANET remains safe after the arrival, displacement or failure of a set k of vehicles. The safety property implies that all groups are independent, include single node roles, and are stable. \(\square \)
Two situations can be distinguished:
When a vehicle \(V_{n}\) arrives or an existing vehicle moves, it integrates an existing group \(GL_{n}\) as a member node.
When a vehicle \(V_{n}\) fails, its treatment depends on its role: if \(V_{n}\) is a GH, \(VGH_{Failure}\) is applied; otherwise, \(VM_{Failure}\) handles failing members.
In both cases, \(V_{n}\) is removed from the VANET. It follows that \((N,GL)\,|-(N1,GL_{1})\,|-...(/\!\!\!{O},GL_{k})\,|-^*\,stop\).
VANETs offering interesting opportunities in traffic safety and road network efficiency while raising several technical issues such as privacy, and the ability to prevent malicious agents from interfering with network operations (e.g. modification of exchanged data, or fraudulent generation of data). In this paper, we proposed TSME, a trust-based security scheme for VANETs to counter such uncertainty. Firstly, we proposed a new grouping algorithm named VGA, associating vehicles into groups and selecting a Group-Head to mediate between the group and the rest of the network. Our grouping algorithm was designed to be highly dynamic and scalable since it can cope with the main situations that a vehicle may face in such a network, including vehicle arrival and departure. Secondly, we introduced a trust management process handling the message exchange into the VANET and built upon the grouping algorithm. Our proposal measured the veracity of a given alert message using a score calculation based on location closeness, time closeness and freshness of the message, as well as the sender's reputation. Reputations were computed and updated based on the closeness of the witness vehicle sending the message to the event, the delay between the event and its associated message, and the number of forwarders. Thirdly, we formally validated the whole proposal, TSME, using an inference system to establish its soundness and its completeness.
In future work, now we have shown the completeness and soundness of TSME, we intend to deal with some real case studies using simple numerical examples as well as large performance simulations to benchmark its efficiency.
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This work was partially supported by the ex-Région Limousin, under a grant for the project "IoTSec"; by the MIRES research federation under grants for projects "SPOCK2" and "CANIoT"; by the Région Nouvelle-Aquitaine under the grant for project "SVP-IoT"; and by the ID-Fix project, an ANR-funded project (ANR-16-CE39-0004).
Digital Security Research Lab, Higher School of Communication of Tunis, SUP'Com, University of Carthage, Tunis, Tunisia
Ryma Abassi & Aida Ben Chehida Douss
MathIS, XLIM (UMR CNRS 7252/Université de Limoges), Limoges, France
Damien Sauveron
Ryma Abassi
Aida Ben Chehida Douss
RA: main contributor of this article, ABCD: main contributor of this article, DS: writing, proofreading, has achieved lot of revisions related to scientific content and its organization to finalize the manuscript. All authors read and approved the final manuscript.
Correspondence to Ryma Abassi.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Abassi, R., Ben Chehida Douss, A. & Sauveron, D. TSME: a trust-based security scheme for message exchange in vehicular Ad hoc networks. Hum. Cent. Comput. Inf. Sci. 10, 43 (2020). https://doi.org/10.1186/s13673-020-00248-4
Accepted: 21 September 2020
VANET | CommonCrawl |
\begin{document}
\begin{abstract} We examine a random structure consisting of objects with positive weights and evolving in discrete time steps. It generalizes certain random graph models. We prove almost sure convergence for the weight distribution and show scale-free asymptotic behaviour. Martingale theory and renewal-like equations are used in the proofs. \end{abstract}
\title{A random model of publication activity} \thispagestyle{empty}
\section{Introduction}
In this paper we examine a dynamic model inspired by scientific publication activity and networks of coauthors. However, the model contains many simplifying assumptions that are not valid in reality. We still use the terminology of publications for sake of simplicity.
The model consists of a sequence of researchers. Each of them has a positive weight which is increasing in discrete time steps. The weights reflect the number and importance of the researcher's publications. One can think of cumulative impact factor for instance.
We start with a single researcher having a random positive weight. At the $n$th step a new publication is born. The number of its authors is randomly chosen. Then we select the authors, that is, one of the groups of that size; the probability that a given group is chosen is proportional to the sum of the weights of its members. After that the weights of the authors of the new publication are increased by random bonuses. Finally, a new researcher is added to the system with a random initial weight.
This is a preferential attachment model; one can see that authors with higher weights have larger chance to be chosen and increase their weights when the new publication is born.
We are interested in the weight distribution of the model. That is, for fixed $t>0$, we consider the ratio of authors of weight larger than $t$, and study the asymptotic behaviour of this quantity as the number of steps goes to infinity.
Our main results (Section 3) include the almost sure convergence of the ratio of authors of weight larger than $t$ under suitable conditions; first, when all weights are integer valued, then assuming that these random variables have continuous distribution. In both cases we describe the limiting sequence or function and determine its asymptotics. They are polynomially decaying under suitable conditions, thus our model shows scale-free behaviour.
The proofs of the almost sure convergence are based on the methods of martingale theory, while the polynomial decay of the asymptotic weight distribution follows from the results of \cite{rek} about renewal-like equations. See Section 4 for the details.
This model generalizes some random graph models. To see this, assume that every publication has only one author, and at each step, when a publication is born, connect its author to the new one with an edge. We get a random tree evolving in time.
In the particular case where the initial weights and author's bonuses are always equal to $1$, we get the Albert--Barab\'asi random tree \cite{ba}. The neighbour of the new vertex is chosen with probabilities proportional to the degrees of the old vertices. Similarly, if the initial weights and the bonuses are fixed, but they are not necessarily equal to each other, we get random trees with linear weights \cite{pittel}, sometimes called generalized plane oriented recursive trees. In these cases the asymptotic degree distribution is well-known.
\section{Notations and assumptions} \subsection{Notations}
Let the label of the only researcher being present in the beginning be $0$; the label of the researcher coming in the $n$th step is $n$.
$X_i$ is the initial weight of researcher $i$ for $i=0, 1,\ldots$. We suppose that $X_0, X_1,\ldots $ are independent, identically distributed positive random variables.
$\nu_n$ is the number of coauthors at step $n$. This is an integer valued random variable for each $n$. Obviously $\nu_n\leq n$ must hold for all $n\geq 1$. On the other hand, for technical reasons we also assume that $\nu_n\geq 1$ for all $n\geq 1$. Since the authors' weights are not necessarily increased, this may be supposed without loss of generality.
Given that $\nu_n=k$, a group of size $k$ is chosen randomly from researchers $0, \ldots, n-1$. The probability that a given group is chosen is proportional to the total weight of the group. The selected researchers will be the authors of the $n$th paper.
Let $Y_{n,1}, Y_{n,2}, \ldots, Y_{n, \nu_n}$ be nonnegative random variables. These are the authors' bonuses at step $n$. That is, the weight of the $i$th coauthor of the $n$th paper is increased by $Y_{n,i}$. The order of the coauthors is the natural order of the labels.
Let $Z_n$ be the total weight of the $n$th paper; that is, $Z_n=Y_{n,1}+Y_{n,2}+ \ldots+ Y_{n,\nu_n}$ for $n\geq 1$.
$W\left( n, i\right)$ denotes the weight of author $i$ after step $n$ for $i=0, \ldots, n$. This is equal to $X_i$ plus the sum of all bonuses $Y_{j,\ell}$ for which author $i$ is the $\ell$th author of the $j$th paper $\left(\ell=1,\ldots, \nu_j, \ j=1,2,\ldots,n\right)$.
Let $S_n$ be the total weight after $n$ steps; namely, \[ S_n=W\left( n,0\right)+\ldots +W\left( n,n\right)=X_0+\ldots+X_n+Z_1+\ldots+Z_n. \]
$X,\ \nu,\ Y_n,\ Y$ and $Z$ are random variables. $X$ is equal to $X_0$ in distribution, and $Y_n$ is equal to $Y_{n,1}$ in distribution for $n\geq 1$. The other random variables will be determined later by the assumptions.
Finally, $\mathcal{F}_n$ is the $\sigma$-algebra generated by the first $n$ steps; $\mathcal{F}_n^+=\sigma\left\lbrace \mathcal{F}_n, \nu_{n+1}\right\rbrace$.
Throughout this paper $\mathbb I(A)$ denotes the indicator of event $A$. We say that two sequences $\left( a_n\right)$, $\left( b_n\right)$ are asymptotically equal $\left( a_n\sim b_n\right)$, if they are positive except finitely many terms, and $a_n/b_n\to 1$ as $n\rightarrow \infty$. A sequence $\left( a_n\right)$ is exponentially small if $\left\vert a_n\right\vert\leq q^n$ holds for all sufficiently large $n\in \mathbb{N}$ for some $0<q<1$.
\subsection{Assumptions}
Now we list the assumptions on the model.
\begin{as} \label{puba1}$X_0, X_1, \ldots$ are independent, identically distributed. The initial weights $X_n$, and the triplets $\left(\mathcal{F}_{n-1},\left(Y_{n,1},\ldots,Y_{n,\nu_n}\right),\nu_n\right)$ are independent $\left( n=1, 2,\ldots\right)$. \end{as}
\begin{as} \label{puba2}$X$ has finite moment generating function. \end{as}
\begin{as}\label{puba3} $\nu_n$ and $\left(Y_{n,1},\ldots,Y_{n,\nu_n}\right)$ are independent of $\mathcal{F}_{n-1}$ for $n\geq 1$. \end{as}
\begin{as}\label{puba4} $\nu_n\rightarrow \nu$ in distribution as $n\rightarrow \infty$; in addition, $\mathbb{E}\nu_n\rightarrow\mathbb{E}\nu <\infty$ and $\mathbb{E}\nu_n^2\rightarrow\mathbb{E}\nu^2 <\infty$ hold. \end{as}
Recall that $\nu_n\leq n$. Assumption 4 trivially holds if $\nu$ is a fixed random variable with finite second moment, and the distribution of $\nu_n$ is identical to the distribution of $\min\left( n,\nu\right)$, or to the conditional distribution of $\nu$ with respect to $\left\lbrace\nu\leq n\right\rbrace$.
\begin{as}\label{puba5} The conditional distribution of $\left(Y_{n,1},\ldots,Y_{n,\nu_n} \right)$, given $\nu_n=k$, does not depend on $n$. Moreover, the components are conditionally interchangeable, given $\nu_n=k$. \end{as}
\begin{as}\label{puba6} $Z_n$ has finite expectation. \end{as}
Now we know that $\left( \nu_n,\ Y_n,\ Z_n\right)\rightarrow \left( \nu,\ Y,\ Z\right)$ in distribution as $n\rightarrow \infty$, where $Y$ and $Z$ are random variables. We need that they also have finite moment generating functions, and they are not degenerate.
\begin{as}\label{puba7}
$Y$ and $Z$ have finite moment generating functions. \end{as}
\begin{as}\label{puba8} $X_n$, $Y_n$, $X$ and $Y$ are positive with positive probabilities for every $n=1,2,\dots$ . In addition, if $Y$ is integer-valued, then the greatest common divisor of the set $\left\lbrace i: \mathbb{P}\left( Y=i\right)>0\right\rbrace$ is equal to $1$. \end{as}
The condition on the positivity of $X_n$ and $Y_n$ is not crucial. The positivity of $X$ and $Y$ implies that the same holds for $X_n$ and $Y_n$ if $n$ is large enough; we may assume this for all $n$ without loss of generality. On the other hand, if $\left( Y_n\right)$ is identically equal to $0$, that is, there are no bonuses at all, then the model only consists of the sequence of independent and identically distributed initial weights $X_n$, and the problem of empirical weight distribution becomes trivial. The last part of this assumption excludes periodicity.
There are two important particular cases satisfying all of our conditions. In the first one the weight of the paper is equally distributed among the authors. That is, $Z_1, Z_2, \ldots$ are independent identically distributed random variables, and $Y_{n,1}=\ldots=Y_{n,\nu_n}=Z_n/\nu_n$. The other option is that every author gets the total bonus, regardless the number of coauthors. More precisely, $Y_1,Y_2,\ldots$ are independent and identically distributed, and $Y_{n,1}=\ldots=Y_{n,\nu_n}=Y_n$, thus $Z_n=\nu_nY_n$.
\section{Main results}
\subsection*{Discrete weight distribution} Suppose first that $X, Y_1, Y_2, \ldots$ are nonnegative integer valued random variables. Let $\xi_n\left( j \right)$ denote the number of researchers of weight $j$ after $n$ steps, that is, \[ \xi_n\left( j\right)=\bigl\vert \left\lbrace 0\leq i\leq n: W\left( n,i\right)= j\right\rbrace\bigr\vert,\quad j, n=1,2,\ldots\,. \]
The first theorem is about the almost sure behaviour of this quantity.
\begin{theorem}\label{pubt1} $\dfrac{\xi_n(j)}{n}\rightarrow x_j$ almost surely as $n\rightarrow \infty$ with positive constants $x_j$, $j=1,2,\ldots$. The sequence $\left( x_j\right)$ satisfies the recursion \begin{equation}\label{pubrec} x_j=\frac{\sum\limits_{i=1}^{j-1} x_{j-i}\biggl[\dfrac{(j\!-\! i) \mathbb{P}(Y=i)}{\mathbb{E} X\!+\!\mathbb{E} Z}+\mathbb{E}\bigl((\nu\!-\! 1)\mathbb I(Y=i)\bigr) \biggr]+\mathbb{P}(X=j)}{\alpha j+\beta+1}\, , \end{equation} where $\alpha=\dfrac{\mathbb{P}(Y>0)}{\mathbb{E} X\!+\!\mathbb{E} Z}$,\ \ $\beta=\mathbb{E}\bigl((\nu\!-\! 1)\mathbb I(Y>0)\bigr)$. \end{theorem}
The second theorem describes the asymptotic behaviour of the sequence $\left( x_j\right)$. \begin{theorem}\label{pubt2} We have $x_j\sim C\,j^{-\gamma}$ as $j\rightarrow \infty$, where $C$ is a positive constant, and \[ \gamma=\frac{\mathbb{E} X\!+\!\mathbb{E} Z}{\mathbb{E} Y}\!+\! 1. \] \end{theorem}
\subsection*{Continuous weight distribution}
Now we assume that the distribution of $X$ and the conditional distributions of $Y_n\mid\nu_n=k$ are continuous for $k=1,2,\dots, n,\ n=1,2,\ldots$\,. This implies that the distribution of $Y_n$ is continuous. Moreover, since the conditional distribution does not depend on $n$ according to Assumption \ref{puba5}, the distribution of $Y$ is also continuous.
Let $F(t)=\mathbb{P}(Y>t)$, $H(t)=\mathbb{E}\bigl((\nu-1)\mathbb I(Y>t)\bigr)$, and \[ L(t,s)=\frac{sF(s)+t(1-F(s))}{\mathbb{E} X+\mathbb{E} Z}-H(s),\quad 0\le s\le t. \] It is clear that $L(t,s)$ is continuous, and, being the difference of two increasing functions, it is of bounded variation for fixed $t$.
This time $\xi_n(t)$ denotes the number of researchers with weight more than $t$ after $n$ steps. \[ \xi_n\left( t\right)=\bigl\vert \left\lbrace 0\leq i\leq n: W\left( n,
i\right)>t\right\rbrace\bigr\vert, \quad t>0,\ n=1,2,\ldots\,. \]
\begin{theorem}\label{pubt3} $\dfrac{\xi_n(t)}{n}\to G(t)$ almost surely, as $n\rightarrow \infty$, where $G(t)$ is the solution of the following integral equation. \begin{equation}\label{pube0} G(t)=\frac{{\displaystyle\int_0^t} G(t-s)\,d_sL(t,s)+H(t)+\mathbb{P}(X>t)} {\dfrac{t}{\mathbb{E} X+\mathbb{E} Z}+\mathbb{E} \nu} \end{equation} for $t>0$, and $G(0)=1$. \end{theorem}
Adding some extra conditions we can obtain results on the asymptotic behaviour of $G$.
\begin{theorem}\label{pubt4} Suppose that the distribution of $Y$ is absolutely continuous. Then we have $G(t)\sim C\,t^{-\gamma}$ as $t\rightarrow \infty$, where $C$ is a positive constant, and \[ \gamma=\frac{\mathbb{E} X+\mathbb{E} Z}{\mathbb{E} Y}. \] \end{theorem}
\begin{remark} The difference of the exponents in the discrete and continuous cases is due to the difference in the definitions. Namely, in the first case $\xi_n$ denotes the weight distribution, while in the second case it stands for the complementary cumulative weight distribution function. \end{remark}
\section{Proofs}
First we prove some propositions we will often use in the sequel.
\begin{lemma}\label{publ1} Let $(\mathcal F_n)$ be a filtration, $(\xi_n)$ a nonnegative adapted process. Let $(w_n)$ be a regularly varying sequence of positive numbers with exponent $\mu>-1$. Suppose that \begin{equation}\label{lemmafelt}
\mathbb \mathbb{E}\bigl((\xi_n-\xi_{n-1})^2\bigm|\mathcal F_{n-1}\bigr) =O\left( n^{1-\delta+2\mu}\right) \end{equation} holds with some $\delta>0$. Let $(u_n)$, $(v_n)$ be nonnegative predictable processes such that $u_n<n$ for all $n\geq 1$.
$(a)$ Suppose that \[ \mathbb \mathbb{E}(\xi_n\mid\mathcal F_{n-1})\le \Bigl(1-\dfrac{u_n}{n}\Bigr)\xi_{n-1}+v_n, \] and $\lim_{n\rightarrow \infty} u_n=u$, $\limsup_{n\rightarrow \infty} v_n/w_n\le v$ with some random variables $u>0,\ v\geq 0$. Then \begin{equation*} \limsup_{n\rightarrow\infty}\frac{\xi_n}{nw_n}\le \frac{v}{u+\mu+1} \quad a.s. \end{equation*}
$(b)$ Suppose that \[ \mathbb \mathbb{E}(\xi_n\mid\mathcal F_{n-1})\ge \Bigl(1-\dfrac{u_n}{n}\Bigr)\xi_{n-1}+v_n, \] and $\lim_{n\rightarrow \infty}u_n=u$, $\liminf_{n\rightarrow \infty} v_n/w_n\ge v$ with some random variables $u>0,\ v\geq 0$. Then \begin{equation*} \liminf_{n\rightarrow\infty}\frac{\xi_n}{nw_n}\ge \frac{v}{u+\mu+1} \quad a.s. \end{equation*} \end{lemma}
This is a stochastic counterpart of a lemma of Chung and Lu \cite{chung}. We will often apply this proposition with the sequence $w_n\equiv 1$ and $\mu=0$.
\textbf{Proof.\ } Suppose first that $v$ is strictly positive. Let $\mathcal F_0$ be the trivial $\sigma$-algebra, $\xi_0=0$, and \[ c_n=\prod_{i=1}^n\Bigl(1-\frac{u_i}{i}\Bigr)^{-1},\quad n\geq 1. \]
We have \[\log c_n=\sum_{i=1}^n\frac{u_i}{i}\bigl(1+o(1)\bigr)= u\sum_{i=1}^n\frac{1+o(1)}{i}\,. \] Hence for all $t>1$ we get that $\lim_{n\rightarrow \infty} (\log c_{[tn]}-\log c_n)=u\log t$. That is, $\left( c_n\right)$ is regularly varying with exponent $u$. It is clear that
\begin{equation}\label{pube1}\mathbb{E}\bigl(c_n\xi_n\bigm|\mathcal F_{n-1}\bigr)\leq c_{n-1}\xi_{n-1}+c_nv_n.\end{equation} Therefore $c_n\xi_n$ is a submartingale. Consider the Doob decomposition $c_n\xi_n=M_n+A_n$, where \[ M_n=\sum_{i=1}^n\left(c_i\xi_i-
\mathbb{E}\bigl(c_i\xi_i\bigm|\mathcal F_{i-1}\bigr)\right) \] is a martingale, and \[
A_n=\sum_{i=1}^n\left(\mathbb{E}\bigl(c_i\xi_i\bigm|\mathcal F_{i-1}\bigr)- c_{i-1}\xi_{i-1}\right). \] From inequality \eqref{pube1} it follows that \[ A_n\leq\sum_{i=1}^n c_iv_i. \]
Consider the increasing process in the Doob decomposition of the square of the martingale $(M_n)$. Using condition \eqref{lemmafelt} we get that \begin{align*}
B_n&=\sum_{i=1}^n\mathop{\textrm{Var}}\bigl(c_i\xi_i\bigm|\mathcal F_{i-1}\bigr)
=\sum_{i=1}^n\mathop{\textrm{Var}}\bigl(c_i(\xi_i-\xi_{i-1})\bigm|\mathcal F_{i-1} \bigr)\\
&\leq\sum_{i=1}^n c_i^2\,\mathbb{E}\bigl((\xi_i-\xi_{i-1})^2\bigm|\mathcal F_{i-1}\bigr)=O\Biggl(\;\sum_{i=1}^n i^{1-\delta+2\mu}c_i^2\Biggr). \end{align*}
Since $n^{1-\delta+2\mu}c_n^2$ is still regularly varying with exponent $2u+1-\delta+2\mu$, it follows that $B_n=O\bigl(n^{2-\delta+2\mu}c_n^2\bigr)$ (see e.g. \cite{bingham,bojanic}). Hence, by Propositions VII-2-3 and VII-2-4 of \cite{[Ne75]}, we have \[ M_n=O(B_n^{1/2+\varepsilon}\bigr)=O\bigl(n^{(2-\delta+2\mu)(1/2+\varepsilon)} c_n^{1+2\varepsilon}\bigr)=o\bigl(nc_nw_n\bigr)\quad a.s., \] for all $0<\varepsilon<\dfrac{\delta}{4(u+1+\mu)}$\,.
On the other hand, using the fact $u+\mu>-1$, and the results of \cite{bingham, bojanic} on regularly varying sequences we obtain that \[ A_n\leq\sum_{i=1}^nc_iv_i\le\bigl(1+o(1)\bigr)\,v\sum_{i=1}^nc_iw_i\sim v\,\frac{nc_nw_n}{u+\mu+1} \] almost surely, as $n\rightarrow \infty$. This implies that \[ c_n\xi_n\le\bigl(1+o(1)\bigr)\frac{v}{u+\mu+1}\,nc_n w_n, \] thus the proof of part $(a)$ is complete for positive $v$.
The general case of nonnegative $v$ can be deduced from the positive case by noticing that \[ \mathbb \mathbb{E}(\xi_n\mid\mathcal F_{n-1})\le \Bigl(1-\dfrac{u_n}{n}\Bigr)\xi_{n-1}+\max\left( v_n, \varepsilon\right) \] for arbitrary $\varepsilon>0$.
The proof of part (b) is similar. In this case \[ A_n\ge\sum_{i=1}^nc_iv_i\sim\frac{v}{u+\mu+1}\,nc_nw_n, \] a.s. on the event $\{v>0\}$. Hence, using $c_n\xi_n\sim A_n$, we get that \[ c_n\xi_n\ge\frac{v}{u+\mu+1}\,nc_nw_n\bigl(1+o(1)\bigr). \] On the event $\left\lbrace v=0\right\rbrace$ the inequality trivially holds. \qed
\begin{lemma}\label{publ2} The conditional probability that an author of weight $j$ is chosen, given $\mathcal{F}_n^+$ and $\nu_{n+1}=k$, is equal to \[ \frac{k-1}{n}+\frac{n+1-k}{n}\cdot\frac{j}{S_{n}} =\frac{k-1}{n}\lef1-\frac{j}{S_n}\right)+\frac{j}{S_n}. \] \end{lemma}
\textbf{Proof.\ } Consider those groups of size $k\ge 2$ that contain researcher $i$ ($0\leq i\leq n$). There are $\binom{n}{k-1}$ of them, because the total number of researchers is $n+1$. Researcher $i$ belongs to all of them, while the other researchers belong to $\binom{n-1}{k-2}$ of those groups. Therefore the total weight of these groups can be obtained in the following way.
\begin{align*}
\sum_{\substack{H\subset\left\lbrace 0,\ldots,n\right\rbrace\\|H|=k,\,i\in H}} \ \sum_{j\in H}W(n,j)&=\binom{n}{k-1}W(n,i)+ \sum_{j\neq i}\binom{n-1}{k-2}W(n,j)\\ &=\binom{n-1}{k-1}W(n,i)+\binom{n-1}{k-2}S_n. \end{align*}
On the other hand, the total weight of all groups of size $k$ is given by \[ \binom{n}{k-1}S_n. \]
Hence the conditional probability that researcher $i$ participates in the $\left( n+1\right)$st paper given that it has $k$ authors is equal to \[ \frac{k-1}{n}+\frac{n-k+1}{n}\cdot\frac{W(n,i)}{S_n} =\frac{k-1}{n}\left( 1-\frac{W(n,i)}{S_n}\right)+\frac{W(n,i)}{S_n}. \] This obviously holds for $k=1$ as well. \qed
\subsection*{Proof of Theorem \ref{pubt1}.}
Recall that in Theorem \ref{pubt1} we assumed that $X, Y_1, Y_2, \ldots$ are integer valued random variables. Let us introduce \[ H\left( i \right)=\mathbb{E}\left( \left( \nu-1\right) \mathbb I\left( Y=i\right)\right), \] then $\beta=\sum_{i=1}^{\infty} H(i)$.
We prove the theorem by induction on $j$. The following argument is valid for all $j=1, 2, \ldots$. For $j>1$ we will use the induction hypothesis.
At each step the number of authors of weight $j$ may change due to the following events. \begin{itemize} \item A given author of weight $j$ is chosen and he gets positive bonus. \item A given author of weight $j-i$ is chosen and his bonus is equal to $i$. \item The initial weight of the new author is $j$. \end{itemize} Therefore Lemma \ref{publ2} implies that \begin{multline}\label{pube1.1}
\mathbb{E}\bigl(\xi_n(j)\bigm|\mathcal F_{n-1}^+\bigr)\\
=\xi_{n-1}(j)\biggl[1-\mathbb{P}\bigl(Y_n>0\bigm|\mathcal F_{n-1}^+\bigr) \Bigl(\frac{\nu_n-1}{n-1} + \frac{n-\nu_n}{n-1}\cdot \frac{j}{S_{n-1}}\Bigr)\biggr]\\
+\sum_{i=1}^{j-1}\xi_{n-1}(j-i)\mathbb{P}\bigl(Y_n=i\bigm| \mathcal F_{n-1}^+\bigr)\Bigl(\frac{\nu_n-1}{n-1}+ \frac{n-\nu_n}{n-1}\cdot\frac{j-i}{S_{n-1}}\Bigr)\\ +\mathbb{P}(X_n=j). \end{multline} Recall that $\nu_n\geq 1$ is assumed.
We introduce the time-dependent versions of the already defined quantities. Namely, \[H_n(i)=\mathbb{E}\bigl((\nu_n-1)\mathbb I(Y_n=i)\bigr); \ \beta_n=\sum_{i=1}^n H_n(i)=\mathbb{E}\bigl((\nu_n-1)\mathbb I(Y_n>0)\bigr). \]
Let us take conditional expectation given $\mathcal F_{n-1}$ in both sides of \eqref{pube1.1}. Then we get that \begin{multline}\label{pube1.2}
\mathbb{E}\bigl(\xi_n(j)\bigm|\mathcal F_{n-1}\bigr)\\ =\xi_{n-1}(j)\biggl[1-\frac{\beta_n}{n-1}-\Bigl(\mathbb{P}(Y_n>0)- \frac{\beta_n}{n-1}\Bigr)\frac{j}{S_{n-1}}\biggr]\\ +\sum_{i=1}^{j-1}\xi_{n-1}(j-i)\biggl[\frac{H_n(i)}{n-1}+ \Bigl(\mathbb{P}(Y_n=i)-\frac{H_n(i)}{n-1}\Bigr)\frac{j-i}{S_{n-1}}\biggr]\\ +\mathbb{P}(X_n=j) \quad (j,n=1,2,\ldots). \end{multline}
We are going to apply Lemma \ref{publ1} to the sequence $\left( \xi_n(j)\right)$ with $w_n\equiv 1$ and $\mu=0$. It is clear that
$|\xi_n(j)-\xi_{n-1}(j)|\le \nu_n+1$, hence \[
\mathbb{E}\bigl((\xi_n(j)-\xi_{n-1}(j))^2\bigm|\mathcal F_{n-1}\bigr)\le \mathbb{E}(\nu_n+1)^2=O(1). \] Thus, condition \eqref{lemmafelt} on the differences of the sequence $\xi_n(j)$ is satisfied. Moreover, as $n\rightarrow \infty$, we have \[ u_n=n\biggl[\frac{\beta_n}{n-1}+\Bigl(\mathbb{P}(Y_n>0)-\frac{\beta_n}{n-1}\Bigr) \frac{j}{S_{n-1}}\biggr]\to \beta+\alpha j. \] Note that $\alpha>0$ because of Assumption \ref{puba8}.
Though the random variables $Z_1, Z_2,\ldots$ are not necessarily identically distributed, they satisfy the following conditions. \[ \sum_{n=1}^{\infty}\frac{\mathop{\textrm{Var}}\left( Z_n\right)}{n^2}<\infty,\quad \lim_{n\rightarrow \infty} \frac1n\sum_{i=1}^n \mathbb{E} Z_i=\mathbb{E} Z. \]
Therefore Kolmogorov's theorem (Theorem 6.7. in \cite{petrov}) can be applied. We get that $S_n\sim n\left( \mathbb{E} X+\mathbb{E} Z\right)$ almost surely as $n\rightarrow \infty$. Using this, and also the induction hypothesis when $j>1$, we conclude that \begin{multline*} v_n=\sum_{i=1}^{j-1}\xi_{n-1}(j-i)\biggl[\frac{H_n(i)}{n-1}+ \Bigl(\mathbb{P}(Y_n=i)-\frac{H_n(i)}{n-1}\Bigr)\frac{j-i}{S_{n-1}}\biggr] +\mathbb{P}(X_n=j)\\ \to \sum_{i=1}^{j-1}x_{j-i}\biggl[H(i)+\mathbb{P}(Y=i)\, \frac{j-i}{\mathbb{E} X+\mathbb{E} Z}\biggr]+\mathbb{P}(X=j), \end{multline*} as $n\to\infty$.
From equations \eqref{pube1.1} and \eqref{pube1.2} one can see that $\left( u_n\right)$ and $\left( v_n\right)$ are nonnegative predictable processes. Moreover, $u_n<n$ if $n$ is large enough, because then $\nu_n<n$ and $j<S_{n-1}$. We have also seen that the limit of $\left( u_n\right)$ is positive. Hence, by Lemma \ref{publ1}, the induction step and the proof of Theorem \ref{pubt1} is complete.
$\square$
\subsection*{Proof of Theorem \ref{pubt2}.}
Write recursion \eqref{pubrec} in the following form. \[ x_j=\sum_{i=1}^{j-1}w_{j,i}x_{j-i}+r_j, \] where for $i,j\geq 1$ we set \[ w_{j,i}=\frac{\biggl[\dfrac{(j\!-\! i)\mathbb \mathbb{P}(Y=i)} {\mathbb \mathbb{E} X\!+\!\mathbb \mathbb{E} Z}+\mathbb \mathbb{E}\bigl((\nu\!-\! 1) \mathbb I(Y=i)\bigr)\biggr]}{\alpha j+\beta+1}\,, \] and \[ r_j=\frac{\mathbb{P}\left( X=j\right)}{\alpha j+\beta+1}\,. \]
In order to apply Theorem 1 of \cite{rek} we try to find sequences $\left( a_i\right)$, $\left( b_i\right)$, $\left( c_{j,\,i}\right)$ such that $w_{j,i}=a_i\!+\! \frac{b_i}{j}\!+\! c_{j,\,i}$ holds, then we have to check that $a_i$, $b_i$, $c_{i,j}$, $r_i$ satisfy the following conditions. \begin{enumerate}[(i)] \item $a_i\ge 0$ for $i\geq 1$, and the greatest common divisor of the set $\left\lbrace i: a_i>0\right\rbrace$ is $1$; \item $r_i$ is nonnegative, and not identically zero; \item\label{iii} there exists $z>0$ such that \begin{gather*} 1<\sum_{i=1}^\infty a_iz^i<\infty,\qquad
\sum_{i=1}^\infty|b_i|z^i<\infty,\\
\sum_{i=1}^\infty\sum_{j=1}^{i-1}|c_{i,j}|z^j<\infty,\qquad \sum_{i=1}^\infty r_iz^i<\infty. \end{gather*} \end{enumerate}
Therefore we set \begin{equation*} a_i=\lim_{j\to\infty}w_{j,i}=\frac{\mathbb{P}(Y=i)}{\alpha(\mathbb{E} X+\mathbb{E} Z)}= \mathbb{P}(Y=i\mid Y>0), \quad i=1, 2,\dots\,, \end{equation*} then we define \[ b_i=\lim_{j\to\infty}j(w_{j,i}-a_i)=\frac{1}{\alpha} \Bigl[H(i)-(\alpha i+\beta+1)a_i\Bigr]. \]
Finally, we introduce \[ c_{j,i}=w_{j,i}-a_i-\frac{b_i}j=-b_i\cdot\frac{\beta+1}{j(\alpha j+\beta+1)}\,. \]
Since $(a_i)$ is a probability distribution, for (\ref{iii}) it suffices to show that $(a_i)$, $(b_i)$, $(c_{j,i})$, and $(r_i)$ are exponentially small.
According to Assumption \ref{puba7}, $Y$ has finite moment generating function. This implies that $\left( a_i\right)$ is exponentially small. The same holds for $\left( b_i\right)$, because \[ \sum_{i=1}^\infty H(i)e^{\varepsilon i}= \mathbb{E}\bigl((\nu-1)e^{\varepsilon Y})\bigr)\le\Bigl[\mathbb{E}(\nu-1)^2\; \mathbb{E}\bigl(e^{2\varepsilon Y}\bigr)\Bigr]^{\!1/2}<\infty \] if $\varepsilon>0$ is small enough. Finally, \begin{multline*}
\sum_{j=1}^\infty\sum_{i=1}^{j-1}|c_{j,i}|e^{\varepsilon i}=
\sum_{j=1}^\infty\sum_{i=1}^{j-1}|b_i|\,\frac{\beta+1}{j(\alpha j+\beta+1)}\,e^{\varepsilon i}\\ \le\sum_{j=1}^\infty \frac{\beta+1}{j(\alpha j+\beta+1)}\;
\sum_{i=1}^\infty |b_i|e^{\varepsilon i}<\infty. \end{multline*}
The sequence $\left( r_j\right)$ is also exponentially small, because $X$ has finite moment generating function by Assumption \ref{puba7}.
$w_{j,i}$, $a_j$, $r_j$ are nonnegative. Assumption \ref{puba8} guarantees that the greatest common divisor of the set $\left\lbrace j: a_j>0\right\rbrace$ is equal to 1, and $r_j>0$ for some $j$.
We have checked all conditions of Theorem 1 of \cite{rek}. Since $X$ is not identically $0$, there exists a $k$ with $x_k>0$. On the other hand, by Assumption \ref{puba8}, $P\left( Y=\ell\right)>0$ for some $\ell$. Now, one can see from the recursion that $x_k, x_{k+l}, x_{k+2l},\ldots$ are all positive, hence the sequence $\left( x_n\right)$ has infinitely many positive terms. Therefore, applying the theorem we obtain that $x_j\sim C\;j^{-\gamma}$ as $j\rightarrow \infty$, where \[ \gamma=-\frac{\sum_{i=1}^\infty b_i}{\sum_{i=1}^\infty ia_i}\,. \] It is easy to see that \begin{gather*} \sum_{i=1}^\infty ia_i=\sum_{i=1}^{\infty} i\,\mathbb{P}\left( Y=i\mid Y>0\right)=\frac{\mathbb{E} Y}{\mathbb{P}(Y>0)}\,;\\ -\sum_{i=1}^\infty b_i=-\frac{\beta}{\alpha}+\sum_{i=1}^\infty ia_i+ \frac{\beta+1}{\alpha}=\frac{\mathbb{E} X+\mathbb{E} Z+\mathbb{E} Y}{\mathbb{P}(Y>0)}\,. \end{gather*} Hence the statement of Theorem \ref{pubt2} follows. \qed
\subsection*{Proof of Theorem \ref{pubt3}}
We will use the results of the discrete part, namely, Theorem \ref{pubt1}. Let $h$ be sufficiently small positive number. We will consider limits as $h\rightarrow 0$.
Let $F_n(t)=\mathbb{P}(Y_n>t)$ and $H_n(t)=\mathbb{E}\bigl((\nu_n-1)\mathbb I(Y_n>t)\bigl)$, as before. Furthermore, for a decreasing function $\varphi$ let $\Delta_h\varphi(t)=\varphi(t-h)-\varphi(t)$.
By Lemma \ref{publ2}, the conditional probability of the event that an author of weight between $t-ih$ and $t-(i-1)h$ is chosen, and his bonus is at least $(i-1)h$, given $\mathcal F_{n-1}^+$, is bounded from above by \begin{equation}\label{pube2} \biggl[\frac{t-(i-1)h}{S_{n-1}}+ \biggl(1-\frac{t-(i-1)h}{S_{n-1}}\biggr)\frac{\nu_n-1}{n-1}\biggr]
\mathbb{P}\bigl(Y_n>(i-1)h\bigm|\mathcal F_{n-1}^+\bigr). \end{equation} Hence the conditional probability with respect to $\mathcal F_{n-1}$ is at most \begin{equation*} u_i:=\frac{t-(i-1)h}{S_{n-1}}\;F_n\bigl((i-1)h\bigr) +\frac{1}{n-1}\biggl(1-\frac{t-(i-1)h}{S_{n-1}}\biggr) H_n\bigl((i-1)h\bigr). \end{equation*} Note that $u_i$ depends on $n$, which is fixed at the moment. We get that \begin{multline*}
\mathbb{E}\bigl(\xi_n(t)\bigm|\mathcal F_{n-1}\bigr)\\ \le\xi_{n-1}(t)+\sum_{i=1}^{\lceil t/h\rceil} \Bigl[\xi_{n-1}(t-ih)-\xi_{n-1}\bigl(t-(i-1)h\bigr)\Bigr]u_i +\mathbb{P}(X>t). \end{multline*} After rearranging we obtain that \begin{multline}\label{pube4}
\mathbb{E}\bigl(\xi_n(t)\bigm|\mathcal F_{n-1}\bigr)\le\xi_{n-1}(t)(1-u_1)\\ +\sum_{i=1}^{\lceil t/h\rceil}\xi_{n-1}(t-ih)(u_i-u_{i+1}) +nu_{\lceil t/h\rceil+1}+\mathbb{P}(X>t). \end{multline}
Here \begin{multline*} u_1=\frac{t}{S_{n-1}}+\frac{1}{n-1}\biggl(1-\frac{t}{S_{n-1}}\biggr) \mathbb{E}(\nu_n-1)\\ =\biggl(\frac{t}{\mathbb{E} X+\mathbb{E} Z}+\mathbb{E}\nu-1\biggr)\frac{1+o(1)}{n}\,, \end{multline*} and \begin{multline*} u_i-u_{i+1}=\frac{h}{S_{n-1}}\,F_n\bigl((i-1)h\bigr)+ \frac{t-ih}{S_{n-1}}\,\Delta_hF_n(ih)\\ -\frac{1}{n-1}\;\frac{h}{S_{n-1}}\,H_n\bigl((i-1)h\bigr) +\frac{1}{n-1}\biggl(1-\frac{t-ih}{S_{n-1}}\biggr) \Delta_hH_n(ih). \end{multline*}
This implies that \[ n(u_i-u_{i+1})\to\frac{h}{\mathbb{E} X+\mathbb{E} Z}\,F\bigl((i-1)h\bigr) +\frac{t-ih}{\mathbb{E} X+\mathbb{E} Z}\,\Delta_hF(ih)+\Delta_hH(ih), \] as $n\to\infty$. Finally, \[ nu_{\lceil t/h\rceil+1}\le\frac{n}{n-1}\Bigl(1+\frac{h}{S_{n-1}}\Bigr) H_n(t), \] hence \[ \limsup_{n\to\infty}nu_{\lceil t/h\rceil+1}\le H(t). \] Let \[ G_u(t)=\limsup_{n\to\infty}\frac{\xi_n(t)}{n}\, \] (subscript $u$ stands for ``upper''). $G_u(t)$ is a decreasing random function, and \begin{multline*} \limsup_{n\to\infty}\sum_{i=1}^{\lceil t/h\rceil} \xi_{n-1}(t-ih)(u_i-u_{i+1})\\ \le \sum_{i=1}^{\lceil t/h\rceil}G_u(t-ih)\biggl[ \frac{F\bigl((i-1)h\bigr)}{\mathbb{E} X+\mathbb{E} Z}\,h +\frac{t-ih}{\mathbb{E} X+\mathbb{E} Z}\,\Delta_hF(ih)+\Delta_hH(ih)\biggl]. \end{multline*}
Denote the sum on the right hand side by $\Sigma_u(t,h)$. We want to apply Lemma \ref{publ1} to the sequence $\xi_n(t)$. It satisfies \eqref{pube4}, and, similarly to the discrete case, \[
\mathbb{E}\bigl((\xi_n(t)-\xi_{n-1}(t))^2\bigm|\mathcal F_{n-1}\bigr)\le \mathbb{E}(\nu_n+1)^2=O(1) \] holds again. The other assumptions are also easy to check. Hence \[ G_u(t)\le\Bigl[\Sigma_u(t,h)+H(t)+\mathbb{P}(X>t)\Bigr]\;\biggl[ \frac{t}{\mathbb{E} X+\mathbb{E} Z}+\mathbb{E}\nu\biggr]^{\!-1}. \] One can readily verify that $\Sigma_u(t,h)$ converges to \begin{multline*} \frac{1}{\mathbb{E} X+\mathbb{E} Z}\Biggl[\int_0^t G_u(t-s)F(s)\,ds -\int_0^t G_u(t-s)(t-s)\,dF(s)\Biggr]\\ -\int_0^t G_u(t-s)\,dH(s)=\int_0^t G_u(t-s)\,d_sL(t,s) \end{multline*} as $h\to 0$, since the Riemann--Stieltjes integrals in the expression exist. This implies that \begin{equation} G_u(t)\le\biggl[\int_0^t G_u(t-s)\,d_sL(t,s)+H(t)+\mathbb{P}(X>t)\biggr] \biggl[\frac{t}{\mathbb{E} X+\mathbb{E} Z}+\mathbb{E}\nu\biggr]^{\!-1}.\label{pube5} \end{equation}
Therefore the solution of the corresponding integral equation \eqref{pube0} with initial condition $G_u(0)=1$ is an upper bound for $G_u(t)$. That is, $G_u(t)\le G(t)$, where $G(t)$ is the deterministic function given in the theorem.
Now we give lower bounds by analogous argumentation.
We estimate from below the conditional probability that an author with weight between $t-ih$ and $t-(i-1)h$ is chosen and his bonus is at least $ih$, given $\mathcal F_{n-1}^+$. Similarly to \eqref{pube2}, we have that it is greater than or equal to \[ \biggl[\frac{t-ih}{S_{n-1}}+ \biggl(1-\frac{t-ih}{S_{n-1}}\biggr)\frac{\nu_n-1}{n-1}\biggr]
\mathbb{P}\bigl(Y_n>ih\bigm|\mathcal F_{n-1}^+\bigr). \] Hence the lower bound of the conditional probability with respect to $\mathcal F_{n-1}$ is the following. \[ \ell_i:=\frac{t-ih}{S_{n-1}}\;F_n(ih) +\frac{1}{n-1}\biggl(1-\frac{t-ih}{S_{n-1}}\biggr)H_n(ih). \] We obtain that \begin{multline*}
\mathbb{E}\bigl(\xi_n(t)\bigm|\mathcal F_{n-1}\bigr)\\ \ge\xi_{n-1}(t)+\sum_{i=1}^{\lceil t/h\rceil} \Bigl[\xi_{n-1}(t-ih)-\xi_{n-1}\bigl(t-(i-1)h\bigr)\Bigr]\ell_i +\mathbb{P}(X>t). \end{multline*} After rearranging we get a formula similar to \eqref{pube4}. \begin{multline}\label{pube6}
E\bigl(\xi_n(t)\bigm|\mathcal F_{n-1}\bigr)\ge\xi_{n-1}(t)(1-\ell_1)\\ +\sum_{i=1}^{\lceil t/h\rceil}\xi_{n-1}(t-ih)(\ell_i-\ell_{i+1}) +n\ell_{\lceil t/h\rceil+1}+\mathbb{P}(X>t). \end{multline}
Here \begin{multline*} \ell_1=\frac{t-h}{S_{n-1}}\,F_n(h)+\frac{1}{n-1} \biggl(1-\frac{t-h}{S_{n-1}}\biggr)H_n(h)\\ =\biggl(\frac{t-h}{\mathbb{E} X+\mathbb{E} Z}\,F(h)+H(h)\biggr)\frac{1+o(1)}{n}\,, \end{multline*} and \begin{multline*} \ell_i-\ell_{i+1}=\frac{h}{S_{n-1}}\,F_n(ih)+ \frac{t-(i+1)h}{S_{n-1}}\,\Delta_hF_n\bigl((i+1)h\bigr)\\ -\frac{1}{n-1}\;\frac{h}{S_{n-1}}\,H_n(ih) +\frac{1}{n-1}\biggl(1-\frac{t-(i+1)h}{S_{n-1}}\biggr) \Delta_hH_n\bigl((i+1)h\bigr). \end{multline*} This implies that $n(\ell_i-\ell_{i+1})$ converges to \[ \frac{h}{\mathbb{E} X+\mathbb{E} Z}\,F(ih) +\frac{t-(i+1)h}{\mathbb{E} X+\mathbb{E} Z}\,\Delta_hF\bigl((i+1)h\bigr)+ \Delta_hH\bigl((i+1)h\bigr) \] as $n\to\infty$. Finally, \[ n\ell_{\lceil t/h\rceil+1}\ge-\frac{2nh}{S_{n-1}}+ H_n(t+2h), \] therefore \[ \liminf_{n\to\infty}n\ell_{\lceil t/h\rceil+1}\ge- \frac{2h}{\mathbb{E} X+\mathbb{E} Z}+H(t+2h). \] Let \[ G_\ell(t)=\liminf_{n\to\infty}\frac{\xi_n(t)}{n}\,; \] then $G_\ell(t)$ is also a decreasing random function. On the right hand side of \eqref{pube6} we have \[ \liminf_{n\to\infty}\sum_{i=1}^{\lceil t/h\rceil} \xi_{n-1}(t-ih)(\ell_i-\ell_{i+1})\ge \Sigma_\ell(t,h), \] where \begin{multline*} \Sigma_\ell(t,h)=\sum_{i=1}^{\lceil t/h\rceil}G_\ell(t-ih)\biggl[ \frac{F(ih)}{\mathbb{E} X+\mathbb{E} Z}\,h\\ +\frac{t-(i+1)h}{\mathbb{E} X+\mathbb{E} Z}\,\Delta_hF\bigl((i+1)h\bigr) +\Delta_hH\bigl((i+1)h\bigr)\biggl]. \end{multline*}
Applying Lemma \ref{publ1} we get that \begin{multline*} G_\ell(t)\ge \biggl[\Sigma_\ell(t,h)-\frac{2h}{\mathbb{E} X+\mathbb{E} Z}+H(t+2h)+\mathbb{P}(X>t) \biggr]\\ \times\biggl[\frac{t-h}{\mathbb{E} X+\mathbb{E} Z}\,F(h)+H(h)+1\biggr]^{\!-1}. \end{multline*}
Let $h$ go to zero again. The sum $\Sigma_\ell(t,h)$ converges to the same Riemann--Stieltjes integral as $\Sigma_u(t,h)$ does. Thus the right hand side of the inequality above converges to the right hand side of \eqref{pube5}. Hence we obtain that $G_\ell(t)\ge G(t)$. This, together with the estimation for $G_u(t)$, implies the statement of the theorem.
$\square$
\subsection*{Proof of Theorem \ref{pubt4}}
Let the density function of $Y$ be denoted by $f$. From the absolute continuity of $F$ the same follows for $H$. Let $h$ be defined by \[ H(t)=\int_t^\infty h(s)\,ds. \] Differentiating $L$ with respect to $s$ we obtain that \[ \frac{\partial}{\partial s}L\left( t, s\right)=\frac{F\left( s \right)-sf\left( s\right)+tf\left( s\right)}{\mathbb{E} X+\mathbb{E} Z}+h\left( s\right) \quad \lef0\le s\le t\right)\,. \]
Hence equation \eqref{pube0} may be written in the following form. \[G\left( t\right)=\int_0^tG\left( t-s\right) w_{t,s}ds+r\left( t\right),\] where \begin{align*} w_{t,s}&=\frac{\dfrac{F\left( s\right)+\left( t-s\right) f\left( s\right)} {\mathbb{E} X+\mathbb{E} Z}+h(s)}{\dfrac{t}{\mathbb{E} X+\mathbb{E} Z}+\mathbb{E} \nu}\\ &=\frac{F\left( s\right)+\left( t-s\right) f\left( s\right)+h\left( s\right)\left(\mathbb{E} X+\mathbb{E} Z\right)} {t+\left( \mathbb{E} X+\mathbb{E} Z\right)\mathbb{E} \nu}\,;\\ r\left( t\right)&=\frac{H\left( t\right)+\mathbb{P}\left( X>t\right)} {\dfrac{t}{\mathbb{E} X+\mathbb{E} Z}+\mathbb{E} \nu}. \end{align*}
In order to apply Theorem 2 of \cite{rek} write $w_{t,s}$ in the following form. \begin{align*} w_{t,s}&=f\left( s\right)+\frac{F\left( s\right)-\left( s+\left( \mathbb{E} X+\mathbb{E} Z\right) \mathbb{E} \nu \right) f\left( s\right)+h\left( s\right)\left( \mathbb{E} X+\mathbb{E} Z\right)} {t+\left( \mathbb{E} X+\mathbb{E} Z\right)\mathbb{E} \nu}\\ &=f\left( s\right)+\frac{b\left( s\right)}{t+d}\,, \end{align*} where \begin{gather*} b\left( s\right)=F\left( s\right)-\bigl(s+\left( \mathbb{E} X+\mathbb{E} Z\right)\mathbb{E} \nu \bigr) f\left( s\right)+h\left( s\right)\left( \mathbb{E} X+\mathbb{E} Z\right);\\ d=\left( \mathbb{E} X+\mathbb{E} Z\right)\mathbb{E} \nu. \end{gather*}
Next we check that all assumptions required in \cite{rek} hold. Since $f$ is a probability density function, $G$ is clearly decreasing and $w$ is nonnegative, all we need is the following three facts. \begin{enumerate}[(i)] \item\label{2-i} $d$ is a positive constant, \item $r$ is a nonnegative, continuous function, \item\label{2-iii} there exists $z>1$ such that \begin{gather*} \int_0^{\infty} f\left( t\right) z^t dt<\infty, \qquad \int_0^{\infty} \left\vert b\left( t\right) \right\vert z^t dt<\infty, \end{gather*} and $r\left( t\right) z^t$ is directly Riemann integrable on $\left[ 0,\infty\right)$. \end{enumerate}
Here (\ref{2-i}) follows from Assumption \ref{puba8}. From the continuity of $F$ and $H$ the same follows for $r$. Finally, the first part of condition (\ref{2-iii}) easily follows from Assumptions 2 and 7. In addition, using that $r$ is monotonically decreasing we get that \[ \sum_{n=1}^{\infty}\sup_{0\leq \theta\leq \tau} r\left( t+n\tau+\theta\right) z^{t+n\tau+\theta}\leq \sum_{n=1}^{\infty}\left[ r\left( t+n\tau\right) z^{t+n\tau}\right] z^{\tau} \] for $z>1$. The right hand side is finite for almost all $t$, because $\int_0^{\infty}r\left( s\right) z^s ds$ is finite. Therefore $r\left( t\right) z^t$ is directly Riemann integrable.
Thus Theorem \ref{pubt4} follows from Theorem 2 of \cite{rek}. Using the continuity of $G$ and the method of the discrete case it is easy to see that $G$ is not identically $0$ for large $t$, thus it is polynomially decaying. What is left is to determine the exponent, that is, \[ \gamma=-\frac{\int_0^{\infty}b\left( s\right) ds}{\int_0^{\infty} s f \left( s\right) ds}. \] The denominator is equal to $\mathbb{E} Y$. In the numerator we have \begin{align*} &\int_0^{\infty}b\left( s\right) ds\\ &=\int_0^{\infty}\Bigl(F\left( s\right)-\bigl(s+\left( \mathbb{E} X+\mathbb{E} Z\right)\mathbb{E}\nu\bigr) f\left( s\right)+ h\left( s\right)\left( \mathbb{E} X+\mathbb{E} Z\right)\Bigr)ds\\ &=\mathbb{E} Y-\mathbb{E} Y- \left( \mathbb{E} X+\mathbb{E} Z\right)\mathbb{E} \nu +H(0)\left( \mathbb{E} X+\mathbb{E} Z\right)\\ &=- \left( \mathbb{E} X+\mathbb{E} Z\right)\mathbb{E} \nu+\mathbb{E}\left( \nu-1\right) \left( \mathbb{E} X+\mathbb{E} Z\right)\\ &=-\left(\mathbb{E} X+\mathbb{E} Z\right). \end{align*}
Therefore we got that \[ \gamma=\frac{\mathbb{E} X+\mathbb{E} Z}{\mathbb{E} Y}, \] and the proof of Theorem \ref{pubt4} is complete. \qed
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Cancer Occurrences in Laboratory Rats from Exposure to RF and Microwave Radiation
James C. Lin
Take-Home Messages
This paper provides a critical and analytical synopsis and assessment of the current status of research on cancers in rats
exposed lifelong to RF and microwave radiation.
There were 18 carcinogenic and co-carcinogenic investigations in rats exposed to RF and microwave radiation
from a diverse range of mobile and wireless communication devices and systems.
A recent U.S. government announcement that rat cancer results from its large RF animal health risk study is
an important occurrence.
The impact of RF exposure on carcinogenesis remains tentative. The discrepancies continue to pose
uncertainty in assessing public health hazards from RF radiation.
The question of whether RF exposure from wireless and mobile devices and systems poses a health risk would
likely remain equivocal and controversial for some time to come.
Link to Xplore
Review on Advanced Short-Range Multimode Continuous-Wave Radar Architectures for Healthcare Applications
José-María Muñoz-Ferreras, Zhengyu Peng, Roberto Gómez-GarcíaChangzhi Li
Short-range radars can be effectively applied in biomedical/healthcare environments, such as monitoring of
vital signs or detection of fall incidents of elderly at home.
Advanced short-range multi-mode radars have improved features for biomedical/healthcare applications.
Applications of advanced short-range multi-mode radars range from monitoring of vital signs to humanaware
localization scenarios.
The proposed hybridizations of the Doppler and frequency-modulated continuous-wave (FMCW) operation
modes leads to minimum-hardware radar architectures with advanced features for biomedical/healthcare
Theoretical analyses and simulations for the Doppler-plus-FMCW, multi-FMCW, and tone-ranging-inspired
radar architectures are provided
Take Home Messages
An Original Research Paper
Compact Broadband Planar Resonator with a Viaed Double Spiral for Robust Wireless Power Transfer
Wenshen Zhou, Pengde Wu, Wen Chuan Mu, Wenwei Yu, Shaoying Huang.
A common problem for most strongly coupled magnetic resonance wireless power transfer systems is the dramatic efficiency drops by surrounding high-dielectric objects,
e.g. people, caused by the narrow bandwidth of the systems due to a high quality factor (Q). In this paper, a compact broadband planar double-spiral resonator with a via is proposed to obtain a robust SCMR system with a stable efficiency where interference from high-dielectric surrounding objects is mitigated. The increased bandwidth is obtained without compromising power transfer efficiency by introducing coupling enhancement to the structure while Q is reduced. The bandwidth of the system is increased by over 15% compared to a conventional system with single-sided resonators, and the efficiency is comparable. In this paper, it is found that the location of the via affects both the efficiency and bandwidth of the SCMR system. Meanwhile, this paper reports an experiment with human hand phantom and a simulation study with multi-layered tissue model, both of which mimic a real human-involved environment, and successfully demonstrate the stability and high efficiency of power transfer of the proposed broadband resonators in a WPT system. Moreover, the proposed structure is tested to be less sensitive to misalignment between the transmitter and the receiver, and its advantages are further shown by comparing it to the systems proposed for similar human-involved environments. This proposed viaed double-spiral resonator is a promising candidate for a robust WPT system for human-involved environments.
A 2020 COST Special Issue
Dielectric Characterization of Ex Vivo Ovine and Human Adrenal Glands for Microwave Thermal Ablation Applications
Anna Bottiglieri, Atif Shahzad, Padraig Donlon, Micheal Conall Dennedy, Aoife Lowery, Martin O'Halloran, Laura Farina.
Historically, adrenal glands diseases causing hypertension, such as Primary Aldosteronism (PA), have been treated through pharmacotherapy or surgical resection.
Given the shortcomings of the available treatment options, the interest in alternative and less invasive treatment modalities such as microwave ablation (MWA), has increased. In order to develop and optimize this novel electromagnetic-based therapy, an accurate knowledge of the dielectric properties of human adrenal glands, as well as preclinical animal models, is crucial. In particular, ovine models represent a feasible animal model to test the safety and performances of MWA. In this study, the dielectric properties of ovine adrenal glands and of normal and diseased human adrenal glands are characterized ex vivo in the microwave frequency range. The dielectric properties of the two functional tissues (cortex and medulla) composing ovine adrenal glands are measured using the open-ended coaxial probe technique and represented with a two pole Cole-Cole model in the frequency range from 0.5 GHz to 8 GHz. This paper presents the first dielectric data of normal and diseased human adrenal tissues, including a functioning adenoma responsible for PA and it compares the human data with data from the animal model.
MILLIMETER-WAVES BREAST CANCER IMAGING VIA INVERSE SCATTERING TECHNIQUES
Martina Teresa Bevacqua, Simona Dimeo, Lorenzo Crocco, Tommaso Isernia, Marco Pasian.
Breast cancer represents one of the main reasons of death among women. As an alternative to the gold standard techniques for breast cancer diagnosis,
microwave imaging has been proposed from research community and many microwave systems have been designed mainly to work at low microwave frequencies. Based both on the results of recent dielectric characterization campaigns on human breast ex-vivo tissues up to 50 GHz and on the promising feasibility studies of mm-wave imaging systems, in this article, we propose and test inverse scattering techniques as effective tool to process mm-wave data to image breast cancer. Differently from the radar techniques so far adopted in conjunction with mm-wave imaging system, inverse scattering techniques turn out to be more versatile and robust with respect to the reduction of the number of processed frequencies and eventually also able to characterize the anomaly in terms of electromagnetic properties. In particular, in the above, two image reconstruction techniques, the Linear Sampling Method and the Born Approximation, are proposed and compared against both simulated and experimental data.
Stable and Lifelong Head Phantoms Using Polymer Composition Mimicking Materials to Test Electromagnetic Medical Imaging Systems
Beada'a Mohammed, Konstanty Bialkowski, Steve Hill, Anthony Edgar Stancombe, Abdulrahman Al-qadami, Michael Heitzman, Amin Abbosh.
Phantoms are critical items for testing and evaluating new prototypes of electromagnetic medical imaging systems.
Realistic tissues distribution, size, shape, and dielectric properties of tissue-mimicking materials across the desired frequency band; in addition, long-term stability is required. In this study, a polymer composition that use to develop head tissues-mimicking materials that satisfy the aforementioned requirements. Polyepoxides (Epoxy), and assorted types of micro scale additive, including graphite, aluminium oxide, carbon black, and brass powders, are used to fabricate the phantom. Different mixing ratios are used to mimic four healthy head tissues; white matter, grey matter, skull, and skin. Blood-mimicking material is also included in the unhealthy phantom to represent stroke (hemorrhagic) at different locations and sizes. Also, water-based mimicking material is used to emulate cerebrospinal fluid (CSF) tissue. The measurements confirm close agreement to properties of actual head tissues across the frequency band 0.5- 5 GHz, which has been used in the ongoing research activities of electromagnetic head imaging systems. Stability over time is investigated and compared with the widely used gelatin-in water-based mimicking materials. The results show the superiority of the developed phantom compared to currently using gelatin-based phantoms.
Assessing a Microwave Imaging System for Brain Stroke Monitoring via High Fidelity Numerical Modelling
David O. Rodriguez-Duarte, Jorge A. Tobon Vasquez, Rosa Scapaticci, Lorenzo Crocco, Francesca Vipiana.
This work presents the outcomes of a numerical analysis based on a 3-D high fidelity model of a realistic microwave imaging system for the clinical follow-up of brain stroke.
The analysis is meant as a preliminary step towards the full experimental characterization of the system, with the aim of assessing the achievable results and highlight possible critical points. The system consists of an array of twenty-four printed monopole antennas, placed conformal to the upper part of the head; each monopole is immersed into a semi-solid dielectric brick with custom permittivity, acting as coupling medium. The whole system, including the antennas and their feeding mechanism, has been numerically modeled via a custom full-wave software based on the finite element method. The numerical model generates reliable electromagnetic operators and accurate antenna scattering parameters, which provide the input data for the implemented imaging algorithm. In particular, the numerical analysis assesses the capability of the device of reliably monitoring the evolution of hemorrhages and ischemias, considering the progression from a healthy statet o an early-stage stroke.
On the Design of a Microwave Imaging System to Monitor Thermal Ablation of Liver Tumors
Mengchu Wang, Lorenzo Crocco, Marta Cavagnaro.
Thermal ablation treatment of cancer is increasingly adopted in the clinical practice, being minimally invasive and highly specific.
However, a significant drawback of the technique is the lack of effective imaging modalities for monitoring the changes undergoing in the thermally treated tissue. In this respect, microwave imaging has been proposed as a possible candidate, owing to its portability, low-cost, non-ionizing nature, and capability to detect changes in dielectric properties of tissues induced by the temperature. The goal of this paper is to provide initial guidelines for the design of a microwave imaging system for thermal ablation monitoring. To this end, an analytical study is performed to determine the proper working conditions, in terms of frequency band and matching medium. Then, three antipodal Vivaldi antennas on different dielectric substrates are designed and numerically assessed. Among those antennas, the Vivaldi antenna on RT/duroid 6010LM substrate proved to be the most suitable choice. The results of this study pave the way to an experimental assessment of the potential of microwave imaging as a modality to monitor thermal ablation treatments.
On The Optimal Matching Medium and The Working Frequency in Deep Pelvic Hyperthermia
Gennaro G. Bellizzi, Kemal Sumser, Martina T. Bevacqua.
A necessary step when designing electromagnetic-based medical devices is the choice of an optimal matching medium,
standing between the patient and the phased array applicator, and the operating frequency. This is crucial to improve the efficiency both from a technical and clinical points of view. In this paper, we propose a new approach, based on the propagation theory, to support the selection of the matching medium properties and the working frequency in a robust way by accounting for patient body shape and properties variability. The case of adjuvant hyperthermia treatment administered to patient with tumors in the pelvic region has been used as a numerical validation in both 2D and 3D patient specific. For this case, the proposed approach suggests an optimal range of working frequencies (130MHz)
Original Research Paper
Human RF-EMF Exposure Assessment due to Access Point in Incoming 5G Indoor Scenario
Marta Bonato, Laura Dossi, Emma Chiaramello, Serena Fiocchi, Silvia Gallucci, Gabriella Tognola, Paolo Ravazzani, Marta Parazzini.
The study aimed at expanding the knowledge about the assessment of radio-frequency electromagnetic fields (RF-EMF)
exposure, considering the novelties introduced by the incoming 5G networks. Specifically, a possible future case of indoor exposure scenario is investigated, where the presence of a 5G access point (AP) in a room is simulated. The AP was modelled by two different indoor uniform planar array (UPA) antennas at 3.7 GHz and at 14 GHz, to evaluate how the beamforming and the higher frequency use could impact the exposure levels. Different scenarios were evaluated, considering the maximum antenna gain, two different human computational models, an adult model and a child one, and by varying the distance and the orientation between the UPA antenna and the two models head. All the simulations were conducted using the Sim4Life platform and in particular the exposure levels were expressed by the specific absorption rate averaged on 10 g of tissue ( SAR10g ), which was analyzed for the skin and for some specific tissues. The work underlined that the highest SAR10g values were obtained in the head area for all scenarios, with the skin SAR10ghighest peaks when the UPA is placed laterally to the human model (195.73 mW/kg and 223.29 mW/kg for the adult and child model, respectively, for 100 mW input power). Furthermore, the work permitted to highlight that the SAR10g exposure levels are slightly higher for the child model, compared to the adult one and that the distance between the UPA antenna and the human models could greatly lower the SAR10g levels. At last, it was found that the SAR10g exposure levels obtained with the UPA antenna at 14 GHz were lower than the ones at 3.7 GHz, although further investigations will be necessary.
Microwave Imaging for the Diagnosis of Cervical Diseases: A Feasibility Analysis
Chiara Dachena, Alessandro Fedeli, Alessandro Fanti, Matteo Bruno Lodi, Matteo Pastorino, Andrea Randazzo.
An inverse scattering method working at microwave frequencies for cervical diagnostics is proposed in this work.
The aim is the diagnosis of cervical myelopathy, which is a disease that affects the first part of the spinal cord (between the C3 and C7 vertebra). A preliminary feasibility analysis oriented toward the development of an imaging system is reported. The system prototype includes a set of antennas that illuminate the neck and retrieve samples of the scattered electric field. The related inverse scattering problem is solved by using a nonlinear Newton-type reconstruction procedure, which provides two-dimensional images of the dielectric parameters of a neck cross section. A simplified cylindrical phantom mimicking the human neck has been designed for assessing the feasibility of the envisioned microwave measurement system and processing technique. Numerical results are reported to evaluate the capabilities of the proposed approach. Moreover, initial experimental results have been obtained by using cylindrical containers and a 3D printed version of the developed neck phantom.
Noise-robust Microwave Breast Imaging Applied to Multi-frequency Contrast Source Inversion
Hiroki Sato, Shouhei Kidera.
Ultra-wideband (UWB) microwave quantitative imaging offers less-painful examination, where a significant dielectric contrast between the malignant tumor and normal tissue is exploited.
However, there are some difficulties in providing an accurate dielectric profile of breast media. This is because additive noises severely contaminate scattered signals. In this study, we apply a post-processing multi-frequency integration scheme to contrast source inversion (CSI) data to suppress image fluctuations. These are caused by the additive noise expected in a UWB system, which is also applicable to other inversion schemes. To deal with the dispersive dielectric properties in the CSI scheme, we introduce a first-derivative model of the Debye dispersion model to compensate for the dispersive effect. The FDTD numerical validations, using realistic breast phantoms with dispersive properties, show that a multi-frequency integration scheme considerably upgrades noise-robustness in complex permittivity reconstruction tissues.
Impact of Textile on Electromagnetic Power and Heating in Near-Surface Tissues at 26 GHz and 60 GHz
Giulia Sacco, Stefano Pisa, Maxim Zhadobov.
With the development of 5th generation (5G) networks the operating frequencies have been progressively expanding towards millimeter waves (MMW).
In some exposure scenarii, presence of textiles impacts the interaction of the electromagnetic field radiated by wireless devices with human tissues. We investigate the impact of a textile layer in contact or in proximity of skin on the power transmission coefficient, absorbed power density and temperature rise using a near-surface tissue model at 26GHz and 60 GHz. Cotton and wool are considered as representative textiles. Our results demonstrate that the textile in contact with skin increases the absorbed power density up to 41.5% at 26 GHz and 34.4% at 60 GHz. The presence of an air gap between a textile and skin modifies the electromagnetic power deposition in the tissues depending on the thicknesses and permittivity. The temperature rise increases compared to the nude skin by up to 52% at 26 GHz and 46% at 60 GHz with the textile in direct contact with skin. With an air gap, for typical textile thicknesses, the temperature variations range from -3.5% to 20.6% and from -11.1% to 20.9% at 26 GHz and 60 GHz, respectively.
2020 COST Special Issue
Hyperthermia Treatment Planning: Clinical Application and Ongoing Developments
H. P. Kok, J. Crezee.
Hyperthermia is a proven clinical anti-cancer treatment, used in combination with radiotherapy and/or chemotherapy.
During hyperthermia, tumour tissue is heated to 40-43°C using radiofrequency or microwave antennas, which strongly enhances effectiveness of radiotherapy and chemotherapy. Hyperthermia treatment quality depends on tumour temperatures achieved and treatment planning (i.e. simulation and optimization of absorbed power and temperature distributions) could be very useful to ensure and improve treatment quality. Hyperthermia treatment planning was mainly a research tool for decades, because of high computational costs and limited quantitative accuracy of treatment planning predictions due to a lack of patient-specific tissue properties. Thanks to developments over the past decade, treatment planning becomes increasingly important in the clinical workflow. Presently, main clinical applications of hyperthermia treatment planning are 1) applicator selection, 2) heating ability evaluation and 3) on-line treatment guidance. To improve the reliability and further increase applicability of treatment planning, ongoing developments focus on 1) dielectric imaging to derive patient-specific dielectric properties, 2) advanced thermal modelling including discrete vasculature and 3) biological modelling to predict the radiosensitizing effect of hyperthermia in terms of equivalent radiation dose. The increased clinical application and ongoing efforts will further improve treatment quality.
A Breath Monitoring Approach based on Electrical Impedance Measurements
Emanuele Tavanti, Gianluca Gambari, Federico Boero, Alessandro Fedeli, Matteo Pastorino, Andrea Randazzo.
An approach for monitoring the respiratory rate based on impedance measurements is presented in this paper.
In order to possibly obtain a minimally-invasive wearable device, the electrodes are located on the head, near mastoid bones, and measure the variations induced in the electrical impedance by the physiological changes in the pharynx produced by respiration. The feasibility of the adopted configuration has been assessed by means of electromagnetic simulations involving a simplified model. Moreover, an ad-hoc data processing algorithm has been developed for extracting the breath rate from the measured signals. Preliminary experimental results are provided for investigating the capabilities of the developed setup.
IEEE 2020 COST Special Issue
Anthropomorphic Calcaneus Phantom for Microwave Bone Imaging Applications
Bilal Amin, Atif Shahzad, Daniel Kelly, Martin O'Halloran, Adnan Elahi.
Recent studies have found a significant dielectric contrast between healthy and osteoporotic human trabecular bones.
This dielectric contrast can be exploited by microwave imaging for monitoring human bone health. The tissue mimicking phantoms play a vital role for preclinical testing of microwave imaging system. This paper presents anatomically realistic multi-layered 3D printed and carbon black based human calcaneus structure. The liquid and solid based tissue mimicking mixtures are also proposed to mimic the dielectric properties of skin, muscle, cortical bone, and trabecular bone. The liquid tissue mimicking mixtures are composed of Triton X-100, water, and salt, whereas the solid tissue mimicking mixtures are composed of carbon black, graphite, polyurethane, and isopropanol. The dielectric properties of the tissue mimicking mixtures were measured using an open-ended coaxial probe measurement technique across 0.5 8.5 GHz. The average percentage difference between the relative permittivity and conductivity of reference data and proposed liquid tissue mimicking mixtures was found to be 7.8% and 9.6% for skin, 0.38% and 14% for muscle, 9.6% and 5% for cortical bone, and 3.4% and 2.4 % for trabecular bone, respectively, across 0.5 8.5 GHz. For solid tissue mimicking mixtures, this difference was found to be 3.93% and 0.64% for skin, 6.13% and 9.21% for cortical bone, and 10.66% and 41.82% for trabecular bone, respectively for relative permittivity and conductivity. The proposed tissue mimicking mixtures along with 3D printed structures can be used as a valuable test platform for microwave bone imaging system development.
IEEE Sensors Special Issue
Implantable Antennas for Bio-medical Applications
Nabeel Ahmed Malik, Paul Sant, Tahmina Ajmal, Masood Ur-Rehman.
Biomedical telemetry has gained a lot of attention with the development in the healthcare industry.
This technology has made it feasible to monitor the physiological signs of patient remotely without traditional hospital appointments and follow up routine check-ups. Implantable Medical Devices(IMDs) play an important role to monitor the patients through wireless telemetry. IMDs consist of nodes and implantable sensors in which antenna is a major component. The implantable sensors suffer a lot of limitations. Various factors need to be considered for the implantable sensors such as miniaturization, patient safety, bio-compatibility, low power consumption, lower frequency band of operation and dual-band operation to have a robust and continuous operation. The selection of the antenna is a challenging task in implantable sensor design as it dictates performance of the whole implant. In this paper a critical review on implantable antennas for biomedical applications is presented.
Microwave Breast Screening Prototype: System Miniaturization with IC Pulse Radio
Lena Kranold, Mohammad Taherzadeh, Frederic Nabki, Mark Coates; Milica Popovic.
In this study, we report on system advancements for an ultrawideband (UWB) microwave radar prototype for early breast cancer detection.
We introduce an integrated circuit (IC) pulse radio as a suitable replacement for both a bulky and expensive off-the-shelf pulse generator and a clock. To assess the system's suitability as a breast screening prototype, we compare measurements on two different experimental breast models (phantoms) with the established and previously reported prototype using off-the-shelf components and the here-introduced IC. We test both systems on a homogeneous fat-mimicking phantom with a 2-mm skin layer as well as a phantom with a 2-mm skin layer and glandular insertions, while the antennas, antenna housing, and sampling oscilloscope are the same for both systems. Additionally, we advance with the IC to higher frequencies, aiming to comply with the band intended for microwave imaging devices with medical applications. Furthermore, we compare the economic requirements of the IC and of the previously reported system by evaluating their cost and compactness. The objective of this study is to investigate if the IC pulse radio can replace bulky off-shelf components to allow us to implement the pulse generation circuitry in one flexible circuit board with the switching and antennas.
Non-Contact Multi-Subject Human Gait Analysis using A State-Space Method with Enhanced 1-D Block Representation
Farnaz Foroughian, Farhan Quaiyum, Paul Theilmann; Bardia Ghajari, Jean E. Piou, Ozlem Kilic, Aly Fathy.
A stepped-frequency continuous-wave (SFCW) radar system with adequate pulse repetition frequency (PRF) is developed to track relatively fast motions from various parts of the human body, separately.
Robust signal processing technique is utilized, where a 1-D block processing technique, based on state-space method (SSM) is developed in two dimensions to form an enhanced Hankel matrix to track human motions from two-dimensional (2-D) data collected using the developed SFCW radar system in single- or multi-subject motion scenarios. Experimental datasets are used to track body parts' motions and velocities with high accuracy. The significant improvement, besides de-noising the data, is due to the fact that applying the 1-D block processing technique to 2-D dataset matrix accounts for the correlation between motion's estimates from consecutive frames. Results agree well with our reference, the human Boulic model.
RF Radar Breast Health Monitoring: System Evaluation with Post-Biopsy Marker
Lena Kranold, Milica Popovic.
This work reports on the first systematic experimental study on tissue phantoms that evaluates the efficacy of the radiofrequency (RF)
multistatic radar for breast health monitoring in the presence of a titanium site marker, typically inserted post-biopsy. We show results of breast phantom measurements for a homogenous fat phantom in several scenarios: all-fat phantom simulating a simple healthy case, a case with an added small 1 cm diameter tumor, a case with a glandular insertion, and finally, a scenario of a tumor embedded in gland. To evaluate the system's performance with respect to post-biopsy site markers, we then show investigations of the system performance with the same phantom and a biopsy marker attached to a glandular insertion with and without an embedded tumor, and compare the imaging results to those phantoms without the biopsy marker. We conclude that our RF radar can detect the tumor despite the presence of a biopsy site marker, and that the conductive titanium marker does not interfere with the system's intended function.
Anthropomorphic Durable Realistic Knee Phantom for Testing Electromagnetic Imaging Systems
Kamel S. Sultan, Beada'a Mohammed, Paul Mills, Amin Abbosh.
A knee phantom with realistic dielectric and anatomical properties has been fabricated using proper molds and equivalent mixtures.
The fabricated phantom is based on a composite material of polymer and additive materials such as aluminium-oxide, and graphite. These materials are selected to achieve high dielectric properties and realistic distribution of knee tissues in addition to long-life stability. A positive mold of muscle tissue and a negative mold of skin tissue are extracted from MRI data, whilst the positive molds for bones, tendons, ligaments, and tibia are extracted from a 1:1 commercial knee joint model. Due to a lack of data about dielectric properties of human knee ligaments in microwave frequency (0.5-10 GHz), dog's ligament tissues are characterized. The fabricated tissues of the knee phantom are stable and accurately match the dielectric properties of knee tissues across the wideband 0.5 GHz to 10 GHz. The phantom will open the door for a portable, low cost, and onsite electromagnetic imaging techniques to detect knee injuries.
A Non-Invasive Flexible Glucose Monitoring Sensor using a Broadband Reject Filter
Moussa Bteich, Jessica Hanna, Joseph Costantine, Rouwaida Kanj, Youssef Tawk, Ali Ramadan, Assaad Eid.
In this paper, a novel, highly accurate, non-invasive glucose-monitoring sensor that is based on a flexible broadband reject filter is presented.
The filter topology comprises a tapered feed line at a top layer that excites four modified log-periodic open loop resonators on the bottom layer, achieving a broadband reject response. Size reduction techniques are applied on the embedded resonators that are optimized to exhibit an enhanced sensitivity to track the variations of the glucose level across a frequency span from 1.25 GHz to 2.65 GHz. The proposed flexible filter is tested pre-clinically and clinically, where a high correlation between its scattering parameters and the variations in glucose levels is attained. Regression models are also developed using experimental data obtained from healthy patients that are subjected to glucose tolerance tests. Results demonstrate less than 4% mean absolute relative difference between the reference and estimated glucose levels, and the predicted glucose levels lie 100% within the clinically acceptable zones as shown by the Clarke Error Grid analysis.
Towards the Robust and Effective Design of Hyperthermic Devices: Case Study of Abdominal Rhabdomyosarcoma with 3D Perfusion
Matteo Bruno Bruno Lodi, Giacomo Muntoni, Alessandro Ruggeri, Alessandro Fanti, Giorgio Montisci, Giuseppe Mazzarella.
This work addresses the challenge of deriving a simple effective multiphysic model useful for the design and simulation of hyperthermia devices for high-quality treatment.
An existing compact patch antenna working at 434 MHz is re-designed using the proposed methodology. The proposed general approach is used to investigate the peculiar case of the hyperthermia treatment of abdominal rhabdomyosarcoma. Instead of patient-specific geometries with discrete vascular tree models, a surface phantom with a 3D blood perfusion model of tumors is used. The antenna is reworked to be robust. The effectiveness of the antenna is evaluated simulating the treatment with a recent non-linear multi-physic model, considering the different descriptions of tumor vasculature. A more robust and effective design is obtained, with respect to its previous version. The antenna bandwidth is increased of about 7%. The treatments with the old version of the antenna were unsatisfactory (40°C after 60 min), whilst the novel design could successfully treat the target region, reaching 42.5°C for 60 min of treatment. To enhance the effectiveness of the treatment, the use of a time-modulated power is studied. The proposed model could be extended to different body region and used to develop an application-oriented design of antennas for hyperthermia treatment.
Design of Hyperthermia Applicator to Heat Multi Brain Tumors Simultaneously based on Adaptive Beamforming Technique
Korany R. Mahmoud, Ahmed Montaser.
Recently, hyperthermia therapy is considered as one of the key treatment principles due to its importance and effectiveness in healing the deep-seated tumors.
However, for brain tumors, it is difficult to heat due to high perfusion and thermal conductivity of the head. Therefore, in this research, the technique of non-invasive heat focalization in multiple tumors, simultaneously, without affecting healthy tissue based on adaptive beamforming was investigated and presented. It is done by controlling the feeding of the antenna array surrounding the brain using a modified hybrid version of gravitational search algorithm and particle swarm optimization (MGSA-PSO). An antenna system in the form of a head helmet was designed and evaluated with 48 antenna elements each of them has a separate excitation that controls the field intensity and beamforming direction towards the tumors. Many scenarios considering a single tumor in different positions with different volumes or multiple tumors are studied to evaluate the performance of the applicator. The helmet was tested on the challenging scenario of a very mature and dense brain with realistic thermal and dielectric properties. The results confirmed the ability of the helmet technology and the proposed antenna system to use a microwave power of 65 W to lift the neoplasm temperature to over 42oC while keeping healthy tissue safe at 37 degrees with none hot spots. Furthermore, the results showed the capability of the proposed model to treat multiple tumors simultaneously.
Full Beta-Dispersion Region Dielectric Spectra and Dielectric Models of Viable and Non-Viable CHO Cells
Samaneh Afshar, Azita Fazelkhah, Katrin Braasch, Elham Salimi, Michael Butler, Douglas Thomson, Greg Bridges.
The dielectric properties of biological cells can be used to gain information on their physiology and morphology.
This paper reports the first measurements of the dielectric spectra of viable and non-viable cells over the full beta-dispersion (interfacial) frequency range. Dielectrophoresis (DEP) single cell in-flow techniques were employed to quantitatively measure the Clausius-Mossotti factor spectrum of individual cells over the 300 kHz 400 MHz range, covering both the MF and UHF DEP cross-over frequencies. Experiments were performed on Chinese hamster ovary (CHO) cells, one set cultured in growth media, the other in nutrient depleted media to induce apoptotic cells. Both cell states were measured using multi-frequency DEP flow cytometry, which provides the equivalent complex dielectric permittivity of individual cells. The measured dielectric spectra facilitate determination of a cell's morphology and the dielectric properties of its intracellular compartments, and are used to develop multi-shell dielectric models of viable and non-viable cells. The developed dielectric models can aid in biosensor design, in interpretation of bulk biomedia measurements where the heterogeneity in cell population can be masked, and in relating measured dielectric responses of cells to stimuli with changes in cellular physiology.
Vessel Sealing Device Using Microwave and High Frequency Current
Aditya Rakhmadi, Kazuyuki Saito, Masashi Sekine, Masashi Sugiyama.
In recent years, various types of medical applications of microwave energy have widely been investigated and reported.
We present a surgical device which employs the thermal effect produced by microwave energy. High power microwave energy can generate a coagulated region at the surface of the biological tissue such as organs. By coagulating organs, bleeding can be stopped. However, a microwave surgical device cannot cut the tissue without any device support, such as a blade. On the other hand, radio frequency (RF) current capable of cutting tissue without any support. This study presents a combination of a forceps type microwave surgical device combined with RF current for biological tissue cutting mechanism. Furthermore, ten sealed porcine blood vessels, sealed by the device, capability to withstand pressure were measured. Sealed blood vessels can withstand up to 200 mmHg of pressure and sufficient to withstand human blood pressure.
Wireless Double Micro-Resonator for Orientation Free Tracking of MR-Catheter During Interventional MRI
Omar Nassar; Dario Mager; Jan G. Korvink.
Interventional magnetic resonance imaging (iMRI) using MR-catheters has been explored during the past decade because of its potential impact on the field of minimally invasive medical procedures,
especially applied to vascular diseases. Tracking the catheter's tip during an iMRI procedure using active electronic components has mayor benefits but still faces challenges regarding safety and the quality of visualization, which has prevented its clinical use up to now. Here we propose a novel micro detector with a total length of 8 mm built upon a flexible substrate with a total thickness of less than 60 um. The design of the detector is based on two perpendicularly oriented saddle coils that together create a homogeneous magnetic field when wrapped onto the catheter tube, thus maintaining constant visibility of the catheter under rotation, with practically no dead angle. Being self-resonant, the proposed detector allows wireless tracking of the catheter position, whilst preventing any heating hazard due to the absence of radiofrequency cables. The micro-resonator was fabricated using a multilayer flexible electronics fabrication process.
Numerical Assessment of RF Human Exposure in Smart Mobility Communications
Gabriella Tognola; Barbara Masini; Silvia Gallucci; Marta Bonato; Serena Fiocchi; Emma Chiaramello; Marta Parazzini; Paolo Ravazzani.
Cars are rapidly evolving into smart connected objects that can communicate not only with the infrastructure but also with other cars through vehicle-to vehicle (V2V) communication.
By the end of 2023, more than 72 million vehicles worldwide will be equipped with devices and technologies that enable to exchange data and communicate with other cars. This challenging scenario is raising cross-cutting issues, such as those related to new radio-frequency exposures of the human body also when travelling. We evaluate the Specific Absorption Rate (SAR) induced in a realistic smart mobility communication scenario operated at 5.9 GHz. V2V antennas were modeled and placed on a realistic 3D model of a city-car to numerically estimate SAR in the body regions and tissues of a human phantom (adult male) inside the car. We found that both local and whole-body average exposures were below the ICNIRP and IEEE limits for the general public in the 100 kHz-6 GHz band, being equal in the worst case scenario to 1.58 W/kg (head) and 0.008 W/kg, respectively. The highest SAR was found in the most superficial tissues (the skin) of body regions very close to the sources. The distance of the passenger from the antennas played an important role in the resulting SAR. This research has a potentially great clinical impact as it contributes to new and realistic knowledge on the exposure scenario in smart mobility communication to assess possible health effects and for the design of policies for public health management.
APS/URSI 2019 Special Issue paper
Design of an Interstitial Microwave Applicator for 3D Printing in the Body
Kaitlin Hall ; Huanan Zhang ; Cynthia Furse.
This paper describes a method for 3D printing in the body using a coaxial applicator to heat a biopolymer that solidifies when heated.
This method is proposed for creating conductive lines in the body for focusing electromagnetic fields on a very small (4mm on a side) implantable medical device. A 2450 MHz sleeved slot coaxial applicator with a pointed, extended, hollow tip is designed to heat 400 mm3 of polymer from body temperature to its crosslinking temperature of 40°C in 10s.
Broadband Implantable Antenna for Wireless Power Transfer in Cardiac Pacemaker Applications
Mengfan Wang, Haixia Liu, Pei Zhang, Xuefang Zhang, Hong Yang, Guofei Zhou, Long Li.
A miniaturized implantable antenna with bandwidth enhancement is proposed for cardiac pacemaker application in MICS band (402-405 MHz)
and ISM band (433-434.8 MHz, 902-928 MHz) in this paper. By introducing split resonant rings, the proposed antenna has achieved ultra-wideband characteristics, which can cover from 272 MHz to 1504 MHz, and the relative bandwidth is 138.7%. Additionally, a new wireless power transfer (WPT) system is designed by integrating with a miniaturized metasurface to enhance the WPT efficiency and extend the lifetime of the implantable device. The transmission coefficient S21 of the WPT system loaded with metasurface is 11 dB higher than the initial system at the resonance point. Furthermore, biosafety is taken into consideration for practical applications. The experiments in the equivalent body environment are performed to demonstrate the reliability of the proposed antenna. The measured and simulated results are in good agreement, which shows that the proposed antenna is appropriate to be applied in wireless body area network communication and power transfer systems simultaneously.
A Wireless Wearable RF Sensor for Brumation Study of Chelonians
Jianlin Zhou, Pragya Sharma, Xiaonan Hui, Edwin Kan.
We present a low-power wireless radio-frequency (RF) sensor to perform continuous vital-sign monitoring for chelonians in various stages of brumation.
Due to their unique body structure, vital signs of chelonians cannot be recorded without substantial animal handling in previous methods, which would severely bias the brumation condition. In contrast, our sensor on harness can couple significant RF energy into various body parts to evaluate the heartbeat, respiration and activity levels. The self-contained unit is lightweight for ease of wearable deployment and low power to avoid unnecessary heat generation as well as frequent battery replacement. We recorded the shell temperature, heart rate, respiration rate and activity levels of a Russian tortoise during the entire brumation cycle and found that the heart rate correlated with the ambient temperature well, while the breath rate did not significantly reduce during brumation. This experimental study is minimally invasive and thus least biased.
Wireless In vivo Biofuel Cell Monitoring
Luigi Di Trocchio, Cristina Carucci, K.R. Sindhu, Chloe Morel, Jean Luc Lachaud, Sabrina Bichon, Sebastien Gounel, Nicolas Mano, Claudine Boiziau, Corinne Dejous, Alexander Kuhn, Simon Hemour.
Enzymatic reactions involving glucose hold the potential for building implantable biosensors and embedded power generators for various medical applications.
While Biofuel cells (BFCs) such as enzymatic glucose/O2 are ensured to benefit from abundant chemical resources that can be harvested in the immediate environment of the human body, the highly critical in vivo kinetics of biofuel cell is not yet fully understood. Unfortunately, existing solutions for real-time monitoring of the reaction on rodents are not possible today, or too bulky, which has a biasing impact on the animal behavior. This work presents a light, battery-less, and wireless strategy to continuously monitor a BFC implanted in a laboratory rat using a Frequency Identification (RFID) link. An extremely lightweight and flexible tag antenna of footprint lower than 10 cm is presented with communication capability above 60 cm in field environment. The operational capabilities are demonstrated with a 24-hour continuous monitoring of an enzymatic glucose/O2 reaction, both in vitro and in vivo.
A Multifunction Dense Array System with Reconfigurable Depth of Penetration
Matthew Charles Smith, Aobo Li, Daniel Sievenpiper.
Noninvasive neuromodulation techniques such as Transcranial Magnetic Stimulation are enabled by electromagnetic induction with
applications in neuroscience research and the treatment of neurological disorders. Current systems are effective but research continues on improving key performance metrics such as precise targeting, focality, and depth of penetration. Additional functions are also being studied; tunable excitation field patterning, reconfigurable depth of penetration, multiple excitation sites, power efficient waveforms and reduced extraneous excitation. Multichannel, multi-coil arrays show promise in achieving many of these parameters in one multifunctional array as opposed to the single-function commercial coils (circular, figure-8 or H coil) being used today. As such, our central focus is to determine whether multifunction dense arrays, despite their many challenges, are a tractable technical approach for future neuromodulation systems. Therefore, the results in this paper are twofold. First, we report the design, fabrication and demonstration of a scalable multichannel (12 channels) or multifunction dense array system to assess the potential to perform these functions in one system. Second, we demonstrate that the depth of penetration of the magnetic field can be reconfigured by varying current magnitude and phase of the smaller coil diameters in the array to achieve the same decay profile performance of a larger diameter coil. Only simulations exist in the literature to date, to the best of our knowledge, we report the first measurements of hexagonal shaped coils in multi-coil arrays have increased depth of penetration over circular shaped coil-based arrays.
Stroke Classification in Simulated Electromagnetic Imaging Using Graph Approaches
Guohun Zhu, Alina Bialkowski, Lei Guo, Beada'a Mohammed, Amin Abbosh.
Identifying stroke subtypes from electromagnetic imaging systems is usually based on frequency domain using radar or tomography algorithms which is computationally expensive.
This paper presents a novel graph degree mutual information (GDMI) approach to distinguish Intracranial Haemorrhage (ICH) from Ischemic Stroke (IS). A total of 50 ICH and 50 IS signals simulated using a 16-antenna electromagnetic head imaging system are analysed to evaluate GDMI. The data collected from each model consists of 256 reflected and received signal. Subsequently, noise is injected into the collected signals to generate three groups of signals with different signal-to-noise ratios (40~dB, 25~dB and 10~dB SNR), to emulate measurement noise and to test the algorithm's robustness}. Each signal is converted into a graph to avoid the variable signal amplitudes. Then, the relationship between each pair of graph degrees is calculated by mutual information and forwarded to a support vector machine to identify stroke type. The results indicate that signals from ICH subjects exhibit a significantly higher GDMI compared to the IS group (p<0.01). An accuracy of 88% is achieved in identifying ICH from IS without the need to use time- and resource-expensive brain image reconstruction algorithms even under 25~dB signal-to-noise levels. The execution time for graph feature extraction and classification is less than one minute on a PC. Such a short time is suitable for stroke emergency requirements.
APS/URSI 2020 Invited paper
Review of Self-Injection-Locked Radar Systems for Noncontact Detection of Vital Signs
Fu-Kang Wang, Chung-Tse Michael Wu, Tzyy-Sheng Horng, Chao-Hsiung Tseng, Shiang-Hwua Yu, Chia-Chan Chang, Pin-Hsun Juan, Yichao Yuan.
n recent decades, wireless healthcare detection has been demonstrated to be an efficient method for monitoring the health condition of targets;
it has involved the detection of motion and vital sign signals with minimal discomfort. Self-injection-locked (SIL) radar systems exhibit several useful advantages over conventional continuous-wave (CW) radar systems in healthcare applications, including a low system complexity, high sensitivity, and high clutter immunity. Several approaches to detecting vital sign signals and locations using SIL radar systems have been proposed. This paper summarizes developments in vital sign detection and localization. Finally, some results of detecting animals using SIL radar systems are presented.
Innovative Stochastic Modeling of Residential Exposure to Radio Frequency Electromagnetic Field Sources
Emma Chiaramello, David Plets, Serena Fiocchi, Marta Bonato, Gabriella Tognola, Marta Parazzini, Laurent Le Brusquet, Luc Martens, Wout Joseph, Paolo Ravazzani.
This study focused on the assessment of radio-frequency electromagnetic fields (RF-EMF) exposure in a realistic apartment due to the presence of a WiFi source deployed in uncertain position.
In order to describe the 2D spatial distribution of electric field induced in the whole apartment for whatever position of the WiFi source, an innovative approach that combines Principal Component Analysis (PCA) and Gaussian process regression (Kriging method) was applied. The 2D surrogate model was used to investigate the exposure in three different usage scenarios of the WiFi sources, i.e. surfing to a new web site, using a Skype video call and watching a You Tube video at 1080p, evaluating the electric field E induced at each location of the apartment for 10,000 different positions of the source. Across all the examined conditions, we found E values distributions with median values in the range 2.2 ? 96.1 mV/m and 90th percentiles in the range 4.9-209.3 mV/m. The 2D surrogate model allowed obtaining a complete description of the exposure for any positions of the WiFi source in the apartment, with a computational effort equal to about 10% of the one needed by using only the WiCa Heuristic Indoor Propagation Prediction (WHIPP) network planner.
An Electrical Impedance Tomography System for Brain Stroke Imaging based on a Lebesgue-Space Inversion Procedure
Andrea Randazzo, Emanuele Tavanti, Mantas Mikulenas, Federico Boero, Alessandro Fedeli, Andrea Sansalone, Giorgio Allasia, Matteo Pastorino.
An electrical impedance tomography (EIT) system for brain stroke monitoring is presented in this paper.
The developed setup is composed by an ad-hoc measurement prototype equipped with an efficient imaging method for the reconstruction of the distribution of the electric conductivity of the body under test. In particular, a time-difference formulation is adopted, and the resulting ill-posed equation is solved by means of an iterative procedure performing a regularization in the framework of Lebesgue spaces. The performance of the method has been assessed by means of several numerical simulations. Moreover, a preliminary validation with experimental data has been performed, too. The obtained results confirm that the approach is able to effectively detect inclusions with different sizes and locations inside the considered head models.
Modeling Electromagnetic Exposure in Humans inside a Whole-Body Birdcage Coil Excited by a Two-Channel Parallel Transmitter Operated at 123 MHz
Mikhail Kozlov ; Marc Horner ; Wolfgang Kainz ; Nikolaus Weiskopf ; Harald Möller.
The influence of the accuracy level for whole-body birdcage coil models on electromagnetic exposure estimations was evaluated using three
anatomical human body models located at the head landmark position. Both generic and vendor-specific birdcage coil models were created and analyzed using a radio-frequency (RF) circuit and a three-dimensional electromagnetic (EM) co-simulation approach to evaluate EM properties of the coil used in a commercial 3T magnetic resonance imaging (MRI) scanner. The fidelity of the coil geometry and excitation, as well as consideration of permissible variations of the coil electrical components and capacitor losses, had a significant impact on the estimated electric field distributions inside the human models. Therefore, the variety of electric fields generated in humans should be carefully considered for a reliable RF safety assessment of a patient with a passive or active implant undergoing a scan in a modern MRI scanner with dual-channel transmit RF coils.
Microwave imaging provide an in expensive, non-ionizing and nondestructive evaluation of the cell tissues for clinical analysis and medical diagnosis.
We demonstrate that the antenna performs well even in close proximity to the phantoms and operationally covers the Federal Communications Commission (FCC) range of the Ultra-Wide Band (UWB) spectrum.
Our target application is centered on the detection of breast cancer at early stage, which serve as a key factor in the successful treatment of the disease.
This work has demonstrated the use of Parallel Surrogate Assistance Differential Evolution Algorithm (PSADEA) optimisation technique in reducing the size of the antennas considerably.
The optimization techniques used in this our work provide the sensor with a good return loss in the UWB frequencies of 3.1 to 10.6 GHz and maintains its bandwidth UWB operation without detuning when placed in closed contact with the human body or breast mimicking tissue (phantom).
Design and Optimization of a Slotted Monopole Antenna for Ultra-wide Band Body Centric Imaging Applications
Isah Musa Danjuma, Mobayode Olusola Akinsolu, Chang Hwang See, Raed Abd-Alhameed, Bo Liu.
This paper presents a cost-efficient design, optimization and physical implementation of a compact slotted ultra-wideband (UWB) monopole antenna for body-centric imaging applications.
The proposed antenna is initially modelled and designed with the aid of commercial software (CST-Microwave Studio). To ensure that the proposed designs are meeting the required specifications with reduced design time, the parallel surrogate model-assisted hybrid differential evolution for antenna optimization (PSADEA) is proposed to optimize the design. Based on the best set of geometry parameter for the optimum antenna performance, the antenna prototype is realized on an FR-4 substrate and analyzed in terms of bandwidth, gain, efficiency, and radiation pattern with and without the tissue models. All measured results are found to be in good agreement with the simulated results. The antenna provides a good reflection coefficient (S11 <-10 dB) in the UWB frequency band from 3.1 GHz to 10.6 GHz and maintains its bandwidth UWB operation without detuning when placed in closed contact with the human body or breast mimicking tissues (phantoms).
Near-field communication (NFC) technique is used to design an on-chip implantable WPT system for neuromodulation applications.
Neuromodulation approach such as optogenetics is a revolutionary approach for the treatment of various neural diseases by stimulating the genetically modified neurons. A non-invasive or minimally invasive and miniaturized implantable system is the most desirable for optogenetic stimulation techniques.
The proposed on-chip coil reduced the size of the receiver (RX) module by 96% compared to the state-of-theart while achieving similar power transfer efficiency (PTE) performance and the best figure of merit (FOM) performance.
The analysis of the Electric field (E-field), Magnetic field (H-field), Specific Absorption Rate (SAR) and temperature increase through the different brain tissue layers is presented to identify the working range of the system.
A case-study of the proposed on-chip coil integrated with the commercial off-the-shelf (COTS) component based rectifier and μLED for optogenetic neuromodulation is presented to validate the system.
A 0.09 mm2 On-Chip Coil Designed in 0.5 μm CMOS process for Brain Neuromodulation Applications
Dipon K. Biswas, Ifana Mahbub.
Chronic neuropathic pain is one of the most debilitating medical conditions that impact the lives of almost 20% of the population in the United States.
Optogenetic neuromodulation is proven to be one of the possible cures for this chronic disease. To stimulate multiple neurons simultaneously, distributed miniaturized implants are needed to cover a wide range of areas. In this paper, an on-chip design of an inductively coupled wireless power transfer (WPT) system for an optogenetic implant is presented. A $0.3 \text{mm} \times 0.3 \text{mm}$ on-chip spiral coil is designed using a standard $0.5 \mu m$ CMOS process and characterized as the receiver of the WPT system. The EM field distribution through different tissue layers is investigated which helps to further evaluate the maximum Specific Absorption Rate (SAR) and temperature through the brain tissue. The system achieves a power transfer efficiency (PTE) of 0.65% and 0.96% with matching networks through 10 mm tissue layer and air links respectively. At 10 dBm transmitted power, the maximum SAR value is simulated to be 1.51 W/kg with the maximum temperature increase of 0.73 °C through the skin layer due to the EM field exposure. A case study of the proposed on-chip coil based WPT system shows the different light intensity level that can be achieved for optognetic neuromodulation application. Thus, the proposed miniaturized system proves to be a good candidate for the future distributive neural interfacing application.
The simultaneous inversion algorithm has been proposed for the purpose of specific absorption rate characterization for the first time.
The simultaneous inversion approach is a robust algorithm for noninvasive specific absorption rate applications.
The proposed algorithm is applicable to the characterization of the specific absorption rate in human phantoms.
The proposed algorithm is robust in terms of noise resistance.
The noninvasive approach allows for the use of solid and inhomogeneous phantoms and allows for the use of existing field measurement hardware to be adapted for SAR applications.
Electromagnetic Inversion for Noninvasive Specific Absorption Rate Characterization
Mario Phaneuf, Puyan Mojabi.
The inverse source framework, which comprises a subset of electromagnetic inversion, is applied to the noninvasive specific absorption rate (SAR) characterization problem.
An algorithm is developed and presented which takes field measurements external to the phantom and provides the electromagnetic sources required to obtain the SAR distribution. The unique aspect of this inverse source algorithm is that it casts the problem as the simultaneous inversion (SI) of two sets of equivalent currents: one for the device under test (DUT), and the other for the phantom. The dependency of these two sets of currents is then incorporated as an explicit regularization term in the resulting algorithm. The method is proposed to be relatively robust in terms of measurement noise. A simplified two-dimensional problem is presented to support this proposition.
We present a novel method of electromagnetic field focusing applicable to 3D implantable devices.
Compared to the case of no focusing device, the achieved focused power is 8 and 16 times greater when the proposed designs are used.
It could improve telemetry or wireless power transfer for future miniaturized implantable medical devices (IMDs).
The paper presents a novel field focusing method, which is simple and biocompatible applicable to 3D implantable devices.
The proposed design can be modified to focus fields with arbitrary polarizations at different depths inside human tissue.
Field Focusing for Implanted Medical Devices
Hossein Mehrpour Bernety, Huanan Zhang, David Schurig, Cynthia M. Furse.
In this paper, we present a novel method of electromagnetic field focusing applicable to 3D implantable devices.
Focusing is achieved using two lines that guide the incident wave deeper into the body and concentrate it around their respective tips. The small gap between the tips of the lines provides constructive coupling and augments the focused field. Compared to the case where no lines are used, the focused power is 8 times and 16 times larger if one line and two lines are utilized, respectively. The proposed design can be modified to focus fields with arbitrary polarizations at different depths inside the body. We propose a method to create this focusing structure by extrusion inside the body using heat-activated polymer-based conductors.
This study deals with safety thermal issues related to the application of spinal tDCS.
Combined resolution of Laplace and bio-heat equation allows to assess temperature increase due to electric stimulation induced by spinal tDCS.
This is the first study addressing the tissues temperature changes induced by tsDCS.
The very low heating induced by tDCS is not likely to activate metabolic changes in the target tissues here considered or to contribute to the few side effect that the applications of spinal tDCS protocols have shown.
Findings of this work respond to the need of evaluating the safety of the spinal tDCS application on different subjects (young male, female, and pregnant women).
Modelling of the temperature changes induced by transcutaneous spinal direct current stimulation (tsDCS)
Serena Fiocchi, Emma Chiaramello, Alberto Priori, Paolo Ravazzani, Marta Parazzini.
Transcutaneous spinal direct current stimulation is a neuromodulation technique recently exploited to regulate the activity and enhance plasticity of spinal neural structure.
Despite the few side-effects reported by clinical studies, the assessment of potential localized temperature increases in tissues is still not solved. In this study, the temperature increase induced by a 3 mA stimulation in the tissue target (i.e. the spinal cord), in its surrounding tissues and in other tissues possibly more vulnerable to temperature increase were assessed through a computational approach. That solves Laplace equation and BioHeat equation for electric field and temperature distribution, respectively, on six whole body high resolution anatomical models, including three models of pregnant women at different gestational ages. Results show that temperature increase distribution in those targets is guided by a complex interaction between different mechanism in which electric stimulation plays a secondary role, in particular when blood perfusion is active. The very low heating assessed (below 1.5 m°C) is not consistent with the hypothesis that the induced temperature increase would critically activate metabolic changes in the target tissues here considered or would contribute to the few side effect that the applications of spinal direct current stimulation protocols have shown.
Utilizing microwave energy, a deeper, four-quadrant ablation area for transcatheter renal denervation (RDN) can be realized without causing complicated vascular lesions as opposed to widely used radio frequency current devices.
An ablation depth of 8 mm or more with a lower maximum temperature of 65 ºC was achieved by using microwave energy as opposed to radio frequency currents 4-5 mm depth and 87 ºC maximum temperature, shown by numerical calculation and heating experiments.
The potential application of this microwave technology is the RDN, aimed at reducing resistant hypertension.
Main contribution of this paper is by comparing microwave energy to radio frequency current based devices in RDN treatment, both in numerical calculations and heating experiments, it is possible to overcome performance shortcomings of widely used devices right now.
In addition, muscle phantoms at 500 kHz and 2.45 GHz respectively, was produced and both phantoms properties were measured to match parameters used in the numerical calculations and the heating experiments.
Comparison of Radio Frequency Current and Microwave Energy for Transcatheter Renal Denervation
Aditya Rakhmadi, Kazuyuki Saito, Shohei Matsuhara, Tomoyuki Tajima, Nobuyoshi Takeshita.
Transcatheter renal denervation (RDN) is a catheter-based procedure for resistant hypertension treatment using radio frequency (RF) ablation.
However, RF ablation is proven to have an inconsistent result of reducing hypertension, for the reason that it has a limited heating capability. We present the feasibility of using microwave energy as a different energy source for RDN ablation treatment, by comparing to RF current energy source. In this paper, we designed a coaxial-slot antenna for microwave (2.45 GHz) and an electrode for RF current (500 kHz), with phantoms that match human muscle dielectric properties at each respective working frequency. Both phantoms thermal properties and material density were measured, and then used in simulation. A total of 10 ablation experiments were conducted to validate the numerical simulation results. Each temperature points of the experiments were measured at 1 mm interval between thermometer probes. The results of the experiment temperature distributions agree well with the simulation results, RF achieved 5-6 mm while microwave achieved 7-8 mm. Additionally, numerical calculations with different input power were conducted to understand the temperature characteristics of both energy source. Microwave energy offers higher input power for a more profound and broader ablation area as opposed to RF current.
A new electromagnetic formulation enables an innovative boundary element (BEM) solution of the electroencephalography (EEG) forward problem which allows to efficiently take into account skull and white matter anisotropies.
BEM approaches, very popular in the EEG medical imaging community, can be empowered with the hybrid contributions of this work which eliminate the need for oversimplifying assumptions on the head-skull-brain conductivity profile.
The solution of the EEG forward problem is a cornerstone for high resolution electroencephalographies, for brain source localization, and for their applications in epilepsy treatment, electric impedance tomography, and transcranial brain stimulation.
Up to today, Boundary Element Methods (BEM), quite popular in EEG medical imaging circles, were not able to handle white matter and skull inhomogeneities and anisotropies. To include these features, full volumetric discretizations of the Finite Element Method (FEM) were required. This work lifts this constraint for the first time by allowing an efficient BEM scheme to handle realistic white matter and skull conductivity profiles.
A Hybrid Volume-Surface-Wire Integral Equation for the Anisotropic Forward Problem in Electroencephalography
Maxime Monin, Lyes Rahmouni, Adrien Merlini, Francesco P. Andriulli.
Solving the electroencephalography (EEG) forward problem is a fundamental step in a wide range of applications including biomedical imaging techniques based on inverse source localization.
State-of-the-art electromagnetic solvers resort to a computationally expensive volumetric discretization of the full head to account for its complex and heterogeneous electric profile. The more efficient, popular in biomedical imaging circles, but unfortunately oversimplifying Boundary Element Method (BEM) relies instead on a piecewise-uniform approximation that severely curbs its application in high resolution EEGs. This contribution lifts the standard BEM contraints by treating the local anisotropies with adequate fiber and thin volume integral equations that are tailored to specific structures of the fibrous white matter and the inhomogeneous skull. The proposed hybrid integral equation formulation thereby avoids the full volumetric discretization of the head medium and allows for a realistic and efficient BEM-like solution of the anisotropic EEG forward problem. The accuracy and flexibility of the proposed formulation is demonstrated through numerical experiments involving both canonical and realistic MRI-based head models.
Switch-based low intermediate frequency (IF) system of a vital sign radar is proposed for simultaneous multitarget and multidirectional detection.
The signal to noise ratio (SNR) can be improved by choosing IF properly to reduce the flicker noise, implying more tolerance in multitarget or multiple-interferer environments.
Multitarget and multidirectional detection can be achieved using a frequency-modulated continuous-wave (FMCW) radar with an RF switch and using ensemble empirical mode decomposition (EEMD) algorithm without a complicated antenna array or analog beamforming.
IMBioC 2019 Special Issue paper
Switch-Based Low Intermediate Frequency System of a Vital Sign Radar for Simultaneous Multitarget and Multidirectional Detection
Guan-Wei Fang, Ching-Yao Huang, Chin-Lung Yang.
This paper proposes a RF-switch-based radar system for detecting multi-target and multi-direction vital sign simultaneously.
Compared to traditional continuous wave (CW) radars which suffer mixed and distorted phases from multiple reflections, frequency modulation continuous wave (FMCW) radars have the nature of suitable range resolution and the inherent capabilities of multi-target monitoring. By using FMCW radars to extract vital signs separately based on our proposed algorithm, the vital sign of the two targets can be detected simultaneously even within the limitation of the resolution bin. Furthermore, adding a RF switch, as a multiplexer, enables a single transceiver to detect the vital signs from multiple directions and multiple targets straightforward. Moreover, this RF switch provides the mixing function. A low-IF architecture is proposed and demonstrated for multi-direction vital sign detection simultaneously, and the signal to noise ratio (SNR) can be improved by choosing IF properly to reduce the flicker noise. The demodulation of heart rate (HR) is challenge due to low SNR and the harmonic of respiratory rate (RR) for FMCW vital sign radars. We apply an ensemble empirical mode decomposition (EEMD) algorithm to extract intrinsic mode functions of RR and HR. Experiments show the proposed algorithm improve SNR and accuracy significantly.
Thermoacoustic tomography (TAT) is applied to detect and monitor the formation of osteoporosis for the first time, and to further explore the physiological mechanism of osteoporosis formation from the perspective of tissue dielectric properties.
Significant differences in thermoacoustic signal intensities between normal bone growth and osteoporotic bone formation are observed, suggesting that TAT has the potential to detect and monitor osteoporosis.
TAT can provide useful information for diagnosis of osteoporosis, prediction of fracture risk, and monitoring of disease progression.
This study represents the first for TAT to in vivo image osteoporosis and provides initial facts that TAT may become a new tool for noninvasive detection and monitoring of osteoporosis.
This work is an exploratory experimental study of TAT for imaging osteoporosis, using micro-CT to validate the TAT findings.
Microwave Thermal Acoustics Special Issue paper
Detection and Monitoring of Osteoporosis in a Rat Model by Thermoacoustic Tomography
Zihui Chi, Xiao Liang, Xue Wang, Lin Huang, Huabei Jiang.
Objectives: Osteoporosis is a bone disease associated with decreased bone mineral density (BMD) and bone quality factors.
Tissue dielectric properties, which can be obtained by thermoacoustic tomography (TAT), are considered as potential parameters correlated with both BMD and bone quality. Therefore, here we propose to use TAT to detect osteoporosis and to monitor the formation of osteoporosis over a long period. Technology or Method: This study used the bilateral ovariectomy to obtain an osteoporotic rat model (n=4) along with a sham control. During the 100 days after the operation, the right tibia of each rat was in vivo thermoacoustically imaged at 5 time points. After the last TAT imaging, micro computed tomography (Micro-CT) was performed on each rat to validate the TAT findings. Visual observation and semi-quantitative methods were used to analyze the thermoacoustic data. Results: During the monitoring period, the thermoacoustic signal intensity of the tibia of the sham-operated rat continued to increase, while the thermoacoustic signal intensities of the tibia of osteoporotic model rats showed fluctuations. The TAT findings were verified by Micro-CT. Conclusions: Osteoporosis can be clearly detected by TAT. Significant differences in thermoacoustic signal intensities between normal bone growth and osteoporotic bone formation are observed. Clinical or Biological Impact: This study provides initial facts that TAT may become a new tool for noninvasive detection and monitoring of osteoporosis.
A time-multiplexed hyperthermia treatment planning technique aiming at focusing the tumor heating while protecting the healthy tissue is evaluated with temperature-dependent tissue properties.
The time-multiplexed hyperthermia via MOGA optimization can successfully intensify the heating into the target region while suppressing pre-defined hotspots when either constant thermal properties or temperature dependent tissue properties are assumed.
The targeted medical application is hyperthermia treatment planning in order to maximize heating in the tumor while minimizing heating in surrounding tissues.
This work demonstrates the robustness of the time-multiplexed hyperthermia approach to the variation of tissue properties due to temperature increases and ensures the clinical benefit of the method.
This work demonstrates that time-multiplexed hyperthermia is effective, regardless of the thermal model used.
Robustness of Time-Multiplexed Hyperthermia to Temperature Dependent Thermal Tissue Properties
Grazia Cappiello, Margarethus Paulides, Tomas Drizdal, Declan Oloughlin, Martin Ohalloran, Martin Glavin, Gerard Van Rhoon, Edward Jones.
Microwave hyperthermia is a promising cancer treatment used in combination with radio- and chemotherapy.
Typically, hyperthermia systems involve several antennas that transfer electromagnetic energy into the tissue. The principal need in hyperthermia treatment is to optimally focus the heating into the target while minimising heating in the surrounding healthy tissue. Patient-specific treatment planning is done to optimize the specific absorption rate and the resulting temperature distribution. Uncertainties associated with the thermal model used for temperature simulations represent an important challenge. Our previous work has demonstrated that the occurrence of hotspots can be reduced and target heating enhanced using time-multiplexed steering procedures. In this paper, the robustness of time-multiplexed hyperthermia against temperature dependent thermal tissue properties is investigated. Temperature simulations are used to predict the time-dependent heating achieved by multiple antenna phase and amplitude configurations that are generated by a multi-objective genetic algorithm and applied sequentially. The proposed strategy is compared with the heating obtained using one single heating setting obtained by particle swarm optimization as typically used in clinical hyperthermia. Thermal performance of the static and time-multiplexed methods are assessed by applying two thermal models, one that uses constant properties of blood perfusion and thermal conductivity of tumor, muscle and fat, and a second one that uses temperature dependent perfusion values. This study shows that time-multiplexed hyperthermia enhances target heating and limits the hotspot appearance regardless of the thermal model used in thermal simulations.
Theoretically and experimentally evaluate important parameters such as current and stimulating frequencies in repetitive transcranial magnetic stimulation (rTMS) that could modulate heart rhythm.
Our system generated an eddy current of 25.4 μA/mm2 in the mouse brain regions and produced the maximum heart rhythm modulating effect at 20 Hz.
The combined modeling and experimental approach is applicable to explore the potential adverse effects of exogenous electromagnetic fields on heart rhythm.
Our study provides novel insights into the mechanism of heart rhythm modulation through rTMS and demonstrates the quantitative and morphological aspects of ECG alteration in such outcome.
The rTMS dominant frequency of 20 Hz induced the most pronounced heart rhythm prolongation, causing the heart rate to decrease by 58.65 % compared to that before rTMS.
Cardiac Influence of Repetitive Transcranial Magnetic Stimulation in Small Animals
Ting-Wei Wang, Yen-Ling Sung, Shien-Fong Lin.
Repetitive transcranial magnetic stimulation (rTMS) system is an important therapeutic tool used in non-invasive brain stimulation.
The electric field induced by the time-varying magnetic field in a stimulating coil could activate nerve fibers in the brain, resulting in depolarization or hyperpolarization of the neurons. However, the potential adverse effects of rTMS on heart rhythm have not been extensively investigated. This study aims to develop an optimized design of rTMS system to evaluate the potential adverse effects of rTMS on mouse heart rhythm via vagus nerve modulation for pre-clinical application. The rTMS-induced electric field in the vagus nerve of brain produced by the strong rate of current change of 1.04×108 A/s in a stimulating coil, which was directly determined by circuit design in charging voltage of the capacitor bank and inductance value of a stimulating coil. A finite element method (FEM) mathematical simulation indicated that the maximum eddy current was 25.4μA/mm2, which was greatly exceeded the vagus nerve activation threshold of 5.6μA/mm2. The animal experiment results also verify that the induced electric field activates the RR-interval prolonging effect might be attributed to vagus nerve stimulation (VNS) from rTMS, and the most pronounced heart rhythm prolonging effect at 20Hz magnetic field treatment, causing the average heart rate decreased to 58.65% of that before rTMS in 10 mice. In conclusion, above-threshold rTMS at 20 Hz could produce maximum adverse effect on heart rhythm through direct vagus nerve activation for pre-clinical applications such as safety screening.
The stochastic dosimetry approach permitted to evaluate the assessment of children exposure variability due to the position of a low frequency near-field source with low computational efforts
The method was useful for individuating the source positions area, where the source could cause the highest levels of exposure
The target biological application is the evaluation of children exposure level due to the common use of domestic appliances, considering the variability of a real exposure scenario
The work permitted to expand the knowledge about the low frequency near-field sources children exposure, not limiting it only on some worst-case scenario hypothesis
Influence of low frequency Near-Field Sources Position on the Assessment of Children Exposure Variability using Stochastic Dosimetry
Marta Bonato, Emma Chiaramello, Serena Fiocchi, Gabriella Tognola, Paolo Ravazzani, Marta Parazzini.
The objective of the present work was to assess the children exposure variability due to low frequency near-field sources using an approach based on stochastic dosimetry.
These scenarios represent a topic of high interest, because it was found that domestic appliances could be relevant for children exposure level. In details, in this paper the exposure of two child models to a hairdryer model was evaluated. Following the ICNIRP guidelines, the electric field amplitudes induced in specific tissues composing the central nervous system and the peripheral nervous system were analyzed. The analysis of the results permitted to highlight a high exposure variability depending on the near-field source position and to individuate the regions where the source could cause the highest levels of exposure, not limiting the analysis only to some worst-case exposure scenarios.
A human-body-mimicking electrolyte, a Pt electrode, and a clinical electrode form an accurate and low-cost in vitro validation method for brain-computer interfaces.
The proposed in vitro setup provides a more clinically accurate assessment of brain-computer interface performance than prior methods.
Brain-computer interfaces can assist the medical field in better understanding and treating neurological disorders (e.g. epilepsy, Alzheimer's, depression, etc.), and accurate assessment of device performance is key to ensuring the ability to record the relevant neural signals.
Prior in vitro validation methods do not closely replicate an in vivo recording environment creating the potential for overestimation of device sensitivity, thus increasing the cost and number of animals required during in vivo testing.
The developed electrode model and impedance characterization can be used to better inform implanted sensor design in the future.
An in vivo-mimicking In vitro testbed for brain-computer interfaces
Katrina Guido, Asimina Kiourti.
To reduce the amount of animal testing associated with brain-computer interfaces (BCI) therefore reducing the associated monetary cost and cost of animal lives,
we report a novel in vitro test setup to validate BCIs. Past validation techniques used stand-alone 50-Ω function generators or function generators connected to high-valued resistors to replicate recording from clinical electrodes; however, neither of these two methods accounts for an electrode's complex, frequency-dependent impedance, and both underestimate the normally high electrode impedance. As such, optimizing and testing via function generators and resistors overestimates BCI sensitivity by at least 250 times and 2 times, respectively. To more closely replicate a clinical recording environment and accurately assess BCI performance, we utilize a function generator to mimic neural signals sent to a human-body-mimicking electrolyte fluid via a Pt electrode. These signals are then recorded via a neural electrode connected to the BCI under test. To validate the proposed method, we compare sensitivity of a previously reported fully passive BCI via the three methods. The presented results demonstrate that recording via the proposed clinically accurate setup when the BCI has been optimized with one of the two other test methods gives the lowest sensitivity in the frequency domain and worst signal stability in the time domain. Overestimating BCI sensitivity during in vitro testing results in the potential inability to record desired neural signals in vivo. The proposed method provides a means by which to accurately assess BCI performance, therefore reducing errors during animal trials and saving time, money, and lives.
Controlling current distribution and applying thermal distribution comparison of causative EM radiation facilitate prior understanding of the required electromagnetic field that can create conformal heating to the targeted cancerous lesion.
Synthesizing the desired radiator with self-imbedded choke operating at low input power can minimize consequent return currents along antenna shaft which alleviates overheating problems and damaging surrounding healthy tissues along antenna shaft.
End-fire highly directed radiation is found to be more efficient in ablating tumors using less input power than that associated with omnidirectional (broadside) applicator using high input power to force homogeneous and fast ablation which contributes in providing confined homogeneous heating required for full ablation.
BSB microwave antenna can provide confined heating of approximately 30 mm diameter tumor at only 3W input power which is comparable to that obtained at much higher input power.
Maintain low reflection over wide frequency band can provide wideband mapping of heterogeneous dielectric and thermal properties of biological tissues operating at the same low power level.
Realization and Experimental Assessment of Baseball-Bat Microwave Antenna for Low Power Cancer Ablation
Eman G. M. I. Hassan, Haifa Takruri, Amira Zaki.
Experimental assessment of using low power microwave ablation for treating focal tumor is presented in this article.
Confinement of heating generated by microwave radiation is one of the major concerns in cancer treatment to maintain the acceptable functionality of the organ and alleviate radiation exposure towards surrounding tissues. Development of baseball-bat shaped (BSB) antenna has been studied using electromagnetic and thermal simulations and evaluated experimentally. Numerical simulations showed less than -10-dB reflection stability is attained for more than 20 GHz. Electromagnetic simulation showed that highly directed end-fire radiation achieves confined power deposition within targeted model and yields in higher SAR attained. Nearly spherical ablated lesions are achieved with no healthy tissues being destroyed in the backward direction. Proposed antenna was fabricated and tested in ex-vivo bovine liver sample and egg-white solution. Good agreement between simulated and measured results where confined ablated lesions attained at only 1W were comparable to that obtained at much higher power ranges (20-60W). Efficacy of BSB antenna to efficiently radiate in different dielectric mediums is noticeably attained. The proposed antenna model may help improving the precision of microwave ablation associated with commonly broadside radiators previously used in literature and provide homogenous SAR and confined heating to overcome the limitations found in treating spherical tumors with heterogeneous properties using much high power with narrow-band feature.
A non-contact and non-invasive method for measurement of glycerol is an exceptional feature of the sensor presented in this work enabled by employing a microwave coupled tag-reader structure.
A printable tattoo shaped passive tag sensor with zero power consumption is presented which as an ideal candidate for smart wearable sensor applications according to its distant and non-invasive measurement capabilities.
Targeted biomedical application of this work is measurement of glycerol in blood and interstitial fluid for indication of hyperglyceridaemia and coronary heart disease as well as the determination of glycerol level in drug-delivery applications.
The major breakthrough of this work is development of a highly sensitive non-invasive non-contact distant sensor with zero power consumption on the sensing part with the capability of detection of a small amount of glycerol in nanoliter volume over a microfluidic channel.
Achieving to 139.5 kHz of frequency shift and 0.2dB of amplitude variation in the resonance profile of the sensor for 2% concentration of glycerol in serum with about 960nlit of sample confined in a microfluidic channel exposed to the passive tag placed 25 mm far from the reader.
Real-Time Non-Contact Integrated Chipless RF Sensor for Disposable Microfluidic Applications
Zahra Abbasi, Masoud Baghelani, Mehdi Nosrati, Amir Sanati-Nezhad, Mojgan Daneshmand.
Glycerol concentration measurement is one of the most important biomarkers for many diagnostic applications such as indication of hyperglyceridaemia and coronary heart disease.
A new distant microwave sensing platform to monitor the concentration of glycerol in a water and serum solution in a microfluidic channel is presented. The sensor is based on a microstrip line reader with defected ground plane coupled to a chipless microwave resonator tag. Due to the strong coupling between the reader and the tag, the distance between them can be increased up to 45 mm. The high level of sensitivity of the structure makes it suitable for nanolitre volume scale chemical sensing inside the microfluidic channel. The sensor demonstrates a frequency shift of 6 MHz when the volumetric percentage (V%) of glycerol in DI water changes from 70% to 0%. The proposed solution provides the opportunity of integrating the chipless passive tag with the microfluidic channel while the reader is distant from the microfluidic system.
This work quantifies the influence of dehydration on dielectric properties and proposes a dehydrationmitigating technique to establish a more stable environment for longer measurement protocols.
Due to the effect of dehydration, a relative change up to 9% in dielectric properties was established in a relevant temperature range during a relevant measurement period of 35 minutes.
More precautions should be taken for dielectric measurement of biological tissue with regards to environmental conditions and other relevant metadata.
For several applications relying on the difference in dielectric properties between healthy and malignant tissue, this error of 9% in measured data is on the same order of magnitude as the effects of interest. Thus, the effect of dehydration could severely obscure the expected difference in dielectric contrast.
Until an agreement on a standard operating procedure with an acknowledged metadata form, we can only encourage authors to report all metadata in their work to allow future comparisons between studies of dielectric properties of biological tissue.
Effect of Dehydration on Dielectric Measurements of Biological Tissue as Function of Time
Gertjan Maenhout, Adam Santorelli, Emily Porter, Ilja Ocket, Tomislav Markovic, Bart Nauwelaers.
The lack of a standardized operating protocol for dielectric measurements of biological tissues makes it difficult to objectively compare
reported dielectric properties since these properties are affected by multiple environmental confounders. This work investigates the effect of dehydration on dielectric properties as a function of time at different temperatures. Additionally, it proposes a dehydration mitigating technique. The dielectric properties of porcine liver samples were measured during a 35 minute measurement interval at different temperatures. A first set of six experiments was conducted in standard laboratory conditions. A second set of two experiments was conducted with a modified measurement setup with reduced air flow. Under standard conditions, relative changes in the dielectric properties up to 9% were observed. With the modified setup, the relative change was halved compared to standard conditions. This reduction indicates that the environmental conditions have a considerable influence and that modifying these conditions extends the stability of the dielectric properties over time. For several applications relying on the difference in dielectric properties between healthy and malignant tissue, an error of 9% in measured data is on the same order as the effects of interest. The expected difference in dielectric contrast could be obscured due to dehydration.
A simple approach that can be used to generate a useful data of the dielectric properties of biological tissues for the development of the microwave imaging and therapeutic system and implantable medical devices.
The dielectric properties of healthy and unhealthy tissues at any age stage can be generated using information of the dielectric properties of solid fraction (DPSF) and water content (WC) of tissues.
This work is beneficial in the development of microwave imaging system for brain stroke, skin cancer detection and fatty liver diseases.
This manuscript addresses the importance of using the DPSF to generate the DPs of any unhealthy tissues that needed in the development of any successful microwave-based medical systems.
Using Dielectric Properties of Solid Fraction and Water Content to Characterize Tissues at Different Health and Age Conditions
Beadaa Mohammed, Mohamed Manoufali, Syed Akbar Raza Naqvi, Konstanty Bialkowski, Paul Mills, Amin Abbosh.
Precise dielectric properties (DPs) of biological tissues play an essential role in the development of non-invasive microwave diagnostic and therapeutic systems.
There is much information in the literature about DPs of healthy tissues, however, there is a gap in DPs of unhealthy tissues and DPs of tissues at different ages. DPs of any healthy tissue can be described by DPs of its solid fraction and water content values. The observed DPs of a tissue change with the level of tissue's water content (increase or decrease) according to the condition of the tissue (i.e., pathology and age). For the existing study, DPs of solid fraction and water content of 19 different tissues from 61 pigs at different ages were measured and used to generate DPs of those tissues under different conditions using a simple mathematical model. The dry-weighing method was used to determine the water content of tissues and DPs of their solid fractions were measured. Then, DPs of those tissues were generated using the proposed method and found to be in a good agreement with some previously published data as well as with the measured data in this study. This study confirms the potential of using DPs of the solid fraction and water content to calculate DPs of tissues at any age or pathology.
We proposed a hierarchical model to utilize radar as an 'Enhancer' to complement with the PSA (Pressure Sensor Array) in improving the static gestures recognition rates, on the contrary, in the dynamic gesture case scenario the PSA acts as an 'Enhancer' to boost the radar performance.
Sequential forward selection (SFS) significantly reduces the computational intensity in terms of less features and improves the classification performance.
For the second-stage of the hierarchical model, soft and hard fusion methods are implied respectively to promote the classification accuracy and eliminate the false alarms. Different weights of the 'Enhancer' output are verified and compared in terms of the accuracy in the soft fusion process.
Soft fusion improves the accuracy by 16.7% and 11.1% with respect to static and dynamic gesture identification, whereas hard fusion reduces the accuracy variance across all the participants and produces a subsequent improvement about 5.5% in the dynamic gestures.
Future work involves more gestures and more participants with neural network-based algorithm and additional sensors configurations and fusion approaches.
IEEE Sensors Special Issue paper
Hierarchical Sensor Fusion for Micro-Gestures Recognition with Pressure Sensor Array and Radar
Haobo Li, Xiangpeng Liang, Aman Shrestha, Yuchi Liu, Hadi Heidari, Julien Le Kernec, Francesco Fiorane.
This paper presents a hierarchical sensor fusion approach for human micro-gesture recognition by combining an Ultra Wide Band (UWB) Doppler radar and wearable pressure sensors.
First, the wrist-worn pressure sensor array (PSA) and Doppler radar are used to respectively identify static and dynamic gestures through a Quadratic-kernel SVM (Support Vector Machine) classifier. Then, a robust wrapper method is applied on the features from both sensors to search the optimal combination. Subsequently, two hierarchical approaches where one sensor acts as 'enhancer' of the other are explored. In the first case, scores from Doppler radar related to the confidence level of its classifier and the prediction label corresponding to the posterior probabilities are utilized to maximize the static hand gestures classification performance by hierarchical combination with PSA data. In the second case, the PSA acts as an 'Enhancer' for radar to improve the dynamic gesture recognition. In this regard, different weights of the 'Enhancer' sensor in the fusion process have been evaluated and compared in terms of classification accuracy. A realistic cross-validation method is chosen to test one unknown participant with the model trained by data from others, demonstrating that this hierarchical fusion approach for static and dynamic gestures yields approximately 16.7% improvement in classification accuracy in the best cases.
This study investigates the potential use of biogenic magnetosomes as a contrast agent for microwave imaging techniques for breast cancer detection.
We propose a functionalized contrast agent which is capable of decreasing the permittivity and increasing the electrical conductivity of breast tumor tissue based on the concentration of the magnetosome solution.
In particular, the focus of the system is an innovative and effective therapy of superficial tumors, as, for instance, melanoma and breast cancer.
The targeted medical application of this study is a potential new contrast agent to be used to improve microwave imaging for breast cancer detection.
Toward Magnetosomes for Breast Cancer Theranostics
Abas Sabouni, Jakob Short, William Terzaghi.
This study investigated the potential use of biogenic magnetosomes as a contrast agent for breast cancer detection using microwave imaging techniques.
We propose a functionalized contrast agent which is capable of decreasing the permittivity and increasing the electrical conductivity of the breast tumor based on the magnetosome concentration. Our results show that biogenic magnetosomes can increase relative permittivity contrast between 9-25% in the microwave frequency range from 1-10 GHz.
Using an electromagnetic wave sensor, that is, radar, we demonstrated that autonomic nervous system activities and heart rate variability can be measured successfully during sleep in a noncontact manner.
Our radar-based system measured the heart inter-beat interval with an average error of 25.9 ms and autonomic nervous system index with an average correlation coefficient of 0.93 for reference electrocardiogram (ECG) data, which indicates the sufficient accuracy and reliability of our proposed techniques.
Our proposed techniques can be applied to healthcare and clinical applications that require long-term and unobtrusive monitoring of a person's physical and mental health.
Our proposed techniques are the first to achieve the accurate measurement of an autonomic nervous system index using a radar system, and increased the correlation coefficient with reference ECG data by 1.7 times on average compared with a conventional system.
Our proposed techniques evaluated the reliability of the heart rate estimated using a radar-based noncontact measurement system, which is indispensable in practice, but has never been achieved by existing techniques.
Noncontact Measurement of Autonomic Nervous System Activities Based on Heart Rate Variability Using Ultra-Wideband Array Radar
Takuya Sakamoto, Kosuke Yamashita.
The noncontact measurement of vital signs using ultra-wideband radar has been attracting increasing attention because
it can unobtrusively provide information about the physical and mental condition of people. In particular, the continuous measurement of a person's time-varying instantaneous heart rate can estimate the activity level of the autonomic nervous system without the person wearing any sensors. Continuous heart rate measurement using radar is, however, a difficult task because accuracy is compromised by numerous factors, such as the posture and motion of the target person. In this study, we introduce techniques for increasing the accuracy and reliability of the noncontact measurement of heart rate variability. We demonstrate the performance of the proposed techniques by applying them to radar measurement data from a sleeping person, and we also compare its accuracy with electrocardiogram data.
This paper proposed a new Affective Virtual Reality System (AVRS), and assessed arousal with Electroencephalography (EEG), Heart Rate (HR), Galvanic Skin Reaction (GSR) and Self-Assessment Manikin (SAM).
VR emotion materials could deliver a better emotion elicitation effect than 2D video on negative emotional scenes according to an intergroup experiment.
AVRS was proved as an effective material capable of eliciting emotion for psychological research mental illness diagnosis and virtual reality interaction research.
This system can be applied to psychological research, mental illness diagnosis and virtual reality interaction research.
Design and Evaluation of Affective Virtual Reality System Based on Multimodal Physiological Signals and Self-Assessment Manikin
Dan Liao, Lin Shu, Guodong Liang, Yingxuan Li, Yue Zhang, Wenzhuo Zhang, Xiangmin Xu.
Affective materials have always been an important factor in improving the efficiency of affect related research for their ecological validity.
In order to achieve high ecological validity in emotion elicitation, this paper proposed a novel Affective Virtual Reality System (AVRS) by utilizing the immersion, privacy and design flexibility of virtual reality. Design elements including subject features, sound features, motion features and color features were extracted by referring to the referenced emotion materials, art works and the existing VR scenes. Affective VR scenes were then designed by Unreal Engine 4.12 and their effectiveness was validated by Self-Assessment Manikin (SAM). Furthermore, arousal was used as an indicator to compare the difference of ecological validity between VR and video emotional materials with an intergroup experiment through Electroencephalography (EEG), Heart Rate (HR), Galvanic Skin Response (GSR), and SAM. Results proved that VR scenes could achieve the same emotion elicitation as video. Especially, there was significant difference in measures of Fearful in SAM evaluation, indicating that VR emotion materials were expected to deliver a good effect on negative emotional scenes. The proposed AVRS as well as the multimodal physiological signal database with valence, arousal and dominance (VAD) labels could help the development of emotion related studies.
We present a novel radiofrequency radiating system for focused hyperthermic applications with magnetic nanoparticles.
Safe, efficient and targeted treatments of hyperthermia with magnetic nanoparticles can be achieved through the proposed RF radiating system.
Although the challenging low frequency range (hundreds of kHz), the system accomplishes a precise and delimited radiofrequency magnetic field distribution, avoiding indiscriminate tissue exposure.
The synergy between a careful design of the radiating system and research on innovative magnetic nanoparticles can pave the way towards more efficient and safer magnetic hyperthermia treatments in clinical applications.
A Radiating System for Low Frequency Highly Focused Hyperthermia with Magnetic Nanoparticles
Danilo Brizi, Nunzia Fontana, Giulio Giovannetti, Luca Menichetti, Laura Cappiello, Saer Doumett, Costanza Ravagli, Giovanni Baldi, Agostino Monorchio.
In this paper, we propose a radiofrequency (RF) radiating system for highly focused hyperthermia with magnetic nanoparticles.
We firstly designed a coil able to focus the magnetic field at the frequency (around 340 kHz) that maximizes the heat release from the chosen nanoparticles. Then, through an electromagnetic software based on the Method of Moments (MoM), we performed numerical simulations of the coil that resulted in excellent agreement with the experimental measurements conducted at the workbench on a fabricated prototype. After that, a radiating system consisting of the coil, a high-power radiofrequency signal generator and a set of magnetic nanoparticles have been set up. We carried out several experimental trials with different samples of magnetic nanoparticles in order to demonstrate the focusing property of the proposed system. The results we obtained suggested that a careful design of the radiating system could pave the way towards more efficient and safer magnetic hyperthermia treatments in clinical applications.
The paper studies the significant impact of multipath wave propagation (surface and diffraction waves) on electromagnetic (EM) sensing systems, which is often overlooked in the design and development of EM sensors.
System simulations and experimental results with a millimeter (mm)-wave sensor demonstrate the need to suppress these unwanted signals in order to increase EM sensing sensitivity.
Experimental measurements demonstrate that sensor's sensitivity to glucose concentrations is almost doubled by suppressing multipath waves with appropriate use of absorbers.
Our study focuses on sensing glucose changes with mm-waves, but this analysis can be useful for any application in EM biomedical sensing which requires the detection of weak signals propagating through lossy tissues.
Study and Suppression of Multipath Signals in a Non-Invasive Millimeter Wave Transmission Glucose Sensing System
Maria Koutsoupidou, Helena Cano-Garcia, Roberto L. Pricci, Shimul C. Saha, George Palikaras, Efthymios Kallos, Panagiotis Kosmas.
Electromagnetic (EM) biomedical sensors in the mm-wave frequency range must detect small changes in signals in the presence of tissues, which are correlated to a pathological condition.
These signals, however, can suffer from artifacts due to complex EM wave interactions such as diffraction and surface wave propagation, which are often overlooked in the design phase of these sensors. This paper studies the impact of these wave phenomena on the signals transmitted and received from a pair of antennas designed to sense glucose changes via changes in transmission through a sample. Numerical simulations and controlled experiments with glucose solutions demonstrate for the first time that unwanted signal contributions from mm surface waves along the tissue can dominate the received signals but can be reduced with the use of appropriately placed absorbers around the antenna sensors. As a result, the sensitivity of such a sensing system to glucose changes is increased. This finding can be very useful in the design and development of the glucose sensor under study, as well as for other EM-based diagnostic medical applications.
Four quantities, namely 1) the net dissipated power around an electrode of an active implantable medical device (AIMD), 2) the net temperature increase, 3) the current flowing from the lead into the electrode, and 4) the net specific absorption rate (SAR) increase, were numerically compared for a set of leads with straight and helical wires to evaluate the lead electromagnetic model (LEM) with respect to radio frequency energy-induced heating.
The most suitable sensor locations were positions from the electrode tip along the first half of the electrode axial axis. A temperature sensor was essential if the net electrode temperature increase required evaluation.
The targeted medical application is an evaluation of heating induced by radio frequency energy that appears in human tissue near an AIMD during magnetic resonance imaging.
For leads with helical wire, our results indicate that 1) to achieve a good validation of the transfer function, that is, the linear regression coefficient of determination R2 to be close to 1, the temperature sensor results must be obtained as fast as possible, but 2) the total transient time must be longer than 360 seconds for evaluation of the LEM calibration factor.
Utilization of result analysis made for generic leads with straight wire can be significantly misleading for predicting results for leads with helical wires.
Comparison of Different Assessment Quantities to Evaluate Lead Electromagnetic Model for Radio Frequency Energy-Induced Heating
Mikhail Kozlov ; Wolfgang Kainz.
To prevent tissue damage near an active implantable medical device (AIMD), the radio frequency energy-induced heating must be estimated for all possible
clinical scenarios in patients undergoing magnetic resonance imaging. Four quantities, namely 1) the net dissipated power around an electrode of an AIMD, 2) the net temperature increase, 3) the current flowing from the lead to the electrode, and 4) the net specific absorption rate (SAR) increase were numerically compared for a set of leads to evaluate the analytical lead electromagnetic model (LEM) with respect to radio frequency energy-induced heating. The set included 32 single electrode leads with straight and helical wires. The LEMs were obtained and validated with 3D electromagnetic and thermal co-simulations at 128 MHz. A shift of the position of temperature or SAR sensor from the electrode tip to the electrode pedestal resulted in a decrease of the linear regression coefficient of determination for the LEM calibration factors. Behavior of different assessment quantities in terms of sensitivity of sensor location significantly varied between leads with the helical and straight wires. In conclusion, utilization of analysis made for generic leads with straight wire can be significantly misleading for the prediction of results for leads with helical wires.
This is first time to model the performance of implantable PV cells in different layers of tissue. We demonstrate how the electrical characteristics are influenced by the implanting location of the device.
A PV cell implanted in the dermis layer can harvest the greatest amount of power.
We propose implanting our energy harvesting PV cells in the hypodermis layer.
Our proposed PV device harvests enough energy to supply power for low-cost implants such as cardiac pacemakers, retinal implants or biomedical sensors.
PV cells implanted in the adipose layer can harvest nearly 11.84 mW using an 850 nm light source.
Photovoltaic Power Harvesting Technologies in Biomedical Implantable Devices Considering the Optimal Location
Jinwei Zhao, Rami Ghannam, Man Kay Law, Muhammad Ali Imran, Hadi Heidari.
There are still many challenges in effectively harvesting and generating power for implantable medical devices.
Most of today's research focuses on finding ways to harvest energy from the human body to avoid the use of batteries, which require surgical replacement. For example, current energy harvesters rely on piezoelectricity, thermoelectricity and solar electricity to drive the implantable device. However, the majority of these energy harvesting techniques suffer from a variety of limitations such as low power output, large size or poor efficiency. Due to their high efficiency, we focus our attention on solar photovoltaic cells. We demonstrate the tissue absorption losses severely influence their performance. We predict the performance of these cells using simulation through the verified experimental data. Our results show that our model can obtain 17.20% efficiency and 0.675 V open-circuit voltage in one sun condition. In addition, our device can also harvest up to 15 mW/ cm2 in dermis and 11.84 mW/ cm2 in hypodermis by using 100 mW/ cm2 light source at 800 nm and 850 nm, respectively. We propose implanting our device in hypodermis to obtain a stable power output.
An inductively-coupled resonator assembly is shown to allow the detection of significantly sub-wavelength diameter biological cells by combining the strong field confinement provided by a split ring resonator with the high quality factor resonance of a dielectric resonator.
Measurements of single, free-flowing cells in a natural aqueous environment at ~10 GHz have been carried out using a coupled resonator sensor, without the need for trapping, immobilizing, culturing or fixing cells in high-field areas.
The coupled resonator approach proposed in this work shows potential as a method of discrimination of cells based on hydration levels, which in other works has been linked to carcinogenesis, as well as cancer aggressiveness grade; therefore the sensor described herein may represent an alternative method of cancer diagnosis or disease progression monitoring via non-invasive liquid biopsies.
In this paper, measurements of living, free-flowing, single cells in aqueous buffer solution, at a frequency sensitive to cell water content, have been made.
The inductive coupling employed in this sensor allows for physical separation of the sensing elements from microwave electronics, allowing for cheap, disposable chips to be used with biological fluids.
Microwave Dielectric Sensing of Free-Flowing, Single, Living Cells in Aqueous Suspension
Clare Watts, Stephen Hanham, James Armstrong, Munir Ahmad, Molly Stevens, Norbert Klein.
Dielectric measurements offer the possibility of highly sensitive detection of physical cell properties,
and are of interest for clinical applications due to their non-destructive nature and the lack of need for cell labelling. Here we report sensitive measurements on single, living, free-flowing cells (not electrostatically or dielectrophoretically trapped, cultured or fixed directly on sensing elements) in aqueous medium at ~9.8 GHz taken using a coupled dielectric-split ring resonator assembly. Inductive coupling between the two resonators enabled separation of microfluidic chips from RF connectors and allowed for time-resolved continuous-wave measurements on flowing single cells via the coaxial ports of a dielectric-loaded microwave cavity. Analysis via an equivalent circuit model showed that the novel resonator assembly maintained the permittivity-dependent sensitivity of a split ring resonator while operating at quality factors >1000 with lossy aqueous media (typically ~1900). Using a microfluidic channel with a 300 x 300 μm cross section, at a water-loaded resonant amplitude of ~-22 dB at 0 dBm input power level, shifts in amplitude due to individual cells passing through the sensing region of up to -0.0015 dB were observed. Correlations between averaged amplitude shifts and cell size as well as material properties demonstrate the diagnostic potential of this technique.
Microwave-induced thermoacoustic imaging (MITAI) is applied to monitor microwave power deposition distribution in human breast during the process of focused microwave breast hyperthermia (FMBH).
Compressive sensing (CS) based MITAI technique is able to provide reliable power deposition monitoring for the iterative focusing process of the FMBH approach under the condition that the obtainable focusing is good enough.
Focused microwave breast hyperthermia (FMBH) for treating breast tumors noninvasively.
This work presents the first systematic computational study for assessing performance and robustness of the MITAI monitored FMBH modality, referred to as FMBH-MITAI modality, utilizing realistic human breast phantoms with different densities and tumor locations.
Focused Microwave Breast Hyperthermia Monitored by Thermoacoustic Imaging: A Computational Feasibility Study Applying Realistic Breast Phantoms
Lifan Xu, Xiong Wang.
Focused microwave breast hyperthermia (FMBH) represents a newly emerging technique endowed with advantages of high accuracy and low side effect for treating breast tumors.
Practical application of the FMBH approach requires monitoring of microwave power deposition in the breast. Microwave-induced thermoacoustic imaging (MITAI) is naturally feasible for such power deposition monitoring task. This work conducts a computational study to evaluate feasibility of the novel FMBH-MITAI modality using realistic breast phantoms. Basic configuration and rationale of both FMBH and MITAI are introduced. Com-pressive sensing (CS) technique has to be applied in MITAI for sparse acoustic measurement in the FMBH-MITAI modality. Procedure of the computational study consists of microwave simulation, thermoacoustic numerical simulation, CS imaging, and performing the iterative optimization. Simulated results show that CS based MITAI is able to serve as a reliable monitor-ing mechanism to efficiently guide the iterative optimization toward the best obtainable focusing condition in most of the sim-ulated scenarios. Finally obtained thermoacoustic images agree well with simulated power deposition distribution well. This work offers valuable performance evaluation and is of significant meaning for potential clinical applications of the FMBH-MITAI modality.
In this manuscript, body temperature and sweat sensors are integrated with a textile NFC antenna, which eliminates the need for external batteries and realizes real-time wireless monitoring.
This paper has presented design, fabrication implementation, measurements and real-life applications of smart textile NFC antennas and a battery-free wireless NFC body temperature and sweat sensing device, aiming for truly ubiquitous wireless health and wellbeing monitoring.
The proposed device targets at body temperature and sweat loss monitoring for daily healthcare, systemic hyperthermia from fever, sweating symptoms caused by various kinds of infection, inflammation and trauma and wound healing monitoring.
Different from conventional battery enabled and wire connected sensors, the significance of this work is by applying textile NFC as a communication interface as well as a wireless power harvester, battery-free real-time body temperature and sweat monitoring has been realized simultaneously.
Apart from the device itself, an App has also been developed on Android system for the sensor data to be accessed by smart phones.
Smart Textile Integrated Wireless Powered Near Field Communication (NFC) Body Temperature and Sweat Sensing System
Yutong Jiang, Kewen Pan, Ting Leng, Zhirun Hu.
Near Field Communication (NFC) is a short-range wireless communication technique that has become attractive devices for healthcare and wellbeing monitoring.
The work reported here demonstrates the development of a battery free wearable sensing system with temperature and sweat sensors embedded into and powered by a smart textile NFC antenna. The NFC antenna is seamlessly integrated with closed-body garments, and sensor data can be easily acquired by NFC readers and smart phones in order to achieve real time and wireless monitor of health status in a convenient and non-intrusive way. A Dickson charge pump circuit has been designed and implemented in order to pump up the voltage and ensure a steady voltage supply for the sweat sensor. The maximum read range for accessing sensor data is 6 cm. The on-body measurement accuracy of the temperature sensor and sweat sensor are able to achieve ± 0.14°C and ± 0.2%, respectively. The presented system can provide wearable battery-free ubiquitous wireless connectivity for point-of-care and any time healthcare and wellbeing monitoring.
A wearable device was developed with electromagnetic sensors in order to non-invasively monitor the progress of brain atrophy and lateral ventricle enlargement as a result of Alzheimer's disease.
The developed wearable RF device is capable of detecting the progression of brain atrophy and lateral ventricle enlargement successfully.
The work in this study targets Alzheimer's disease and aims to develop a non-invasive device for monitoring the progression of the disease in patients.
The breakthrough in this work is the development of a wearable device that uses RF sensors for detecting changes in the brain as a result of Alzheimer's disease.
Non-Invasive Wearable RF Device towards Monitoring Brain Atrophy and Lateral Ventricle Enlargement
Imran Saied, Tughrul Arslan.
Alzheimer's disease is the most common form of neurodegenerative disease and a leading cause of dementia today.
A pathological effect of Alzheimer's disease is brain atrophy, which is the progressive shrinkage of the brain volume and weight. Another effect of Alzheimer's disease is the enlargement of lateral ventricles in the brain. Currently, MRI and CT scanners can detect and show images of the brain during different stages of Alzheimer's disease. However, its limited accessibility, high costs, and static structure make it inconvenient for some to use. This paper presents the design and novel application of a wearable device comprising of flexible microwave antennas, with an operating frequency range of 800 MHz to 2.5 GHz, that detects the progression of brain atrophy and lateral ventricle enlargement in patients with Alzheimer's at the earliest stage possible. The operating principle of the antennas are simulated in near field using CST and the device is experimentally validated using lamb brain samples and samples representing cerebral spinal fluid (CSF). The measured reflection coefficients and transmission coefficients were found to correlate with changes in brain volume and changes in CSF volume successfully, thus giving an indication of the progression of Alzheimer's disease in a patient.
This proposed protocol evaluates the ability of a microwave imaging system to provide sufficient data quality.
The protocol enables the identification of the system-specific resolution, which in practice is worse than the theoretical estimate.
The approach is applicable to various biomedical microwave-imaging applications through modifications of the measured phantoms.
The approach helps identify faults in the imaging setup and is suggestive of hardware modifications that remedy these faults.
The flexibility of the protocol enables its application (with minor modifications) to any acquisition surface (e.g. planar, cylindrical, hemispherical antenna orientation).
Quality Control of Microwave Equipment for Tissue Imaging
Daniel Tajik, Jessica Trac, Natalia Nikolova.
While the development of microwave imaging technology for biomedical applications has been ongoing for many years, no clinical devices are currently in use.
A major challenge is achieving data quality that would ensure adequate image resolution for the specific diagnostic application. Imaging systems are typically designed with a theoretical resolution limit in mind, which is rarely achieved in practice due to measurement uncertainties, background clutter and system noise. Uncertainties and background clutter are particularly prominent in medical diagnostic imaging. This manuscript proposes a method for data quality assessment of an experimental imaging system that aims at a specific image resolution. It utilizes two measurements, one of a uniform background medium and one of the same medium with a small scattering probe embedded within it. The probe's size and permittivity reflect the desired application-specific resolution. The method extracts the system point-spread function (PSF) from the two measurements and computes the PSF contrast-to-noise ratio (CNR). A case study is presented, demonstrating the quality control protocol and its ability to identify data sets of inadequate quality and provide an evaluation metric. The protocol also highlights possible sources of error and enables data filtering that increases significantly the reconstructed image quality.
We exploit a low-cost and inkjet printed UHF RFID tag as a sensor by modifying the equivalent circuit of the antenna to mitigate the effects of water, blood, and the human body.
The targeted biological and medical applications are intravenous (IV) level sensing, blood storage management, and wound healing detection.
The proposed RFID tag antenna features impedance match with Impinj R6 RFID from 890 MHz – 937 MHz and has a read range of 3 m, 2.5 m and 1.5 m on the surface of a water bottle, IV solution and blood bag, respectively.
As compared with traditional designs, this tag antenna provides 26 % more read range with relatively small size 40 14 mm2 and has a specialty of water proximity sensing, leading to a compact and low-cost solution which is ideal for mass production.
These features make this tag ideal for healthcare application in hospitals, which can add better facilitation for patient monitoring and also reduces the cost.
2019 APS Special Issue paper
Low-Cost Ink-Jet Printed RFID Tag Antenna Design for Remote Healthcare Applications
Abubakar Sharif, Jun Ouyang, Yi Yan, Ali Raza, Muhammad Ali Imran, Qammer Hussain Abbasi.
This paper presents a low-cost, inkjet printed radio frequency identification (RFID) tag antenna for remote health care applications.
The electrically small tag consists of nested-slot configuration and parallel strips. The tag antenna is exploited as a sensor by modifying its equivalent circuit to mitigate the effects of water, blood sample phantom, and the human body. As a result, the proposed RFID antenna features impedance match with Impinj R6 RFID chip from 880 MHz – 937 MHz with compact dimensions of 40 x 14 mm. Moreover, this tag has a read range of 3 m, 2.5 m and 1.5 m on the water bottle, intravenous (IV) solution and blood bag, respectively. However, the read range of RFID tag on an empty water bottle or IV solution bag is 0.5 m. By comparing the read range of tag on empty and solution filled IV bags, the proposed tag is used as a water proximity sensor. Experimental testing of the tag is performed for sensing the level of the IV solution. Also, this tag is tested by mounting on liquid mixture (a mixture of salt and sugar is used as a phantom to mimic the blood) filled plastic bags, which leads to a low-cost solution for blood storage management. Experimental results show a good agreement of proposed tag towards its use in healthcare applications, which leads to better healthcare facilitation regarding cost, time and care.
A supervised machine-learning approach offers a calibration-free and computationally efficient solution for extracting heart rates from radar-measured signals in real time.
The algorithm offers a practical solution for learning individual heartbeat signatures from composite vital sign radar signals in a timely manner.
The technique can be applied to vital sign radars offering a non-intrusive way of monitoring one of the most widely sought after indicator of health and performance, the heartbeat.
The results demonstrated that, by using the proposed machine-learning algorithm, there is no need for multiple optimization schemes, tuning steps, and intermediate signal processing steps for static observation of vital sign radar.
A Supervised Machine Learning Algorithm for Heart-rate Detection Using Doppler Motion-Sensing Radar
Justin Johnson Saluja, Joaquin J. Casanova, Jenshan Lin.
The advancement of vital sign radar technology has proven to be a useful tool in assessing various physiological dynamics including heartbeat and respiration.
There remains several signal processing challenges in this field, which include overcoming the nonlinearities and harmonics that populate the power spectrum. Respiration harmonics distort and overwhelm the measurement of heartbeat due to the large signal amplitude. A supervised machine learning algorithm, the gamma filter, offers an efficient, calibration-free solution to model the time series heartbeat signal given respiration and respiration artifacts. The measured signal is provided by a 5.8-GHz quadrature Doppler radar and a modified ECG signal is used as the ground truth for training the filter. Experimental results show that the heartbeat is independent and separable from respiration and the algorithm can be implemented in real time.
Brain strokes are one of the leading causes of disability and mortality in adults in developed countries. We investigate how microwave tomography reveals changes in the brain' tissues and thereby enable to detect and identify the type of stroke.
We demonstrate that hemorrhagic stroke can be automatically identified with microwave tomography.
Because of portability and cost effectiveness, microwave imaging systems may significantly improve the medical care of cerebrovascular accidents.
Stroke identification including images reconstruction and automatic detection lasts less than 5 minutes.
We implement massive parallel computing to solve the electromagnetic inverse problem and to speed up the reconstruction process.
Detection of Simulated Brain Strokes Using Microwave Tomography
Vanna Lisa Coli, Pierre-Henri Tournier, Victorita Dolean-Maini, Ibtissam El Kanfoud, Christian Pichot, Claire Migliaccio, Laure Blanc-Féraud.
Brain strokes are one of the leading causes of disability and mortality in adults in developed countries.
The ischemic stroke (85% of total cases) and hemorrhagic stroke (15%) must be treated with opposing therapies, thus the nature of the stroke must be determined quickly in order to apply the appropriate treatment. Recent studies in biomedical imaging have shown that strokes produce variations in the complex electric permittivity of brain tissues, which can be detected by means of microwave tomography. Here we present some synthetic results obtained with an experimental microwave tomography-based portable system for the early detection and monitoring of brain strokes. The determination of electric permittivity first requires the solution of a coupled forward-inverse problem. We make use of massive parallel computation from domain decomposition method and regularization techniques for optimization methods. Synthetic data are obtained with electromagnetic simulations corrupted by noise, which have been derived from measurements errors of the experimental imaging system. Results demonstrate the possibility to detect hemorrhagic strokes with microwave systems when applying the proposed reconstruction algorithm with edge preserving regularization.
Live cells were characterized individually in a fast, compact and label-free manner, and the dynamic range of the impedance spectroscopy was greatly increased by 2-port instead of 1-port measurements.
A lumped equivalent circuit of nondispersive resistances and capacitances was found sufficient to explain the impedance spectrum of a cell between 9 kHz to 9 GHz, and the sensitivity of the cell impedance to microwave scattering (S) parameters was analyzed for the first time.
The equivalent circuit parameters were found the most sensitive to the insertion loss of a series-trapped cell and the return loss of a shunt-trapped cell on the coplanar waveguide.
The equivalent circuit parameters could be reliably extracted because low-frequency S-parameters were mainly governed by the membrane resistance, high-frequency S-parameters were mainly governed by the cytoplasm capacitance, and intermediate-frequency S-parameters were mainly governed by the membrane capacitance and cytoplasm resistance.
The theory of ultrawideband impedance spectroscopy was carefully derived and documented for the first time.
Sensitivity Analysis for Ultra-wideband 2-port Impedance Spectroscopy of a Live Cell
Xiao Ma ; Xiaotian Du, Lei Li, Hang Li, Xuanhong Cheng, James Hwang.
For ultra-wideband (9 kHz–9 GHz) 2-port impedance spectroscopy of a biological cell, sensitivity analysis was carried out for extracting lumped cell characteristics such as
membrane resistance (as large as 1 MΩ) and cytoplasm capacitance (as small as 10 fF) from the scattering parameters. The scattering parameters were measured on a coplanar waveguide with a Jurkat cell trapped by dielectrophoresis either in a series or shunt configuration. The sensitivity analysis validated our previous empirical observation that the insertion loss of a series-trapped cell and the return loss of a shunt-trapped cell were most sensitive to the cell impedance. Additionally, the membrane resistance and cytoplasm capacitance were most sensitive to low- and high-frequency scattering parameters, respectively. In the future, the analysis can be used to optimize the test setup and protocol for fast, compact and label-free characterization of a cell at the subcelluar level.
The innovative feature of RF electromagnetic fields shown here, consists in offering protection to human cells against the damaging action of a chemical agent.
The protective effect of RF electromagnetic fields occurs in presence of modulated signals and depends on signal bandwidth, as well as on SAR level.
RF-induced protective effect has been here proved, on an in vitro model, to be a tunable phenomenon depend-ing on the electromagnetic conditions adopted, which provides a step towards applications targeting cells lo-cated at different depths in the body.
These findings suggest the existence of a complex interaction between modulated RF fields and biological systems. Such interaction becomes detectable when samples are appropriately sensitized, such as when RF exposure is combined with other chemical or physical treatments.
Effects of Radiofrequency Exposure and Co-Exposure on Human Lymphocytes: the Influence of Signal Modulation and Bandwidth
Stefania Romeo, Anna Sannino, Olga Zeni, Leopoldo Angrisani, Rita Massa, Maria Rosaria Scarfi.
The occurrence of modulation-specific effects after co-exposures to Radiofrequency (RF) and other agents has been discussed in the literature.
In this paper, the influence of modulation and bandwidth in eliciting the DNA damage of RF alone and in combination with mitomycin-C (MMC), is analyzed in human lymphocytes. Blood cultures from healthy donors were exposed to 1950 MHz, and Continuous Wave (CW), Wideband Direct-Sequence Code Division Multiple Access (WCDMA, 4.5 MHz bandwidth), and Additive White Gaussian Noise (AWGN, 9 MHz bandwidth) signals were considered. For each signal, SAR values of 0.15, 0.3, 0.6, 1.25 W/kg were tested. RF exposure alone never induced DNA damage in the micronucleus assay. When RF exposure was followed by MMC treatment, the effect depended on modulation and bandwidth. CW exposure never altered the MMC-induced DNA damage, while such damage was reduced when either signals WCDMA at 0.3 W/kg SAR or AWGN at 0.15 and 0.3 W/kg were applied. These results indicate the influence of modulation for the occurrence of the protective effect, with a relation between the bandwidth and the power absorbed by samples. If confirmed invivo , clinical applications using modulated RF signals could be devised, to protect cells from side effects of therapeutic treatments.
This paper provides a detailed survey related to the possible health hazards linked with EMF exposure based on what empirical studies suggest and the different metrics that are currently used for evaluating, limiting and mitigating the effects of this type of exposure on the general public.
Brain tumour is still the main cause of concerns, which may be related to the extensive use of wireless devices, even though the effects of EMF exposure is now being investigated in other parts of the body (e.g. eyes, reproductive system). Moreover, some studies advocate a modification of the guidelines to better take into account the duration of exposure (i.e. long-term exposure).
Generic/composite metrics (based on existing metrics) have recently been designed to better evaluate the exposure of large geographical area. A generic metric for measuring the individual exposure would also be of interest.
Key 5G enabling technologies, such as densification, massive MIMO, and mmWave, will surely have an impact on the ambient level of EMF exposure in the near future, but they will also provide new opportunities to reduce it, e.g. context-aware beamforming or low exposure spatial modulation schemes.
A Survey on Electromagnetic Risk Assessment and Evaluation Mechanism for Future Wireless Communication Systems
Muhammad Ali Jamshed, Fabien Heliot, Tim Brown.
The accurate measurement of electromagnetic exposure and its application is expected to become more and more important in future wireless communication systems,
given the explosion in both the number of wireless devices and equipments radiating electromagnetic-fields(EMF)and the growing concerns in the general public linked to it. Indeed, the next generation of wireless systems aims at providing a higher data rate,better quality of service(QoS), and lower latency to users by increasing the number of access points,i.e.densification, which in turn will increase EMF exposure. Similarly, the multiplication of future connected devices,e.g. internet of things(IoT)devices, will also contribute to an increase in EMF exposure. This paper provides a detailed survey relating to the potential health hazards linked with EMF exposure and the different metrics that are currently used for evaluating,limiting and mitigating the effects of this type of exposure on the general public. This paper also reviews the possible impacts of new wireless technologies on EMF exposure and proposes some novel research directions for updating the EMF exposure evaluation framework and addressing these impacts in future wireless communication systems. For instance, the impact of mmWave or massive-MIMO/beamforming on EMF exposure has yet to be fully understood and included in the exposure evaluation framework.
Microwave techniques may provide a convenient method of noninvasive hydration assessment due to the strong relationship between dielectric properties and water content in biological tissues.
The feasibility of microwave hydration assessment is demonstrated through a physiologically-driven framework for modeling changes in microwave properties at the extremities during dehydration.
Our target application is hydration assessment in athletes and older adults due to the elevated risk of dehydration from their activity, environment, and/or physiological changes.
It is found that permittivity is a more sensitive metric than conductivity when assessing water-loss dehydration.
The modeling techniques developed in this paper provide the framework for interpreting in vivo experimental measurements in the context of hydration assessment.
Original Paper, Part 1
Feasibility Study of Hydration Monitoring using Microwaves Part 1: A Model of Microwave Property Changes with Dehydration
David Christopher Garrett, Elise Fear.
Dehydration is a prevalent condition which can have profound health consequences. If detected early, it can often be treated by oral fluid replacement (drinking or eating).
However, existing assessment techniques lack the accuracy and/or convenience for ongoing monitoring, motivating the development of novel methods. We propose using low-power microwave measurements (2-12 GHz) at the extremities to monitor human hydration, relying on the strong relationship between dielectric properties of tissues and water content. Electromagnetic simulations of realistic models are used to explore changes in microwave signals transmitted through the forearm to changes in hydration. Tissue properties are adjusted according to expected changes in water content, and average dielectric properties are estimated from signals transmitted through the arm by ultra-wideband antennas placed in contact with the tissues. A causal relationship between weight loss due to water loss and dielectric permittivity is found in human simulation models. Little relationship is found with conductivity. The theoretical groundwork for hydration assessment with microwaves is developed through a model which relates changes in total body water content with changes in microwave properties at the extremities. This model could be useful for monitoring hydration in at-risk populations such as older adults and athletes.
Microwaves are inherently sensitive to water content, and may therefore provide a clinically-relevant method of noninvasive hydration monitoring.
This study presents a validation study of microwave techniques for hydration assessment in athletes by performing measurements prior to and following acute water loss due to exercise.
We find changes in estimated permittivity with water loss, but no relationship with measured attenuation.
Athletes may find applications of this technique in guiding fluid replacement for maximal performance and safety during exercise and recovery.
Feasibility Study of Hydration Monitoring using Microwaves Part 2: Measurements of Athletes
David Christopher Garrett, Jared R. Fletcher, David B. Hogan, Tak Shing Fung, Elise Fear.
Hydration is a key consideration for athletes, where even mild levels of dehydration may negatively impact performance.
Existing methods of assessment are either inaccurate or impractical for use in field settings, motivating the search for new approaches. Extremity microwave measurement is a promising method for ongoing hydration monitoring, owing to the inherent differences in dielectric properties with varying tissue water content. This paper reports on a feasibility study of \textit{in vivo} hydration assessment using microwave measurements in athletes undergoing acute water loss during exercise. We developed and then tested a system for performing reliable microwave property estimation at the forearm. This system was used to measure hydration status in varsity wrestlers before and after a training session. A relationship between estimated permittivity and body weight change due to water loss was found, showing promise for the use of microwaves to assess hydration status. No significant relationship with attenuation was found. A novel method of assessing changes in hydration status is described, which may be of practical use for athletes in guiding fluid replacement during and after exercise.
Sophisticated electromagnetic modeling and experiments were performed to assess the effect of the electrode structure on the radio frequency (RF) -induced heating.
For this case study, we found that increasing the electrode size tends to reduce the magnitude of the lead transfer function, subsequently reducing the RF-induced heating near the electrode.
To present a lead electrode which can reduce the RF-induced heating for active implantable medical devices (AIMDs) under magnetic resonance imaging (MRI).
We could demonstrate that the electrode structure of the lead tip has an impact on the MRI RF-induced heating of the AIMDs.
Both simulations and experiments demonstrate that increasing the electrode size reduces the RF-induced heating for an implanted lead during MRI scanning.
APS Special Issue
Impact of Electrode Structure on RF-induced Heating for an AIMD Implanted Lead in a 1.5-Tesla MRI System
Rui Yang, Jianfeng Zheng, Yu Wang, Ran Guo, Wolfgang Kainz, Ji Chen.
In this paper, the impact of the electrode structure on radio frequency (RF) induced heating near an active implantable medical device (AIMD) implanted
lead under magnetic resonance imaging (MRI) exposure is investigated. The specific absorption rate (SAR) distributions and the temperature rises of the leads with different electrodes were assessed. It is shown that increasing the size of lead electrode reduces the SAR distribution and temperature rise near the electrode. Our results indicate that a larger electrode size will reduce the magnitudes of the lead transfer function, and subsequently reduces the heating effect near the electrode. Both numerical simulations and experimental measurements were performed to verify the effectiveness of a large electrode in mitigating the RF-induced heating.
This research aims to solve the issue with localization with near field of electromagnetism and in the lossy human tissues.
The results show that, the position of in-body medical instrument can be determined within 1 cm and predict the orientation with error of maximum 10 degree.
The proposed method is a novel way to determine location and orientation of an in-body medical-instrument.
High accuracy localization in the challenging near field with relatively low frequency.
This method can be performed from DC to up to frequencies that goes through the human tissues with negligible losses.
Determining the Position and Orientation of In-body Medical Instruments Using Near-Field Magnetic Field Mapping
Vedat Cavlu, Paul Brennan.
Abstract:There is a increasing demand for localizing medical implants in-body, such as wireless capsule endoscope (WCE) and Nasogastric tube (NGT).
Some studies have been conducted to solve this issue using either permanent magnets, static current sources or RF fields. The permanent magnet fails due to low power, and static current source requires relatively high power source. The RF field source requires high frequencies to get enough precision, which undergoes high attenuation in the body. At low frequency, when the distance between the source and the receiver array is shorter than the wavelength, the far field assumption fails for localization methods. Therefore, we propose a novel method of mapping the magnetic field vector in the near field region, with which wavelength independent localization is done. We did extensive MATLAB and CST Microwave simulations followed by practical experiments. The proposed method has achieved localization accuracy of less than 1 cm in Y-Z plane, 2 cm in depth (in X-axis) and the maximum orientation error remained 10° in 3-D.
Neuromodulation approach such as optogenetics helps regain the functionality of the paralyzed limbs due to stroke and neural diseases.
An optogenetic implant requires a fixed amount of power to turn on the μLED and stimulate the neurons via inductive coupling based wireless power transfer method.
Coaxial and lateral displacements and angular misalignment cause degradation of inductive coupling and thus reduce the delivered power to the implants. Our approach of deriving the misalignment tolerance range based on the modeled, simulated and measured path loss and the link efficiency through the tissue media is a unique approach to estimate the performance reliability of the implant.
The proposed approach and methodology of designing a wireless power transfer system for optogenetic application with the aim to maximize the link efficiency given the constraints of the sizing and Specific Absorption Rate (SAR) would be highly valuable for the biomedical implant research community.
IMBioC Special Issue
Effects of Coaxial-lateral and Coaxial-angular Displacements on Link Efficiency of a Wirelessly Powered Optogenetic Implant: Design, Modeling and Experimental Validation
Dipon Kumar Biswas, Nishat Tarannum Tasneem, Ifana Mahbub.
Abstract:In recent years, wireless power transfer (WPT) system has evolved tremendously as a means to deliver power to miniaturized implantable sensors.
Efficiently delivering power to implants is a challenge due to the loose coupling between the transmitter (TX) and receiver (RX) coils because of the various displacements (coaxial, lateral and angular). The coupling coefficient deteriorates significantly due to the displacements thus decreasing the overall power transfer efficiency (PTE) of the system. In this paper, we present an analysis and modeling of the effects of various displacements on the efficiency and the overall performance of a miniaturized WPT system designed for an optogenetic implant. To emulate the tissue media inside a human head, skin, skull and gray matter layers are theoretically modeled using dielectric properties and simulation models are developed using the Ansys High Frequency Structure Simulator (HFSS) software. The propagation loss and link efficiency are modeled and simulated as a function of various displacement combinations. To validate the theoretical and simulation models, the WPT system is characterized in various displacement conditions using chicken breast as the media. The measurement results also show a good agreement with the simulation results, thus providing an estimation for the misalignment tolerance range for given specifications. The efficiency performance analysis of the proposed WPT system for various worst-case scenarios also provides a preliminary model for designing a closed-loop wireless power delivery regulation scheme in the future.
This work presents the design, construction, and testing of an RF injection network for MR-conditional medical device testing of devices for use within 1.5 T MRI scanners to reduce the risks to patients with an active implantable medical device (AIMD) in this electromagnetic environment.
A directional lumped element coupler, power splitter, an attenuator/isolator, low pass filter and high pass filter were designed and implemented as part of the network and at the end a neuromodulation system was tested using the developed RF injection network for conductive emission testing.
Application target is active implantable medical devices exposed to 63.4 MHz RF field in 1.5 T MRI systems.
MRI scanner is well-known to pose a series of risks to patients with an active implantable medical device (AIMD). The anticipated risks to both the patient and the implanted device are described in ISO/TS 10974:2018(E).
In this work, RF injection network was developed for conductive emission testing of AIMDs to reduce the risk of loss of device functionality such as, but not limited to, a failure to deliver the intended therapy, re-programming, device reset, permanent damage, and tissue stimulation due to RF rectification.
RF Injection Network Development for Testing of Active Implantable Medical Devices Exposed to RF Fields in 1.5 T MRI Systems
Ali Attaran, William B. Handler, Blaine A. Chronik.
Abstract:This work presents the design, construction, and testing of an RF injection network for MR-conditional medical device testing of devices for use within 1.5 T MRI scanners (i.e. frequency of 63.4 MHz).
The system was developed to meet the requirements of ISO/TS 10974:2018(E). A directional lumped element coupler, power splitter, an attenuator/isolator, low pass filter and high pass filter were designed and implemented as part of the network. The RF injection network was developed in both a compact version implemented in a single PCB and discrete PCB version for use in different situations. The performance of each designed component was simulated and compared to measurement results. As an application example, a neuromodulation system was tested using the developed RF injection network for conductive emission testing.
For the first time the compressive sensing theory has been applied and clinically tested in hyperthermia treatment planning.
The proposed approach allows to improve treatment quality for head & neck tumors with the HYPERCollar3D (despite its general formulation) exploiting both sparsity promotion concepts and FOCO (a convexprogramming- based SAR optimizer).
The approach proposed in this work deals with the optimal planning of an hyperthermia treatment.
The proposed SP-FOCO allows to optimally and case-specifically select a subset of antennas from an oversized applicator. Oversized applicators represent a way to exploit additional degrees of freedom and consequently dealing with arbitrary located tumor.
Results suggested the development of similar approaches for applicator design.
APS Special Issue Paper
Selecting the Optimal Subset of Antennas in Hyperthermia Treatment Planning
Gennaro G. Bellizzi, Maarten Paulides, Tomas Drizdal, Gerard Van Rhoon, Lorenzo Crocco, Tommaso Isernia.
Abstract:Hyperthermia treatment planning is a deeply patient-specific task which includes the optimal determination of the excitations of an array applicator.
To enhance flexibility, various solutions exploiting different frequencies, antenna element, number and applicator geometries have been proposed in the literature. Among them, increasing the frequency and the number of radiating elements has shown effective for achieving more conformal heating. However, as each radiating element requires a power amplifier to control it, increasing the number of antennas considerably impacts the overall cost and complexity of the system. Accordingly, a procedure capable of selecting an optimal patient-specific subset of antennas from an oversized phased array applicator (with more antenna elements than available amplifiers) could help improving cost-effectiveness. In this study, we present an original approach which allows improving performance by adaptively selecting the optimal subset of antennas to be activated for a given (redundant) applicator and a given patient. The proposed approach takes inspiration from the compressive sensing theory by embedding the sparsity promotion paradigm into a treatment planning procedure which casts power-deposition as a constrained convex optimization. Performance were demonstrated for the case of head and neck hyperthermia, and benchmarked against the antenna selection procedure presently used in the clinical practice.
This paper describes a simple test bed made from pork skin, fat or lard, and solid or ground pork for testing subdermal antennas. Fat and pork loin provide realistically varying tissue electrical properties. Lard and ground pork provide consistent tissue properties.
It mimics human tissues sufficiently well to enable effective design of subdermal antennas.
This allows testing of subdermal antennas for next-generation implantable medical devices (IMDs), and we demonstrate initial tests on a 3D focusing antenna design.
These IMDs are likely to be much smaller than IMDs today, requiring a new type of wireless telemetry system, of which subdermal antennas are a likely component.
This paper considers both the average and standard deviation of the electrical properties of the tissues.
A Layered Pork Model for Subdermal Antenna Tests at 433 MHz
Zachary Deneris, D. Eldon Pe'a, Cynthia M. Furse.
Abstract:Clinical or Biological Impact: Antennas for next generation implantable medical devices are likely to be distributed throughout the body,
particularly in the near-surface subdermal regions, rather than tethered to the surface of the implantable device. A simple test bed for evaluating subdermal antennas during the design phase is needed. Objectives: The objectives of this work are to create a simple biological test bed for subdermal implantable antennas that includes the normal expected variation of the tissues Technology or Method: A layered model using pork products with skin, pork fat or lard, and pork loin or ground pork is used. Results: The ex vivo porcine tissues are similar to in vivo human tissues. Pork fat and loin have more variability than lard and ground pork but are more difficult to imbed subdermal antennas in. This model provides an easy to use platform for testing subdermal antennas in the lab. This test bed was developed for 433 MHz, the ISM band closest to the MedRadio band (402-405 MHz). Initial tests are demonstrated on a two-wire passive system for focusing power within the muscle region.
We report the first-ever study that explores unwanted electromagnetic energy coupling to the mouth retractor used during tonsillectomy.
In vitro measurement results demonstrate that unintentional RF energy coupling is indeed a real issue, leading, in turn, to unwanted temperature increase in the surrounding tissues.
Our ultimate goal is the prevention of related post-operative tonsillectomy complications, including dysgeusia that currently affects one-third of patients.
This is the first time that RF energy leakage is confirmed during tonsillectomy, and identified as a possible cause of post-operative dysgeusia.
Both monopolar electrosurgery and coblation tonsillectomy procedures are explored and contrasted at typically used power levels.
Unintentional RF Energy Transfer During Tonsillectomy: An In Vitro Investigation
Satheesh Bojja Venkatakrishnan, Vigyanshu Mishra, Maria Koenigs, Tendy Chiang, Asimina Kiourti.
Abstract:One-third of tonsillectomy patients experience post-operative taste disturbances (dysgeusia), yet the underlying cause is unknown.
We hypothesize that unwanted radio-frequency (RF) energy couples to the metal-based mouth retractor during tonsillectomy and could be a possible cause of dysgeusia. To validate our hypothesis, in vitro studies are performed in a ground beef phantom with sensors measuring: a) the unwanted current coupled to the mouth retractor, and b) the unwanted temperature rise in the tissues that surround the retractor. The simulated surgery was performed using two separate surgical techniques: monopolar electrosurgery and coblation. Results indicate that unintentional RF energy transfer is indeed a real issue. During electrosurgery, peak-to-peak unwanted currents vary from 80.53 to 181.48 mA for typical power levels ranging from 10W to 30W. Tissue temperature unintentionally increases by 1.3°C and 1.8°C, respectively. Coblation indicates smaller coupling effects, with peak-to-peak currents on the mouth retractor capped at 12.33 mA for a typical 7 W setting. Concurrently, tissue temperature is reduced by 0.73°C, as attributed to the saline solution inherent to coblation. As the first of its kind, this study illuminates possible causes of post-tonsillectomy dysgeusia and intends to trigger future studies. The ultimate goal is safe and complication-free tonsillectomies
Radio-Frequency backscattering is employed to enable batteryless and wireless brain implants that are: a) matched to high-impedance clinical electrodes, and b) tolerant to DC voltage.
As compared to previous wireless and batteryless brain implants, the proposed approach offers a remarkable improvement in sensitivity by 25 times.
Unobtrusive monitoring of deep brain signals may significantly improve the individual's physical and mental well-being (e.g., for patients with epilepsy, Alzheimer's, Parkinson's, and more).
Batteryless brain implants matched to high-impedance electrodes can readily be employed to clinical applications.
Improvements on the interrogator side help suppress the phase noise and improve the demodulated signal integrity.
Passive Impedance Matching For Implanted Brain-Electrode Interfaces
Wei-Chuan Chen, Katrina Guido, Asimina Kiourti.
Abstract:We propose a new technique for matching the high impedance of sub-cranial electrodes to wireless brain implants that is:
a) passive, b) highly tolerant to the DC offset voltage caused by the electrochemical reaction in the recording electrode, and c) complemented by an improved external interrogator design that exhibits reduced phase noise. As compared to previous wireless and batteryless brain implants, the proposed approach offers a remarkable improvement in sensitivity by 25 times. The proposed system consists of an external interrogator and a neuro-recorder implanted under the scalp. For operation, the interrogator sends a 2.4 GHz carrier signal to "turn on" the implant. This carrier self-biases a PNP Bipolar Junction Transistor (BJT) that enables matching to the recording electrode at frequency fneuro in a batteryless manner. Concurrently, the recorded neuropotentials (at frequency fneuro ) pass through a Schottky diode that allows them to mix with the carrier and generate a 4.8GHz±fneuro modulated signal. The latter is then transmitted back to the interrogator for demodulation. To verify the implant's operation, in-vitro measurements are presented. Measurement results demonstrate that emulated neuropotentials as low as 200μVpp can be detected at a 33kΩ electrode impedance. As such, the proposed system presents a game-changing capability for a wide range of applications.
Self-powered and ultra-compact NanoNeuroRFID system for Brain Computer Interfaces.
Ultra-miniaturized (<200 μm diameter) magnetoelectric antennas for brain implantable devices.
Wireless Implantable devices based on magnetoelectric antennas.
Sub-mm size brain implantable devices using ultra-compact magnetoelectric antennas.
NanoNeuroRFID: A Wireless Implantable Device Based on Magnetoelectric Antennas
Mohsen Zaeimbashi ; Hwaider Lin ; Cunzheng Dong ; Xianfeng Liang ; Mehdi Nasrollahpour ; Huaihao Chen, Neville Sun, Alexei Matyushov, Yifan He Xinjun Wang, Cheng Tu, Yuyi Wei, Yi Zhang, Sydeny Cash, Marvin Onabajo, Aatmesh Shrivastava, Nian-xiang Sun Sun.
Abstract:A major obstacle during the design of brain-computer interfaces is the unavailability of a neural implantable device that is μ-scale in size, wireless, self-powered, and long-lasting.
Current state-of-the-art implantable devices suffer from various limitations. Electromagnetic-based wireless devices are big in size because of their large antenna, which must be larger than onetenth of the wavelength of the operational frequency. Ultrasound-based wireless devices, in addition to their low data rate, have a massive loss in the skull and need an intermediate electromagnetic transceiver under the skull. Furthermore, almost all state-ofthe- art wireless devices use micro-electrodes for neuronal recording, which are not reliable in long-term monitoring applications because of direct contact between the tissue and metal electrodes. In this paper, we propose a novel wireless and ultra-compact implantable device termed NanoNeuroRFID. At the core of this device there is a Magnetoelectric (ME) antenna array. ME antennas are smart and ultra-miniaturized (<200μm diameter), and can perform multiple tasks: 1) They can harvest electromagnetic energy to power the NanoNeuroRFID system. Their limit of detection for RF magnetic fields is 40pT. 2) They can sense quasi-static neuronal magnetic fields as small as 200pT without a direct contact to the tissue, allowing a long lifetime and reliable neural recording. 3) They can communicate with an external transceiver, and their operational frequency could be 10s to 100s of MHz where tissue loss is small.
We present a non-invasive helmet-restraint integrated with an RF coil for awake, unanesthetized non-human primate magnetic resonance imaging (MRI).
The non-invasive helmet coil was designed and constructed using three dimensional (3D) modeling and additive manufacturing based upon computed tomography (CT) images.
Electromagnetic simulations of the helmet coil loaded with a computer-based Rhesus macaque model provided accurate predictions of coil performance.
A Rhesus macaque successfully completed behavioral training based on positive reinforcement and was able to sit in the MRcompatible primate chair and wear the helmet for MR imaging studies.
The non-invasive helmet coil eliminates the need for permanently implanting non-human primates with a head post (which has significant complications), allows using of more subjects in experiments, provides good immobilization for magnetic resonance imaging, and provides improved signal to noise ratio (SNR) compared to a non-integrated loop coil.
Subject-specific, Non-invasive Helmet-restraint RF Coil for Awake, Non-human Primate MR Imaging
Bahareh Behzadnezhad, Jacob Andreae, Samuel A. Hurley, Caitlynn Filla, Ellie Mueller, Bruce D. Collick, Nader Behdad, Luis Populin, Alan B. McMillan.
Abstract:The purpose of this study was to develop a non-invasive restraint helmet integrated with an RF coil for awake-behaving non-human primate MR imaging.
To prevent image-corrupting motion, the head needs to be immobilized which is currently achieved via invasive surgical headpost implants. In this work, an RF coil holder was integrated into the design of a subject-specfic helmet to place the coil in closest distance to the head for improved signal-to-noise ratio (SNR). Additive manufacturing was used to print the helmet based upon CT images of the subject. A single channel transmit/receive loop coil was designed using electromagnetic simulations loaded with a voxel-based monkey head model. The Rhesus macaque used in this study underwent behavioral training based on positive reinforcement before engaging in MR imaging. Imaging was performed using the helmet coil which was successful in immobilizing the macaque head in an awake, unanesthetized subject. Results showed improved SNR by approximately 28% compared to a loop coil used with an implanted head post, and minimal motion artifact in structural imaging. The non-invasive helmet coil eliminates the need for permanently implanting monkeys with a headpost, provides the necessary head immobilization, and allows the use of more subjects for neuroimaging studies.
A novel multistatic head imaging system utilising a software-defined radio, solid-state switching network, and static antenna array is proposed.
The system is highly compact, lightweight, and inexpensive, attributes that could make it simple to transport to medical emergencies and could enable better accessibility for disadvantaged communities.
Head imaging applications are the focus, with the system being verified using a simplified head phantom and targets emulating cancerous tumours and bleeds.
The imaging accuracy of the proposed system is comparable to a Vector Network Analyser system using the same imaging algorithm, while greatly reducing size and cost.
The system could produce images of the head phantom within less than a minute, making it feasible for use in time-critical applications.
Portable Microwave Head Imaging System Using Software-Defined Radio and Switching Network
Anthony Edgar Stancombe, Konstanty Bialkowski, Amin Abbosh.
Abstract:Head imaging plays an imperative role in the detection and localisation of conditions affecting the brain, such as strokes, cancerous tumours, and haemorrhages caused by trauma.
Current systems require the patient to be transported to the imaging device which increases the time taken to apply treatments, allowing the patient's condition to worsen. This paper proposes a portable microwave head imaging system that can easily be taken to the patient. The system utilises a novel combination of a software-defined radio and solid-state switching network to collect the imaging data, and operates in the frequency band of 0.85 – 2 GHz. It can capture input signals over a range of approximately 106 dB, which is suitable for detecting realistic brain injuries. The concept has been validated through experimental data collection and confocal image generation, and was capable of producing images in less than a minute. A simplified head phantom was developed for gathering the verification data and proved that the system was capable of locating targets with dielectric properties similar to brain tumours and bleeds. The presented design is highly accessible, achieved through being small, light, and inexpensive: key factors that could help save lives in time-critical medical emergencies.
This paper presents analytic design equations for the optimization of whole-body and head MRI RF coils.
The purity of the fundamental mode guarantees transverse-plane illumination uniformity.
The presented technique aims at improving the image quality in MRI.
The presented analysis shows that the transverse-plane illumination uniformity is frequency independent, and the obtained results are applicable to ultra-high-field operation.
The presented idealized design equations are crucial for initiating exact iterative FDTD-based or FEM-based optimization.
Design Considerations for Radiofrequency Whole-Body and Head Coils
Abbas Omar.
Abstract:It is shown that the transverse-plane B+1 homogeneity in TEM whole-body and head RF coils, which is responsible for acquiring uniformly illuminated MR images,
is frequency independent and solely depends on achieving high excitation purity of the fundamental m=1 mode. This necessitates that the capacitive coupling introduced by the interconnecting lumped capacitors should maximize the separation between the resonance frequency of the fundamental mode and that of all other higher order modes. Alternatively, full-orthogonal excitation of the coil, if possible, would mitigate this design challenge, especially if high-Q operation cannot be guaranteed. In this contribution a field theoretical technique for the optimization of the capacitive coupling is introduced and applied to the birdcage configuration as representative of TEM coils. A fairly general analysis of these coils is also introduced and/or revisited, which underlies the proposed technique and offers simple analytic design equations. Simulation results are presented to validate the considered approach. The presented analysis deals with the homogeneously filled coils and therefore does not consider the B+1 inhomogeneities caused by the loading. The latter will be the subject of future publications.
Electromagnetic fields are utilized for enhanced drugs delivery into cells with controlled and on-chip electroporation.
High efficient molecular delivery into mammalian cells is demonstrated with a microdevice able to provide both controlled cell electroporation and microwave sensing for cellular characterization.
The evaluation of drugs, genes or proteins into cells for therapeutic applications is targeted.
On-chip electroporation enables high molecular delivery into cells, while maintaining a high viability rate and therefore overcoming some of the drawbacks of conventional electroporation systems.
Evaluation of a Microwave Biosensor for On-chip Electroporation and Efficient Molecular Delivery into Mammalian Cells
Amar Tamra; David Dubuc, Marie-Pierre Rols, Katia Grenier.
Abstract:In this study, a microwave biosensor based on interdigitated electrode layout is evaluated as an on-chip electroporation system applicable to electroporation of adherent cell monolayers.
The present device is designed in a manner that allows electrical microwave spectroscopy measurements in subsequent studies. We demonstrate the applicability of our device for electroporation-mediated molecule transfer of adherent cells in standard laboratory conditions. Application of electric pulses (6 V, 100μs, 1 Hz) to the cells induces successful delivery of a fluorescent probe (∼98%), while maintaining a high viability rate (∼100%). The comparable delivery rate to existing systems along with the improved viability suggest that our proposed electroporation system has great potential in the research fields, considering that our system is conceived to perform electrical microwave spectroscopy measurements after electroporation.
Our microwave microfluidics device paired with on-chip calibrations enable admittance measurements of fluids over a six-decade frequency range (40 kHz to 67 GHz).
We use our microwave microfluidics devices to characterize weak ion-pairing interactions in nanoliter volumes of common buffer solutions in situ and non-destructively.
Buffer solutions are ubiquitous in biological systems and quantifying their electrical and ionic properties enables future studies of ion dynamics in biomolecular systems.
These broadband measurements can inform more narrowband measurements of biological, biochemical, and pharmaceutical fluid systems, which may be more cost-effective and lead to real-time assessment of biological systems.
Measurement of Ion-Pairing Interactions in Buffer Solutions with Microwave Microfluidics
Charles Little, Angela Stelson, Nathan Orloff, Christian Long, James Booth.
Abstract:Microwave microfluidic spectroscopy is an emerging technique for quantifying the frequency-dependent electrical response of fluids.
This technique can access important physical properties including ion mobility and hydration, which are directly applicable to biochemistry. One critical step towards quantifying these effects is to develop accurate models for the behavior of buffer solutions containing mobile ions. Here, we show that ions in buffer solutions produce a weak ion-pairing response. We used microfluidic channels integrated with coplanar waveguides in combination with a hybrid microwave calibration protocol to extract the broadband microwave admittance spectra of a standard TAE-Mg2+ buffer solution between 100 kHz and 67 GHz. To characterize the ion-pairing response, we fit the calibrated admittance data with two models: a conventional model without ion-pairing and with a water relaxation described by a 'Cole-Cole' function, to our alternative model that includes ion-pairing and a single Debye-type water relaxation. Including ion-pairing improved the goodness of fit across the entire frequency range. In the higher concentration buffer solution, we saw a reduction in the max systematic error in the fit residuals from 10% to less than 4%. The measurement and fitting techniques are widely applicable, providing critical information about the behavior of solvated ions.
Compensated mixed coupling coil (MCC) design is proposed and reduce the performance degradation caused by either lateral or angular misalignment
Power transfer efficiency drops slightly even when lateral or angular misalignment occurs (almost flat angular misalignment for high resonant mode).
For subcutaneously implantable devices, such as deep brain stimulator (DBS), a stable WPT power supply can be available to suppress the misalignment issues caused by motions.
Not only frequency split issues is conquered, but also misalignment problems can be mitigated for by using dual frequency MCC for frequency shift keying modulation in wireless power and data transfer systems.
Detailed MCC model design and optimization are proposed to predict and analyze the performance of WPDT systems.
Design of Dual Frequency Mixed Coupling Coils of Wireless Power and Data Transfer to Enhance Lateral and Angular Misalignment Tolerance
Tao-Cheng Yu, Wei-Hsiang Huang, Chin-Lung Yang.
Abstract:A dual-resonant mixed coupling structure is proposed to enable lateral and angular misalignment immunity in wireless power and data transfer systems.
By considering the duality property of electrical coupling and magnetic coupling, mixed coupling coils make the efficiency of wireless power transfer nearly insensitive to axial and angular misalignment. A complete, sophisticated equivalent circuit is proposed and demonstrates the features of the dual-resonant characteristic and lateral and angular misalignment tolerance. Comparing to traditional printed spiral coil systems, the proposed mixed coupling structure reduces decline rate of S21 by 26.4 % and 78.1 % at 35 mm axial misalignment and 90° angular misalignment at low resonant frequency, respectively. Moreover, at high resonant frequency, S21 only drops 16.5% with 60 mm lateral misalignment and is almost flat for 90° angular misalignment. Last but not least, data communication is also validated by transmitting a 200 kHz frequency shift keying signal, and the probability errors remain at acceptable level when misalignment occurs.
The proposed dual-band wireless capsule endoscopic (WCE) antennas utilize impulse radio (IR) signals to realize WCE medical image transmission with low attenuation in the MHz band.
Through a rough estimation from the simulated and measured results, proposed antennas can realize a total data rate of 2.5 Mbps within 100 mm antenna distance if two antennas are aligned, or within 50 mm antenna distance if two antennas are misaligned to 60 mm.
Potential medical applications include the diagnosis of the gastrointestinal tract (GI), in particular the ability to diagnose the entire small intestine area.
Main contribution of this paper is the fabrication and measurement of a dual-band transmitting antenna that can be placed in the WCE capsule, and a modified receiving antenna that can be mounted on the abdomen of the human body for MHz band WCE medical image transmission.
In addition, the increase of data rate will improve the received image quality, and MHz band communication will reduce signal attenuation and extend WCE battery life.
Dual-Band Antenna Design for Wireless Capsule Endoscopic Image Transmission in the MHz Band Based on Impulse Radio Technology
Yunxiao Peng, Kazuyuki Saito, Koichi Ito.
Abstract:Low frequency communication such as MHz band communication is a feasible method to reduce signal attenuation fromhuman body,
although slow data rates limit its range of applications. In this paper, we designed two coil antennas for wireless capsule endoscopic (WCE) medical image transmission, including a small-sized dual-band transmitting antenna that can be placed inside the WCE, and a corresponding receiving antenna that can be mounted on the abdomen of the human body. The proposed antennas can be fed by a battery-powered impulse radio (IR) transceiver that contains five signal peaks in the MHz band, and two signal peaks (38.5 and 57.6 MHz) are selected for antenna design to realize dual-band communication with higher data rates. Dimensions of the transmitting antenna and the receiving antenna are π ×(5.5)2 × 2mm3 and π ×(40)2 × 4.8mm3, respectively. The antenna distance that satisfies the WCE medical image transmission requirements is roughly estimated by the received powerfrom the transceiver. With the antenna distance of 50 mm, the attenuation is 33 and 47 dB at 38.5 and 57.6 MHz, respectively. With the antenna distance of 100 mm, the attenuation is 39 and 59 dB at 38.5 and 57.6 MHz, respectively. It can be estimated thatproposed antennas can realize a total data rate of 2.5 Mbps within 100 mm antenna distance if two antennas are aligned, or within 50 mm antenna distance if two antennas are misaligned to 60 mm.
This paper underlines the strong application potential of using high frequency electric fields and intracellular dielectric spectroscopy to identify, discriminate and isolate the highest aggressiveness and resistance cells from a tumor.
This paper shows that significant cell UHF-DEP signature change can be measured between undifferentiated and differentiated cell subpopulations.
In the frame of high recurrence cancer as Glioblastoma, emergence of new therapies able to target and neutralize highly tumorigenic cells is required with new approaches and technologies allowing fine characterization of such cells.
Sensitive to cell dielectric specificities, UHF-DEP could provide an efficient and label-free solution for cellular analysis and aggressiveness potential diagnosis: an innovative and complementary marker to conventional biological phenotypic and functional characterizations.
UHF-Dielectrophoresis Crossover Frequency as a New Marker for Discrimination of Glioblastoma Undifferentiated Cells
Remi Manczak, Sofiane Saada, Thomas Provent, Claire Dalmay, Barbara Bessette, Gaelle Begaud, Serge Battu, Pierre Blondy, Marie-Odile Jauberteau, Canan Baristiran Kaynak, Mehmet Kaynak, Cristiano Palego, Fabrice Lalloue, Arnaud Pothier.
Abstract:This article introduces the first results of dielectric spectroscopy characterization of glioblastoma cells; measuring their crossover frequencies in the Ultra High Frequency range (above 50 MHz) by dielectrophoresis technics.
Experiments were performed on two glioblastoma lines: U87-MG and LN18 that were cultured following different conditions, in order to achieve different phenotypic profiles. We demonstrate here that the presented dielectrophoresis electrokinetic method can be used to discriminate the undifferentiated from the differentiated cells. In this study, microfluidic lab-on-chip systems implemented on Bipolar-Complementary Oxide Semiconductor (BiCMOS) technology are used allowing single cell handling and analysis. Based on characterizations of their own intracellular features, both selected glioblastoma cell lines cultured in distinct culture conditions have shown clear differences of DEP crossover frequency signatures compared to differentiated cells cultured in normal medium. These results support the concept and validate the technique efficiency for cell characterization in glioblastoma pathology.
We present a novel, adaptive method, based on robust L1-LDA features, for indoor human motion recognition from micro-Doppler measurements.
The proposed method exhibits remarkable resistance against training data corruptions (e.g., due to mislabelings), as well as the ability to adapt/improve as more labeled measurements from the subject of interest become available.
The proposed method enables privacy-aware remote monitoring of patients and elderly for a variety of healthcare applications, including rehabilitation and aging-at-home.
To the best of our knowledge, this work presents the first method in the literature for adaptive L1-LDA.
Adaptive Radar-Based Human Activity Recognition with L1-Norm Linear Discriminant Analysis
Panos Markopoulos, Sivan Zlotnikov, Fauzia Ahmad.
Abstract:We present a novel radar-based indoor human gross motor activity classifier, which employs L1-norm Linear Discriminant Analysis (L1-LDA) to
identify low-rank subspaces whereon micro-Doppler signatures from distinct motions are most differentiable. Both non-adaptive and adaptive implementations of the proposed classifier are presented, with the latter providing refinement and adaptation to the specific activity patterns of the human subject of interest. In contrast to standard LDA, L1-LDA exhibits resistance against outliers that may lie among the training data, e.g., due to mislabeling. We use real-data from four motion classes to experimentally compare the performance of the proposed methods with standard (L2-norm-based) LDA. The results corroborate that the proposed methods markedly outperform LDA when the training datasets are corrupted with mislabeling, while they provide similar performance under nominal training data.
The first prototype RF head coil using Metamaterial Zeroth Order Resonator structure on a thin substrate for 10.5T MRI is discussed.
This work is targeted for the development of an efficient RF coil for human head imaging at Ultra-High Magnetic fields.
The coil utilizes a periodic structure that is physical length independent, has high unloaded to loaded Q-factor ratio, and is efficient for the RF coil applications
The proposed coil is proven to be safe for clinical use and has a good coil efficiency.
IMBIOC 2017 Special Issue
Metamaterial Zeroth Order Resonator RF Coil for Human Head: Preliminary design for 10.5T MRI
Vijayaraghavan Panda, Lance Delabarre, Gregor Adriany, Thomas Vaughan, Anand Gopinath.
Abstract:The objective is to develop a highly efficient RF head coil on a thin substrate for the Ultra-high magnetic field MRI systems.
The Metamaterial Zeroth Order Resonator is investigated for this purpose. Simulation and experimental results are provided for an 8-channel M-ZOR based RF coil in comparison with a standard high performance 8-channel dipole based RF coil for the 10.5T MRI system. Each element is 18 cm long, identical, evenly spaced along the circumference of the cylindrical phantom, loaded with dielectric material, and referred to as inverted Metamaterial Zeroth Order Resonator. The resonator elements are open circuited, matched, and tuned to 447.06 MHz with the phantom. An unloaded to loaded Q-factor ratio of 2.97 is obtained from the scattering matrix of the proposed design. The length independent nature of the proposed design and the flexibility of the lumped elements have provided an optimized element with a substrate thickness of roughly 3 mm. With the proposed design, there is a similar RF magnetic field strength (B1+) to SAR ratio with a reduced 10g averaged SAR of 2.892 for the same input power compared to that of a dipole coil. This could make the coil acceptable for the clinical high-quality imaging.
Design of a Microwave Global Endometrial Ablation Device
Hojjatollah Fallahi, Punit Prakash.
Abstract:Endometrial ablation is a minimally invasive treatment employed for thermal coagulation of the endometrial lining of the uterus for treatment of menstrual heavy bleeding.
Here, we present an applicator employing a microwave loop antenna for global endometrial ablation enabling conformal ablation of uterine cavities of varying size in the range 4 × 6.5 cm in length and 2.5 × 4.5 cm in width, with a single positioning of the device. For treating large cavities, conformal microwave radiation is achieved by coupling the loop antenna with a passive element. A 3D-coupled FEM electromagnetic and heat transfer simulator was employed to optimize the antenna geometry with the goal of maximizing return loss at the 915 MHz operating frequency, and achieving adequate ablation depths for a range of uterine cavity sizes. Proof-of-concept devices were fabricated and experimentally evaluated in ex vivo tissue. The simulated and measured return loss of the optimized design was 20 dB at 915 MHz. Experiments in ex vivo tissue demonstrated the ability of the presented device to achieve mean ablation depths of 7.3 mm. We demonstrated a technique for creating planar ablation patterns suitable for global endometrial ablation of different uterine cavity sizes by employing a loop antenna with a passive element. Our design provides a planar ablation pattern by using a loop antenna with a passive element that will allow for ablation applications not realizable with conventional coaxial monopole, dipole, and slot antennas.
The wireless neurosensing system (WiNS) offers a revolutionary implantable wireless fully-passive device, with an RF sensitivity of -135 dBm, for recording of cortical neural activation.
The ability of this system to sense signals generated by the brain, with accuracy comparable to a wired system, is confirmed through a series of biopotential recordings.
The target application of the neurosensing system is neuropathology research, mainly epilepsy studies along with possible brain-computer interfacing (BCI) and brain-machine interfacing for prosthesis control.
Here, for the first time, this technology is used to record a series of electrophysiological signals in an actual animal; this not only validates our system but also opens the door for a spectrum of applications.
An instrumental component to achieving the in vivo validation of WiNS was the development of low impedance probes to closely match the input impedance of the implant.
Fully-Passive Wireless Implant for Neuropotential Acquisition: An In Vivo Validation
Carolina Moncion, Lakshmini Balachandar, Satheesh Bojja Venkatakrishnan; Jorge J. Riera, John Volakis.
Abstract:Implantable systems are often employed to perform continuous high-resolution recordings of neural activity.
These systems frequently require invasive procedures when implanting and maintaining effective operation. This causes major interruptions to daily life. Previous work demonstrated an in vitro minimum detectable signal (MDS) of 15 μV in amplitude and RF sensitivity down to – 135 dBm. This suggests the possibility of detecting diminutive biopotentials in a wireless fully-passive manner. Here, for the first time, we validate the wireless neurosensing system (WiNS) through a series of in vivo electrophysiological recordings including both spontaneous cardiac activity and sensory evoked neural activity, with amplitudes ranging from a few microvolts to millivolts and across a spectrum of frequencies. We also present design considerations and the development of probes for neurosensing to accomplish detectability of biopotentials in the tens of microvolts in rats. The developed probes show improved impedance matching with the neurosensing system. Specifically, the new probes showed an impedance several orders of magnitude lower than those commercially available, thereby significantly improving signal detection. Notably, the presented in vivo validation of this technology has great future clinical implications in neuroscience as it offers a wireless and unobtrusive device for neurological research, monitoring, and therapeutic purposes.
Using a wideband test setup of well-controlled impedance, both the lower and higher crossover frequencies for dielectrophoresis of a single biological cell was experimentally measured.
The measured lower crossover frequency was found in general agreement with that calculated by using the Clausius-Mossotti function in conjunction with experimentally extracted cell characteristics such as membrane resistance and capacitance, as well as cytoplasm resistance and capacitance.
New formulas were derived to evaluate the Clausius-Mossotti function from cell resistances and capacitances, instead of cell permittivity with assumed cell size and shape. The sensitivity of the crossover frequencies on the cell characteristics was analyzed, too.
The measured higher crossover frequency was found lower than that calculated. The difference can be attributed to the field being highly nonuniform in single-cell dielectrophoresis, especially at higher frequencies. Additionally, with closely spaced electrodes in single-cell dielectrophoresis, adhesive force may have to be considered even for a relatively nonadherent Jurkat cell.
The result suggests that the classical Clausius-Mossotti function, originally derived from the Maxwell-Wagner mixture model of a cell suspension, may not apply to single-cell dielectrophoresis in a straightforward manner, especially at high frequencies.
Validation of Clausius-Mossotti Function in Wideband Single-Cell Dielectrophoresis
Xiaotian Du, Xiao Ma, Hang Li, Lei Li, Xuanhong Cheng, James Hwang.
Abstract:For the first time, both lower and upper crossover frequencies of the real part of the Clausius-Mossotti function were calculated by using cell parameters
measured on the same ultra-wideband setup as that for single-cell dielectrophoresis. The calculation suggests that the lower crossover frequency can be from 0 to 127 kHz and the upper crossover frequency can be from 45 to 108 MHz, in the unlikely case when uncertainties in the cell parameters all add up. The calculated lower crossover frequency was found to be in general agreement with the measured values of 28 ± 4 kHz. However, the calculated upper crossover frequency was significantly different from the measured values of 326 ± 35 MHz The difference can be attributed to the field being highly nonuniform in single-cell dielectrophoresis, especially at higher frequencies. Additionally, with closely spaced electrodes in singlecell dielectrophoresis, adhesive force may have to be considered even for a relatively nonadherent Jurkat cell. In any case, the difference between the calculated and measured crossover frequency suggests that the classical Clausius-Mossotti function, originally derived from the Maxwell-Wagner mixture model of a cell suspension, may not apply to single-cell dielectrophoresis in a straightforward manner, especially at high frequencies.
The diagnosis of lung diseases such as, for example, pneumothorax, requires a continuous tracking of their air/liquid content. This latter influences the conductivity value of the chest and it can be inferred in real-time by solving the EIT inverse problem through LBE methodologies.
The conclusion in this manuscript is that, thanks to the numerical and comparative assessment, it is possible to state the LBE technique at hand yields instantaneous and robust conductivity predictions starting from a low size training set.
The targeted biological and/or medical application is the continuous real-time tracking of the lungs ventilation/status for patients under mechanical ventilation in intensive care units.
The significance/breakthrough of this work is the numerical assessment of the reliability and the effectiveness of the LBE technique at hand as applied to solve the EIT inverse problem in real-time to faithfully inferring the lungs status.
The numerical assessment presented in this paper has proven that the LBE method at hand overcomes representative state-of-the-art techniques thanks to the joint exploitation of the noise-filtering capabilities of the PLS and of the adaptive generation/refinement of the training database.
Real-Time Electrical Impedance Tomography of the Human Chest by means of a Learning-by-Examples Method
Marco Salucci, Giacomo Oliveri, Andrea Massa.
Abstract:The real-time inversion of electrical impedance tomography (EIT) data for human chest monitoring is dealt with.
More specifically, the paper is concerned with a numerical comparative assessment of an innovative Learning-by-Examples (LBE) method, which has been provisionally introduced and preliminarily validated by the authors, aimed at solving the inverse problem (IP) arising when estimating the status of the lungs through EIT.
The complete investigation of VO2 thin films interaction with extremely high frequency waves, as well as nanoparticles interaction with optical waves was conducted.
The optical properties of VO2 nanosphere and size effect on these characteristics in insulator and metallic phases were investigated.
VO2 in the form of nanoparticles and thin films reveals controllable properties. Thus, VO2 is a very proper material to use in sensors at different electromagnetic wave frequencies.
The VO2 nanoparticles can be suggested as a heat transfer agent for cancer cells ablation or hyperthermia treatment.
The intrinsic radiation of VO2 film in the 28-32 GHz band in the vicinity of metal-insulator phase transition was observed.
AIM Special Issue
Interaction of Optical and EHF Waves with VO2 Nanosized Films and Particles
Alexander P. Kamantsev, Victor V. Koledov, Vladimir G. Shavrov, Dmitriy S. Kalenov, Mikhail P. Parkhomenko, Svetlana V. von Gratowski, Nooshin V. Shahmirzadi, Tavakol Pakizeh, Artemy V. Irzhak, Vladimir M. Serdyuk, Iuliia P. Novoselova, Anton A. Komlev, Andrey E. Komlev, Dmitriy A. Kuzmin, Igor V. Bychkov.
Abstract:In this paper VO2 film on quartz substrate was prepared and investigated in extremely high frequency (EHF) band 27–37 GHz.
The study of EHF response of the nanosized VO2 films reveals strong anomalies in the temperature range of metal-insulator transition (MIT). The intrinsic radiation of VO2 film in the 28-32 GHz band in the vicinity of MIT was observed. Optical Raman spectra of VO2 film perforated by micron size holes arrays were studied. The micron holes and arrays show strong change of the Raman spectra at wavelength 532 nm due to the heating by laser beam. Optical properties of homogeneous VO2 nanospheres (NSs) were studied theoretically as well. The size effect on the optical properties of VO2 NSs was investigated. Transition into the metallic phase caused by heating of VO2 -NSs leads to formation of localized surface plasmon resonance which red-shifts slightly while its size increases. Increasing of NS's diameter in insulator state leads to the appearance of a peak in the visible wavelength. The optical spectra of VO2 -NS are much broader than that of Ag-NS. This is associated with the fact that localized electric field in form of dipolar mode is more intensive for Ag than in case of VO2.
This paper proposed a new biolabeling system using ferromagnetic resonance properties of multiple magnetic nanowire types for in-vitro cancer type diagnosis.
The exterior biofunctionalization that bonds ligands to specific cell types enables multiplexed cell labeling; while the intrinsic FMR properties of MNWs enables the spontaneous identification of multiple labels.
The feasibility of the proposed biolabeling system is validated by applying the same MNW characterization and identification approach on MNW array measurements.
The work confirms that distinct FMR signals can be detected in a mixed system of individual nanowire types using through transmission response. To account for low FMR signals and close FMR B-field spacing, a fitting algorithm can be used to validate the presence of the specific material types.
This proposed biolabeling system has the capability to expand cancer cell detection throughput and reduce processing time; it can also be combined with other labeling methods to enhance the testing range of the current methods.
IMBioC Special Issue Paper
Development of a Biolabeling System Using Ferromagnetic Nanowires
Wen Zhou, Joseph Um, Yali Zhang, Alexander Nelson, Zohreh Nemati, Jaime Modiano, Bethanie Stadler, Rhonda Franklin.
Abstract:This paper presents the development of a biolabeling system using different ferromagnetic nanowire types.
The proposed biolabeling system concept includes a cell separation procedure description using magnetic nanowires and how nanowires can be used and identified as biolabels. A study on single magnetic nanowire behavior is performed, and the algorithm used to characterize and identify each nanowire type separately and from a mix of multiple types is described. The algorithm is verified using measurement data from individual nanowire array samples and stacked combinations of cobalt, iron and nickel. From the strong correlation between the measured transmission coefficient data of magnetic nanowire arrays and a mathematical model, the potential of interpreting multiple magnetic nanowire types inside cells is confirmed.
A versatile magnetic exposure system able to reach intensities in the order of mT has been theoretically designed, with the aim of using it for different biomedical applications of low intensity magnetic fields, from a few Hz to 20 kHz.
The biological applications for which this exposure system has been designed are very different: a cuvette for drug delivery applications; a chamber for ex vivo experiments on brain slices, and a rat phantom for in vivo animal studies.
The system is designed to reach a magnetic field of 1.4 mT with a homogeneity of 95% in the volume between the coils where the target will be placed.
The novelty of the proposed exposure system mainly relies on its versatility, which permits in vitro, ex vivo and in vivo laboratory experiments in a wide frequency range and with negligible thermal increase induce.
AIM Special Issue Paper
A Versatile Magnetic Exposure System for In-Vitro, Ex-Vivo and In-Vivo Experiments Finalized to Therapeutic Applications in the IF Range
Elena Della Valle, Micaela Liberti, Francesca Camera, Alessandra Paffi, Stefania Petralito, Vincenzo Roncace, Costanza Burattini, Giorgio Aicardi, Francesca Apollonio.
Abstract:The use of magnetic fields in therapeutic applications has considerably increased in recent years.
In particular, many researchers have focused their attention on the use of low intensity magnetic fields, either alone or in combination with nanoparticles for drug delivery systems in nanomedicine. Laboratory experiments aimed at defining in vitro and in vivo outcomes are required, and reliable low intensity magnetic field exposure systems are needed. In the present study, we have performed the analytical and numerical design of a novel magnetic exposure system suitable for different biological applications, such as magnetoliposome drug delivery, ex vivo experiments on brain slices and in vivo studies. This system is be able to generate intensities of the order of mT in a frequency range from ELF to 20 kHz.
A compact, multi-purpose broadband architecture for integrated complex permittivity sensors, utilizing a patch element and a multiharmonic downconversion for fast and energy-efficient readout.
The architecture can be embedded in systems performing GHz frequency range permittivity footprint measurement for material characterization as well as permittivity imaging.
Applications range from traditional clinical and point-of-care scenarios to the increasingly emerging area of wearable devices. Examples include in-vivo tissue hydration monitoring, label-free malignant tissue inspection as an assisting tool in removal surgery, evaluation of drug penetration through skin and blood
glucose concentration measurement.
The proposed sensor readout core has the smallest known area and is the first to demonstrate permittivity imaging capabilities at microwave frequencies.
Miniaturized Broadband Microwave Permittivity Sensing for Biomedical Applications
Gerasimos Vlachogiannakis, Zhebin Hu, Harshitha Thippur Shivamurthy, Andrea Neto, Michiel A.P. Pertijs, Leo de Vreede, Marco Spirito.
Abstract:We present a compact, scalable and broadband architecture for the implementation of complex microwave permittivity sensors in CMOS technology.
The proposed architecture consists of a patch sensor embedded in a programmable balanced readout bridge, and performs third and fifth harmonic downconversion for fast multifrequency readout. Circuits designed can act as the basic building block for a wide span of biomedical applications, ranging from wearables to permittivity imaging. Experimental results of manufactured prototypes demonstrate measurement noise reduction through bridge balancing, debye model parameter estimation of independent material with a 1.6% error using full frequency dataset and 5.3% in high energy efficiency mode, as well as image construction based on material permittivity differences.
We derive and test the two-dimensional discrete dipole approximation (2D DDA) method for use in microwave imaging.
The two-dimensional electric field forward solution of the microwave imaging system is numerically simulated for a simplified breast tumour model, and it has been compared to finite element solution using COMSOL Multiphysics.
Sufficient sampling size for the imaging domain of our microwave breast imaging has been proposed for which the solution accuracy with respect to the sampling, inclusion, size, and property contrast has been demonstrated.
The simulation results and the measurements show good agreement and we conclude that we can utilize the 2D discrete dipole approximation as an alternative, fast and reliable forward solver for microwave tomography.
Application of Two-Dimensional Discrete Dipole Approximation in Simulating Electric Field of a Microwave Breast Imaging System
Samar Hosseinzadegan, Andreas Fhager, Mikael Persson, Paul Meaney.
Abstract:The two-dimensional electric field distribution of the microwave imaging system is numerically simulated for a simplified breast tumour model.
The proposed two-dimensional discrete dipole approximation (DDA) has the potential to improve computational speed compared to other numerical methods while retaining comparable accuracy. We have modeled the field distributions in COMSOL Multiphysics as baseline results to benchmark the DDA simulations. We have also investigated the adequate sampling size and the effect of inclusion size and property contrast on solution accuracy. In this way, we can utilize the 2D DDA as an alternative, fast and reliable forward solver for microwave tomography. From a mathematical perspective, the derivation of the 2D DDA and its application to microwave imaging is new and not previously implemented. The simulation results and the measurements show that the 2D DDA is a well-grounded forward solver for the specified microwave breast imaging system.
The magnetic fields exposure device was newly designed and produced for generating magnetic fields in a CO2-incubator culturing of human cells.
The results suggested that 60 Hz, 50 mT magnetic fields enhance the efficacy of anticancer drug to human cancer cells.
Three types of anticancer drugs used in this study are affected differently by the magnetic fields. The magnetic fields increase the effects for all the drugs, by which the numbers of viable cells are decreased by 40% than those with only the drugs alones.
If our finding can be applied clinically, magnetic fields exposure to cancer site might allow for an effective target chemotherapy, reducing dosage and suppressing side effects.
Combined Effect of 60 Hz Magnetic Fields and Anticancer Drugs on Human Hepatoma HepG2 Cells
Makiko Kakikawa, Tetsuya Maeda, Sotoshi Yamada.
Abstract:Our previous data using bacterial cells Escherichia coli, have showed that 60 Hz, 50 mT magnetic fields enhance the potency of antibiotics.
By using magnetic fields in clinical cancer chemotherapy, it is presumed to be able to enhance an anticancer drug potency on the target region, reduce dosage, and suppress side effects. However, since the prokaryotic bacterial cells, E.coli, differ from human cells which are eukaryotes and multicellular organisms in many ways, it was not clear whether the magnetic fields affect a potency of anticancer drug against human cancer cells. In this study, we designed and produced magnetic fields exposure device for human cancer cells in culture, and investigated whether 60 Hz, 50 mT magnetic fields affect the potency of anticancer drugs against human hepatoma HepG2 cells. The results of experiments with an anticancer drug, cisplatin indicated that the quantity of viable HepG2 cells become decreased significantly by the combination of cisplatin and magnetic fields as compared to that by cisplatin alone. This suggested that 60 Hz, 50 mT magnetic fields increase the cytocidal activity of cisplatin to human hepatoma cells. The efficiency of the anticancer drugs, mitomycin C and doxorubicin against HepG2 cells was also increased significantly by exposure to magnetic fields, although the time associated with the greatest enhancement of the drugs potency achieved by magnetic fields differed among drugs. These results suggest that 60 Hz, 50 mT magnetic fields strengthen the effect of anticancer drugs on human cancer HepG2 cells.
We present an easy-to-use, portable, autonomous RF electronic readout circuit as a first proof-of-concept for a passive biosensor network platform that integrates passive acoustic wave biosensors and their remote interrogation.
The acoustic device can detect low-weight biochemical targets in low-volume samples and represents an important step toward a biosensor network platform for cancer diagnosis and monitoring of environmental health.
The association of a Love wave device with functionalized porous matrices could improve the performance of the sensor in biological media for highly sensitive detection of low weight molecular targets while facilitating the development of an easily regenerable system.
The main innovation is related to the modelling of the device and simulation using the Finite Element Method (FEM), which is a good way to take into account the physical properties of porous 3D-layers which would also make it possible to design very sensitive layers adapted to the detection targets, such as cancer bio-markers and toxins.
This novel approach has potential applications low molecular weight biochemical detection for early cancer diagnosis and environmental monitoring, among others.
Mesoporous titania-coated biosensor and FEM model design for highly sensitive detection of low molecular weight targets
Ollivier Tamarin, Hamida Hallil Abbas, Wassim Ouelhazi, Maxence Rube,Jean Luc Lachaud, Vincent Raimbault, Cedric Boissiere, Marie Paule Bonnet,Dominique Rebiere & Corinne Dejous.
Abstract:This paper presents the interest of a highly sensitive biosensor coated with a TiO2 mesoporous film as sensitive layer.
The main novelty is related to the modelling of the device and simulation by using Finite Element Method, as a good way to take into account the physical properties of porous 3D-layers. The strategy of using such Love wave devices, with 3D porous layers, offering further easy functionalization, aims not only to increase the amount of targets caught on the sensor surface, but also to enhance the detection mechanism by a higher perturbation of the Love wave acoustic energy which could be trapped inside the 3D sensitive layer. First, as a proof of concept, experimental devices with a 3D titania mesoporous layer were realized, and they have shown a good agreement with simulated results. Furthermore, experimental test with several Newtonian liquids are investigated, in a range of viscosities from 1 to 7 cP, typical of those concerned by our biochemical applications. The sensitivity with a 300 nm thick porous sensing layer was 10 times that of the bare device, with interesting dynamical issues to be further studied, giving rise to the great potentialities for biological detection of low weight biochemical targets.
The developed EM model for human motions creates repeatable and realistic reference data which when coupled with the signal processing technique can be useful for obtaining a full understanding of human limb joint motion analysis.
The proposed method is successful in extracting different limb joint trajectories from a complex human motion but with some limitations such as difficulty in tracking hands for a walking subject and reliance on reference data to identify the desired motion details.
The targeted applications of this work are treating patients with joint problems, athlete performance analysis, motion classification, and so on.
The significance of this work is the development of a limb joint tracking technique suitable for use with low cost and simple Doppler radar in a typical non-controlled environment.
A by-product of this work is the use of a portable and flexible software-defined transceiver system as the Doppler radar utilized in the experiments.
Non-Contact Human Gait Analysis and Limb Joint Tracking Using Doppler Radar
Farhan Quaiyum, Nghia Tran, Jean E. Piou, Ozlem Kilic, Aly Fathy.
Abstract:A full understanding of normal motion from human limb joint trajectory tracking could be essential to develop and establish a scientific basis for correcting any abnormalities.
Practicality of using continuous wave (CW) radars, being simple and low cost, has been successfully investigated for non-contact gait monitoring, but it is still challenging to identify the motions of different limb joints. Here, we investigate the feasibility of extending the 1-D block processing algorithm to distinctly track specific limb joints and discuss the advantages and limitations of the technique for CW radar. To establish a repeatable reference data, we run a full wave EM analysis on customized Boulic human model emulating specific video recorded body motions. Results based on measured data are in agreement with simulated ones for simple motions like swinging one hand or one leg only. The proposed technique is also successful in extracting lower body parts while the whole body is in motion, however it is still hard to clearly extract upper body parts like swinging hands.
Tissue constituents, air and water can be used as inputs to dielectric mixture models to predict dielectric properties of liver and lung at different hydration and inflation states, respectively.
Maxwell mixture theory is more successful than Maxwell-Fricke mixture theory for tissue dielectric property modelling in both low and high water content tissues.
Mixture models can be coupled with Debye and Cole-Cole equations to construct wideband tissue dielectric property models that can be used for multiple tissue types that have various water contents as well as different hydration states of the same tissue.
These models will potentially increase the accuracy of microwave ablation simulations of liver and lung by accounting for changes in tissue constituents due to temperature elevation and water vaporization.
Development of Water Content Dependent Tissue Dielectric Property Models
Sevde Etoz, Christopher L. Brace.
Abstract:We propose dielectric tissue property models dependent on both water and air content covering the microwave frequency range.
Water is the largest constituent of biological tissues and its effect on the dielectric properties of biological tissue has been studied. However, dehydration effects due to thermal heating have not been fully characterized. We combined 1) Maxwell-Fricke mixture theory with a four-pole Cole-Cole equation to include water and air content dependency and as the second approach a different 2) Maxwell mixture model was coupled with a Debye function. The proposed approaches (1 and 2) were able to predict the permittivity ( ϵ′ ) and conductivity ( σ ) of bovine liver and swine lung tissues at different hydration and inflation states from 1-15 GHz. A second approach coupling Maxwell and Debye models required fewer assumptions and modelled tissue properties with higher accuracy (less than 15% mean percent error in all tissue types). These models may help improve the accuracy of microwave ablation simulation when tissue water content changes as a result of vaporization, and may facilitate personalized treatment planning.
High magnetic field magnetic resonance imaging requires a new look at radio frequency structures for excitation and reception of the imaging signals from nucleons. This paper provides an overview of some of the current work being done for high frequency, high magnetic field magnetic resonance imaging.
New transmit and receiver structures are utilizing electromagnetic phenomenon rather than simple inductive effects as the frequency increases and wavelength (λ) decreases, allowing the use of metamaterials and wireless technologies.
These new radio frequency structures will provide enhanced signal to noise ratio, leading to increased contrast and higher resolution images
Improved, high resolution images will lead to better medical diagnoses.
RF Aspects of High and Ultra High Field Magnetic Resonance Imaging [(U)HF-MRI]: Recent Advances
Robert Caverly.
Abstract:High field magnetic resonance imaging (HF-MRI) brings superior imaging with high contrast and signal to noise ratio (SNR).
For scanning systems higher than 3.0 T and Larmor frequencies above 128 MHz, wavelength and higher tissue conductivity can cause distortion of the electromagnetic fields, leading to degraded image quality. This paper will show current trends in HF-MRI to mitigate some of these issues and provide a linkage between traditional MRI components and novel structures for UHF-MRI. Transmit and receive structures as well as the use of metamaterials for improved excitation and SNR will be covered. Other examples of these recent trends will be potential antenna structure to replace the current c.oil-based technology, and then finally, receiver/patient protection circuits will be covered.
This paper presents a new method to increase the robustness and accuracy of Doppler radar-based vital signs monitoring sensors.
It is shown that using the chest wall acceleration signal yields a better result compared to the chest wall displacement. The heartbeat rate detection accuracy is improved by more than 10% on average.
A novel mathematical representation for the heartbeat mechanical signal is provided. The model is quite useful in the analysis and understanding of the human heartbeat vibration on the chest wall.
The new model also confirms our observation that the chest wall acceleration provides a higher detection accuracy than its displacement.
Accurate Doppler Radar-Based Cardiopulmonary Sensing Using Chest-Wall Acceleration
Mehrdad Nosrati, Negar Tavassolian.
Abstract:This paper presents the theory and experimental results of a new method to significantly increase the detection accuracy of the human heartbeat rate using a continuous-wave Doppler radar.
Traditionally, it is assumed that the chest wall displacement signal which is recorded by the radar and used for heartbeat rate estimation is a pure sine wave. In this paper, we use the fact that that the displacement signal is a complex Gaussian function rather than a pure sine wave. This function shows a declining amplitude versus frequency; therefore, the heartbeat signal will be much weaker than the respiration signal and can be easily buried in the respiration's harmonics. However, by exploiting the chest wall acceleration instead of its displacement, the heartbeat signal is greatly amplified, leading to a significantly higher heartbeat rate detection accuracy. Recorded data from 12 healthy human subjects show an average heartbeat rate detection accuracy of more than 95% when compared with reference electrocardiogram (ECG) recordings. The proposed technique is robust, simple, and requires minimum calculation resources which is important for online monitoring and power consumption reduction. Measurement results indicate its potential for being used in reliable non-contact heartbeat rate monitoring systems.
Virtual dielectric spectroscopy of biomolecular samples is possible using computation methods.
Molecular models of water affect complex permittivity predicted by molecular dynamics simulations.
Our method enables rationalization of microwave biosensor design.
Molecular dynamics simulation predicts and interprets complex permittivity of biosamples.
Water models in molecular dynamics simulation prediction of dielectric properties of biomaterials
Michal Cifra, Jiri Prusa, Daniel Havelka, Ondrej Krivosudsky.
Abstract:To develop and reliably use diagnostic and therapeutic methods employing microwaves, we need to have an accurate knowledge of biological dielectric properties.
Traditionally, dielectric properties of biosamples are determined experimentally. However, such measurements require dedicated hardware and physical availability of sufficient volume of biological samples of interest. Instead, here we demonstrate the prediction of complex permittivity of a simple biomolecular sample using computational molecular dynamics simulations. We focus here on the role of a molecular model of water since it is the major compound determining microwave dielectric properties of biological tissues and wet samples. Here, for the first time, we analyze how the common molecular water models (SPCE, TIP3P, and TIP4P) affect complex permittivity of biomolecular solutions predicted by molecular dynamics simulations. We found that the type of the molecular water model used in the simulation affects not only water contribution but also biomolecule contribution to the permittivity spectra. Our results contribute to in silico prediction and understanding of dielectric properties of biomaterials
TWRI systems can remotely monitor thieves or robbers inside a building or subjects under the rubble.
Delay and sum (DAS) and range migration (RM) algorithms are among the most used techniques for image reconstruction and in this paper their pros and cons are investigated.
The two inversion algorithms have been compared based on analytical, numerical and experimental data acquired for realistic scenarios.
DAS and RM have similar resolution and dynamics with the former having a better field of view and the latter being faster.
Comparison between Delay and Sum and Range Migration Algorithms for Image Reconstruction in Through-the-Wall Radar Imaging Systems
Stefano Pisa, Emanuele Piuzzi, Erika Pittella, Paolo D'Atanasio, Alessandro Zambotti, Giulia Sacco.
Abstract:Through-the-wall radar imaging (TWRI) systems allow police, fire personnel and defense forces to detect, identify and track subjects inside buildings or under rubble.
In this paper, the delay and sum (DAS) and range migration (RM) algorithms are compared as imaging techniques for a multiple input multiple output (MIMO) stepped-frequency (SF) radar system. These algorithms have been applied to analytical, simulated, and measured data both in the absence and in the presence of a wall between the antenna and the target. Both techniques were able to accurately reconstruct the position of targets behind a wall. The DAS presents a wider angle of non-ambiguity while the RM is faster. An improvement of the DAS, in terms of accuracy in the target positioning, is achieved applying the Fermat's principle.
The electromagnetic behaviour of a radiator has been coupled with two other physical phenomena, and with a space-dependent blood perfusion model of a tumor, to better understand the role of each process and how they relate together to accomplish an hyperthermia tumour therapy.
The proposed multi-physic analysis based on the blood perfusion model is an efficient solution to have a deep insight on the hyperthermia treatment.
This work can be a powerful supporting tool in oncology applications where the hyperthermic treatment is often used as adjuvant therapy alongside with radiotherapy and chemotherapy.
To the best of the authors' knowledge, this is the first time that a complete analysis on the coupling of physical phenomena for a hyperthermia application of a tumour based on a realistic blood perfusion model has been done.
A Blood Perfusion Model Of A RMS Tumor In A Local Hyperthermia Multi-Physic Scenario: A Preliminary Study
Giacomo Muntoni, Alessandro Fanti, Giorgio Montisci, Marta Muntoni.
Abstract:Hyperthermia concerns about the heating treatment of cancerous tissues, mostly using a radiator, i.e. an antenna, as a heat source.
The main goal of this therapy, primarily used as adjuvant therapy along with radio and chemotherapy, is to reach a suitable temperature inside the neoplastic mass – about 7-8 degrees above the normal body temperature – able to kill the cancerous cells without compromising the healthy tissues. Many factors contribute to this goal, such as the radiation characteristics of the antenna, the thermal profiles of the healthy and cancerous tissues and the dynamic of the water flux inside the bolus. Moreover, despite being often overlooked, an important role is played by the perfusion characteristics of the tumor. In this work, we present a multi-physic analysis in a local hyperthermia scenario, considering a simple patch antenna resonating at 434 MHz as a heat source, a water bolus, a bi-layered body phantom and, most importantly, a tumor placed inside the phantom described employing a realistic space-dependent blood perfusion model. The results of this study show the effectiveness of the hyperthermia treatment using the physiopathology-driven perfusion model, and they may be useful in a real local hyperthermia case.
What are the innovative features of utilizing electromagnetics for biomedical applications in this manuscript, in one sentence?
Infrared transmission and Radio-Frequency backscattering techniques are combined to enable batteryless and wireless brain implants with multi-channel capabilities.
What is the conclusion in this manuscript, in one sentence?
We demonstrate the first-ever batteryless and wireless multi-channel brain implant with sensitivity as high as 20 μVpp at all channels
What are the targeted biological and/or medical applications, in one sentence?
Unobtrusive monitoring of deep brain signals may significantly improve the individual's physical and mental well-being for patients with epilepsy, Alzheimer's, Parkinson's, and more.
What is the significance/breakthrough of this work?
Deep brain signals from multiple locations within the brain can now be concurrently recorded via batteryless and wireless implants.
Accomplishments in this manuscript you would like to highlight that are not mentioned above, for our readers, in one sentence?
As compared to previous batteryless multi-channel brain implants, the proposed design exhibits 28 times higher sensitivity, 2 times smaller footprint, and improved scalability.
2018 APS Special Issue
A Multi-Channel Passive Brain Implant for Wireless Neuropotential Monitoring
Wei-Chuan Chen, Cedric W. L. Lee, Asimina Kiourti, John L. Volakis.
Abstract:We propose a novel multi-channel (8 channels) passive neuro-sensing system for wireless acquisition of brain signals as low as 20 μVpp.
Compared to previous batteryless multi-channel neuropotential sensors, the proposed design exhibits: a) 28 times better sensitivity, b) ∼ 2 times smaller footprint, and c) scalability to 100s or even 1000s of channels. The proposed system consists of an external interrogator and a neuro-recorder implanted inside the scalp. For operation, the interrogator sends a) a 2.4 GHz carrier signal to "turn on" the implant, and b) an infrared control signal for channel selection. The latter activates the desired channel via a photo-activated multiplexer. For this channel, the carrier signal is mixed with the neural signal ( fneuro ) to generate a 4.8GHz±fneuro modulated signal. The latter is then transmitted back to the interrogator. To verify the implant's operation inside biological tissues, in−vitro measurements are presented using pig skin. Experimental results show that the proposed neuropotential recorder exhibits 20 μVpp sensitivity at all eight channels (viz. it can record any signal generated by the human brain). The system is also in compliance with the strictest Federal Communications Commission standards for patient safety. Notably, the proposed approach is scalable to a much higher number of channels. As such, the proposed.
A dielectric coat, if properly designed, may reduce the electromagnetic interaction between an elongated metallic prosthesis and the Magnetic Resonance Imaging (MRI) radiofrequency (RF) antenna moderating the RF-artefact rise at 64 MHz and 128 MHz.
We described, by means of an equivalent circuit, the interaction between the MRI RF-antenna and an elongated metallic prosthesis explaining the optimal relation between the thickness and electric permittivity of a coat whose aim is to reduce the rise of RF-artefacts in an MRI exam.
The targeted biological and medical applications are the elongated metallic prostheses worn by patients subjected to an MRI exam.
An electromagnetic cloaking application tailored on a specific MRI field is introduced. The obtained results represent a key point for the design and realization of a coat material at 128 MHz whose effects are to strongly reduce the interaction between a metallic elongated prosthesis and the RF MRI antenna. Furthermore, results obtained at 64 MHz suggest that it is possible to cover a generic hip prosthesis through a proper ordinary biocompatible material to achieve the desired effects.
2018 AIM Special Issue
A Near Field Cloaking Study to Reduce MRI RF-Artefacts in Presence of Elongated Prostheses
Umberto Zanovello, Luca Zilberti, Ladislau Matekovits.
Abstract:Objective: To analyze a near-field electromagnetic cloaking to reduce the radiofrequency (RF) magnetic field inhomogeneities
(responsible for the RF-artefacts onset) in Magnetic Resonance Imaging (MRI) in presence of elongated metallic hardware. Technology or Method: A lumped circuit is considered to explain the role that a dielectric coat has on hiding a metallic cylinder to the RF antenna. The theoretical assumptions are proved by means of full-wave simulations that are also applied to a realistic hip prosthesis considering a frequency equal to 64 MHz and 128 MHz. Results: The numerical results confirm the theoretical assumptions. Both the theoretical analysis and the numerical simulations highlight the different role that the coat thickness and electric permittivity have in the definition of a proper dielectric coat. Clinical or Biological Impact: A particular cloaking approach leads to a dielectric coat whose constitutive electrical parameters may be simple enough to fit the considered application reducing the interaction between an elongated prosthesis and the RF antenna. Furthermore, results obtained at 64 MHz suggest the possibility to employ an existing biocompatible material to achieve the envisaged purposes.
Deep Transcranial Magnetic Stimulation (dTMS), administered through H4 coil, has been recently proposed for the addiction treatment and it's aimed to stimulate bilaterally the prefrontal cortex and to activate the reward pathway.
Computational electromagnetic models help in gaining knowledge on the mechanism laying behind neurostimulation, by providing a detailed electric field distribution induced in cerebral tissues.
Simulations demonstrates that H4 induces the highest electric fields at cortical level, targeting preferentially prefrontal cortex and the anterior cingulate cortex and then supporting its use for addiction treatment.
This work represents, in contrast with prior works based on homogenous tissue phantoms, a powerful and informative tool for both planning, optimization and outcomes evaluation of clinical protocols based on dTMS systems for addiction treatment.
Deep TMS coil H4 can be specifically used to target cortical and subcortical structures involved in food craving related disorders.
Deep transcranial magnetic stimulation for the addiction treatment: Electric field distribution modeling
Serena Fiocchi, Emma Chiaramello, Livio Luzi, Anna Ferrulli, Marta Bonato, Yiftach Roth, Abraham Zangen, Paolo Ravazzani, Marta Parazzini.
Abstract:Deep Transcranial Magnetic Stimulation (dTMS) is a neurostimulation technique for deep brain structures that has recently been successfully applied in the clinic for treatment of addiction.
In contrast to conventional magnetic stimulation, which uses planar coils (figure-of-8) to target specific superficial regions of the brain, dTMS requires the design of complex three-dimensional coils in order to induce deeply penetrating fields. Recent clinical studies have focused on the use of H4 coils, which utilizes a left-right symmetric structure for bilateral stimulation of the prefrontal cortex, and demonstrated efficacy for therapy such as smoking cessation. The mechanism of activity, however, remains poorly understood, in part because the affected regions of the brain are not known in detail. To this purpose, computational techniques applied to highly detailed inhomogeneous tissue phantoms, provide a powerful tool for testing coil efficacy. In this work we quantified both electric field E distribution and its penetration depth in the prefrontal cortex, induced by a specific Hesed-coil, H4, designed for the addiction treatment and by the traditional figure-of-8 coil for comparison. Results show that H4 coil preferentially targets insula and cingulate cortex. Moreover, it can induce in the deepest tissues E amplitude ranging between the 20-40% of the cortical peak and it can penetrate the cortex up to 4 cm with a E>50% of the cortical peak, thus noticeably increasing the penetration depth of the traditional TMS systems.
A full-scale computer-based optimization of a system of permanent magnets for magnetic drug targeting is presented.
A new methodology for designing magnetic drug targeting systems is proposed.
Our methodology can be employed in any medical application which uses magnetic drug delivery.
The presented methodology of magnetic drug targeting optimization can be applied to systems where the placement of permanent magnets in close proximity to the targeted organ or tissue is complicated or even impossible.
Further optimization of the magnetic system is necessary based on the desirable configuration of the magnetic force field in the subject of study.
Magnetic Targeted Drug Delivery to the Human Eye Retina: an Optimization Methodology
Sergey Erokhin, Dmitry Berkov.
Abstract:We present a new optimization method for permanent magnet systems aimed for magnetic targeted drug delivery.
On the example of a human eye, a special attention is paid to the challenging situation, where the placement of magnets in close vicinity of the targeted area is impossible. In this paper we demonstrate how a system of magnets can be optimized to provide the maximal magnitude of the magnetic field gradient with a prescribed orientation in an extended area (vitreous body). The presented methodology is applicable for all tasks involving a magnetic targeted drug delivery to biological objects.
Subdermal (tattoo) antennas made from gold nanoparticle ink may be used to create antennas at the body surface which could be used to re-radiate telemetry signals from a smaller, implantable device.
Current research in polymer engineering is moving towards materials that can be injected as fluids that turn to soft, conductive solids at body temperature; this paper anticipates using these materials for tattooed subdermal antennas.
Even with voids, typical of what would occur with a subdermal tattoo, the antennas can still be effective, as shown from comparing the current distributions for solid, mesh, and segmented strip dipoles
Measurements confirm the feasibility of subdermal antennas.
A Comparison of Solid, Mesh, and Segmented Strip Dipoles in a Subdermal Environment
Andrew Chrysler, Kaitlin L. Hall, Cynthia M. Furse.
Abstract:The objective of this paper is to evaluate the feasibility of subdermal (tattoo) antennas in the fat layer, which use the low conductivity of the fat to electrically insulate a dipole antenna.
Current research in polymer engineering is moving towards materials that can be injected as fluids that turn to soft, conductive solids at body temperature; this paper anticipates using these materials for tattooed subdermal antennas. Simulations and measurements were used to evaluate the current distributions that are shared between antennas with and without voids (solid, segmented, and meshed strip dipole antennas) and surrounding body tissues to give insight into the performance of subdermal antennas and their coupling to the body. The body tissues play a strong role in adapting the current distributions. The high dielectric materials electrically shorten the antenna. The high conductivity muscle conducts or guides current into the body. Any voids in the antennas (e.g. gaps between segments or holes in the mesh) are particularly important, as they generate stronger coupling to the tissues. The feasibility of using fat as insulation is verified in simulation and confirmed with measurement.
This paper presents critical-path characterizations of implantable electrode arrays for next generation neural interfacing circuits, laying the foundation for fully implantable electrode characterization.
Implantable electrode arrays have a substantial increase in noise and impedance when implanted with additional low-frequency biological noise unexplained by local cortical activity.
These characterizations provide a foundation for advanced neural interfacing circuits that will require wideband noise and impedance characterizations currently unavailable in the literature.
Detailed characterizations of the Tucker-Davis Technologies microwire array and the Utah electrode array have been presented, particularly for wide-band applications. Typical characterizations cite impedance only at 1 kHz, but this is not descriptive of the wide-band characteristics nor the low frequency noise and are thus insufficient for neural interfacing circuit design.
IEEE Sensors 2017 Special Issue
Impedance and Noise Characterizations of Utah and Microwire Electrode Arrays
Avery Tye Gardner, Hunter S. Stratham, David J. Warren, Ross M. Walker.
Abstract:This work presents an in-depth noise and impedance characterization of two of the most widely used microelectrode arrays
(the Utah Electrode Array and the TDT Microwire Array) and provides quantitative analysis of how properties change when implanted in rodent cortex. Custom low-noise circuits and de-embedding methods were designed to acquire $\mathrm{nV} /\sqrt {\mathrm{Hz}}$ noise power spectral densities from high impedance electrodes. A total of 80 electrodes were implanted across five rats and measured under deep anesthesia, demonstrating a 1.5x to 3x increase in noise and 2.25x to 9x in impedance compared to in vitro measurements. Low frequency biological noise was also observed and studied through post mortem measurements. These results are informative for designing neural interfacing systems for both neuroscience and medical applications.
High energy transfer efficiency and wide coverage range are desired to power implanted medical devices (IMDs) with wireless power transfer (WPT) systems.
A 200 mm × 300 mm rounded rectangular transmitting coil and a double-layer circular receiving coil with an outer diameter of 24 mm have been optimized for a 2-coil IMD-WPT system.
With the optimized coils, experimental results show that high energy efficiency higher than 40% can be achieved even the implanted receiver is located deep in the body with a wide coverage range of 18 cm × 10 cm.
2018 WPT Special Issue
Optimized Design of Coils for Wireless Power Transfer in Implanted Medical Devices
Yufeng Zeng, Dongyuan Qiu, Xiangtian Meng, Bo Zhang, Sai Chun Tang.
Abstract:Implanted medical device (IMD) wirelessly powered by magnetic resonant coupling has attracted wide attention recently.
In this study, an optimized 2-coil wireless power transfer (WPT) system is adopted to eliminate the requirement of embedded battery in an IMD. In order to deliver stable power to implantable devices with wide coverage range and high efficiency, coil optimization is investigated, including the consideration of the coil structure, pitch and number of turns. By using finite element analysis (FEA), both the transmitting and receiving coils have been optimized at 6.78MHz. A 200 mm $\times$ 300 mm rounded rectangular transmitting coil and a novel double-layer circular receiving coil with an outer diameter of 24 mm were developed, and the transmitting coil was segmented by multiple resonant capacitors to significantly reduce the coil voltage to a safe level. Experiment results show that stable power transfer efficiency over 40% can be achieved at a distance of 5 cm with the optimized transmitting and receiving coils.
The developed numerical workflow is appropriate for evaluating power deposition and temperature rise on electrodes of a coax lead located in a homogeneous medium in appliances with ISO/TS 10974 Tier4.
Small lead electromagnetic model uncertainty does not require that the spatial distribution of power deposition and temperature rise in close proximity to lead electrodes be independent on an incident electric field.
Determination of the hot spots described in Clause #8 of ISO/TS 10974 Ed2 can depend on the end user's selection of pathways, that is, variety of incident tangential electric field applied during investigation, the use of specific absorption rate or temperature rise quantities, as well as the duration of RF-induced heating when the approach based on temperature rise is used.
The power injection approach based on the comparison of temperature increase along the lead tip electrode axis can result in substantial underestimation or overestimation of the power deposition around the lead tip electrode, as well as maximum temperature rise in close proximity to the tip electrode, ring electrode, or both.
Lead Electromagnetic Model to Evaluate RF-Induced Heating of a Coax Lead: A Numerical Case Study at 128 MHz
Mikhail Kozlov, Wolfgang Kainz
Abstract:One of the major components of magnetic resonance imaging safety for patients with an active implantable medical device is the evaluation of in vivo radio frequency induced heating of tissue near a lead electrode, which can result in tissue damage.
This numerical case study investigated a number of the recommendations, assumptions, and requirements of Clause 8 of the technical specification ISO/TS10974. For this, a lead electromagnetic model (LEM) of a generic coax lead at 128 MHz was evaluated with 3D electromagnetic and thermal co-simulations of the entire lead. Two sets of 120 incident electric fields with different profiles were generated in a homogenous medium using the electrical properties of blood by an array of four antennas. Substantial dependence of power deposition and temperature profiles around lead electrodes on the incident electric field did not reduce the quotient of the variances of the fitted LEM values, observed values of power deposition, and the net temperature increase, above background, with the presence of the generic coax lead. The power injection approach based only on the comparison of the temperature increase in the medium along the lead tip electrode axis can result in substantial underestimation or overestimation of power deposition around lead electrodes.
Functionalization of bone scaffolds using magnetic nanoparticles allows hyperthermia of bone tumors in an effective way.
The possibility of employing innovative magnetic scaffolds as therapeutic tool in orthopaedic oncology is analyzed via numerical simulations. Using a Cole-Cole model, non-linear material properties are evaluated to define external field parameter to perform an effective treatment for bone tumors such as Fibrosarcoma and Osteosarcomas.
Accurate electromagnetic and thermal modeling of scaffolds and nanoparticles, in the whole range of involved temperature, is required to design effective and safe treatments.
Different tumoral tissues and qualitative features such as the presence, size and type of surgical fracture, affect in a significant way the hyperthermia treatment.
Numerical Investigation Of Bone Tumer Hyperthermia Treatment Using Magnetic Scaffolds
Alessandro Fanti, Matteo Bruno Bruno Lodi, Giuliano Vacca, Giuseppe Mazzarella.
Abstract:This works claims to define, via numerical simulations, magnetic field parameters to perform an effective in situ bone tumor hyperthermia treatment using magnetic scaffolds.
A Cole-Cole model to describe the frequency response of the magnetic susceptibility of nanoparticles embedded in novel magnetic biomaterials is explored. The heating phenomena is investigated considering both the ischemic and inflamed state of the fracture gap at the bone/implants interface. Both Osteosarcoma and Fibrosarcoma tumors are analyzed. Magnetic hydroxyapatite and poly-ε-caprolactone scaffolds are investigated. From the thermal analysis, it is found that the fracture behaves as a resistance to heat conduction, therefore strength and frequency of external magnetic field has to be tuned to perform the treatment taking the fracture status into account. Moreover, numerical experiments indicate that low perfused Fibrosarcoma can be treated using moderate-strength field, whereas more intense external fields are required to treat strongly vascularized Osteosarcoma without damaging healthy bone tissue. Magnetic hydroxyapatite stands out to be the most performant and versatile material to treat both tumors. These simulations can be regarded as a starting point to analyze possible clinical use of magnetic scaffolds for in situ bone hyperthermia.
We propose an experimental-computational technique for low frequency dosimetric assessments that reduces the experimental burden while maintaining accuracy and robustness.
The proposed technique can be used when the magnetic source is unknown or not suitable to be modeled.
By adopting surface measurements, the proposed technique allows to characterize any low frequency magnetic source in a very convenient way.
By adopting the boundary element method for extrapolating surface measurements as well as a curl inversion operator for magnetic vector potential evaluation, the proposed technique significantly reduces the noise from the input data.
The positive features of the proposed technique have been put in evidence by testing it in a transcranial magnetic stimulation dosimetric application.
Computational Low Frequency Electromagnetic Dosimetry Based on Magnetic Field Measurements
Alessandro Arduino, Oriano Bottauscio, Mario Chiampi, Ilkka Laakso, Luca Zilberti.
Abstract:This paper compares different experimental-computational strategies for the estimation of electric fields induced in human bodies by low frequency magnetic sources
characterized by a set of magnetic field measurements. The analysis is carried out by considering three alternative procedures, which use, as the first input, the distribution of the magnetic flux density in a volume containing the studied body or on a surface surrounding the sources. The comparison is performed on a realistic model problem, related to transcranial magnetic stimulation (TMS), in which numerically simulated "virtual measurements" are employed. The comparative analysis is developed in terms of both result accuracy and robustness against noisy input due to unavoidable experimental uncertainties. It results that by performing the measurements on a surface surrounding the sources, a significant reduction of the experimental burden is found with respect to the case of volume measurements, without affecting neither the accuracy nor the robustness of the procedure. In particular, when whole body electric field evaluation must be carried out, the advantage of surface measurements with respect to volume ones becomes significant. Moreover, a preferable scheme obtained as hybridization of previously proposed strategies is identified. Besides the adoption of a TMS model problem in the comparison procedure, the achieved result can be extended to any low frequency dosimetric assessment where the magnetic sources are difficult to model or not completely known.
Core@shell Fe-oxide@SiO2 nanoparticles: a system for delivering heat selectively to cancer cells through an alternating electromagnetic field. This study provides a critical, anatomy-informed dielectric study of the properties of the bladder.
The most effective heat transfer is obtained through a maximisation of the energy losses: a careful investigation of the nanoparticles system is required to comply with the constraints imposed by hyperthermia applications.
An almost superparamagnetic system that minimises magnetostatic interactions (e.g. agglomeration) while keeping dynamic hysteresis losses for heat delivery.
Magnetic and Thermal Characterization of Core-Shell Fe-oxide@SiO2 Nanoparticles for Hyperthermia Applications
Gabriele Barrera, Marco Coïsson, Federica Celegato, Elena Sonia Olivetti, Luca Martino, Ivana Miletto, Paola Tiberto.
Abstract:Nanoparticles for magnetic hyperthermia pose significant constraints in their size and composition to ensure cellular uptake and biocompatibility,
while still requiring significant hysteresis losses exploitable at electromagnetic field values and intensities not exceeding safety limits for the human body. In this paper, core-shell Fe-oxide@SiO $_2$ nanoparticles have been synthesized and their size has been controlled so that the blocked-to-superparamagnetic transition is close to room temperature. Their size remains therefore as small as possible, while still displaying significant hysteresis losses in dynamic conditions (electromagnetic fields up to 48 kA/m at 100 kHz). Static loops measured by vibrating sample magnetometry and dynamic loops measured by a custom B-H tracer are used to characterize the particles magnetic properties, as well as a custom-built, fully modelled, hyperthermia setup. The specific absorption rate is obtained either from static and dynamic loops areas, and from direct hyperthermia measurements. Dynamic loops are shown to be a good estimator of specific absorption rate values.
In MIMO wireless systems, multiple antenna elements are adopted to work as transmitter and receiver to improve capacity such as the enhancement in the data transmission rates of high resolution images from the capsule to outside base station/gateway.
The proposed four-element implantable MIMO antenna features a wide impedance bandwidth of 18.64% (2.14–2.58 GHz) with a maximum gain of –15.18 dBi, and has the compact dimensions of 18.5 mm × 18.5 mm × 1.27 mm. The mutual coupling between different antenna elements is less than –15.99 dB.
The ex-vivo measured results illustrate a merit agreement with the simulated ones in a three-layer phantom. The radiation characteristic of the designed antenna follows the healthy protection standards, and the performance of diversity and far-field link budget indicates merit channels characteristics.
A Miniaturized Four-Element MIMO Antenna with EBG for Implantable Medical Devices
Yi Fan, JinHong Huang, TianHai Chang, Xiongying Liu.
Abstract:A miniaturized four-element multiple-input-multiple-output (MIMO) antenna operating in the band of In-dustrial, Scientific, and Medical (ISM) 2.4-2.48 GHz, is investigated to supply sufficient transmission rate and fight against multipath fading.
A single antenna integrates with dual radiators, and each radiator is owned by a pair of sub-antenna units, greatly diminishing the whole dimensions of the implanted MIMO device. A cross-shaped slot etched in the radiator, a branch extending on the conductor ground, and a pair of electromagnetic band gaps (EBGs) are integrated to achieve high isolation. The proposed MIMO antenna has the compact dimensions of 18.5 mm × 18.5 mm × 1.27 mm. The simula-tion in the three-layer phantom indicates that the impedance matching is good with a bandwidth of 18.64% (2.14-2.58 GHz) and a maximum gain of -15.18 dBi and the mutual coupling is reduced to less than -15.99 dB at the ISM band. An ex-vivo test was implemented in a fresh pork slab and the measurement results are well matched with the simulation ones. Health safety considerations and link budget are discussed to validate the antenna's availability in biomedical telemetry, and the envelope correlation coefficient is computed, illustrating the high independence between antenna elements.
Accurate knowledge of the dielectric properties of tissues is the basis for electromagnetic (EM) medical device design, development, and optimisation, and therefore, lack of knowledge on the dielectric properties of the bladder is a stumbling block for the application of bladder-related technologies, including EM therapeutic and diagnostic tools.
This study provides a critical, anatomy-informed dielectric study of the properties of the bladder.
Through a large number of tissue measurements, this study examines for the first time the impact of the dielectric properties of the bladder in light of common measurement confounders (time from excision, temperature), and uniquely examines the properties of both the inside and the outside of the bladder wall.
This study provides reliable dielectric data of the bladder over the microwave frequency range, and fitted dielectric models, that can be used to support the design of bladder-related medical technologies.
This study has been performed in line with modern best practices standards for the reporting of experimental metadata along with experimental data, and the data and metadata will be made openly available online.
Characterisation of the Dielectric Properties of the Bladder over the Microwave Range
Emily Porter, Saqib Salahuddin, Alessandra La Gioia, Adnan Elahi, Atif Shahzad, Arun Kumar, David Kilroy, M. O'Halloran,
Abstract:The dielectric properties of tissues characterise the interaction of electromagnetic fields with the body. Accurate knowledge of these properties facilitates the design of effective electromagnetic-based
medical technologies. A number of such technologies have been proposed to target the diagnosis or treatment of bladder conditions, including urinary incontinence and bladder cancer. However, available dielectric data for the bladder is extremely limited in scope and has not been updated in line with modern knowledge, including best practices in measurement procedures to reduce the impact of confounders, along with improved reporting of experimental metadata. For these reasons, in this work we present a study of dielectric measurements on the bladder of freshly excised bovine and porcine tissues over the microwave frequency range. We examine the properties of the inside and outside of the bladder wall, and carefully control and record confounders during the measurements. The obtained data is analysed in terms of confounders (temperature, time from excision, inter-species differences), and compared to data from the literature. The resulting dataset, composed of 52 measurements from 10 animals, provides a thorough and quantitative understanding of the dielectric properties of the bladder. The results of this study will benefit the development of related electromagnetic medical technologies, and thereby positively impact on patient care over the long-term.
This paper presents the design and the implementation of an on-chip magnetoresistive sensors array for cell detection and localisation. Giant magnetoresistance (GMR) sensors have been used due to their high sensitivity and resolution.
A novel calibration and localisation algorithm has been coded and implemented. In order to generate the required homogenous magnetic field, a custom 3D printed Hallbach cylinder has been simulated and characterised.
Sensory chips could detect an average magnetic sensitivity of 2 V/T at room temperature.
The ferrofluid and a customised 3D printed Halbach cylinder were employed to simulate the magnetic field change in the cell.
The implemented algorithm helps in achieving high sensitivity and positioning speed, also thanks to an accurate calibration of the GMR sensors.
2017 IEEE Sensors Special Issue
Magnetoresistive Biosensors for On-Chip Detection and Localisation of Paramagnetic Particles
Zhaochen Yin, Edoardo Bonizzoni, Hadi Heidari
Abstract:This paper presents the design and the implementation of an on-chip magnetoresistive sensors array for cell detection and localisation.
Giant magnetoresistance (GMR) sensors have been used due to their high sensitivity and resolution. A new calibration and localisation algorithm has been coded and implemented. In order to generate the required homogenous magnetic field, a custom 3D printed Hallbach cylinder has been simulated and characterised. The system includes sensory and electronic boards to collect the data and to transfer them to a computing server. The experimental results are displayed in a visual interface. Ferrofluid is used to model and simulate the magnetic field change of the cell. This paper demonstrates a 4×4 sensors array and provides a step towards the miniaturised on-chip magnetoresistive based cell detection and localisation for portable diagnostics applications.
An insulated cable implanted in a lossy material and submitted to a radiated radiofrequency field can be modeled by a modified transmission line with a distributed excitation along the line.
This model gives a better physical insight into the problem of the compatibility of implanted cables in Magnetic Resonance Imaging (MRI) such as pacemaker, defibrillator or neurostimulator leads.
This work shows the equivalence between the transfer function model usually used to model the interaction of an implanted lead with the radiofrequency field of MRI and the modified transmission line model.
An idea that can be derived from this model is to reduce the heating at the electrode of an implanted lead by creating a big reflection coefficient before the end of the lead using a multi-section approach
Transmission Line Model Of An Implemented Insulated Cable For Magnetic Resonance Imaging Radiofrequency Hazard Evaluation
Alexia Missoffe, Julie Kabil, IADI, Pierre-Andre Vuissoz, Jacques Felblinger
Abstract:This work demonstrates that one can model the power deposited at the electrode of an implanted insulated cable submitted to the radiofrequency field of a 1.5 T Magnetic Resonance Imaging modality (64MHz) with a transmission line model.
This offers an alternative which is more related to physics to the usually used transfer function model. The equivalence between the models is shown through a finite difference model and a new analytical formula for the transfer function as a function of transmission line parameters. First, the possibility of modeling an insulated cable with a transmission line model was analyzed through full-wave numerical simulations. The assumption of a transmission line model underlying the transfer function model was shown to be right for a simple cable embedded in tissue imitating gel and the transmission line parameters extracted from this analysis were consistent with analytical formulas derived from the laws of physics. The transmission line model predictions were first compared to experimental and simulated data of the cable transfer function and then, to experimental and simulated data of the resonant behavior as a function of length of cables with different termination conditions. The measured and simulated transfer functions fit perfectly a transmission line model with an analytical expression of the propagating constant. The transmission line model extracted from the transfer function allows to predict the resonant behavior of two cables with different termination conditions
This paper proposes a 2×2 magnetic resonance charging system considering both magnetic transmit beamforming and receive beamforming for charging wearable medical devices.
Both numerical results and simulation results in COMSOL model are provided to demonstrate the effectiveness of the proposed system and optimization algorithm.
The target medical applications are wearable medical devices using for Alzheimer's disease, ergonomics, rehabilitation and neurology which are inconvenient or sometimes infeasible to replace or change batteries for the elderly or patients.
Such a magnetic resonance charging system provides a promising way to make wearable medical devices permanently unplugged.
Magnetic Transceiver Beamforming for A 2 × 2 Magnetic Resonance Charging System
Xiaoqing Liu, Bingqing Mei, Xiaodong Wang, Zhigang Wen
Abstract:In this paper, we consider a magnetic resonance charging system consisting of two transmit coils and two receive coils that is designed to wirelessly charge
biomedical wearable diagnostic and therapeutic devices using for ergonomics, rehabilitation and neurology. Since it is proved that the power transfer efficiency will be maximized when the phase difference between the transmit currents equals to zero, an optimization problem is formulated to maximize the received power at both receive coils by adjusting the amplitude of transmit coil currents and the receive coil resistances equipped on wearable devices. The formulated problem is nonconvex and we propose an efficient iterative solution by solving a series of geometric programs that approximate the original problem. Both numerical results and simulation results in COMSOL model are provided to demonstrate the effectiveness of the proposed biomedical wearable system and optimization algorithm.
This work analyses the sensing radius of a coaxial probe for accurate dielectric characterisation of heterogeneous tissues.
The probe sensing radius can be smaller than the probe radius and depends on the histology of the tissue sample.
Accurate knowledge of the sensing radius has the potential for improving the design of novel microwave imaging devices and hyperthermia systems.
This work demonstrates that a lack of knowledge of the probe sensing radius leads to errors in the interpretation of dielectric data acquired from heterogeneous tissues, and thereby to inaccurate medical device design.
Despite the assumption made in previous dielectric studies, this work shows that the dielectric contribution of a particular tissue depends on both its location within the sensing volume and its dielectric properties.
IEEE APS Special Issue
Quantification of the Sensing Radius of a Coaxial Probe for Accurate Interpretation of Heterogeneous Tissue Dielectric Data
Alessandra La Gioia, Saqib Salahuddin, Martin O'Halloran, Emily Porter
Abstract:Accurate tissue dielectric measurements are crucial for the development of electromagnetic diagnostic and therapeutic devices that are designed based on estimates of the dielectric properties of diseased and healthy tissues.
Although the dielectric measurement procedure is straightforward, several factors can introduce uncertainties into dielectric data. Generally, uncertainties are higher in the dielectric measurement of heterogeneous tissues, due to the fact that there is no standard procedure for acquiring and interpreting the dielectric data of heterogeneous tissues. Uncertainties related to tissue heterogeneity can be minimised by estimating the probe sensing volume, defined by the sensing depth and radius, and characterising the tissue distribution within that volume. While several studies have investigated the sensing depth, this work focuses on examining the sensing radius. Both dielectric measurements and numerical simulations with heterogeneous porcine tissues in the microwave range of 0.5-20 GHz have been conducted to quantify the sensing radius and the dielectric contribution of each tissue within the sensing volume. Experiments demonstrate that the sensing radius, which depends on the individual dielectric properties of the constituent tissue types, can be smaller than the probe radius. This work further quantitatively demonstrates that the dielectric contribution of a particular tissue depends on both its location within the sensing volume and its dielectric properties. This study provides fundamental knowledge for accurately interpreting dielectric data of heterogeneous tissues, with the aim of supporting medical device development.
Real-time Microwave Imaging of a Compressed Breast Phantom with Planar Scanning
Daniel Tajik, Farzad Foroutan, Denys S. Shumakov, Aaron D. Pitcher, Natalia K. Nikolova
Abstract:Two real-time reconstruction algorithms, quantitative microwave holography and scattered-power mapping, have been shown to be successful in the imaging of compressed tissue of
relatively small thicknesses such as 1 cm and 2 cm. In both cases, planar data acquisition of frequency-swept transmission coefficients has been employed. Despite the fact that these algorithms are based on a linear forward model of scattering, they have been capable of providing quantitative estimates of the tissue permittivity due to the experimentally derived kernel of the scattering integral. Here, we demonstrate similar performance with a thicker (about 5 cm) compressed-breast phantom. This thickness is greater or comparable to the median thickness employed in mammography, depending on the view (craniocaudal or mediolateral oblique). The two methods are described in a common mathematical framework for the first time. The importance of the system calibration and the choice of a host medium are discussed through experiments. A new method for focusing onto suspect regions is demonstrated. The limitations of real-time imaging are highlighted along with an outlook to improving the image resolution and suppressing artifacts without sacrificing the reconstruction speed.
What are the innovative features of using electromagnetics for biomedical applications in this manuscript (in one sentence)? The modulation of light reflected by optical gratings is used to measure temperature changes during laser ablation for cancer removal.
What is the conclusion drawn in this manuscript (in one sentence)? Carbon fibers based on fiber Bragg gratings allow performing distributed temperature monitoring during image-guided laser ablation.
What are the targeted biological and/or medical applications (in one sentence)? The medical application here presented is the laser ablation therapy for cancer removal in soft organs, e.g., liver.
What is the significance/breakthrough of this study? This study brings useful tools for real-time monitoring of laser ablation effects.
What are the accomplishments you would like to highlight in this manuscript to our readers (which are not mentioned above, in one sentence)? As confirmed by tests in the presented pre-clinical scenario, the proposed probe is suitable for temperature monitoring during CT-guided laser ablation, and no artifact is produced on the diagnostic images.
Solutions to improve the outcomes of thermal treatments in oncology: multi-point temperature monitoring
Emiliano Schena, Federico Davrieux, Paola Saccomandi, Daniele Tosi, Riccardo Gassino, Carlo Massaroni, Daniela Lo Presti, Guido Costamagna, Guido Perrone, Alberto Vallan, Michele Diana, Jacques Marescaux
Abstract:Over the last few decades, minimally invasive treatments have gathered a large interest as alternatives to surgical resection.
Among others, laser ablation has gained a broad clinical acceptance in the treatment of a certain number of solid tumors (e.g. liver, lung, and prostate). In this context, the knowledge of temperature during treatment may be useful to better control the amount of damaged tissue and to subsequently improve clinical outcomes. The objective of this work is to assess the feasibility of two multi-point probes for temperature monitoring during laser ablation. The probes consist of a needle made up of a carbon fiber tube. Each probe embeds an array of 7 fiber Bragg grating sensors. Experiments performed in in vivo animal models (pig livers) show that the probe can reach deep-seated organs and offer the possibility to monitor tissue temperature in seven different positions. This information may be crucial to guide clinicians in the optimization of treatment settings and to improve the accuracy of theoretical models which will pilot future studies to design new heating devices and to develop patient-specific treatments
The reported work utilizes an open-stub as the sensing element, demonstrates a fully integrated complex dielectric sensor in K-band frequencies and results in flexible DC output.
The proposed sensor can be useful as a compact, label-free and low-power all-electrical sensing approach for relative dielectric sensing of biological and chemical materials with minimal invasion.
It can offer continuous glucose monitoring, human body hydration sensing as implants or wearable device, also it can be useful as a lab-on-a-chip for DNA sequencing, malignant cell growth observation, cell cultivation monitoring etc.
The sensor demonstrates complex permittivity within 3.7% accuracy for the real part and 4.3% for the imaginary part for dielectric chemical samples like methanol and ethanol.
A Fully Integrated Low-Power 30 GHz Complex Dielectric Sensor in a 0.25-μm BiCMOS Technology
Farabi Ibne Jamal, Subhajit Guha, Mohamed Eissa, Jan Wessel, Dietmar Kissinger
Abstract:This paper presents an integrated low-power dielectric sensor in K-band frequencies implemented in a 0.25 μm SiGe BiCMOS process including the sensing front-end and readout circuits.
The sensor enables the measurement of both real and imaginary part of permittivity of the material under test (MUT). The MUT is exposed on the resonator component in a sensing oscillator and the oscillator results in permittivity and conductivity dependent change in the output frequency and output power, respectively. The frequency information is translated into DC voltage using a frequency discriminator and the output power is detected using a power detector. The sensor has been calibrated using iso-propanol, ethanediol and acetone solutions. Methanol-ethanol mixture solutions, in steps of 25% of concentration change, have been used to demonstrate the functionality of the sensor. The selectivity is showed using methanol-ethanol mixtures with concentration differences of 5% around a mixture ratio of 50:50. The chip is 2.3 sq. mm in size and consumes 60 mW power. The sensor measures complex permittivity within 3.7% accuracy for the real part and 4.3% for the imaginary part. As a compact and low-power solution the sensor is a potential candidate for minimal invasive investigations of chemicals and bio-materials at mm-wave frequencies.
Invasive surgery is a problem for biomedical implants due to infection at the incision site and reactions to anesthesia.
The use of stents embedded with electronics is an attractive alternative method for implanting minimally invasive biomedical devices via angiographic catheter delivery.
The use of stents precludes batteries. Hence, wireless power transfer to the device is a requirement.
This work employs inductive and capacitive coupling to demonstrate two different methods for transferring power to a stent-based biomedical device safely. The ideal power transfer method depends on the location of the implant.
Such devices could be used to monitor biological indicators and vital signs such as blood pressure, blood glucose and even neural signals.
Near Field Wireless Power Transfer to Stent-Based Biomedical Implants
Ammar Aldaoud, Jean-Michel Redoute, Kumaravelu Ganesan, David Garrett, Steven Prawer
Abstract:Safe wireless power transfer is an essential requirement for biomedical implants. Emerging technologies are becoming smaller, less invasive and consuming less power.
Moreover, stent-based devices are being recognized as a minimally invasive alternative to traditional surgery. Hence, the idea of using the body of the stent as the power receiving element is becoming increasingly attractive. The objective of this work is to analyze two near field wireless power transfer methods to stent-based devices, viz., inductive and capacitive coupling. The methods used are lumped element modelling, ac circuit theory, finite-element analysis and experiments to validate the model with excised bovine muscle tissue. Capacitive coupling is proposed as an alternate method due to the transmitter design that can be worn anywhere on the body. It achieves power transfer efficiencies of 2.6% and 1% when placed at depths in muscle tissue of 15 mm and 30 mm respectively. Safety requirements are also met. The capacitive link can accept an input power of 53 mW before exceeding the safe specific absorption rate limit of 1.6 W/kg averaged over 1 g of tissue.
In this paper, a low-cost super-tiny senor node with temperature sensing function is presented together with fully integrated mm-wave frequency wireless power transfer technique in 65 nm CMOS technology, which can increase temperature monitoring accuracy for a tiny location such as a solution for low-cost disposable skin sensor.
In this paper, a mm2 sized, low cost and no battery sensor with temperature sensing is achieved in a 65 CMOS technology which can be applied in medical treatment.
In this paper, the sensor node is targeted at the low-cost, low-sized, disposable sensing application in medical/biological.
This paper presents an mm-wave wireless powered sensor node which is fully integrated in a CMOS technology, including on-chip antenna, on silicon mm-wave wireless power receiver, energy storage, energy monitoring, and a low power temperature sensing with a transmitter.
Power Reduced Monolithic Wireless Sensor
Hao Gao, Marion Matters, Peter Baltus
Abstract:Future wireless sensor networks require reliable, battery-less, miniaturized, low-cost sensor nodes with ultra-low-power consumption.
Remote RF-powering provides a reliable wireless power source for such monolithically integrated sensor nodes. Combining highly-integrated ultra-low-power millimeter wave (mm-wave) sensing, wireless power transfer (WPT) and on-chip antenna is a path towards battery-less, fully monolithically integrated, millimeter-sized sensor nodes with only a few milligram of weight. In this work, a passive fully integrated monolithic wireless sensor for mm-wave sensing is presented in a 65 nm CMOS technology, including an on-chip wireless power receiver, on-chip antenna, and a low power transmitter.
We propose to combine radar and wearable sensors (accelerometer, gyroscope, and magnetometer) data to simultaneously improve human activities classification rates and minimise false alarms when detecting critical events such as falls.
Feature selection and fusion methods, at both feature level and decision level, are tested on experimental data collected simultaneously using radar and wearable sensors.
Sequential forward selection (SFS) significantly helps reduce the dimension of the feature subset and improve the classification accuracy.
For the feature level fusion, we were able to increase the classification accuracy by 12.2% and 3.3%, compared to the use of radar and wearable sensors on their own.
Decision level fusion also improves to 94.8% and 96.7% classification accuracy with Fuzzy logic-based decision making and LOGP soft fusion. A novel voting system was also proposed by combining two SVM and KNN classifiers trained by radar and inertial sensor independently. This is demonstrated to achieve 97.8% classification accuracy and 100% fall detection specificity for the 'fall event' class.
Future work will explore the robustness of this multi-sensory approach with additional data collection involving more subjects and more sensors types and configurations.
A multi-sensory approach for remote health monitoring of older people
Haobo Li, Aman Shrestha, Hadi Heidari, Julien Le Kernec, Francesco Fioranelli,
Abstract:Growing life expectancy and increasing incidence of multiple chronic health conditions are significant societal challenges.
Different technologies have been proposed to address these issues, to detect critical events such as stroke or falls, and to monitor automatically human activities for health condition inference and anomalies detection. This paper aims to investigate two types of sensing technologies proposed for assisted living: wearable and radar sensors. First, different feature selection methods are validated and compared in terms of accuracy and computational loads. Then, information fusion is applied to enhance activity classification accuracy combining the two sensors. Improvements in classification accuracy of approximately 12% using feature level fusion is achieved with both Support Vector Machine and K Nearest Neighbor classifiers. Decision-level fusion schemes are also investigated, yielding classification accuracy in the order of 97-98%.
REM Behavior Sleep Disorder is a serious, sometimes dangerous condition that causes people to "act out" their dreams; thus, monitoring and diagnosis of such disease are significant.
This paper demonstrates the design and implementation of an innovative system for monitoring patients suffering from REM sleep behavior disorder.
Continuous monitoring of RBD patients is crucial to prevent injury; the proposed identify RBD episodes with accuracy higher than 90%.
The work leverages the finer-grained information to monitor patients suffering from RBD and fully exploits the amplitude and phase information.
Monitoring of Patients Suffering from REM Sleep Behavior Disorder
Xiaodong Yang, Syed Aziz Shah, Aifeng Ren, Nan Zhao, Jianxun Zhao, Fangming Hu, Zhiya Zhang, Wei Zhao, Masood Ur Rehman, Akram Alomainy
Abstract:The Rapid Eye Movement (REM) Sleep Behavior Disorder (RBD) is a parasomnia that involves involuntary, unwanted and random movements of a dreaming patient.
Typically, these dreams contain violent activities. There is a high likelihood of the patient being injured or hurt his bed-partner as a result of these enactments. Continuous monitoring of sleeping RBD patients can prevent these harmful events through timely intervention. This work presents a novel method for continuous observation of RBD patients exploiting fine-grained amplitude and phase information of the wireless channel response. The variations in the wireless channel response as a result of different patient movements are assessed and used to identify RBD episodes. The data obtained is classified using support vector machine and delivers accuracy level of more than 90%.To the best of authors' knowledge, it is a first attempt at using radio frequency signals to sense RBD in real-time.
This application utilizes wireless communication to transmit biological data from the transplanted liver to provide constant and real-time monitoring after transplantation.
This work shows that it is feasible to establish wireless communication from the liver area to the body surface and thus to implement such intended applications in the described scenario using ultra-wideband (UWB) technology.
The potential medical applications include wireless monitoring and diagnostics of internal organs of the human body as well as drug delivery at specific locations.
This work offers, for the first time, the fundamental understanding of UWB channel characteristics at the liver location considering respiration-induced organ movements.
With the potential utilization of ultra-wideband technology for liver implanted wireless communications, this would enable other future innovative healthcare applications.
Channel Characteristics and Wireless Telemetry Performance of Transplanted Organ Monitoring System Using Ultra-wideband Communication
Pongphan Leelatien, Koichi Ito, Kazuyuki Saito, Manmohan Sharma, Akram Alomainy
Abstract:Recently, wireless implanted applications have attracted considerable interest in the medical domain due to their capability to offer healthcare sensing and monitoring.
One such promising application is the wireless monitoring of transplanted organs, particularly transplanted liver. Ultra-wideband technology is well suited for wireless implanted applications since it allows physical size reduction as well as increased longevity of the implanted devices. However, because of the high attenuation of UWB signals inside the human body, it is necessary to examine their propagation characteristics and demonstrate the feasibility of such application. In this paper, an investigative study involving the liver implanted wireless telemetry link using UWB technology is presented. Measurements using multilayer phantoms and simulations using a human digital model have been conducted within the frequency band of 4.5-6.5 GHz. Multilayer phantom measurements have demonstrated the attenuations ranging between -50 dB and -100 dB over the considered frequency band. A path loss model at the liver area has been obtained from the simulations. Also, the numerical results have shown that the attenuation variation due to respiration-induced organ movements was within 30 dB range with respect to the largest organ movement distance of 40 mm which emphasized the influence of organ movements on the in-body attenuations. Our preliminary link budget evaluation of liver-skin surface communication link including the effect of shadowing and organ movements estimated that it is possible to achieve a high data rate of 10 Mbps with a bit error rate of 10−3 at a distance of about 40 mm.
We assess the viability of having a dual-band skin-attached repeater antenna for in- and off-body communication for the transmission of medical data in the context of Body Area Networks. It is the first time a single-port antenna with both ISM (2.4 GHz) and UWB (4-10.6 GHz) operation bands is proposed for BAN applications.
The study demonstrates the implementation of a skin-adhesive antenna and the feasibility of both in- and off-body communication links with a passive implant and an external base station, respectively.
The proposed repeater antenna can be used in continuous monitoring applications, in order to detect health conditions in advance.
We study for the first time the possibility of having a skin-attached repeater antenna with capability of establishing an in-body communication link with multiple implants and further relay the collected information through a UWB off-body link, thus increasing the off-body communication range and also diminishing the transmission time.
The present paper includes the complete study of the in- and off-body links not only in the time-domain, in which we transmit a QPSK-modulated ECG signal, but also in the frequency-domain, where we assess the power transfer and link budget.
Dual-Band Skin-Adhesive Repeater Antenna for Continuous Body Signals Monitoring
Joao Felicio, Carlos A. Fernandes, Jorge R. Costa
Abstract:We present a 35-mm diameter skin-adhesive antenna intended to operate as a gateway to relay the low power signals from a sensor implanted in the body to an external base station (BS).
Such device would be useful in the context of Body Area Networks (BANs) for continuous body-signal monitoring. The proposed adhesive repeater is low profile, and it is designed for dual-band operation at the ISM band (2.4 GHz) and in the ultrawideband (UWB) spectrum (4-10.6 GHz) using a single excitation port. The ISM band is used for in-body communication with the implants and the UWB band for off-body burst communication with the BS. The antenna consists of three layers that grant it compactness and performance robustness. We assess the in-body link between the repeater and a custom-designed miniaturized implantable probe antenna. The study is performed in the frequency-domain with results showing adequate input impedance matching (s11 ≤ -10 dB) and robustness to different body parts. We extend the analysis to time-domain by transmitting a synthetized medical signal between the two antennas. In addition, we evaluate the feasibility of an off-body link for burst communication. Again, the results indicate very good performance by the repeater antenna both in terms of impedance matching (s11 ≤ -10 dB) and preservation of time-signals (fidelity ≥ 75%), which is relevant in impulse radio systems. To the authors' best knowledge, this is the first time this kind of dual-band adhesive antenna is being proposed for BANs, complete with performance tests using frequency- and time-domain figures-of-merit.
A novel approach able to deeply optimize clinical treatments of magnetic hyperthermia with nanoparticles.
Innovative determination of the complex magnetic permeability of magnetic colloidal fluid by simply using in
vitro SAR measurements.
A rapid, alternative and broadband approach for determining the electromagnetic properties of magnetic
nanoparticles.
Possibility to test the efficiency of nanoparticles in a tissue-like environment: optimized and realistic
treatment planning for magnetic hyperthermia.
A Novel Approach for Determining the Electromagnetic Properties of a Colloidal Fluid with Magnetic Nanoparticles for Hyperthermia Applications
Danilo Brizi, Nunzia Fontana, Giulio Giovannetti, Alessandra Flori, Luca Menichetti,Saer Doumett, Giovanni Baldi, Agostino Monorchio,
Abstract:The paper presents a general analytical method for evaluating the magnetic properties of colloidal fluid with magnetic nanoparticles and agar through in vitro Specific Absorption Rate (SAR) measurements.
The approach for the determination of magnetic complex susceptibility herein presented reveals itself as simple, rapid, broadband and accurate enough to compete with alternative conventional direct methods requiring complex and expensive instrumentation. In particular, it makes use of indirect equations based on the single order Debye model combined with a punctual set of in vitro SAR measurements. The procedure has a general validity and it can be easily applied in the up-growing field of magnetic hyperthermia studies.
We have used our microwave-imaging system in order to successfully reconstruct the Supelec-breast phantom and evaluate its usability.
Our system is capable of recover images of the studied phantoms, although the interior of the Supelec phantom is challenging due to the high plastic content.
The studied phantom was developed to be a tool within the microwave-imaging community.
These are the first published reconstructed images of the Supelec phantom and shows that the phantom is a useful tool when developing a microwave-imaging system.
It is also shown that our system is capable of reconstructing an image of the phantom without utilizin | CommonCrawl |
Entropic Forces
In 2009, Erik Verlinde argued that gravity is an entropic force. This created a big stir—and it helped him win about $6,500,000 in prize money and grants! But what the heck is an 'entropic force', anyway?
Entropic forces are nothing unusual: you've felt one if you've ever stretched a rubber band. Why does a rubber band pull back when you stretch it? You might think it's because a stretched rubber band has more energy than an unstretched one. That would indeed be a fine explanation for a metal spring. But rubber doesn't work that way. Instead, a stretched rubber band mainly has less entropy than an unstretched one—and this too can cause a force.
You see, molecules of rubber are like long chains. When unstretched, these chains can curl up in lots of random wiggly ways. 'Lots of random ways' means lots of entropy. But when you stretch one of these chains, the number of ways it can be shaped decreases, until it's pulled taut and there's just one way! Only past that point does stretching the molecule take a lot of energy; before that, you're mainly decreasing its entropy.
So, the force of a stretched rubber band is an entropic force.
But how can changes in either energy or entropy give rise to forces? That's what I want to explain. But instead of talking about force, I'll start out talking about pressure. This too arises both from changes in energy and changes in entropy.
Entropic pressure — a sloppy derivation
If you've ever studied thermodynamics you've probably heard about an ideal gas. You can think of this as a gas consisting of point particles that almost never collide with each other—because they're just points—and bounce elastically off the walls of the container they're in. If you have a box of gas like this, it'll push on the walls with some pressure. But the cause of this pressure is not that slowly making the box smaller increases the energy of the gas inside: in fact, it doesn't! The cause is that making the box smaller decreases the entropy of the gas.
To understand how pressure has an 'energetic' part and an 'entropic' part, let's start with the basic equation of thermodynamics:
What does this mean? It means the internal energy of a box of stuff changes when you heat or cool it, meaning that you change its entropy but also when you shrink or expand it, meaning that you change its volume Increasing its entropy raises its internal energy at a rate proportional to its temperature Increasing its volume lowers its internal energy at a rate proportional to its pressure
We can already see that both changes in energy, and entropy, can affect Pressure is like force—indeed it's just force per area—so we should try to solve for
First let's do it in a sloppy way. One reason people don't like thermodynamics is that they don't understand partial derivatives when there are lots different coordinate systems floating around—which is what thermodynamics is all about! So, they manipulate these partial derivatives sloppily, feeling a sense of guilt and unease, and sometimes it works, but other times it fails disastrously. The cure is not to learn more thermodynamics; the cure is to learn about differential forms. All the expressions in the basic equation are differential forms. If you learn what they are and how to work with them, you'll never get in trouble with partial derivatives in thermodynamics—as long as you proceed slowly and carefully.
But let's act like we don't know this! Let's start with the basic equation
and solve for First we get
This is fine. Then we divide by and get
This is not so fine: here the guilt starts to set in. After all, we've been told that we need to use 'partial derivatives' when we have functions of several variables—and the main fact about partial derivatives, the one that everybody remembers, is that these are written with with curly d's, not ordinary letter d's. So we must have done something wrong. So, we make the d's curly:
But we still feel guilty. First of all, who gave us the right to make those d's curly? Second of all, a partial derivative like makes no sense unless is one of a set of coordinate functions: only then we can talk about how much some function changes as we change while keeping the other coordinates fixed. The value of actually depends on what other coordinates we're keeping fixed! So what coordinates are we using?
Well, it seems like one of them is and the other is… we don't know! It could be or or or perhaps even This is where real unease sets in. If we're taking a test, we might in desperation think something like this: "Since the easiest things to control about our box of stuff are its volume and its temperature, let's take these as our coordinates!" And then we might write
And then we might do okay on this problem, because this formula is in fact correct! But I hope you agree that this is an unsatisfactory way to manipulate partial derivatives: we're shooting in the dark and hoping for luck.
Entropic pressure and entropic force
So, I want to show you a better way to get this result. But first let's take a break and think about what it means. It means there are two possible reasons a box of gas may push back with pressure as we try to squeeze it smaller while keeping its temperature constant. One is that the energy may go up:
will be positive if the internal energy goes up as we squeeze the box smaller. But the other reason is that entropy may go down:
will be positive if the entropy goes down as we squeeze the box smaller, assuming
Let's turn this fact into a result about force. Remember that pressure is just force per area. Say we have some stuff in a cylinder with a piston on top. Say the the position of the piston is given by some coordinate and its area is Then the stuff will push on the piston with a force
and the change in the cylinder's volume as the piston moves is
gives us
So, the force consists of two parts: the energetic force
and the entropic force:
Energetic forces are familiar from classical statics: for example, a rock pushes down on the table because its energy would decrease if it could go down. Entropic forces enter the game when we generalize to thermal statics, as we're doing now. But when we set these entropic forces go away and we're back to classical statics!
Entropic pressure—a better derivation
Okay, enough philosophizing. To conclude, let's derive
in a less sloppy way. We start with
which is true no matter what coordinates we use. We can choose 2 of the 5 variables here as local coordinates, generically at least, so let's choose and Then
and similarly
Using these, our equation
If you know about differential forms, you know that the differentials of the coordinate functions, namely and form a basis of 1-forms. Thus we can equate the coefficients of in the equation above and get:
and thus:
which is what we wanted! There should be no bitter aftertaste of guilt this time.
That's almost all I want to say: a simple exposition of well-known stuff that's not quite as well-known as it should be. If you know some thermodynamics and are feeling mildly ambitious, you can now work out the pressure of an ideal gas and show that it's completely entropic in origin: only the first term in the right-hand side above is nonzero. If you're feeling a lot more ambitious, you can try to read Verlinde's papers and explain them to me. But my own goal was not to think about gravity. Instead, it was to ponder a question raised by Allen Knutson: how does the 'entropic force' idea fit into my ruminations on classical mechanics versus thermodynamics?
It seems to fit in this way: as we go from classical statics (governed by the principle of least energy) to thermal statics at fixed temperature (governed by the principle of least free energy), the definition of force familiar in classical statics must be adjusted. In classical statics we have
is the energy as a function of some coordinates on the configuration space of our system, some manifold But in thermal statics at temperature our system will try to minimize, not the energy but the Helmholtz free energy
is the entropy. So now we should define force by
and we see that force has an entropic part and an energetic part:
When the entropic part goes away and we're back to classical statics!
I'm subject to the natural forces. – Lyle Lovett
This entry was posted on Wednesday, February 1st, 2012 at 4:29 am and is filed under information and entropy, physics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.
18 Responses to Entropic Forces
Suggestion for next post: Deriving the force-extension curve of a freely-jointed chain model for a polymer ("rubber band"). Then the worm-like-chain model, and compare with single-molecule DNA-stretching experiments.
Mike Stay says:
So there should be a similar splitting of the momentum, with a part due to the free action and a part due to quantropy.
1 February, 2012 at 10:37 pm
Darn, you beat me to it. Shh!
Yes, the nice thing about having two analogies to play with (classical statics versus thermal statics, thermal statics versus quantum dynamics) is that one can generate a lot of ideas; it takes longer for both analogies to 'saturate' than if you have just one.
I'm busy writing a post on quantropy, where I try to work it out in an example so we can explore in detail ideas like the one you mentioned. It's hard to develop a good intuition for quantropy without looking at some examples. Of course one can follow the analogies and make a lot of very good guesses about it. But the hands-on feel for entropy that I've built up through many calculations, I'm still lacking for quantropy.
Arrow says:
Shouldn't there be a + sign in equation for entropic force?
Anyway I always have trouble with entropy and especially with the notion of it as a fundamental quantity (same goes for information).
For example let's look at the simplest case I can think of – of one dimensional piston of length L with just one molecule of ideal gas going back and forth between the walls. The molecule will hit walls with certain average frequency dependent on the average momentum (ie temperature). So if I understand it correctly in this case entropy is directly related to the length of a piston since to describe the microscopic state we have to specify the position of the molecule and it's direction. So decreasing the piston length L while keeping temperature (and therefore avg momentum) constant will decrease the entropy and also result in the molecule hitting the walls more frequently so the avg. force exerted by the molecule on the walls will increase.
Ok, so one could say that the average force increased because of decrease in entropy, but while correct that is an abstract statement which seems (to me anyway) much less informative then stating that the average force increased due to decrease in piston length. Here the piston length seems like a fundamental parameter of the problem and entropy is just an abstract concept derived from it.
Now I understand the usefulness of entropy when talking about macroscopic processes since it allows us to abstract from the details of microscopic behavior so we can calculate useful quantities even when we don't have good grasp of the details of microscopic behavior in our problem. But I don't see it's usefulness at the microscopic level where quantities like space, time, momentum and energy seem much more fundamental and relevant.
This is also why the notion of "gravity as an entropic force" seems much less appealing to me then gravity as spacetime curvature (if only other forces could be derived from spacetime geometry…btw I've seen papers that show EM can be seen as a manifestation of spacetime torsion, is this a valid approach?).
Arrow wrote:
No, not if we're talking about the same equation. But you may indeed have noticed an inconsistency in what I wrote, due to a typo. I wrote:
But that last minus sign was wrong. In fact
In other words, the entropic force points in the direction of increasing entropy (at least if , which is true except in rather unusual circumstances, which I will ignore henceforth).
So if I understand it correctly in this case entropy is directly related to the length of a piston since to describe the microscopic state we have to specify the position of the molecule and it's direction. So decreasing the piston length L while keeping temperature (and therefore avg momentum) constant will decrease the entropy and also result in the molecule hitting the walls more frequently so the avg. force exerted by the molecule on the walls will increase.
It sounds like you're saying the force decreases if the entropy increases as we expand the piston. The equations I'm throwing around say the force is positive if if the entropy increases as we expand the piston:
It's true that as you expand a piston full of ideal gas, the force pushing on its top decreases. But my blog post is talking about itself, not how this force changes as you change (the length of the piston). Obviously the force changes like this:
I will avoid discussing gravity, except for this:
btw I've seen papers that show EM can be seen as a manifestation of spacetime torsion, is this a valid approach?
I'd never seen such an idea, despite spending an unhealthy amount of time thinking about 'teleparallel gravity', a theory that's almost equivalent to general relativity, but in which gravity is described using torsion rather than curvature. Now that you mention it, I see a paper that claims you can describe gravity coupled to electromagnetism and spinors using torsion. I can see that it's not the work of a crackpot, but I can't assure you that it's correct.
WebHubTelescope says:
This is an excellent post, and a jumping off point for lots of discussion.
Here is one —
If we were to use the rubber band analogy in terms of the greenhouse gas theory, how would it work?
I would suggest that a greenhouse gas serves to limit the outgoing radiation into bands of wavelength. This reduces the space of allowable energy states and thus reduces the entropy of the subsystem. However, we still must maintain an energy balance with the external system, and so the entropic part of the decrease in free energy is exactly compensated by a temperature increase.
At the most elemental level, that is why greenhouse gases raise the temperature of a planet's surface. We can talk all we want about variability in climate dynamics and atmospheric lapse rate, etc, but this is the heart of the argument.
Stretching the rubber band is like putting notches in the emission spectrum. That decreases entropy of the photonic volume, and temperature has to compensate. Mathematically, this is calculated by rescaling the Planck gray-body response.
I bring this up because the complexity of the gravity=entropic force argument makes this look simple in comparison.
So now we have four very similar equations:
The minimal-action path given Hamilton's principal function satisfies
d(Action) = Momentum * d(Position) – Energy * d(Time)
where all of these are functions of time.
The one you talked about here is
d(Energy) = kT * d(Entropy) – Force * d(Position).
If we have a statistical ensemble of paths and need to choose one based on a constraint on the mean action and, say, the mean position at a given time, we have
d(Action) = Lambda * d(Entropy) – Momentum * d(Position)
When we do quantum superpositions rather than statistical ensembles, we get your notion of quantropy.
If we have a rubber band under tension and increase the temperature (like in this heat engine described by Feynman) then the rubber band contracts:
d(Entropy) = Force * d(ThermalExpansionCoefficient) – Energy * d(Coolness)
Can we describe these last three in a similar way to the first? As we change the position of the piston, do the temperature and entropy change as though they were a particle moving in phase space with energy playing the role of Hamilton's principal function? Similarly, if we change the temperature, do the force and thermal expansion coefficient change as though they were a particle moving in a phase space with entropy playing the role of the principal function?
Mike wrote:
As we change the position of the piston, do the temperature and entropy change as though they were a particle moving in phase space with energy playing the role of Hamilton's principal function? Similarly, if we change the temperature, do the force and thermal expansion coefficient change as though they were a particle moving in a phase space with entropy playing the role of the principal function?
Yes, I believe so! Blake mentioned some examples of this phenomenon here, where he wrote:
Here's M. J. Peterson (1979), "Analogy between thermodynamics and mechanics" American Journal of Physics 47, 6: 488, DOI:10.1119/1.11788.
We note that equations of state—by which we mean identical relations among the thermodynamic variables characterizing a system—are actually first‐order partial differential equations for a function which defines the thermodynamics of the system. Like the Hamilton‐Jacobi equation, such equations can be solved along trajectories given by Hamilton's equations, the trajectories being quasistatic processes which obey the given equation of state. This gives rise to the notion of thermodynamic functions as infinitesimal generators of quasistatic processes, with a natural Poisson bracket formulation. This formulation of thermodynamic transformations is invariant under canonical coordinate transformations, just as classical mechanics is, which is to say that thermodynamics and classical mechanics have the same formal structure, namely a symplectic structure.
The boldface sentence is a way of saying 'yes' to your question in a bunch of thermodynamic examples. I'm pretty sure it's a very general fact.
Hello, John! Are your posts (for example this one) available as PDF's? Some of them, like network theory, are on azimuth wiki, which can produce pdf, but not this one. I wanted to read it on a e-book reader, however this html doesn't fit it really well, especially latex as images.
Hi! No, I haven't made them available as PDFs. You can get these series of posts on my website:
• Information Geometry.
• Network Theory.
I think they look better there than here—just click the box on top to get the jsmath set up and the box will go away.
I not put my posts on quantropy or 'thermodynamics versus classical mechanics' onto my website yet, but I will, and I'll let people know when I do. It takes a bit of work. I'll probably put them into a single series, because they belong together. (In fact all this stuff fits together into a big story, but that's going to take a while for me to flesh out!)
I'm writing a paper based on the Network Theory series, and I plan to write a paper on quantropy too. They'll be more polished than these blog posts…
One reason people don't like thermodynamics is that they don't understand partial derivatives…
Well, I do love thermodynamics, but the most difficult thing for me is to decide what is the sign near the work term . And what work is it — done by the system or by the environment. May be there is some trick to remember?
Anyway, I hope what follows will be right. So consider a rubber band of length — let it be the only geometric parameter describing the band. Let be the force that pulls your handwhen you are stretching the band. So if it pulls, it is positive. Then:
Hence :
Thus if you heat the rubber band it will pull harder, it shrinks. I was just curious whether I could prove it :-) Maybe I failed but the fact still holds. One need to know this property of rubber in order to explain the rotating sense of a rubber band heat engine.
Hi! I don't think there's any 'trick' to remembering the sign of work. I agree that it's an annoying issue. But it just means I need to spend a minute deciding whether I'm talking about the work the system is doing on the environment or the work the environment is doing on the system, which has the opposite sign.
I find it much more annoying when people tell me set my watch "forward" one hour when Daylight Saving Time starts in the spring. Do they mean to set my watch to an earlier time, or a later time? The word "forward" is confusing. The "forward" of a book is near the front, but as you read "forwards" through the book you move toward the back. Similarly, ancient history is the study of the time when everything was a lot younger than it is now!
I had to learn category theory to really understand this stuff.
Of course, one can try to choose a convention and stick with it. President Kennedy famously said "ask not what your country can do for you—ask what you can do for your country!" So he preferred to always think about the work the system (you) did on its environment (your country).
Thus if you heat the rubber band it will pull harder, it shrinks. I was just curious whether I could prove it :-)
I think your argument is correct, and it's nice! My argument would be to use the formula I gave:
The force of a rubber band or stretched spring has an entropic part (the first term) and an energetic part (the second term). The entropic part is proportional to temperature, so it gets bigger when it's hot. The energetic part doesn't change.
The entropic part is proportional to temperature, so it gets bigger when it's hot. The energetic part doesn't change.
Before proceeding, note that my is opposite to your $F$ — when the band pulls is negative (pressure is negative here, unlike the gas piston). So, to my point. Actually both parts depend on temperature and they both can change. So from your formula one should carefully find
So both derivations are identical indeed (despite the notion difference ).
But I'd like to emphasize again what really matters — the sign
$\frac{\partial S}{\partial L} < 0 $
For a metal rod or a piston (and I guess for a spring) it is positive. Stretching these systems increases the phase space allowed for the system so the entropy increases. Meanwhile if you heat the systems mentioned they expand. Well, just like we were taught at school "when a substance is heated it expands".
The story is opposite for a rubber band. If it is stretched, its entropy decreases. If it is heated it contracts. So the whole thing was to demonstrate how these two "anomalies" are interconnected.
Thermodynamic identities « Peeter Joot's Blog. says:
[…] John Baez. Entropic forces, 2012. URL https://johncarlosbaez.wordpress.com/2012/02/01/entropic-forces/. […]
Rubber and Rubber Balloons: Paradigms of Thermodynamics | Enteropia says:
[…] The thing with rubber is that the elastic forces you experience are entropic, that is when you stretch a rubber band you (roughly speaking) do not increase its internal energy, you decrease its entropy. That's because rubber molecules are long twisted chains and when you expand rubber you straighten them, thus ordering (decreasing their entropy). A simple kinetic theory of rubber based on entropic reasoning is presented in the book. For quick introduction on rubber thermodynamics I suggest you John Baez's post about entropic forces. […]
amarashiki says:
I has been thinking about this post for a long time, John. The reason is that your expression for the force as the sum of an entropic term plus a potential (energy) term looks pretty similar (but not identical) to the expresion of the force in lagrangian dynamics with dissipation. The big "but" is that the dissipative part is generally assumed to take the form of the so-called Rayleigh dissipative function D:
Therefore, if we identify the entropic part with the dissipative term related to the Rayleigh function, we have
Does this last equation make sense?
I don't think it makes sense to identify an entropic force with a frictional force coming from a Rayleigh function, because a frictional force is almost always velocity-dependent while an entropic force is often not.
Furthermore, the entropic force
involves a partial derivative with respect to while the frictional force
involves a partial derivative with respect to .
Furthermore, the entropic force is proportional to temperature, , while the frictional force is not.
They seem very different.
Life and the Second Law – Yet another Blog says:
[…] Starting with the discussion of the inclined plane in school, most people are used to to energetic forces. These are forces that arise due to the gradient of an energy function, often also called a potential function or just potential. Examples are the electric forces drawing electrons through wires due to the potential difference created by a battery in simple circuits, or the force of gravity. Think again of our red ball on the slope, who's acceleration and de-acceleration is simply due to the gradient in its gravitational potential function, which just happens to be the slope at its current position. However, many seemingly familiar forces actually arise not from the drive to decrease potential energy, but from a gain in entropy. A very familiar example is the force pulling a rubber band back together when you stretch it. In its relaxed state, the long polymers making up the band can curl up and wiggle in many more ways, compared to when they are all stretched in parallel. Thus, the force pulling the band back together is not the potential energy of the molecular bonds, which would have a very different characteristic, but indeed the potential increase in entropy, i.e. the increased volume of the accessible microscopic phase space in the relaxed state. And even seemingly more familiar forces, such as the force created by an expanding gas pushing on a cylinder, are in fact of entropic origin. A very insightful, and mathematically explicit discussion of entropic forces is given in this excellent blog post by John Baez. […] | CommonCrawl |
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Based on the idea and the provided source code of Andrej Karpathy (arxiv-sanity)
X-Ray Polarimetry with the Polarization Spectroscopic Telescope Array (PolSTAR) (1510.08358)
Henric S. Krawczynski, Fabian F. Kislat, Janie Hoormann, Hiromasa Miyasaka, Matthew G. Baring, Jeffrey Booth, Finn E. Christensen, Jason Dexter, Robin J. English, Jeffrey A. Favretto, Brian W. Grefenstette, Thomas Maccarone, Tatehiro Mihara, Lorenzo Natalucci, Steven Pravdo, William W. Zhang Washington University in Saint Louis, Physics Department, McDonnell Center for the Space Sciences Jet Propulsion Laboratory, California Institute of Technology, Cahill Center for Astronomy, Astrophysics, Rice University, Department of Physics, Astronomy, Georgia College, Department of Chemistry, Physics, Astronomy, North-West University, Centre for Space Research, Technical University of Denmark, DTU Space, National Space Institute, University of Virginia, Department of Astronomy, MPI for Extraterrestrial Physics Garching, Durham University, Centre for Extragalactic Astronomy, Department of Physics, North Carolina State University, Department of Physics, Cambridge, Institute of Astronomy, UK, Penn State University, Department of Astronomy, Astrophysics, University of California, Berkeley, Department of Astronomy & Theoretical Astrophysics Center, Purdue University, Department of Physics, Astronomy, Texas Tech University, Physics Department, Nagoya University, Center for Experimental Studies, Kobayashi-Maskawa Institute for the Origin of Particles, the Universe, Univ. of Michigan in Ann Arbor, Astronomy Dept., Harvard-Smithsonian Center for Astrophysics, Istituto di Astrofisica e Planetologia Spaziali, INAF, Lawrence Livermore National Laboratory, Tohoku University, Astronomical Institute)
Oct. 28, 2015 astro-ph.IM, astro-ph.HE
This paper describes the Polarization Spectroscopic Telescope Array (PolSTAR), a mission proposed to NASA's 2014 Small Explorer (SMEX) announcement of opportunity. PolSTAR measures the linear polarization of 3-50 keV (requirement; goal: 2.5-70 keV) X-rays probing the behavior of matter, radiation and the very fabric of spacetime under the extreme conditions close to the event horizons of black holes, as well as in and around magnetars and neutron stars. The PolSTAR design is based on the technology developed for the Nuclear Spectroscopic Telescope Array (NuSTAR) mission launched in June 2012. In particular, it uses the same X-ray optics, extendable telescope boom, optical bench, and CdZnTe detectors as NuSTAR. The mission has the sensitivity to measure ~1% linear polarization fractions for X-ray sources with fluxes down to ~5 mCrab. This paper describes the PolSTAR design as well as the science drivers and the potential science return.
GitHub 0
NuSTAR Detection Of A Cyclotron Line In The Supergiant Fast X-ray Transient IGR J17544-2619 (1407.0112)
Varun Bhalerao, Lorenzo Natalucci, Deepto Chakrabarty, Felix Fuerst, Roman A. Krivonos, George Younes Inter University Centre for Astronomy, Astrophysics, India, INAF, Istituto di Astrofisica Spaziale e Fisica Cosmica, Space Sciences Laboratory, University of California, Berkeley, Physics Department, Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, Cahill Center for Astronomy, Astrophysics, Caltech, DTU Space, National Space Institute, Technical University of Denmark, Lawrence Livermore National Laboratory, Columbia Astrophysics Laboratory, Columbia University, Institute of Astronomy, National Tsing Hua University, Jet Propulsion Laboratory, California Institute of Technology,
Dec. 23, 2014 astro-ph.HE
We present NuSTAR spectral and timing studies of the Supergiant Fast X-ray Transient (SFXT) IGR J17544-2619. The spectrum is well-described by a ~1 keV blackbody and a hard continuum component, as expected from an accreting X-ray pulsar. We detect a cyclotron line at 17 keV, confirming that the compact object in IGR J17544-2619 is indeed a neutron star. This is the first measurement of the magnetic field in a SFXT. The inferred magnetic field strength, B = (1.45 +/- 0.03) * 10^12 G * (1+z) is typical of neutron stars in X-ray binaries, and rules out a magnetar nature for the compact object. We do not find any significant pulsations in the source on time scales of 1-2000 s.
An Ultraluminous X-ray Source Powered by An Accreting Neutron Star (1410.3590)
M. Bachetti, B. W. Grefenstette, A. Beloborodov, A. C. Fabian, S.R. Kulkarni, D. Stern CNRS, Institut de Recherche en Astrophysique et Planétologie, 9, Avenue du Colonel Roche, BP 44346, 31028 Toulouse Cedex 4, France Toulouse, France. Cahill Center for Astrophysics, 1216 East California Boulevard, California Institute of Technology, Pasadena, California 91125, USA. MIT Kavli Institute for Astrophysics, Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Physics Department, Columbia University. 538 W 120th Street, New York, NY 10027, USA 6Space Sciences Laboratory, University of California, Berkeley, CA 94720, USA DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, DK-2800 Lyngby, Denmark Lawrence Livermore National Laboratory, Livermore, CA 94550, USA Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Columbia Astrophysics Laboratory, Columbia University, New York, NY 10027, USA NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA Department of Physics, McGill University, Montreal, Quebec, H3A 2T8, Canada Department of Physics, Texas Tech University, Lubbock, TX 79409, USA Department of Astronomy, University of Michigan, 500 Church Street, Ann Arbor, MI 48109-1042, USA Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA)
Oct. 14, 2014 astro-ph.HE
Ultraluminous X-ray sources (ULX) are off-nuclear point sources in nearby galaxies whose X-ray luminosity exceeds the theoretical maximum for spherical infall (the Eddington limit) onto stellar-mass black holes. Their luminosity ranges from $10^{40}$ erg s$^{-1} < L_X$(0.5 - 10 keV) $<10^{40}$ erg s$^{-1}$. Since higher masses imply less extreme ratios of the luminosity to the isotropic Eddington limit theoretical models have focused on black hole rather than neutron star systems. The most challenging sources to explain are those at the luminous end ($L_X$ > $10^{40}$ erg s$^{-1}$), which require black hole masses MBH >50 solar masses and/or significant departures from the standard thin disk accretion that powers bright Galactic X-ray binaries. Here we report broadband X-ray observations of the nuclear region of the galaxy M82, which contains two bright ULXs. The observations reveal pulsations of average period 1.37 s with a 2.5-day sinusoidal modulation. The pulsations result from the rotation of a magnetized neutron star, and the modulation arises from its binary orbit. The pulsed flux alone corresponds to $L_X$(3 - 30 keV) = $4.9 \times 10^{39}$ erg s$^{-1}$. The pulsating source is spatially coincident with a variable ULX which can reach $L_X$ (0.3 - 10 keV) = $1.8 \times 10^{40}$ erg s$^{-1}$. This association implies a luminosity ~100 times the Eddington limit for a 1.4 solar mass object, or more than ten times brighter than any known accreting pulsar. This finding implies that neutron stars may not be rare in the ULX population, and it challenges physical models for the accretion of matter onto magnetized compact objects.
The Large Observatory For x-ray Timing (1408.6526)
M. Feroci, S. Brandt, A. Santangelo, M. Ahangarianabhari, D. Altamirano, N. Andersson, J.-L. Atteia, S. Balman, A. Baykal, S. Bianchi, F. Bocchino, S. Boutloukos, N. Bucciantini, C. Budtz-Jørgensen, G.A. Caliandro, J. Casares, P. Cerda-Duran, J. Chenevez, T. Courvoisier, A. D'Aì, D. De Martino, M. Del Santo, A. Drago, P. Esposito, Y. Favre, M. Finger, M. Gabler, E. Garcia-Berro, P. Giommi, A. Goldwurm, M. Grassi, C. Guidorzi, F. Hansen, A. Heger, J. Huovelin, K. Iwasawa, T. Johannsen, G. Kanbach, L. Keek, S. Korpela, I. Kuvvetli, P.P. Laubert, F. Longo, S. Mahmoodifar, V. Mangano, A. Martindale, M. Mendez, R. Mignani, G. Miniutti, G. Mouret, T. Muñoz-Darias, P. O'Brien, M. Orlandini, F. Ozel, J. M. Paredes, A. Pellizzoni, C. Pittori, M. Prakash, P. Ramon, I. Rashevskaya, M. Reina Aranda, M. Ribo, P. Rodríguez- Gil, E.M.R. Rossi, L. Sabau-Graziati, S. Scaringi, S. Shore, J.-Y. Seyler, V. Sochora, B. Stappers, T.E. Strohmayer, T. Takahashi, L. Tolos, D.F. Torres, S. Turriziani, P. Varniere, S. Watanabe, H. Wende, C.A. Wilson-Hodge, N. Zampa, F. Zwart, E. Kuulkers INFN, Sez. Roma Tor Vergata, Rome, Italy, ISDC, Geneve University, Switzerland, Astronomical Institute Anton Pannekoek, University of Amsterdam, The Netherlands, INAF-IASF-Bologna, Italy, Faculty of Physical, Applied Sciences, University of Southampton, United Kingdom, ASDC, Rome, Italy, Dipartimento di Chimica e Fisica, Palermo University, Italy, Politecnico Milano, Italy, Dept. of Physics, Astronomy University of Padua, Italy, IAAT Tuebingen, Germany, National Space Institute, Lyngby, Denmark, DAM, ICC-UB, Universitat de Barcelona, Spain, Cagliari University, Italy, Astronomical Institute of the Academy of Sciences of the Czech Republic, Czech Republic, Cambridge University, Cambridge, United Kingdom, Laboratoire d'Astrophysique de Bordeaux, France, MIT, Cambridge, United States, McGill University, Montréal, Canada, Ferrara University, Ferrara, Italy, Department of Medical Biophysics, University of Toronto, Canada, Leicester University, United Kingdom, Universities Space Research Association, Huntsville, United States, Monash Centre for Astrophysics, School of Physics, School of Mathematical Sciences, Monash University, Australia, University of Tasmania, Australia, Radboud University, The Netherlands, Open University, United Kingdom, NASA/Marshall Space Flight Center, Huntsville, United States, Durham University, United Kingdom, University of Iowa, United States, Copernicus Astronomical Center, Warsaw, Poland, NASA/Marshall Space Flight Center, United States, Cornell University, Ithaca, United States, Dipartimento di Fisica, Università degli Studi di Milano, Italy, University of Trieste, Italy, University of California, United States, University of Melbourne, Australia, Kapteyn Astronomical Institute, University of Groningen, The Netherlands, University of Maryland, United States, University of Alberta, Canada, Observatoire Astronomique de Strasbourg, France, INAF-OA Torino, Italy, Space Telescope Institute, United States, Raman Research Institute, India, Czech Technical University in Prague, Czech Republic, Armagh Observatory, United Kingdom, NRL, Washington, United States, Institute for Nuclear Theory, University of Washington, United States, Instituto de Astrofisica de Canarias, Tenerife, Spain, Leiden Observatory, The Netherlands, INAF-IASF-Milano, Italy, Silesian University in Opava, Czech Republic, Institut für Kernphysik, Technische Universität Darmstadt, ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Germany, Leibniz-Institut fuer Astrophysik Potsdam, Germany, National University of Ireland, Ireland, Aristotle University of Thessaloniki, Greece, Goddard Space Flight Center, Greenbelt, United States, University of Alicante, Spain, Physical Institute of the Academy of Sciences of the Czech Republic, Czech Republic, University of Warwick, United Kingdom, INAF-OA Padova, Padova, Italy, University of Rome Tor Vergata, Italy, University of Bologna, Italy, Universidad de La Laguna, Santa Cruz de Tenerife, Spain, APC, Université Paris Diderot, CEA/Irfu, Observatoire de Paris, France, School of Physics, Astronomy, University of Southampton, United Kingdom, Kepler Institute of Astronomy, University of Zielona Gòra, Poland, University of Wisconsin, United States, Wayne State University, Detroit, United States, Foundation for Research, Technology, Heraklion, Greece, CNES, Toulouse, France, Instituto Astrofisica de Andalucia, Granada, Spain, Perimeter Institute for Theoretical Physics, Waterloo, Canada, Università di Napoli Fedelico II, Italy, School of Physics, Astronomy, University of Birmingham, United Kingdom, University of California, Berkeley, Space Sciences Laboratory, United States, Ohio University, United States, Max-Planck-Institut fuer extraterrestrische Physik, Garching, Germany, Max Planck Institute for Gravitational Physics, Germany, Technical University of Catalonia, Barcelona, Spain, Department of Physics, Astronomy, University of Waterloo, Canada, Sapienza University, Rome, Italy, Institute for Astronomy K.U. Leuven, Leuven, Belgium, Texas Tech. University, United States, Tata Institute of Fundamental Research, Mumbai, India, Jorgen Sandberg Consulting, Denmark, Istanbul Kültür University, Turkey, Facultad de Ciencias-Trilingüe University of Salamanca, Spain, University of Surrey, United Kingdom, Oxford University, United Kingdom, European Space Agency, ESTEC, The Netherlands, European Space Astronomy Centre, Madrid, Spain,
Aug. 29, 2014 astro-ph.IM
The Large Observatory For x-ray Timing (LOFT) was studied within ESA M3 Cosmic Vision framework and participated in the final down-selection for a launch slot in 2022-2024. Thanks to the unprecedented combination of effective area and spectral resolution of its main instrument, LOFT will study the behaviour of matter under extreme conditions, such as the strong gravitational field in the innermost regions of accretion flows close to black holes and neutron stars, and the supra-nuclear densities in the interior of neutron stars. The science payload is based on a Large Area Detector (LAD, 10 m 2 effective area, 2-30 keV, 240 eV spectral resolution, 1 deg collimated field of view) and a WideField Monitor (WFM, 2-50 keV, 4 steradian field of view, 1 arcmin source location accuracy, 300 eV spectral resolution). The WFM is equipped with an on-board system for bright events (e.g. GRB) localization. The trigger time and position of these events are broadcast to the ground within 30 s from discovery. In this paper we present the status of the mission at the end of its Phase A study.
The LOFT Ground Segment (1408.6541)
E. Bozzo, P. Binko, J.W. den Herder, L. Guy, E. Kuulkers, R. Rohlfs, P.J. Smith ISDC, University of Geneva, Switzerland, CNRS, IRAP, Toulouse, France, National Space Institute, Technical University of Denmark, Lyngby, Denmark, SRON, The Netherlands Institute of Space Research, Utrecht, The Netherlands, CEA Saclay, DSM/IRFU/SAp, France, ESA/ESAC, Madrid, Spain, Observatoire Astronomique de Strasbourg, France, University of Erlangen-Nuremberg, Germany, Leibniz-Institut fuer Astrophysik Potsdam, Germany, ESA/ESTEC, Noordwijk, Netherlands, INFN, Sez. Roma Tor Vergata, Italy,
LOFT, the Large Observatory For X-ray Timing, was one of the ESA M3 mission candidates that completed their assessment phase at the end of 2013. LOFT is equipped with two instruments, the Large Area Detector (LAD) and the Wide Field Monitor (WFM). The LAD performs pointed observations of several targets per orbit (~90 minutes), providing roughly ~80 GB of proprietary data per day (the proprietary period will be 12 months). The WFM continuously monitors about 1/3 of the sky at a time and provides data for about ~100 sources a day, resulting in a total of ~20 GB of additional telemetry. The LOFT Burst alert System additionally identifies on-board bright impulsive events (e.g., Gamma-ray Bursts, GRBs) and broadcasts the corresponding position and trigger time to the ground using a dedicated system of ~15 VHF receivers. All WFM data are planned to be made public immediately. In this contribution we summarize the planned organization of the LOFT ground segment (GS), as established in the mission Yellow Book 1 . We describe the expected GS contributions from ESA and the LOFT consortium. A review is provided of the planned LOFT data products and the details of the data flow, archiving and distribution. Despite LOFT was not selected for launch within the M3 call, its long assessment phase (> 2 years) led to a very solid mission design and an efficient planning of its ground operations.
NuSTAR Observations of the Magnetar 1E 2259+586 (1408.0768)
Julia K. Vogel, Hongjun An, Steven E. Boggs, Eric V. Gotthelf, Fiona A. Harrison, Michael J. Pivovaroff Physics Division, Physical, Life Sciences Directorate, Lawrence Livermore National Laboratory, Livermore, CA, USA, Physics Department, Columbia Astrophysics Laboratory, Columbia University, New York, NY, USA, Department of Physics, McGill University, Montreal, Quebec, Canada, Space Sciences Laboratory, University of California, Berkeley, CA, USA, DTU Space, National Space Institute, Technical University of Denmark, Lyngby, Denmark, Cahill Center for Astronomy, Astrophysics, California Institute of Technology, Pasadena, CA, USA, Department of Astronomy, Astrophysics, Pennsylvania State University, University Park, PA, USA, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA,
Aug. 4, 2014 astro-ph.HE
We report on new broad band spectral and temporal observations of the magnetar 1E 2259+586, which is located in the supernova remnant CTB 109. Our data were obtained simultaneously with the Nuclear Spectroscopic Telescope Array (NuSTAR) and Swift, and cover the energy range from 0.5-79 keV. We present pulse profiles in various energy bands and compare them to previous RXTE results. The NuSTAR data show pulsations above 20 keV for the first time and we report evidence that one of the pulses in the double-peaked pulse profile shifts position with energy. The pulsed fraction of the magnetar is shown to increase strongly with energy. Our spectral analysis reveals that the soft X-ray spectrum is well characterized by an absorbed double-blackbody or blackbody plus power-law model in agreement with previous reports. Our new hard X-ray data, however, suggests that an additional component, such as a power-law, is needed to describe the NuSTAR and Swift spectrum. We also fit the data with the recently developed coronal outflow model by Beloborodov for hard X-ray emission from magnetars. The outflow from a ring on the magnetar surface is statistically preferred over outflow from a polar cap. | CommonCrawl |
Why does the delta-V for LEO-KSC to LEO-Equatorial orbit seem unusually high?
After reading Q36179 and this table on the Wikipedia page for Delta-V budget (also linked from Q36179), I'm curious:
Why does going from a KSC-launched LEO to an orbit like one launched from the equator seem to require so much $\mathrm{\Delta V}$? $\mathrm{4.24~\frac{km}{s}}$ is almost half of what's required just to get to LEO to begin with, and that's factoring in a prograde launch!
Isn't this just a change in orbital inclination? KSC is "only" 28.57$^{\circ}$ N. latitude.
Related question: Is there a general formula for determining $\mathrm{\Delta V}$ given the initial and final (or change of) inclination? My intuition says that some trig functions would be involved.
orbital-motion
rocket-science
asked Aug 12, 2016 at 2:01
pr1268pr1268
$\begingroup$ I can tell you from playing KSP that plane changes are very expensive. You have to cancel a lot of vertical velocity. $\endgroup$
Isn't this just a change in orbital inclination?
"Just" is one of the most dangerous words in science and engineering.
Plane changes are very expensive in terms of delta V costs. Changing orbital inclination is a plane change. The cost to make an inclination change from a circular orbit to another circular orbit, with no change in orbital altitude, is $\Delta v = 2v\sin(\Delta i / 2)\ $.
For example, the delta V needed to change from a geosynchronous orbit with an inclination of 28 degrees to a geostationary orbit is 1.4 km/s. The delta V cost to successively go from a low Earth orbit inclined at 28 degrees to a geostationary orbit inclined at 28 degrees and then to a geosynchronous orbit is 5.3 km/s.
That three burn sequence (one in LEO to transfer to geosynchronous altitude, a second at geosynchronous altitude to circularize the orbit, and a third at the node to transfer to geostationary) is not how it's done. Delta V costs are nonlinear, thanks to the nonlinear way in which gravitation works. The result is a savings of about one km/s compared to that three burn sequence, but it's still more expensive than launching from the equator.
Regarding the wikipedia page on delta V budget, you have to take that page with a huge grain of salt.
edited Aug 12, 2016 at 13:01
answered Aug 12, 2016 at 2:59
David HammenDavid Hammen
$\begingroup$ Should I assume that the word "only" (in the next sentence) is equally dangerous? Note that I did put scare quotes around it. ;-) $\endgroup$
– pr1268
$\begingroup$ Also, my original question assumed a change in orbital inclination but remaining a low-Earth orbit. Does an orbital plane change remaining at LEO require a transfer to geosynchronous orbit and back? $\endgroup$
Would a rocket burn more fuel to get from Earth's surface to LEO, or to get from LEO to GEO?
What are the temperatures of objects in Low Earth Orbit (LEO)?
Does the length of the sidereal day vary systematically?
Orbital mechanics and rocketry: Is it ever a good idea to intentionally lower periapsis?
What are the steps and energetic requirements for getting from the HST to the ISS?
Why does Pluto's orbit cross Neptune's orbit?
Delta-v required to get to Medium Earth Orbit?
Does launching spacecraft from the southern United States actually give a meaningful energy boost?
Why is the JWST orbit only taking 30 days?
Does the inclination of the orbit affect the orbital period of a celestial body around the Sun?
$\Delta v$ to raise the apogee of an orbit? | CommonCrawl |
\begin{document}
\title{Secure entanglement distillation for double-server blind quantum computation} \author{Tomoyuki Morimae} \email{[email protected]} \affiliation{ASRLD Unit, Gunma University, 1-5-1 Tenjin-cho Kiryu-shi Gunma-ken, 376-0052, Japan} \affiliation{Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom}
\author{Keisuke Fujii} \email{[email protected]} \affiliation{The Hakubi Center for Advanced Research, Kyoto University, Yoshida-Ushinomiya-cho, Sakyo-ku, Kyoto 606-8302, Japan} \affiliation{Graduate School of Informatics, Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto 606-8501, Japan} \affiliation{Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan} \date{\today}
\begin{abstract} Blind quantum computation is a new secure quantum computing protocol where a client, who does not have enough quantum technologies at her disposal, can delegate her quantum computation to a server, who has a fully-fledged quantum computer, in such a way that the server cannot learn anything about client's input, output, and program. If the client interacts with only a single server, the client has to have some minimum quantum power, such as the ability of emitting randomly-rotated single-qubit states or the ability of measuring states. If the client interacts with two servers who share Bell pairs but cannot communicate with each other, the client can be completely classical. For such a double-server scheme, two servers have to share clean Bell pairs, and therefore the entanglement distillation is necessary in a realistic noisy environment. In this paper, we show that it is possible to perform entanglement distillation in the double-server scheme without degrading the security of the blind quantum computing. \end{abstract}
\pacs{03.67.-a} \maketitle
A first generation quantum computer will be implemented in a ``cloud" style, since only limited number of groups, such as huge industries and governments, will be able to possess it. When a client uses such a quantum server via a remote access, it is crucial to protect client's privacy. Blind quantum computation~\cite{BFK,FK,Barz,Vedran,AKLTblind,topoblind,CVblind,topoveri,MABQC,Sueki,composable,composableMA} is a new secure quantum computing protocol which can guarantee the security of client's privacy in such a cloud quantum computing. Protocols of blind quantum computation enable a client (Alice), who does not have enough quantum technologies at her disposal, to delegate her quantum computation to a server (Bob), who has a fully-fledged quantum computer, in such a way that Alice's input, output, and program are hidden to Bob~\cite{BFK,FK,Barz,Vedran,AKLTblind,topoblind,CVblind,topoveri,MABQC,Sueki,composable,composableMA}.
The original protocol of blind quantum computation was proposed by Broadbent, Fitzsimons, and Kashefi (BFK)~\cite{BFK}. Their protocol uses the measurement-based quantum computation on the cluster state (graph state) by Raussendorf and Briegel~\cite{Raussendorf}. A proof-of-principle experiment of the BFK protocol has been also achieved recently with a quantum optical system~\cite{Barz}. The BFK protocol has been recently generalized to other blind quantum computing protocols which use the measurement-based quantum computation on the Affleck-Kennedy-Lieb-Tasaki (AKLT) state~\cite{AKLT,AKLTblind,Miyake}, the continuous-variable measurement-based quantum computation~\cite{CV,CVblind}, and the ancilla-driven model~\cite{Janet,Sueki}.
Since the original BFK protocol was proposed, new protocols have been developed in order for blind quantum computation to be more practical. One direction is making blind protocols more fault-tolerant. While the BFK protocol, which utilizes the brickwork state, would be fault-tolerant, its threshold value is extremely small. The recently proposed topological blind quantum computation~\cite{topoblind} employs a special three-dimensional cluster state~\cite{Raussendorf_topo} and allows us to perform topologically protected blind quantum computation even with a high error probability $0.43\%$ (i.e., fidelity of $99.57\%$) in preparations, measurements, and gate operations.
Another direction is making Alice as classical as possible. In the above BFK-based protocols, Alice emits randomly-rotated single-qubit states, such as single-photon states. Recently, it was shown~\cite{Vedran} that in stead of single-photon states, coherent states are also sufficient. Since coherent states are considered to be more classical than single-photon states, this result suggests that Alice can be more classical.
It is also possible to make Alice completely classical: the double-sever blind protocol was introduced in Ref.~\cite{BFK}, where two Bobs share Bell pairs (but cannot perform classical communication with each other) and perform computational tasks ordered by Alice's classical message. The double-server blind protocol is also fault-tolerant, but Bell pairs of fidelity above 99\% are required even if topological blind quantum computation is employed. Since Bell pairs have to be sent from the third party or Alice herself via public quantum channels, such an ability to generate high-fidelity Bell pairs or encoding them into quantum error correction codes would be too demanding.
In this paper we settle this problem. We show that it is possible to perform entanglement distillation in the double-server scheme without degrading the security of blind quantum computing. As a result, the required fidelity of the Bell pairs is improved dramatically to $81\%$, which is determined by the hashing bound and achieved by quantum random coding~\cite{Bennett_PRL,Bennett_PRA}. Since the Bell pair generation of fidelity higher than $81\%$ is nowadays easily achievable by using, for example, parametric down conversion, the present result is crucial in blind quantum computation to make Alice (or the third party) as classical as possible by using practically noisy Bell pair sources and quantum channels.
\if 0 In this paper, we show that it is possible to perform entanglement distillation in the double-server scheme without degrading the security of blind quantum computing. {\color{red}In Refs.~\cite{Vedran,MABQC}, ways of making Alice more classical were proposed. In Ref.~\cite{topoblind}, a method of making Bob robust was proposed. The result of this paper means that now we can also ease the center's burden. According to the motivation of blind quantum computing, it is desirable that the center does not require any high quantum technology. Therefore, we do not want to assume that the center can encode Bell pairs with a quantum error correcting code and send them to two Bobs. Hence the center has to send bare Bell pairs to two Bobs, and in that case the error threshold is FUJII1. However, as we will show, the entanglement distillation is possible by Bobs, and therefore we obtain the error threshold FUJII2. This drastic improvement of the threshold is the main result of this paper. We also remark that it is also a likely scenario that we remove the center and assume that Alice generates Bell pairs, since generation of Bell pairs is not so difficult in several experimental setups, such as the parametric down conversion. In that case, our result gains more significance since it is ridiculous to assume that Alice can perform the encoding with a quantum error correcting code.} \fi
Before proceeding to our main result, let us briefly review the BFK blind protocol~\cite{BFK}.
Assume that Alice wants to perform the measurement-based quantum computation on the $m$-qubit graph state corresponding to the graph $G$. The quantum algorithm which Alice wants to run is specified with the measurement basis $\{|0\rangle\pm e^{i\phi_j}|1\rangle\}$ for $j$th qubit ($j=1,2,...,m$), where
$\phi_j\in\{\frac{k\pi}{4}|k=0,1,...,7\}$. (Note that such $X-Y$ plain measurements are universal~\cite{Raussendorf}.) The BFK protocol runs as follows (see also Fig.~\ref{single}): \begin{itemize} \item[S1.] Alice tells Bob the graph $G$~\cite{graph}. \item[S2.] Alice sends Bob
$\bigotimes_{j=1}^m|\theta_j\rangle$, where
$|\theta_j\rangle\equiv|0\rangle+e^{i\theta_j}|1\rangle$ and
$\theta_j$ is randomly chosen by Alice from $\{\frac{k\pi}{4}|k=0,1,...,7\}$. \item[S3.] Bob makes
$|G\{\theta_j\}\rangle\equiv\Big(\bigotimes_{(i,j)\in E}CZ_{i,j}\Big)
\bigotimes_{j=1}^m|\theta_j\rangle$, where $E$ is the set of edges of $G$ and $CZ_{i,j}$ is the $CZ$ gate between $i$th and $j$th qubits. \item[S4.]
Alice and Bob now perform the measurement-based quantum computation on $|G\{\theta_j\}\rangle$ with two-way classical communications as follows: when Alice wants Bob to measure $j$th qubit ($j=1,2,...,m$)
of $|G\{\theta_j\}\rangle$, she sends Bob $\delta_j\equiv\theta_j+\phi_j'+r_j\pi$, where $r_j\in\{0,1\}$ is a random binary chosen by Alice and $\phi_j'$ is the modified version of $\phi_j$ according to the previous measurement results, which is the standard feed-forwarding of the one-way model~\cite{Raussendorf}. Bob measures $j$th qubit in the basis
$\{|0\rangle\pm e^{i\delta_j}|1\rangle\}$ and tells the measurement result to Alice. \end{itemize}
\begin{figure}
\caption{ The single-server blind protocol. (a) Alice sends many single-qubit states to Bob. QD is a device which emits randomly rotated single qubits. (b) Bob creates a resource state. Alice and Bob perform the measurement-based quantum computation through the two-way classical channel. CC is a classical computer. }
\label{single}
\end{figure}
We call this protocol the single-server protocol, since there is only a single server (Bob). It was shown~\cite{BFK} that whatever Bob does he cannot learn anything about Alice's input, output, and algorithm.
\if0 Because Alice's computational angle $\phi_j$ is ``one-time padded" with a random angle $\theta_j$, Bob cannot learn $\phi_j$ from $\delta_j$ (for rigorous proofs of the security of the BFK protocol, see Ref.~\cite{BFK,FK,composable}). Furthermore, if Bob is honest, Alice and Bob can perform the correct measurement-based quantum computation, since
$|G\{\theta_j\}\rangle= \Big(\bigotimes_{(i,j)\in E}CZ_{i,j}\Big)
\bigotimes_{j=1}^m|\theta_j\rangle = \Big(\bigotimes_{j=1}^me^{-iZ\theta_j/2}\Big) \Big(\bigotimes_{(i,j)\in E}CZ_{i,j}\Big)
|+\rangle^{\otimes m}$, where $|+\rangle\equiv|0\rangle+|1\rangle$
and $Z\equiv|0\rangle\langle0|
-|1\rangle\langle1|$, and therefore $\theta_j$ in $\delta_j$ is nicely canceled. In other words, what Bob does is effectively the measurement-based quantum computation on the graph state corresponding to the graph $G$ with the measurement angles $\{\phi_j\}_{j=1}^m$. \fi
In the above single-server protocol, Alice has to have the ability of emitting randomly-rotated single-qubit states, $\{|\theta_j\rangle\}_{j=1}^m$. It was shown in Ref.~\cite{BFK} that if we have two servers (Bob1 and Bob2) who share Bell pairs but cannot communicate with each other, Alice can be completely classical. (Alice has only to have a classical computer and two classical channels, one is between Alice and Bob1 and the other is between Alice and Bob2.) We call such a scheme the double-server scheme, since there are two servers. A protocol of the double-server scheme runs as follows~\cite{BFK} (see also Fig.~\ref{double}): \begin{itemize} \item[D1.] A trusted center distributes Bell pairs to Bob1 and Bob2~\cite{center}. Now Bob1 and Bob2 share $m$
Bell pairs, $(|00\rangle+|11\rangle)^{\otimes m}$. \item[D2.]
Alice sends Bob1 classical messages $\{\theta_j\}_{j=1}^m$, where $\theta_j$ is randomly chosen by Alice from $\{\frac{k\pi}{4}|k=0,1,...,7\}$. \item[D3.] Bob1 measures his qubit of the $j$th Bell pair
in the basis $\{|0\rangle\pm e^{-i\theta_j}|1\rangle\}$ $(j=1,...,m)$. Bob1 tells Alice the measurement results $\{b_j\}_{j=1}^m\in\{0,1\}^m$. \item[D4.] After these Bob1's measurements, what Bob2 has is
$\bigotimes_{j=1}^mZ_j^{b_j}|\theta_j\rangle
=\bigotimes_{j=1}^m|\theta_j+b_j\pi\rangle$. Now Alice and Bob2 can start the single-server BFK protocol with the modification $\{\theta_j\}_{j=1}^m\to\{\theta_j+b_j\pi\}_{j=1}^m$. \end{itemize}
\begin{figure}
\caption{ The double-server blind protocol. (a) Bob1 and Bob2 share Bell pairs. Alice sends classical messages to Bob1. Bob1 performs measurements on his qubits of the Bell pairs, and tells the measurement results to Alice. (b) Alice and Bob2 run the single-server blind protocol through the two-way classical channel. CC is a classical computer. }
\label{double}
\end{figure}
In addition to the advantage of the completely classical Alice, the double-server scheme is intensively studied in computer science in the context of the multi-prover interactive proof system, which assumes computationally unbounded and untrusted prover (server), and device-independent quantum key distribution~\cite{BFK,Aharonov,Reichardt}.
Note that the impossibility of the communication between two Bobs is crucial in the double-server protocol. If Bob1 can send some message to Bob2, Bob1 can tell Bob2 $\{\theta_j+b_j\pi\}_{j=1}^m$, and then Bob2 can learn something about $\{\phi_j\}_{j=1}^m$, since Bob2 knows $\{\theta_j+b_j\pi+\phi_j'+r_j\pi\}_{j=1}^m$. On the other hand, if Bob2 can tell Bob1 $\{\theta_j+b_j\pi+\phi_j'+r_j\pi\}_{j=1}^m$, Bob1 can learn something about $\{\phi_j\}_{j=1}^m$, since Bob1 knows $\{\theta_j+b_j\pi\}_{j=1}^m$. In these cases, the security of Alice's privacy is no longer guaranteed.
In order to perform the correct double-server protocol, two Bobs must share clean Bell pairs. Sharing clean Bell pairs is also crucial in many other quantum information protocols such as the quantum teleportation~\cite{teleportation}, the quantum key distribution~\cite{QKD1,QKD2} and the distributed quantum computation~\cite{dist1,dist2,dist3,dist4,dist5}. One standard way of sharing clean Bell pairs in a noisy environment is the entanglement distillation~\cite{Bennett_PRL,Bennett_PRA,Deutsch,Dur}. In entanglement distillation protocols, two people, say Bob1 and Bob2, who want to share clean Bell pairs start with some dirty $n$ Bell pairs. Then they perform local operations with some classical communications, and finally ``distill" $m$ $(m<n)$ clean Bell pairs~\cite{Bennett_PRL,Bennett_PRA,Deutsch,Dur}.
If we consider the application of the entanglement distillation to the double-server blind protocol, one huge obstacle is that two Bobs are not allowed to communicate with each other in the double-server scheme. Hence, message exchanges between two Bobs, which are necessary for the entanglement distillation, must be done through the Alice's mediation, i.e., Bob1 (Bob2) sends a message to Alice, and Alice transfers it to Bob2 (Bob1). It is not self-evident that the security of the double-server blind protocol is guaranteed even if we plug an entanglement distillation protocol into the double-server blind protocol~\cite{donot}. For example, Bob1 might send a message to Alice pretending that it is a ``legal" message for the entanglement distillation. Alice might naively forward that message to Bob2 without noticing Bob1's evil intention and believing that it is a harmless message. In this case, Bob1 can indirectly send some message to Bob2, and hence the security of the double-server protocol is no longer guaranteed.
If the entire entanglement distillation is completed before starting the double-server protocol, and if Alice delegates her computation to Bobs only once, then the communication between two Bobs during the entanglement distillation is harmless, since when they are doing the entanglement distillation, messages related to Alice's computation are not yet sent to Bobs.
However, if Alice delegates more than twice, then two Bobs might exchange information about the previous double-server computation during the entanglement distillation for the next round of the computation as in the case of the ``device-independence" argument of the quantum key distribution with devices having memory~\cite{memory}. Furthermore, the entanglement distillation might be done in parallel with the double-server protocol in order to avoid a decoherence. In these cases, we must be careful about the communication between two Bobs during the entanglement distillation. In terms of the composable security, this means that we are interested in the composable security of the ``distillation $+$ blind computing" protocol~\cite{donot}.
Throughout this paper, we denote four Bell states by
$|\psi_{z,x}\rangle\equiv (I\otimes X^xZ^z)
(|0\rangle\otimes|0\rangle+|1\rangle\otimes|1\rangle)$, where $(z,x)\in\{0,1\}^2$ and
$X\equiv|0\rangle\langle1|+|1\rangle\langle0|$.
{\it Protocol}.---
Now let us show that the entanglement distillation by two Bobs is indeed possible without degrading the security. As in the case of the original BFK double-server protocol, a trusted center (or Alice) generates $n$ Bell states, $|\psi_{00}\rangle^{\otimes n}$,
and distribute them to two Bobs; one qubit of each $|\psi_{00}\rangle$ is sent to Bob1 and the other to Bob2. Due to the noise in the channel between the center and Bobs, each Bell state decoheres, $|\psi_{00}\rangle\to\rho$. Hence two Bobs share $n$ inpure pairs $\rho^{\otimes n}$, where $\rho$ is a dirty Bell state: one qubit of $\rho$ is possessed by Bob1 and the other is by Bob2. Without loss of generality, we can assume that $\rho$ is the Werner state, $\rho= F\psi_{11} +\frac{1-F}{3}( \psi_{00} +\psi_{01} +\psi_{10}
)$, where $\psi\equiv|\psi\rangle\langle\psi|$. If it is not the Werner state, it can be converted into the Werner state by using the twirling operation (after applying $I\otimes XZ$)~\cite{Bennett_PRA}. In order to perform the twirling operation, Alice has only to randomly choose a $SU(2)$ operator, and tell its classical description to two Bobs. Therefore the twirling operation does not affect the security.
Since $\rho$ is Bell-diagonal, $\rho^{\otimes n}$ is the mixture of tensor products of Bell states: \begin{eqnarray*} \rho^{\otimes n} = \sum_{(z_1,x_1,...,z_n,x_n)\in\{0,1\}^{2n}} p(z_1,x_1,...,z_n,x_n)\bigotimes_{j=1}^n\psi_{z_j,x_j}. \end{eqnarray*} Alice randomly chooses a $2n$-bit string $s_1$ and sends it to two Bobs. This $s_1$ is chosen completely randomly being independent of other parameters (such as $\theta_j$, $\phi_j$, etc.). Each Bob then performs certain local unitary operation which is determined by $s_1$. Each Bob next measures a qubit of a single pair in the computational basis, and tells the measurement result to Alice. (The detail of the unitary operation, which is irrelevant here, is given in Ref.~\cite{Bennett_PRA}. Which pair is measured is also determined by $s_1$~\cite{Bennett_PRA}. In brief, these unitary operations and measurements are performed for obtaining $s_1\cdot v$ (mod2) for the hashing, where $v\equiv(z_1,x_1,...,z_n,x_n)$.) From these measurement results by Bobs, Alice can gain a single bit $s_1 \cdot v$ (mod2) of information.
Since a single pair is measured out, now two Bobs share $n-1$ pairs. If Alice and two Bobs repeat a similar procedure (i.e., Alice randomly chooses a $2(n-1)$-bit string $s_2$ and tells it to two Bobs. Two Bobs perform local operations, measure a single pair in the computational basis, and tell the measurement results to Alice), Alice can gain another single bit of information. In this way, they repeat this procedure many times, and Alice obtains enough bits to perform the hashing, which works as follows.
The probability distribution $p(z_1,x_1,...,z_n,x_n)$ has almost all its weight for a set of $\sim 2^{nS(\rho)}$ ``likely" strings, where $S(\rho)$ is the von Neumann entropy of $\rho$. The probability that a $2n$-bit string $(z_1,x_1,...,z_n,x_n)$ falls outside of the set of the $2^{n(S(\rho)+\epsilon)}$ most probable strings is $O(e^{-\epsilon^2n})$~\cite{Bennett_PRA}. Therefore, Alice can (almost) specify $p(z_1,x_1,...,z_n,x_n)$ if she gains $nS(\rho)$ bits of information about $p(z_1,x_1,...,z_n,x_n)$. This means that it is sufficient for Alice and two Bobs to repeat the above procedure for $nS(\rho)$ times. Then, two Bobs spend $nS(\rho)$ pairs for measurements, and therefore at the end of the distillation they share $m\equiv n-nS(\rho)$ pairs,
$\bigotimes_{j=1}^m|\psi_{z_j,x_j}\rangle$, where $(z_j,x_j)\in\{0,1\}^2$. Alice knows the $2m$-bit string $(z_1,x_1,...,z_m,x_m)$.
After the distillation, Alice and two Bobs can start the double-server protocol. Now we modify the double-server protocol as follows: \begin{itemize} \item[D1'.] Two Bobs share
$\bigotimes_{j=1}^m|\psi_{z_j,x_j}\rangle$. \item[D2'.] Alice sends Bob1 classical messages $\{\theta_j'\equiv(-1)^{x_j}\theta_j+z_j\pi\}_{j=1}^m$, where $\theta_j$ is randomly chosen by Alice from
$\{\frac{k\pi}{4}|k=0,1,...,7\}$. \item[D3'.]
Bob1 measures his qubit of the $j$th Bell pair in the basis $\{|0\rangle\pm e^{-i\theta_j'}|1\rangle\}$ ($j=1,...,m$). Bob1 tells Alice the measurement results $\{b_j\}_{j=1}^m\in\{0,1\}^m$. \item[D4'.] The same as D4. \end{itemize} Since D4' is the same as D4, it is obvious that Alice can run the correct single-server blind quantum computation with Bob2.
{\it Bob1 cannot send any message to Bob2}.--- Let us show that Bob1 cannot send any message to Bob2. What Bob2 receives from Alice are bit strings, $s_1,...,s_{n-m}$, and $\{\theta_j+b_j\pi+\phi_j'+r_j\pi\}_{j=1}^m$. Since $s_1,...,s_{n-m}$ are completely uncorrelated with what Bob1 sends to Alice, Bob2 cannot gain any information about Bob1's message from $s_1,...,s_{n-m}$.
Furthermore, $r_j$ is randomly taken by Alice from $\{0,1\}$ being independent of what Bob1 sends to Alice. Therefore, Bob2 cannot gain any information about $b_j$ from $\theta_j+b_j\pi+\phi_j'+r_j\pi$. Bob1 and Bob2 share entangled pairs. However, due to the no-signaling principle, only sharing entangled pairs is useless for message transmission. Hence Bob1 cannot send any message to Bob2.
{\it Bob2 cannot send any message to Bob1}.--- Next let us show that Bob2 cannot send any message to Bob1. What Bob1 receives from Alice are bit strings, $s_1,...,s_{n-m}$, and $\{\theta_j'\equiv(-1)^{x_j}\theta_j+z_j\pi\}_{j=1}^m$. Again, $s_1,...,s_{n-m}$, are useless for the message transmission from Bob2 to Bob1.
Furthermore,
$\theta_j$ is randomly chosen by Alice from $\{\frac{k\pi}{4}|k=0,1,...,7\}$ being independent of what Bob2 sends to Alice and $(z_1,x_1,...,z_m,x_m)$. Therefore, Bob1 cannot gain any information about Bob2's message from $\theta_j'$.
Hence Bob2 cannot send any message to Bob1.
{\it Two Bobs cannot learn Alice's computational information}.--- Finally, let us show the security of Alice's computational information. First, from Bob2's view point, the difference between our protocol (i.e., the distillation plus the modified double-server protocol) and the original BFK double-server protocol is only that Bob2 receives bit strings, $s_1,...,s_{n-m}$, from Alice. Since these bit strings are completely uncorrelated with Alice's computational information, our protocol is as secure as the original BFK double-server protocol against Bob2.
Second, from Bob1's view point, the differences between our protocol and the original BFK double-server protocol are \begin{itemize} \item[(i)] Bob1 receives bit strings, $s_1,...,s_{n-m}$, from Alice. \item[(ii)] Bob1 receives $\theta_j'\equiv(-1)^{x_j}\theta_j+z_j\pi$ instead of $\theta_j$ from Alice ($j=1,2,...,m$). \end{itemize} Again, we can safely ignore (i).
Regarding (ii): since $\theta_j$ is randomly taken from
$\{\frac{k\pi}{4}|k=0,1,...,7\}$ being independent of Alice's computational information and $(z_1,x_1,...,z_m,x_m)$, Bob1 cannot gain any information about Alice's computation from $\theta_j'$.
Hence our protocol is as secure as the original double-server BFK protocol against Bob1.
\if0 {\it Two-way distillation}.--- Next let us consider the two-way distillation protocol~\cite{Bennett_PRL,Bennett_PRA,Deutsch}. For simplicity, we here use the entanglement pumping version~\cite{Dur}. The two-way distillation protocol adopted to the double-server blind protocol runs as follows. Here, $N$ is a sufficiently large integer. \begin{itemize} \item[TE1.]
The trusted center generates $|\psi_{00}\rangle$, which is called the control pair, and sends one qubit of it to Bob1 and the other to Bob2. Set $k=1$. \item[TE2.] If $k=N$, goto TE5. Alice randomly chooses $(y_1^k,y_2^k)$ from $\{0,1\}^2$ and sends it to the trusted center. The center generates \begin{eqnarray*}
(Y^{y_1^k}\otimes Y^{y_2^k})|\psi_{00}\rangle, \end{eqnarray*}
which is called the target pair, and sends one qubit of it to Bob1 and the other to Bob2. Here $Y\equiv i|1\rangle\langle0|-i|0\rangle\langle1|$. \item[TE3.] Two Bobs perform local operations (see Fig.~\ref{circuit} (a)) on the target and control pairs, and then measure the qubits of the target pair in the computational basis. Two Bobs tell Alice their measurement results, $b_1^k\in\{0,1\}$ and $b_2^k\in\{0,1\}$, respectively. \item[TE4.] Alice calculates $\gamma^k\equiv b_1^k\oplus b_2^k\oplus y_1^k\oplus y_2^k$, and sends $\gamma^k$ to two Bobs. If $\gamma^k=0$, two Bobs keep the control pair, set $k=k+1$, and back to TE2. If $\gamma^k=1$, two Bobs discard the control pair, and goto TE5. \item[TE5.] End of the protocol. Now two Bobs share a clean Bell pair or nothing. \end{itemize} By iterating the above protocol sufficiently many times, two Bobs share sufficiently many clean Bell pairs, hence the entanglement distillation is completed.
\begin{figure}
\caption{ ``c" means the control pair and ``t" means the target pair. Black circles are Bob1's qubits and white circles are Bob2's qubits. (a) The distillation circuit for $(y_1^k,y_2^k)=(1,1)$.
$U\equiv(|0\rangle-i|1\rangle)\langle0|+(|0\rangle+i|1\rangle)\langle1|$ and
$V\equiv(|0\rangle+i|1\rangle)\langle0|+(|0\rangle-i|1\rangle)\langle1|$. $M$ is the measurement in the computational basis. (b) This circuit is equivalent to (a), since $UY=-XU$ and $VY=XV$. }
\label{circuit}
\end{figure}
Note that the application of $Y^{y_1^k}\otimes Y^{y_2^k}$
on $|\psi_{00}\rangle$ by the trusted center in step TE2 is crucial. If the center does not do it, and if Alice just sends $b_1^k\oplus b_2^k$ to two Bobs in the step TE4, two Bobs can communicate with each other. For example, Bob1 can send his message to Bob2 in the following way: in the step TE3, Bob1 does not do any measurement, and sends Alice a single bit, which he wants to send to Bob2, as $b_1^k$. Then Alice advertises $b_1^k\oplus b_2^k$. Since Bob2 knows $b_2^k$, he can learn $b_1^k$ from $b_1^k\oplus b_2^k$. In the same way, Bob2 can send Bob1 any single bit of message.
Let us show that in the above our distillation protocol two Bobs cannot communicate with each other. We represent the classical message $\gamma^k$ from Alice by the quantum state $|\gamma^k\rangle$. Then, the state which Bob2 possesses is \begin{eqnarray*} &&\bigotimes_{k,r} \frac{1}{4} \sum_{y_1^{k,r},y_2^{k,r}} \mbox{Tr}_1\Big(\psi_{y_1^{k,r}\oplus y_2^{k,r},y_1^{k,r}\oplus y_2^{k,r}} \Big) \otimes \fbox{$\gamma^{k,r}$}\\ &=&\bigotimes_{k,r} \Big( \frac{I}{2} \otimes \frac{1}{4} \sum_{y_1^{k,r},y_2^{k,r}} \fbox{$b_1^{k,r}\oplus b_2^{k,r}\oplus y_1^{k,r}\oplus y_2^{k,r}$}~ \Big)\\ &=&\bigotimes_{k,r}\Big(\frac{I}{2}\otimes\frac{I}{2}\Big), \end{eqnarray*} where $\mbox{Tr}_1$ is the partial trace over Bob1's system, $I$ is the identity operator, and
$\fbox{$x$}\equiv |x\rangle\langle x|$. Note that we have introduced new index $r$, since we distill $m$ pure Bell states, and therefore the index $r$ distinguishes each round. The above equation shows that what Bob2 possesses is the completely-mixed state. Hence Bob2 cannot gain any information from it. A similar proof can be done for Bob1.
Let us next show the correctness of the above protocol. That is, if two Bobs are honest, they can perform the correct entanglement distillation. Since now we are considering the entanglement distillation, it is natural to assume that the channel from the center to two Bobs is a map such that \begin{eqnarray*} \psi_{s,t} \to \sum_{(\alpha,\beta)\in\{0,1\}^2} p_{\alpha,\beta} \psi_{\alpha,\beta} +(\mbox{off-diagonal terms}) \end{eqnarray*} for all $(s,t)\in\{0,1\}^2$, where $1\simeq p_{s,t}\gg p_{\alpha,\beta}$ for $(\alpha,\beta)\neq (s,t)$~\cite{map}. Such a channel maps \begin{eqnarray*}
(Y^{y_1^k}\otimes Y^{y_2^k})|\psi_{00}\rangle \to (Y^{y_1^k}\otimes Y^{y_2^k})\eta_{y_1^k,y_2^k} (Y^{y_1^k}\otimes Y^{y_2^k}), \end{eqnarray*} where $\eta_{y_1^k,y_2^k}$ is a state such that the coefficient of $\psi_{00}$ is much larger than those of $\psi_{01}$, $\psi_{10}$, and $\psi_{11}$. The circuit of Fig.~\ref{circuit} (b) is equivalent to that of Fig.~\ref{circuit} (a). Therefore, if we take account of the modification of the measurement results, $b_1^k\to b_1^k\oplus y_1^k$ and $b_2^k\to b_2^k\oplus y_2^k$, we can perform the correct distillation. \fi
TM was supported by JSPS and Program to Disseminate Tenure Tracking System by MEXT. KF was supported by MEXT Grant-in-Aid for Scientific Research on Innovative Areas 20104003.
\end{document} | arXiv |
\begin{document}
\title{A Julia Framework for Graph-Structured\\ Nonlinear Optimization}
\author{David L. Cole${}^{\dag}$, Sungho Shin${}^{\ddag}$, and Victor Zavala${}^{\dag\ddag}$\thanks{Corresponding Author: [email protected]} }
\date{\small
${}^\dag$Department of Chemical and Biological Engineering, \\[0in]
University of Wisconsin-Madison, Madison, WI\\[.05in]
${}^\ddag$Mathematics and Computer Science Division, \\
Argonne National Laboratory, Argonne, IL\\[-2in] }
\maketitle
\begin{abstract} Graph theory provides a convenient framework for modeling and solving structured optimization problems. Under this framework, the modeler can arrange/assemble the components of an optimization model (variables, constraints, objective functions, and data) within nodes and edges of a graph, and this representation can be used to visualize, manipulate, and solve the problem. In this work, we present a {\tt Julia} framework for modeling and solving graph-structured nonlinear optimization problems. Our framework integrates the modeling package {\tt Plasmo.jl} (which facilitates the construction and manipulation of graph models) and the nonlinear optimization solver {\tt MadNLP.jl} (which provides capabilities for exploiting graph structures to accelerate solution). We illustrate with a simple example how model construction and manipulation can be performed in an intuitive manner using {\tt Plasmo.jl} and how the model structure can be exploited by {\tt MadNLP.jl}. We also demonstrate the scalability of the framework by targeting a large-scale, stochastic gas network problem that contains over 1.7 million variables. \end{abstract}
{\bf Keywords}: graphs, nonlinear optimization, modeling, scalability
\section{Introduction}
Modeling and solving large nonlinear optimization problems is essential in diverse applications such as stochastic optimization, dynamic optimization, PDE-constrained optimization, and network optimization \cite{shin2021}. The complexity of such problems continuously pushes the boundary of existing computational tools and limits application scope. To overcome these challenges, it is necessary to develop tools that can facilitate the detection, manipulation, and exploitation of problem structure \cite{colombo2009structure,gondzio2009exploiting,gondzio2003parallel, grothey2009structure, jalving2021, wan2019, yoshio2021,zavala2008}. \\
Recently, it has been proposed to represent optimization problem structures in the form of graphs \cite{allman2019,berger2021gboml_tutorial, daoutidis2019decomposition, jalving2019, jalving2021, mitrai2020decomposition,mitrai2021blockmodeling,mitrai2021stochastic,tang2018optimal}. Under a graph representation, the components of an optimization problem (variables, constraints, and objectives) are assigned to nodes and edges \cite{jalving2021,shin2020MadNLP}. Representing the problem structure as a graph has several benefits; specifically, the graph structure can be used to visualize and manipulate the model (e.g., graph partitioning) using powerful tools such as {\tt Metis} \cite{karypis1998} or {\tt KaHyPar} \cite{schlag2016}. Moreover, the graph structure can be communicated to optimization solvers and this facilitates the use of structure-exploiting linear algebra strategies inside nonlinear optimization solvers such as Schur decomposition \cite{bartlett2006qpschur, kang2014,laird2008large, laird2011parallel, rao1998, rodriguez2020,word2014efficient,zavala2008, zhu2009exploiting} and Schwarz decomposition \cite{frommer2001,na2020,shin2020b,shin2020MadNLP,shin2020a}. \\
There are currently a few software frameworks that enable the construction and solution of structured optimization problems: {\tt Pyomo} (available in Python) \cite{hart2017pyomo, hart2011pyomo}, the Graph-Based Optimization Modeling Language, {\tt GBOML} \cite{berger2021gboml_tutorial,berger2021gboml}, and {\tt Plasmo.jl} (available in Julia) \cite{jalving2021}. {\tt Pyomo} uses a modeling extension called a “network” to model structured optimization problems; this approach creates ports (collections of variables) with arcs placed to link objects on separate ports. {\tt Pyomo} also has additional capabilities for working with the graph structure of stochastic optimization problems \cite{watson2012pysp}. {\tt GBOML} builds an optimization problem into blocks, where blocks are either nodes (containing variables, constraints, and/or objectives) or hyperedges (containing constraints) of a hypergraph. {\tt GBOML} is geared more specifically towards mixed-integer linear programs and is designed to facilitate efficient modeling and potentially enable decomposition schemes to exploit problem structure \cite{berger2021gboml_tutorial}. {\tt Plasmo.jl} places the problem components (variables, constraints, objectives, and data) into an abstract modeling object called an OptiGraph that is composed of OptiNodes (containing the modeling components) and OptiEdges (capturing the structural connectivity between components). These packages allow the modeler to define problem structure directly when constructing the model. This approach differs from approaches that aim to detect graph structures after the model has been built \cite{allman2019,daoutidis2019decomposition,mitrai2020decomposition,mitrai2021blockmodeling, mitrai2021stochastic,tang2018optimal}. The internal OptiNode models of {\tt Plasmo.jl} is constructed based on {\tt JuMP.jl} \cite{Dunning2017jump}, which is an algebraic modeling package available in Julia. {\tt JuMP.jl} provides the algebraic modeling user interface, which is inherited by {\tt Plasmo.jl}, and also provides the automatic differentiation capabilities. \\
Graph structures are exploited in nonlinear optimization solvers during the computation of Newton-like search steps (at the linear algebra level) using techniques such as Schur and Schwarz decomposition. Shin and co-workers recently implemented a nonlinear, interior-point optimization solver in {\tt Julia} called {\tt MadNLP.jl} that provides these types of decomposition techniques \cite{shin2020MadNLP}. This solver also offers an extension called {\tt MadNLPGraph.jl}, which directly takes an OptiGraph object from {\tt Plasmo.jl} and exploits this structure during the evaluation of objective/constraint functions and of derivative information and during the Newton step computation. \\
Recent work by Jalving and co-workers showcased {\tt Plasmo.jl} capabilities, discussed the underlying OptiGraph abstraction, and provided an interface to the {\tt PIPS-NLP} solver (implemented in C++) for exploiting graph structures using Schur decomposition \cite{chiang2014, jalving2021}. Recent work by Shin and co-workers \cite{shin2020MadNLP} introduced {\tt MadNLP.jl} and explored the use of Schwarz decomposition for exploiting graph structures provided by {\tt Plasmo.jl}. In this work, we provide an in-depth overview on capabilities that arise from the integration of {\tt Plasmo.jl} and {\tt MadNLP.jl} (see Figure \ref{fig:graphical_abstract}) and present Schur decomposition capabilities that have been implemented in {\tt MadNLP.jl}. Schur decomposition provides a flexible and robust decomposition paradigm for exploiting diverse structures communicated by {\tt Plasmo.jl} (e.g., time, space, or scenario structures). Moreover, we provide a detailed illustrative example to highlight diverse modeling and solution capabilities and we demonstrate scalability of the framework using a large-scale optimization problem that arises in the context of stochastic optimal control of natural gas networks and that contains over 1.7 million variables. All the code needed to reproduce the results of the paper can be found in \url{https://github.com/zavalab/JuliaBox/tree/master/GraphNLP}. \\
The manuscript is structured as follows. Section 2 provides an overview of graph-based modeling and solution as well as implementations in {\tt Plasmo.jl} and {\tt MadNLP.jl}. Section 3 provides an illustrative example to highlight modeling, visualization, and partitioning capabilities of {\tt Plasmo.jl}. Section 4 provides a case study for a stochastic, nonlinear optimization problem arising in gas networks. Section 5 provides conclusions and a perspective on future work.
\section{Overview of {\tt Plasmo.jl} and {\tt MadNLP.jl}}
Graph-based modeling and solution of optimization problems have several benefits made possible by graph analysis tools. We use the packages {\tt Plasmo.jl} for constructing and partitioning problems as a graph-based model and use {\tt MadNLP.jl} for solving the graph-based models by exploiting the structure. In this section, we discuss the implementation of graph-based modeling tool {\tt Plasmo.jl} and structure-exploiting solver {\tt MadNLP.jl} and the interface between the packages. The interplay between {\tt Plasmo.jl} and {\tt MadNLP.jl} is illustrated in Figure \ref{fig:graphical_abstract}.
\begin{figure}
\caption{Visualization of the interaction of {\tt Plasmo.jl} and {\tt MadNLP.jl}. {\tt Plasmo.jl} enables a generic optimization problem to be modeled as a graph-based model, which can then be partitioned and/or aggregated. The graph-based model from {\tt Plasmo.jl} can be passed to the solver package {\tt MadNLP.jl}. {\tt MadNLP.jl} is an interior-point solver which exploits the problem structure that results from graph-based modeling to parallelize function and derivative evaluations and to solve the linear systems in parallel using Schur or Schwarz decompositions.}
\label{fig:graphical_abstract}
\end{figure}
\subsection{{\tt Plasmo.jl}}
A compact representation of a graph-structured nonlinear optimization problem was proposed in \cite{jalving2021} and is given by: \begin{subequations}\label{eq:graph_model} \begin{align} \min_{\{\boldsymbol{x}_n\}_{n \in {\mathcal{N}}}} &\;\; \sum_{n \in \mathcal{N}} f_n(\boldsymbol{x}_n)\\ \textrm{s.t.} &\; c_n^I(\boldsymbol{x}_n) = 0,\quad n \in {\mathcal{N}}\\ &\; c_e^L(\{\boldsymbol{x}_n\}_{n \in \mathcal{N}(e)})=0,\quad e \in \mathcal{E}\\ &\; \boldsymbol{x}_n \geq 0,\quad n \in \mathcal{N} \end{align} \end{subequations} \noindent The graph $\mathcal{G}(\mathcal{N},\mathcal{E})$ has associated node set $\mathcal{N}$ and (undirected) edge set $\mathcal{E}$. The neighborhood of any node $n$ is denoted by $\mathcal{N}(n)$, while the set of nodes supporting an edge $e$ are represented by $\mathcal{N}(e)$. We also have that $\mathcal{E}(n)$ represents the set of all edges that are incident to node $n$. Symbol $\boldsymbol{x}_n$ denotes the decision variables on the node $n$ and $f_n(\cdot)$ represents the objective function on node $n$. Internal equality constraint functions (i.e., constraints contained within a node) are denoted by $c_n^I(\cdot)$. Linking constraint functions (i.e., constraints that embed variables in multiple nodes) are given by $c_e^L(\cdot)$. \\
The graph-based formulation \eqref{eq:graph_model} provides a unifying abstraction to capture structures appearing in diverse applications such as dynamic optimization (graph is a line), stochastic optimization (graph is a tree), and PDE optimization (graph is a discretization mesh). In fact, most sparse nonlinear optimization problems of the form: \begin{subequations}\label{eq:optimization_model} \begin{align} \min_{\boldsymbol{x}} &\;\; f(\boldsymbol{x})\\ \textrm{s.t.} &\; c(\boldsymbol{x}) = 0\\ &\; \boldsymbol{x} \geq 0 \end{align} \end{subequations} can in principle be rearranged in the form \eqref{eq:graph_model} by using a proper allocation/assignment of the problem components into nodes and edges. This allocation can be done manually by the modeler (using domain-specific insight) or automatically using graph analysis tools (e.g., partitioning and community detection). Representing the problem as a graph can be particularly useful for problems that exhibit sparse structures and a high degree of modularity. \\
We should highlight that there is significant flexibility in how components can be assigned to nodes and edges in a graph abstraction; for instance, we could allocate a single variable/constraint to a node or we could embed the entire set of variables/constraints into a single node. Such flexibility can be exploited for developing diverse solution strategies or to facilitate visualization. In principle, even dense/unstructured problems can be modeled as \eqref{eq:graph_model} form. However, for those problems, there is no practical benefit of using graph-based modeling packages and structure-exploiting solution algorithms. \\
The {\tt Julia} package {\tt Plasmo.jl} has been developed to facilitate the representation of optimization problems using graph structures. The implementation details of {\tt Plasmo.jl} are presented in \cite{jalving2021}. {\tt Plasmo.jl} is based on a general modeling abstraction that is called an OptiGraph. An OptiGraph is an abstract object given by an undirected hypergraph that is composed of sets of OptiNodes and OptiEdges. An OptiNode is a self-contained optimization problem with components given by variables, constraints, objectives, and data; the OptiNode internally contains a {\tt JuMP.jl} model object. Connectivity across OptiNodes is captured in the form of OptiEdges, which are linking constraints that connect components of two (or more) OptiNodes (hence creating a hypergraph). The OptiGraph abstraction also enables hierarchical graph representations under which an OptiNode can be an OptiGraph itself; this facilitates the construction of problems with complex nested structures and manipulation of such structures (e.g., via aggregation and partitioning). In the following discussion, we will refer to OptiGraphs, OptiNodes, and OptiEdges as graphs, nodes, and edges. \\
Hierarchical graphs provide substantial modeling flexibility; for instance, the nodes within a graph can be partitions of variables, constraints, objectives, and/or data, and such partitions can be created on-the-fly using the {\tt partition} function. Hierarchical graphs also allow the user to place collections of nodes onto a subgraph, creating a secondary partition of the problem. While both nodes and subgraphs act as partitions, the nodes are defined in the initial problem formulation and do not change, while the subgraphs can be altered and partitioned further using functions within {\tt Plasmo.jl}. \\
Hierarchical graphs containing multiple levels of subgraphs can be aggregated/collapsed into a single node using the {\tt aggregate} function. The {\tt aggregate} function also allows modelers to convert the entire graph (including all nodes, edges, and subgraphs) into a single node containing the entire model components (a generic {\tt JuMP.jl} model). With this, one can aggregate the original problem in different ways to gain insight on the problem structure or to use different types of solution capabilities available (e.g., those that exploit or do not exploit structures).
\subsection{{\tt MadNLP.jl}}
{\tt MadNLP.jl} is a nonlinear programming (NLP) solver implemented in the {\tt Julia} programming language. {\tt MadNLP.jl} implements a state-of-the-art, filter line-search method (as that implemented in {\tt Ipopt}). {\tt MadNLP.jl} has the ultimate goal of facilitating the implementation of algorithmic and linear algebra strategies that can enhance robustness and efficiency. Such flexibility allows for the exploitation of diverse structures present in NLPs (e.g., sparsity patterns or high-level graph structures). Here, we provide a high-level introduction of the interior-point method and of the Schur complement decomposition method implemented in {\tt MadNLP.jl}.
\subsubsection{Interior-Point Method}
In an interior-point method, one attempts to find the solution of \eqref{eq:optimization_model} by solving a sequence of smooth {\it barrier subproblems}. The barrier subproblems are formulated by replacing the inequality constraints by log-barrier terms as: \begin{subequations}\label{eqn:barrier}
\begin{align}
\min_{\boldsymbol{x}} &\;\; f(\boldsymbol{x}) - \sum_{i=1}^n \mu \log \boldsymbol{x}_{(i)}\\
\textrm{s.t.} &\; c(\boldsymbol{x}) = 0,\quad (\boldsymbol{\lambda}),
\end{align} \end{subequations} where $\boldsymbol{x}_{(i)}$ denotes the $i$-th component of $\boldsymbol{x}$. The barrier subproblems are solved using a decreasing sequence of values for the the penalty parameter $\mu>0$. For convenience, we let $\boldsymbol{X}:=\mathop{\textrm{diag}}(\boldsymbol{x})$, $\boldsymbol{Z}:=\mathop{\textrm{diag}}(\boldsymbol{z})$, and $\boldsymbol{1}:=[1;1;\cdots;1]$. \\
To solve \eqref{eqn:barrier}, we compute search (Newton-like) steps by solving a linearized version of the first-order optimality conditions to \eqref{eqn:barrier}. To see this, we write the optimality conditions for \eqref{eqn:barrier}: \begin{subequations}\label{eqn:barrier-kkt}
\begin{align}
\nabla_{\boldsymbol{x}} f(\boldsymbol{x}) - \mu\boldsymbol{X}^{-1} \boldsymbol{1} + \nabla_{\boldsymbol{x}}c(\boldsymbol{x})^\top \boldsymbol{\lambda} &= 0\\
c(\boldsymbol{x})& = 0,
\end{align} \end{subequations} or equivalently, by introducing $\boldsymbol{z} = \mu\boldsymbol{X}^{-1} \boldsymbol{1}$, \begin{subequations}\label{eqn:barrier-kkt}
\begin{align}
\nabla_{\boldsymbol{x}} f(\boldsymbol{x}) - \boldsymbol{z} + \nabla_{\boldsymbol{x}}c(\boldsymbol{x})^\top \boldsymbol{\lambda} &= 0\\
c(\boldsymbol{x})& = 0\\
\boldsymbol{X}\boldsymbol{z} -\mu\boldsymbol{1} &= 0.
\end{align} \end{subequations} The Newton step at iteration $k$ is given by: \begin{align}\label{eqn:barrier-newton}
\begin{bmatrix}
\boldsymbol{W}^{(k)} & {\boldsymbol{A}^{(k)}}^\top & -I\\
\boldsymbol{A}^{(k)}&&\\
\boldsymbol{Z}^{(k)} && \boldsymbol{X}^{(k)}\\
\end{bmatrix}
\begin{bmatrix}
\boldsymbol{d}^{(k)}_x\\
\boldsymbol{d}^{(k)}_\lambda\\
\boldsymbol{d}^{(k)}_z
\end{bmatrix}
=
\begin{bmatrix}
\nabla_{\boldsymbol{x}} f(\boldsymbol{x}^{(k)}) -\boldsymbol{z}^{(k)} + {\boldsymbol{A}^{(k)}}^\top \boldsymbol{\lambda}^{(k)}\\
c(\boldsymbol{x}^{(k)})\\
\boldsymbol{X}^{(k)} \boldsymbol{Z}^{(k)} \boldsymbol{1} - \mu \boldsymbol{1}
\end{bmatrix}, \end{align} where $\boldsymbol{W}^{(k)}:=\nabla^2_{\boldsymbol{x}\boldsymbol{x}} \mathcal{L}(\boldsymbol{x}^{(k)},\boldsymbol{\lambda}^{(k)},\boldsymbol{z}^{(k)})$, $\boldsymbol{A}^{(k)} := \nabla_{\boldsymbol{x}}c(\boldsymbol{x}^{(k)})$. The right-hand side of \eqref{eqn:barrier-newton} comes from the linearization of \eqref{eqn:barrier-kkt} and the right-hand-side is the evaluation of the optimality system \eqref{eqn:barrier-kkt} at the current iteration. By eliminating the third block-row, we obtain \begin{align*}
\begin{bmatrix}
\boldsymbol{W}^{(k)} + \boldsymbol{\Sigma}^{(k)} &{\boldsymbol{A}^{(k)}}^\top\\
\boldsymbol{A}^{(k)}
\end{bmatrix}
\begin{bmatrix}
\boldsymbol{d}^{(k)}_x\\
\boldsymbol{d}^{(k)}_\lambda
\end{bmatrix}=
\begin{bmatrix}
\nabla_{\boldsymbol{x}}f(\boldsymbol{x}^{(k)}) - \mu {\boldsymbol{X}^{(k)}}^{-1}\boldsymbol{1} + {\boldsymbol{A}^{(k)}}^\top \boldsymbol{\lambda}^{(k)}\\
c(\boldsymbol{x}^{(k)})
\end{bmatrix} \end{align*} where $\boldsymbol{\Sigma}^{(k)}:={\boldsymbol{X}^{(k)}}^{-1} \boldsymbol{Z}^{(k)}$. The solution $(\boldsymbol{d}^{(k)}_x,\boldsymbol{d}^{(k)}_\lambda,\boldsymbol{d}^{(k)}_z)$ of \eqref{eqn:barrier-newton} is used to compute the Newton step. Then, a line-search algorithm is used to determine the steplength $\alpha$ and the iterate: \begin{align*}
\boldsymbol{x}^{(k+1)} &= \boldsymbol{x}^{(k)}+\alpha\cdot \boldsymbol{d}^{(k)}_x\\
\boldsymbol{\lambda}^{(k+1)} &= \boldsymbol{\lambda}^{(k)}+\alpha \cdot \boldsymbol{d}^{(k)}_\lambda\\
\boldsymbol{z}^{(k+1)} &= \boldsymbol{z}^{(k)}+\alpha \cdot \boldsymbol{d}^{(k)}_z. \end{align*} The steplength $\alpha$ is chosen so as to improve feasibility or the objective function, see \cite{wachter2006}.
\subsubsection{Schur Decomposition}
Graph modeling greatly facilitates the implementation of decomposition strategies; among these, the Schur complement decomposition approach has shown to be flexible and robust. The details of this approach can be found in \cite{chiang2014,jalving2021,kang2014}. In the context of the graph-structured NLP \eqref{eq:graph_model}, the KKT conditions can be written as: \begin{subequations}\label{eq:KKT_system} \begin{align} \nabla_{\boldsymbol{x}_n} f_n(\boldsymbol{x}_n) + \mu \boldsymbol{X}^{-1}_n\boldsymbol{1} + \nabla_{\boldsymbol{x}_n}c_n^I(\boldsymbol{x}_n)\boldsymbol{\lambda}_{I,n} + \sum_{e \in \mathcal{E}(n)} \nabla_{\boldsymbol{x}_{n }} c_e^L(\{\boldsymbol{x}_{n'}\}_{n' \in \mathcal{N}(e)})^\top \boldsymbol{\lambda}_{L,e} &= 0,\quad n\in\mathcal{N}\\ c_n^I(\boldsymbol{x}_n) &= 0,\quad n \in \mathcal{N} \\ c_e^L(\{\boldsymbol{x}_{n}\}_{n \in \mathcal{N}(e)}) &= 0, \quad e \in \mathcal{E}, \end{align} \end{subequations} where $\boldsymbol{X}_n:=\mathop{\textrm{diag}}(\boldsymbol{x}_n)$, $\mathcal{E}(n)$ is the set of edges one of whose support is $n$, and $\boldsymbol{\lambda}_{I,n}$ and $\boldsymbol{\lambda}_{L,e}$ are the dual variables corresponding to the internal and the linking constraints respectively. The corresponding KKT system can be formulated as:
\begin{align}\label{eq:BBD_system}
\left[\begin{array}{cccc|c} \boldsymbol{K}^{(k)}_{1}& &&& {\boldsymbol{B}^{(k)}_{1}}^\top\\ &\boldsymbol{K}^{(k)}_{2}& && {\boldsymbol{B}^{(k)}_2}^\top\\ &&\ddots&&\vdots\\ &&&\boldsymbol{K}^{(k)}_{n}&{\boldsymbol{B}^{(k)}_n}^\top\\\hline \boldsymbol{B}^{(k)}_1&\boldsymbol{B}^{(k)}_2&\hdots&\boldsymbol{B}^{(k)}_n& \\ \end{array}\right] \left[\begin{array}{c} \boldsymbol{d}^{(k)}_{1}\\ \boldsymbol{d}^{(k)}_{2}\\ \vdots \\ \boldsymbol{d}^{(k)}_{n}\\ \hline \boldsymbol{d}^{(k)}_L \end{array}\right]= -\left[\begin{array}{c} \boldsymbol{r}^{(k)}_{1}\\ \boldsymbol{r}^{(k)}_{2}\\ \vdots \\ \boldsymbol{r}^{(k)}_{n}\\ \hline \ \boldsymbol{r}^{(k)}_L\ \end{array}\right]. \end{align} \noindent Here, $\boldsymbol{d}^{(k)}_{n} := ( \boldsymbol{d}^{(k)}_{x,n}, \boldsymbol{d}^{(k)}_{I,n})$ are the primal-dual steps for all $n \in \mathcal{N}$ where $\boldsymbol{d}^{(k)}_{x,n}$ are the primal steps and $\boldsymbol{d}^{(k)}_{I,n}$ are the dual steps associated with $\boldsymbol{\lambda}^I_{n,k}$; $\boldsymbol{d}^{(k)}_L$ are the dual steps associated with $\boldsymbol{\lambda}^{(k)}_{L,e}$ for $e \in \mathcal{E}$ (the linking constraints); and $\boldsymbol{K}^{(k)}_{n}$, $\boldsymbol{B}^{(k)}_{n}$, $\boldsymbol{r}^{(k)}_{n}$, and $\boldsymbol{r}^{(k)}_L$ are defined as: \begin{align*}
\boldsymbol{K}^{(k)}_{n} &:= \left[\begin{array}{cc} \boldsymbol{W}^{(k)}_{n} + \boldsymbol{\Sigma}^{(k)}_{n} & {\boldsymbol{J}^{(k)}_n}^\top \\ \boldsymbol{J}^{(k)}_{n} & \boldsymbol{0} \end{array} \right],\quad n \in \mathcal{N}\\
\boldsymbol{B}^{(k)}_{n} &:= \left[ \begin{array}{cc} \boldsymbol{Q}^{(k)}_{n} & \boldsymbol{0} \end{array} \right],\quad n \in \mathcal{N}\\
\boldsymbol{r}^{(k)}_{n} &:= \left[ \begin{array}{c} \nabla_{\boldsymbol{x}_{n}} f(\boldsymbol{x}^{(k)}_{n}) \\ c_n^I(\boldsymbol{x}^{(k)}_{n}) \end{array} \right],\quad n \in \mathcal{N} \\
\boldsymbol{r}^{(k)}_L &:= \{c_{e}^L(\{\boldsymbol{x}^{(k)}_{n}\}_{n \in \mathcal{N}(e)})\}_{e \in \mathcal{E}} . \end{align*} \noindent Here, $\boldsymbol{J}^{(k)}_{n} := \nabla_{\boldsymbol{x}_n} c_n^I(\boldsymbol{x}^{(k)}_{n})$ is the Jacobian of the internal constraints, $\boldsymbol{W}^{(k)}_{n}$ is the Hessian of the Langrangian of \eqref{eq:graph_model}, $\boldsymbol{\Sigma}^{(k)}_n:={\boldsymbol{X}^{(k)}_{n}}^{-1} \boldsymbol{Z}^{(k)}_{n}$, and $\boldsymbol{Q}^{(k)}_{n} := \nabla_{\boldsymbol{x}_n} \{c_e^L (\{\boldsymbol{x}^{(k)}_{n'}\}_{n' \in \mathcal{E}(n)})\}_{e \in \mathcal{E}}$ is the Jacobian of the linking constraints. The block-bordered structure in \eqref{eq:BBD_system} naturally results from the graph structure and facilitates the implementation of the Schur decomposition approach. We also note that the matrices $\boldsymbol{K}^{(k)}_{n}$ and $\boldsymbol{B}^{(k)}_{n}$ for $n \in \mathcal{N}$ may have varying sizes and that such sizes depend on the model construction and on the number of linking constraints. We also highlight that the diagonal blocks in the linear system can embed additional structures (that might arise from hierarchical graph structures). \\
Having the block-bordered structure, the decomposition approach proceeds by forming the Schur complement: \begin{subequations}\label{Schur_complement} \begin{align}
\boldsymbol{S}^{(k)} &= -\sum_{n\in \mathcal{N}} {\boldsymbol{B}^{(k)}_{n}}^\top {\boldsymbol{K}^{(k)}_{n}} ^{-1} \boldsymbol{B}^{(k)}_{n} \label{Schur_complement-1}\\
\boldsymbol{S}^{(k)} \boldsymbol{d}^{(k)}_L &= \sum_{n\in \mathcal{N}} {\boldsymbol{B}^{(k)}_{n}}^\top {\boldsymbol{K}^{(k)}_{n}}^{-1} \boldsymbol{r}^{(k)}_{n} - \boldsymbol{r}^{(k)}_L\label{Schur_complement-2}\\
\boldsymbol{K}^{(k)}_{n} \boldsymbol{d}^{(k)}_{n} &= \boldsymbol{B}^{(k)}_{n} \boldsymbol{d}^{(k)}_L - \boldsymbol{r}^{(k)}_{n}, n \in \mathcal{N}.\label{Schur_complement-3} \end{align} \end{subequations} The computation of ${\boldsymbol{B}^{(k)}_{n}}^\top {\boldsymbol{K}^{(k)}_{n}} ^{-1} \boldsymbol{B}^{(k)}_{n} $ can be parallelized since each block operation is independent. Once the blocks ${\boldsymbol{B}^{(k)}_{n}}^\top {\boldsymbol{K}^{(k)}_{n}} ^{-1} \boldsymbol{B}^{(k)}_{n} $ are computed in parallel, they are assembled into matrix $\boldsymbol{S}^{(k)}$, and the dense linear system \eqref{Schur_complement-2} is solved to obtain the dual step direction for the link constraints. From this step, the block steps are computed in parallel via \eqref{Schur_complement-3}. Parallelization can significantly speed up the step computation, especially when the $\boldsymbol{K}^{(k)}_{n}$ blocks are large and the Schur complement $\boldsymbol{S}^{(k)}$ is small.
\subsection{Model-Solver Interface}
{\tt MadNLP.jl} is designed to interface with OptiGraph object models created in {\tt Plasmo.jl}. A couple of benefits come from interfacing these capabilities: (i) {\tt Plasmo.jl} creates a graph-based model where the structural information of the problem is stored, and such information can then be automatically exploited with {\tt MadNLP.jl} and solved with specialized algorithms, and (ii) the modular structure within the OptiGraph model allows parallel function and derivative evaluations. When used in conjuction with {\tt Plasmo.jl}, {\tt MadNLP.jl} detects the partitions as defined by {\tt Plasmo.jl}. \\
If the modeler uses a more general (unstructured) modeling language (such as {\tt JuMP.jl}), a custom partition must be manually set with a vector directing which nodes belong on which partition. Alternatively, if no custom partition is set, {\tt MadNLP.jl} provides capabilities to partition the problem internally using {\tt Metis}. This latter method may be especially useful if the user does not have advanced knowledge of the problem structure or thinks that there may be additional (non-obvious) structures to exploit.
\section{Illustrative Example}
In this section, we highlight differences that arise when modeling problems using a general algebraic package such as {\tt JuMP.jl} and the graph-based package {\tt Plasmo.jl}. We also illustrate how graph-based modeling allows users to formulate and partition a problem and note how this can be useful in solving a problem.
\subsection{Model}
We consider an example that arises in the optimal tuning of a proportional integral derivative (PID) controller. The problem is a stochastic program designed to minimize the error across multiple operational scenarios by choosing tuning parameters. The problem is formulated as: \begin{subequations}\label{eq:grid} \begin{align}
\min_{\{x_s, u_x \}_{s \in S}, K_c, \tau_I, \tau_D}&\;\; \frac{1}{|S|} \sum_{s\in S} \int_0^{t_f}\left(100(x_{sp,s}-x_s(t))^2 + 0.01 u_s(t)^2\right) \label{eqn:PID_objective_function}\\ \textrm{s.t.} &\; \frac{1}{\tau} \frac{dx_s(t)}{dt} + x_s(t) = K u_s(t) + K_d d_s \label{eq:controller_gain}\\ &\; u_s(t) = K_c \left(x_{sp,s}(t) - x_s(t)\right) + \tau_I \int_0^t\left(x_{sp,s}(t) - x_s(t)\right)dt + \tau_D \frac{dx_s(t)}{dt}\label{eq:PID_constraint}\\ &\; x_s(t) = x_0\label{eq:setpoint_constraint}\\ &\; -10 \leq K_c \leq 10 \label{eq:limits_constraint_start}\\ &\; -100 \leq \tau_I \leq 100\\ &\; -100 \leq \tau_D \leq 100\\ &\; -2.5 \leq x_s(t) \leq 2.5\\ &\; -2.0 \leq u_s(t) \leq 2.0 \label{eq:limits_constraint_end}. \end{align} \end{subequations} \noindent Here, $t$ is the time for $t \in [0,t_f]$ where $t_f = 10$, and $s$ is the scenario index for $s\in S$, where $S$ is a set of five scenarios. The state variable is represented by $x_s(t)$ for $s \in S$, and the set-point is represented by $x_{sp,s}$ for $s \in S$. Equation \eqref{eqn:PID_objective_function} is the objective function penalizing the deviation from the set-point and the magnitude of the control actions. Equation \eqref{eq:controller_gain} is the constraint representing a first-order linear dynamical system with a disturbance and without time delay. Here, $\tau$ is the time constant, $K$ is the process gain, $K_d$ is the disturbance gain, and $d_s$ is the disturbance for $s \in S$. Equation \eqref{eq:PID_constraint} is the PID controller formulation, where the tuning parameters $K_c$, $\tau_I$, and $\tau_D$ relate to the proportional, integral, and derivative terms. These tuning parameters are the design variables and will thus have the same value across scenarios. For simplicity, \eqref{eq:setpoint_constraint} requires all scenarios to have the same starting point ($x_0$). Equations \eqref{eq:limits_constraint_start}-\eqref{eq:limits_constraint_end} give upper and lower limits for all decision variables.
\subsection{Implementation}
In implementing the problem as a {\tt JuMP.jl} model object or a {\tt Plasmo.jl} OptiGraph object, we discretize the dynamical system using 100 evenly-spaced time points . Scenario and operating parameters are shown in the code snippet of Figure \ref{fig:code_snippet_PID_IC}. \\
The formulation as a {\tt JuMP.jl} object is given in the code snippet in Figure \ref{fig:code_snippet_PID_JuMP}. Comments within the snippet explain different aspects of the implementation. We highlight that this model uses indices for time and scenario to define process variables or inputs; moreover, \textit{this formulation has no predefined structure}. \\
The corresponding formulation in {\tt Plasmo.jl} is shown in the code snippet of Figure \ref{fig:code_snippet_PID_Plasmo} and contains an inherent structure by defining the problem with nodes and edges. In this construction, nodes represent time points within each scenario. A master node is added to contain the design variables (i.e. the controlled tuning parameters) that are then linked across scenarios. In this formulation, dummy variables are introduced (lines \ref{line:dummy_start} - \ref{line:dummy_end}) to avoid nonlinearities in the linking constraints (this is a lifting procedure). There are several ways in which one can formulate the tuning problem in {\tt Plasmo.jl}; for instance, the dummy variables could have been avoided by placing the nonlinear constraints on a single node (placing a full scenario in each node). However, the more variables and constraints contained in a single node, the fewer options that exist for partitioning the problem into a hierarchical subgraphs. Here, we chose to model this problem with a large number of nodes to allow for additional partitioning. This is an important consideration when formulating a model using a graph structure.
\begin{figure}
\caption{Code Snippet setting parameters for the PID controller tuning problem}
\label{fig:code_snippet_PID_IC}
\end{figure}
\begin{figure}
\caption{Code Snippet defining a PID controller problem formulated in {\tt JuMP.jl}}
\label{fig:code_snippet_PID_JuMP}
\end{figure}
\begin{figure}
\caption{Code Snippet defining a PID controller problem formulated in {\tt Plasmo.jl}}
\label{line:dummy_start}
\label{line:dummy_end}
\label{line:add_subgraphs}
\label{fig:code_snippet_PID_Plasmo}
\end{figure}
\subsection{Partitioning}
The problem in Figure \ref{fig:code_snippet_PID_Plasmo} is partitioned based on scenarios, because each individual scenario is placed on a separate subgraph (Line \ref{line:add_subgraphs}). The high-level OptiGraph consists of a master node connected to five separate subgraphs. Using the {\tt aggregate} function, each subgraph can be combined into a single node (creating a new OptiGraph). Visualizations of the scenario-partitioned model in its aggregated and non-aggregated form, along with their accompanying adjacency matrices, are shown in Figure \ref{fig:Scen_Figure}. These were created using {\tt Plasmo.jl} visualization capabilities \cite{jalving2021}. These visualizations reveal the tree structure seen in typical stochastic optimization formulations and highlight how the modeler can use visualization capabilities to navigate and understand the problem structure.
\begin{figure}
\caption{Visualization of the PID controller problem partitioned by scenario. Adjacency matrix (left) is shown with the graph-based model partitioned by scenario with colors representing each subgraph (middle) and the aggregated form of the graph (right)}
\label{fig:Scen_Figure}
\end{figure}
With the initial OptiGraph created, the problem can be partitioned in other ways. Desired partitioning schemes will depend on the model structure and intended decomposition method. For the PID controller, we show that the problem can also be partitioned in the time domain (rather than the scenario domain). The code snippet in Figure \ref{fig:code_snippet_PID_partition} shows how the formulation in Figure \ref{fig:code_snippet_PID_Plasmo} can be repartitioned over time. Each new partition contains 25 time points from each scenario. We first create a reference map (line \ref{line:refmap}) and a node membership vector (line \ref{line:nmv}). The node membership vector contains an integer value corresponding to each node of the reference map that directs {\tt Plasmo.jl} into which partition to place each node. When the membership vector is filled (lines \ref{line:nmv_fill_start} - \ref{line:nmv_fill_end}), a new partition can be created (line \ref{line:make_partition}) and applied to the original OptiGraph (line \ref{line:make_subgraphs}). The resulting OptiGraph is partitioned in time and is visualized in Figure \ref{fig:Time_Figure}, along with the new adjacency matrix and its corresponding aggregated structure. This structure reveals the typical linear tree structure of a dynamic optimization problem.
\begin{figure}
\caption{Code Snippet for creating manual partition of an OptiGraph}
\label{line:refmap}
\label{line:nmv}
\label{line:nmv_fill_start}
\label{line:nmv_fill_end}
\label{line:make_partition}
\label{line:make_subgraphs}
\label{fig:code_snippet_PID_partition}
\end{figure}
\begin{figure}
\caption{Visualization of the PID controller problem partitioned by time. Adjacency matrix (left) is shown with the graph-based model partitioned by scenario with colors representing each subgraph (middle) and the aggregated form of the graph (right)}
\label{fig:Time_Figure}
\end{figure}
Partitioning a problem can result in quite different structures. Figures \ref{fig:Scen_Figure} and \ref{fig:Time_Figure} contain the same problem but present distinct structures. These structures may be best exploited by using certain decomposition schemes. For example, the aggregated structure shown in Figure \ref{fig:Scen_Figure} can be solved efficiently with Schur decomposition because it is a bilevel tree. Alternatively, the aggregated structure seen in Figure \ref{fig:Time_Figure} may best be exploited with Schwarz decomposition because subproblems could have a degree of overlap without encompassing the whole graph. These modeling and partitioning capabilities of graph-structured optimization can thus be useful for analyzing and solving problems, as evidenced above.
\section{Large-Scale Case Study}
To demonstrate the scalability of {\tt Plasmo.jl} and {\tt MadNLP.jl}, we implemented a large-scale natural gas network problem. This is a two-stage stochastic optimization problem for a system that comprises pipelines, compressors, and junctions (Figure \ref{fig:Gas_Pipeline}). The first-stage variables are given by the compressor power policy, while the second-stage (recourse) variables are the states of the pipelines (e.g., pressures and flows) in different scenarios. Each scenario within the problem contained a different gas demand profile at the end of the network.
\begin{figure}
\caption{Visualization of a natural gas pipeline containing $l$ compressors, $m$ pipelines, and $n$ junctions. Junctions are placed at connections between two pipelines and connections between a pipeline and compressor. $c_i$ represents the $i$th compressor, $p_i$ the $i$th pipeline, and $j_i$ the $i$th junction. Gas is supplied at the first junction and delivered at the $n$th junction.}
\label{fig:Gas_Pipeline}
\end{figure}
\subsection{Model}
To construct this problem, we used the constraints defined by \cite{jalving2021, jalving2018} and the code for a deterministic problem given by \cite{jalving2021} and modified the code to capture uncertainty. There are three object types that lead to modeling constraints: junctions, compressors, and pipelines. There are also additional constraints created by links between these object types. The constraints and objective function are given in subsections \ref{sec:junction_constraints} - \ref{sec:obj_function} below. The formulations presented below follow closely those defined in \cite{jalving2021}. However, we introduce a few minor changes to form a stochastic model and to more closely follow the code used to construct this model. \\
The following sets are employed within the given formulation: $\mathcal{L}_c$ represents the set of all compressors, $\mathcal{L}_p$ represents the set of all pipelines, and $\mathcal{L} := \mathcal{L}_c \cup \mathcal{L}_p$ represents the union of the set of compressors and set of pipelines. We also use $\mathcal{J} := \{1, ..., N_j \}$ for the set of all junctions in the network, where a junction is placed at any connection between pipelines and between any connection of a pipeline with a compressor. Junction $1$ is where gas is supplied to the network and junction $N_j$ is the final junction of the pipeline and the location of gas delivery. We further define the set $\mathcal{J}' := \mathcal{J} \backslash \{1, N_j\}$. Pipelines were also discretized into a set of equally spaced points, $\mathcal{X} := \{1, ..., N_x\}$, and the operating time was discretized into a set of time points, $\mathcal{T} := \{1, ..., N_t\}$. Here, $N_x$ and $N_t$ are the final spacial and time points respectively. We also define the sets $\mathcal{X}' := \mathcal{X} \backslash N_x$ and $\mathcal{T}' := \mathcal{T} \backslash 1$. Lastly, we use $\mathcal{S}$ to represent the set of all scenarios.\\
\subsubsection{Junction Constraints}\label{sec:junction_constraints}
The junction constraints are given in Equation \eqref{eqn:junction_constraints}. $\theta_{j,t,s}$ is the pressure for $j \in \mathcal{J}$, $t \in \mathcal{T}$, and $s \in \mathcal{S}$, with $\ubar{\theta}_j$ as the lower bound and $\bar{\theta}_j$ as the upper bound for $j \in \mathcal{J}$. $\bar{F}_{t,s}$ is the amount of supplied gas at the first junction for $t \in \mathcal{T}$ and $s \in \mathcal{S}$ and bounded by a maximum supply amount, $\bar{\phi}_{supply}$. $F_{t,s}$ is the amount of gas being delivered at the end of the network for $t \in \mathcal{T}$ and $s \in \mathcal{S}$. $\tilde{F}_{t,s}$ is a positive variable that allows for $F_{t,s}$ to exceed the demand ($d_{t,s}$) within a given scenario $s$ at time $t$ (Equation \eqref{eq:slack}). $\tilde{F}_{t,s}$ is also penalized within the objective function (Equation \eqref{eq:objective_function}) to help limit over supplying gas in any scenario.
\begin{subequations}\label{eqn:junction_constraints}
\begin{align}
& F_{t,s} \le d_{t,s} + \tilde{F}_{t,s}, \qquad t \in \mathcal{T}, s \in \mathcal{S} \label{eq:slack}\\
& \ubar{\theta}_j \le \theta_{j,t,s} \le \bar{\theta}_j, \qquad j \in \mathcal{J}, t \in \mathcal{T}, s \in \mathcal{S}\\
& 0 \le \bar{F}_{t,s} \le \bar{\phi}_{supply}, \qquad t \in \mathcal{T}, s \in \mathcal{S}\\
& 0 \le F_{t, s} , \qquad t \in \mathcal{T}, s \in \mathcal{S}\\
& 0 \le \tilde{F}_{t,s}, \qquad t \in \mathcal{T}, s \in \mathcal{S}
\end{align} \end{subequations}
\subsubsection{Compressor Constraints}
The compressor constraints are given in Equation \eqref{eqn:compressor_constraints}. Here, $p^{in}_{\ell, t, s}$ is the suction pressure, $p^{out}_{\ell, t,s}$ is the discharge pressure, $\bar{p}_{\ell, t,s}$ is the boost in pressure between the suction and discharge , $P_{\ell, t,s}$ is the power of the compressor, and $f_{\ell,t,s}$ is the flow passing through the compressor for $\ell \in \mathcal{L}_c$, $t \in \mathcal{T}$, and $s \in \mathcal{S}$. The parameters $c_P$, $T$, and $\gamma$ are the heat capactiy, temperature, and isentropic efficiency, respectively. The parameters $\ubar{\theta}_{in}$, $\ubar{\theta}_{out}$, $\ubar{\theta}_{boost}$, $\bar{\theta}_{in}$, $\bar{\theta}_{out}$, $\bar{\theta}_{boost}$, and $\bar{\psi}_{power}$ represent lower or upper limits on their respective variables. The operation of the compressor is dictated by Equation \eqref{eqn:comp1}. Lastly, we note that the definition of $f^{in}_{\ell,t,s}$ and $f^{out}_{\ell,t,s}$ for $\ell \in \mathcal{L}_c, t \in \mathcal{T}, s \in \mathcal{S}$ in Equation \eqref{eqn:comp_links} is done to simplify the mathematical definition of the linking constraints (see Equation \eqref{eqn:link_constraints}).
\begin{subequations}\label{eqn:compressor_constraints}
\begin{align}
& P_{\ell, t,s} = c_P \cdot T \cdot f_{\ell, t,s} \left[ \left(\frac{p^{out}_{\ell,t,s}}{p^{in}_{\ell,t,s}}\right)^{\frac{\gamma - 1}{\gamma}} -1 \right], \qquad \ell \in \mathcal{L}_c, t \in \mathcal{T}, s \in \mathcal{S} \label{eqn:comp1} \\
& p^{out}_{\ell,t,s} = p^{in}_{\ell,t,s} + \bar{p}_{\ell,t,s}, \qquad \ell \in \mathcal{L}_c, t \in \mathcal{T}, s \in \mathcal{S}\label{eqn:comp2} \\
& 0 \le P_{\ell, t, s} \le \bar{\psi}_{power}, \qquad \ell \in \mathcal{L}_c, t \in \mathcal{T}, s \in \mathcal{S} \label{eqn:power_limit}\\
& \ubar{\theta}_{in} \le p^{in}_{\ell,t,s} \le \bar{\theta}_{in}, \qquad \ell \in \mathcal{L}_c, t \in \mathcal{T}, s \in \mathcal{S}\\
& \ubar{\theta}_{out}\le p^{out}_{\ell,t,s} \le \bar{\theta}_{out},\qquad \ell \in \mathcal{L}_c, t \in \mathcal{T}, s \in \mathcal{S}\\
& \ubar{\theta}_{boost} \le \bar{p}_{\ell,t,s} \le \bar{\theta}_{boost}, \qquad \ell \in \mathcal{L}_c, t \in \mathcal{T}, s \in \mathcal{S}\\
& 0 \le f_{\ell,t,s} , \qquad \ell \in \mathcal{L}_c, t \in \mathcal{T}, s \in \mathcal{S}\\
& f_{\ell,t,s} = f^{in}_{\ell,t,s} = f^{out}_{\ell,t,s}, \qquad \ell \in \mathcal{L}_c, t \in \mathcal{T}, s \in \mathcal{S} \label{eqn:comp_links}
\end{align} \end{subequations}
\subsubsection{Pipeline Constraints}
The pipeline constraints are definied in Equation \eqref{eqn:pipeline_constraints}. Here, $p_{\ell, k,t,s}$ and $f_{\ell,k,t,s}$ are the pressure and flows respectively for $\ell \in \mathcal{L}_P$, $k \in \mathcal{X}$, $t \in \mathcal{T}$, and $s \in \mathcal{S}$. We also define the constants $c_{1,\ell}$, $c_{2,\ell}$, and $c_{3,\ell}$ for $\ell \in \mathcal{L}_p$ which relate to the physical properties of the gas and pipelines (see \cite{jalving2021}). $\Delta t$ is the size of the discretized time intervals and $\Delta x_{\ell}$ is the size of hte discretized space interval on pipeline $\ell \in \mathcal{L}_p$. We also introduce the variables $m_{\ell, t,s}$ for $\ell \in \mathcal{L}_p, t \in \mathcal{T}$, and $s \in \mathcal{S}$ as the line pack of a given pipeline. Equations \eqref{eqn:pipe1} and \eqref{eqn:pipe2} relate to the mass and momentum dynamics of the pipeline, Equations \eqref{eqn:ss1} and \eqref{eqn:ss2} require the problem to start at a steady state, Equations \eqref{eqn:linepack1} and \eqref{eqn:linepack2} define the linepack and require the pipelines to refill at the end of the time period to at least their initial linepack, and Equations \eqref{eqn:pipelinks1} - \eqref{eqn:pipelinks2} define flow and pressure terms that are used within the mathematical definition fo the linking constraints (see Equation \eqref{eqn:link_constraints}).
\begin{subequations}\label{eqn:pipeline_constraints}
\begin{align}
& \frac{p_{\ell, t,k, s} - p_{\ell,t-1, k,s}}{\Delta t} + c_{1,\ell} \frac{f_{\ell, t,k+1, s} - f_{\ell, t,k, s}}{\Delta x_\ell} = 0, \qquad \ell \in \mathcal{L}_p, t \in \mathcal{T}',k \in \mathcal{X}', s \in \mathcal{S} \label{eqn:pipe1}\\
& \frac{f_{\ell, t,k, s} - f_{\ell,t-1, k,s}}{\Delta t} = -c_{2,\ell} \frac{p_{\ell, t, k+1,s} - p_{\ell, t, k, s}}{\Delta x_\ell} - c_{3,\ell} \frac{f_{\ell, t,k,s} |f_{\ell,t,k,s}|}{p_{\ell, t,k,s}}, \qquad \ell \in \mathcal{L}_p, t \in \mathcal{T}',k \in \mathcal{X}', s \in \mathcal{S}\label{eqn:pipe2}\\
& \frac{f_{\ell, 1, k+1, s} - f_{\ell, 1, k, s}}{\Delta x_\ell} = 0, \qquad \ell \in \mathcal{L}_p, k \in \mathcal{X}', s \in \mathcal{S}\label{eqn:ss1}\\
& c_{2, \ell} \frac{p_{\ell, 1, k+1,s} - p_{\ell, 1, k, s}}{\Delta x_\ell} + c_{3,\ell} \frac{f_{\ell, 1, k, s} |f_{\ell, 1, k, s}|}{p_{\ell, 1, k, s}} = 0\label{eqn:ss2}\\
& m_{\ell, t, s} = \frac{1}{c_{1,\ell}} \sum_{k=1}^{N_x - 1} p_{\ell, t, k, s} \Delta x_{\ell}, \qquad \ell \in \mathcal{L}_p, t \in \mathcal{T}, s \in \mathcal{S}\label{eqn:linepack1}\\
& m_{\ell, N_t,s} \ge m_{\ell, 1, s}, \qquad \ell \in \mathcal{L}_p, s \in \mathcal{S} \label{eqn:linepack2}\\
& f_{\ell, t, 1, s} = f^{in}_{\ell, t, s}, \qquad \ell \in \mathcal{L}_p, t \in \mathcal{T}, s \in \mathcal{S}\label{eqn:pipelinks1} \\
& f_{\ell, t, N_x, s} = f^{out}_{\ell, t, s}, \qquad \ell \in \mathcal{L}_p, t \in \mathcal{T}, s \in \mathcal{S} \\
& p_{\ell, t, 1, s} = p^{in}_{\ell, t, s}, \qquad \ell \in \mathcal{L}_p, t \in \mathcal{T}, s \in \mathcal{S}\\
& p_{\ell, t, N_x, s} = p^{out}_{\ell, t, s}, \qquad \ell \in \mathcal{L}_p, t \in \mathcal{T}, s \in \mathcal{S} \label{eqn:pipelinks2}
\end{align} \end{subequations}
\subsubsection{Linking Constraints}\label{sec:linking_constraints} Here, we introduce the notation used by \cite{jalving2021} of $\mathcal{L}_{rec(j)}$ as the set of pipelines or compressors that flow into junciton $j$ and $\mathcal{L}_{snd(j)}$ as the set of pipelines or compressors that receive the flow from junction $j$. Further, we define $\theta_{rec(\ell),t,s}$ and $\theta_{snd(\ell),t,s}$ as the receiving and sending junctions respectively for $\ell \in \mathcal{L}$, $t \in \mathcal{T}$, and $s \in \mathcal{S}$. Equations \eqref{eqn:flow_links} - \eqref{eqn:delivery_link} require that the flow going in and out of a junction is conserved and Equations \eqref{eqn:press_link1} and \eqref{eqn:press_link2} require that the pressure is consistent with the objects to which the junction is connected.
\begin{subequations}\label{eqn:link_constraints}
\begin{align}
& \sum_{\ell \in \mathcal{L}_{rec(j)}} f^{out}_{\ell, t, s} - \sum_{\ell \in \mathcal{L}_{snd(j)}} f^{in}_{\ell,t,s} = 0, \qquad j \in \mathcal{J}', t \in \mathcal{T}, s \in \mathcal{S} \label{eqn:flow_links}\\
& \bar{F}_{t,s} - \sum_{\ell \in \mathcal{L}_{snd(1)}} f^{in}_{\ell,t,s} = 0, \qquad t \in \mathcal{T}, s \in \mathcal{S} \label{eqn:supply_link}\\
& \sum_{\ell \in \mathcal{L}_{rec(N_j)}} f^{out}_{\ell, t, s} - F_{t,s}, \qquad t \in \mathcal{T}, s \in \mathcal{S}\label{eqn:delivery_link}\\
& p^{in}_{\ell, t, s} = \theta_{rec(\ell), t, s}, \qquad \ell \in \mathcal{L}, t \in \mathcal{T}, s \in \mathcal{S} \label{eqn:press_link1}\\
& p^{out}_{\ell, t, s} = \theta_{snd(\ell), t, s}, \qquad \ell \in \mathcal{L}, t \in \mathcal{T}, s \in \mathcal{S} \label{eqn:press_link2}
\end{align} \end{subequations}
\subsubsection{Overall Model}\label{sec:obj_function}
The overall model incorporating the above constraints is given in Equation \eqref{eq:gas_graph_model}. The objective function (Equation \eqref{eq:objective_function}) maximizes the overall profit by taking the minimum of the negative scaled sum of gas delivered plus a scaled compressor power. There is also a penalty term for any amount delivered over a scenario's demand. Positive scalars $\alpha$, $\beta$, and $\kappa$ are weights on their respective terms to give an overall profit. Equations \eqref{eqn:constraint1} - \eqref{eqn:constraint2} are the constraints detailed in sections \ref{sec:junction_constraints} - \ref{sec:linking_constraints} above. We also introduce the variables $\bar{P}_{\ell, t}$ for $\ell \in \mathcal{L}_c$ and $t \in \mathcal{T}$ which are the first stage variables of compressor power. These variables fix the compressor power across all scenarios by requiring that any given compressor operates with the same power at each time point in each scenario (Equation \eqref{eqn:first_stage_variables}).
\begin{subequations}\label{eq:gas_graph_model}
\begin{align}
\min_{\boldsymbol{x}} &\;\; \sum_{\ell \in \mathcal{L}_c, t \in \mathcal{T}, s \in \mathcal{S}} \alpha P_{\ell, t, s} - \sum_{t \in \mathcal{T}, s \in \mathcal{S}} \beta F_{t,s} + \sum_{t \in \mathcal{T}, s \in \mathcal{S}} \kappa \tilde{F}_{t,s} \label{eq:objective_function}\\
\textrm{s.t.} &\; \text{Junction Constraints} \quad (\ref{eqn:junction_constraints}) \label{eqn:constraint1}\\
&\; \text{Compressor Constraints} \quad (\ref{eqn:compressor_constraints})\\
&\; \text{Pipeline Constraints} \quad (\ref{eqn:pipeline_constraints})\\
&\; \text{Linking Constraints} \quad (\ref{eqn:link_constraints})\label{eqn:constraint2}\\
&\; \bar{P}_{\ell, t} = P_{\ell, t, s}, \qquad \ell \in \mathcal{L}_c, t \in \mathcal{T}, s \in \mathcal{S} \label{eqn:first_stage_variables}
\end{align} \end{subequations}
Finally, note that there are several decision variables within this formulation. In Equation \ref{eq:objective_function}, we introduce the variable $\boldsymbol{x}$, which is a vector of all decision variables, including $F_{t,s}$, $\tilde{F}_{t,s}$, $\bar{F}_{t,s}$, $\theta_{j,t,s}$, $p_{\ell, t, s} $, $f_{\ell,t,s}$, $\bar{p}_{\ell,t,s}$, $P_{\ell, t, s}$, $f_{\ell', t,k,s}$, $p_{\ell',t,k,s}$, and $\bar{P}_{t,s}$ for $\ell \in \mathcal{L}_c$, $\ell' \in \mathcal{L}_p$, $j \in \mathcal{J}$, $t \in \mathcal{T}$, $k \in \mathcal{X}$, and $s \in \mathcal{S}$.
\subsection{Graph Structure and Visualizations}
The gas network formulation forms a graph that places each scenario on its own subgraph within {\tt Plasmo.jl}. These scenario subgraphs were further structured by representing each $\ell \in \mathcal{L}_c$, $\ell' \in \mathcal{L}_p$, and $j \in \mathcal{J}$ as a subgraph. The subgraphs for compressors and junctions were comprised of individual nodes for each $t \in \mathcal{T}$. Pipeline subgraphs were also structured in space, resulting in nodes for each $k \in \mathcal{X}$ in addition to $t \in \mathcal{T}$. In our problem formulation, we used 11 compressors, 13 pipelines, and 25 junctions. We applied discretization to the model to obtain 24 points in time and 10 evenly-spaced points in space for each pipeline. The problem was scaled by changing the number of scenarios within the problem and included as many as 150 scenarios. With each scenario containing 11,376 variables, the largest form of this problem with 150 scenarios contained over 1.7 million variables. \\
A visualization of a scenario of this gas network, modeled as a graph, is shown in Figure \ref{fig:GN_one_scenario}. The circular nature of the structure along the time dimension arises due to periodicity constraints that require the final linepack in each pipeline to be greater than or equal to the initial linepack of that pipeline. There are 24 nodes within each ring, with one node for each time step. The spatial discretization of the pipelines is evident by the sets of ten pipeline nodes between each junction. Each node in Figure \ref{fig:GN_one_scenario} contains sets of variables and constraints, and are linked to other nodes through linking constraints. \\
\begin{figure}
\caption{Visualization of a single scenario of the stochastic gas network OptiGraph}
\label{fig:GN_one_scenario}
\end{figure}
The stochastic form of this problem is visualized in Figure \ref{fig:GN_four_scenario} (with four scenarios shown). A master node in the center of the figure contains the first stage variables ($\bar{P}_{\ell, t},$ for $\ell \in \mathcal{L}_c$ and $t \in \mathcal{T}$), and it is connected by linking constraints to each compressor node in every scenario, requiring the same compressor in each scenario to operate with the same power at each time point (see Equation \eqref{eqn:first_stage_variables}). The variables placed on this master node determine the size of the Schur complement. Because the compressors within each scenario are linked to the first-stage variables on the master node (rather than being linked directly across scenarios) the Schur complement size remains constant regardless of the number of scenarios included within the model. In the case of our formulation, {\tt MadNLP.jl} detects the two-stage structure of the problem and does not increase the Schur complement size when additional scenarios are added. \\
\begin{figure}
\caption{Visualization of a stochastic gas network OptiGraph with four scenarios. The first-stage variables on the compressor nodes are linked to a master node}
\label{fig:GN_four_scenario}
\end{figure}
Figures \ref{fig:GN_one_scenario} and \ref{fig:GN_four_scenario} show that graph-structured optimization could be useful for visualization of complex models. These visualizations also illustrate that the user can exploit at least three structures present in the problem: time, space, and scenarios. The time structure is shown by the circular nature of the scenario graphs; the spatial structure is shown by the length of the graph from one set of junctions to the final set of junctions; and the scenario structure is shown by the separate scenario graphs that are then linked to a master node. We partitioned this problem by scenario by placing each scenario on its own subgraph and aggregating each subgraph into a node using {\tt Plasmo.jl} prior to solving with {\tt MadNLP.jl}. This resulted in the two-level tree structure seen in two-stage stochastic optimization. However, it would also be possible to construct and partition the problem by its other structures (time or space). In addition, the visualizations of other graph-based optimization models could lead to insights into additional structures and other non-obvious structures could be illuminated by using a graph-partitioning algorithm. \\
\subsection{Model Results}
To demonstrate the capability of this two-stage stochastic model, we show the optimal solution of this problem under 10 scenarios for a 170 km network (pipe diameter of 0.92 m) under the formulation discussed above. The gas demand at the end of the pipeline varied across each scenario and can be seen in Figure \ref{fig:Gas_Demands}. Each scenario involved a step up and a step down, but the magnitude and location of that step could vary between scenarios. While each scenario operated under the same compressor power policy, the gas demands in each scenario were still able to be met exactly, with the exception of the first time point of scenario 6, which over delivered by less than 2 tonnes per hour. \\
\begin{figure}
\caption{Natural gas demand profiles in 10 different scenarios for a stochastic natural gas network problem. The network consisted of 11 compressors, 13 pipelines, and 25 junctions. When operating with the same compressor power policy, each scenario was able to exactly meet the given demand with the exception of time point 1 of scenario 6, which over delivered by less than 2 tonnes per hour.}
\label{fig:Gas_Demands}
\end{figure}
The ability to meet the varying gas demands while operating under the same compressor power policy is largely made possible by varying the pressure and linepack within each scenario. Figure \ref{fig:linepack_results} shows the linepack across different scenarios for two pipelines within the model (pipelines 3 and 13). The linepack shape over time was similar across scenarios, but the magnitude could differ to meet the required demand. The optimal compressor power policy is shown in Figure \ref{fig:Compressor_Policy}. Most compressors operated at a similar power from hour 2 until hour 20, where powers started to diverge. Compressor 11 saw a sharp increase during the last few time points. These changes in the later time periods were likely due to compressors restoring the linepack to the original values. The linepack in pipeline 13 was depleted over time until about hour 20, after which the linepack was slowly restored (Figure \ref{fig:linepack_results}). Pipeline 13 is one of two pipelines that follows compressor 11, the compressor that saw a sharp peak in power usage at hour 20.
\begin{figure}
\caption{Linepack change over time for a stochastic natural gas network problem containing 11 compressors, 13 pipelines, and 25 junctions. The linepack in 10 different scenarios varied with time to meet varying gas demands at the end of the pipeline. Pipeline 3 (top) and pipeline 13 (bottom) are shown.}
\label{fig:linepack_results}
\end{figure}
\begin{figure}
\caption{Compressor power policy for a stochastic natural gas network problem. The network consisted of 11 compressor, 13 pipelines, and 25 juncitons with varying gas demands at the end of the network.}
\label{fig:Compressor_Policy}
\end{figure}
\subsection{Solver Performance}
To test solver performance, we solved this optimization problem with different numbers of scenarios in {\tt MadNLP.jl}. Here, we used three different solver options in {\tt MadNLP.jl}: MA57, PardisoMKL, and {\tt MadNLP.jl}'s Schur solver equipped with MA57 as the linear solver. When operating with MA57, we tested {\tt MadNLP.jl} in both serial and with 30 parallel threads, and we tested PardisoMKL and Schur using 30 threads each. In each of these tests, the problem was modeled as a graph using {\tt Plasmo.jl}'s OptiGraph. For comparison to a problem not modeled as a graph, we also created a {\tt JuMP.jl} model of the system and solved that model with MA57 where {\tt MadNLP.jl} had access to 30 parallel threads (this helps compare efficiency gains from structured modeling). Each test was run on a shared-memory parallel computing server that contained a 40 core Intel(R) Xeon(R) CPU E5-2698 v4 processor running @ 2.2 GHz. Solution times for different problem sizes are shown in Figure \ref{fig:sol_times} and given in Tables \ref{tab:solution_times} and \ref{tab:solution_times_pit}. The times shown in these figures were collected by running each test four times and averaging the last three solution times. The first time was omitted to exclude compilation time. \\ \begin{figure}
\caption{Total solution time per iteration (top) and linear solution time per iteration (bottom) with different problem sizes for {\tt MadNLP.jl} equipped with solvers MA57, Schur, PardisoMKL. The problem was formulated as either a {\tt Plasmo.jl} OptiGraph or a {\tt JuMP.jl} model as indicated. Values represent an average of three trials}
\label{fig:sol_times}
\end{figure}
\begin{table}[h]
\caption{Solution times with different problem sizes for {\tt MadNLP.jl} equipped with solvers MA57, Schur, and PardisoMKL. The problem was formulated as either a {\tt Plasmo.jl} OptiGraph or a {\tt JuMP.jl} model as indicated. Values represent an average of three trials}
\centering
\begin{tabular}{c rrrrrrrr}
\hline\hline
& & \multicolumn{7}{c}{Problem Size (Thousands of Variables)} \\
Solver & & 12 & 57 & 114 & 285 & 853 & 1,138 & 1,707 \\
\hline
\hline
\multirow{3}{*}{\shortstack{MA57/Plasmo.jl\\ 1 thread}}& iterations & 45 & 93 & 97& 129& 150& 145& 183\\
& total solver time (s) & 2.23 & 116.9 & 292.6& 382.5& 1405.2& 1829.4& 3850.5\\
& linear solver time (s) & 1.86 & 109.9 & 270.0& 291.3& 973.2& 1232.8& 2425.8\\
\hline
\multirow{3}{*}{\shortstack{MA57/Plasmo.jl\\ 30 threads}}& iterations & 45 & 93 & 97& 130 & 150& 145& 183\\
& total solver time (s) & 2.42 & 121.7 & 301.7& 315.4& 1169.4& 1474.3& 3092.5\\
& linear solver time (s) & 1.92 & 118.7 & 290.6& 299.8& 978.7& 1245.1& 2413.4\\
\hline
\multirow{3}{*}{\shortstack{MA57/JuMP.jl\\ 30 threads}}& iterations & 45 & 95 & 100 & 123 & 146 & 145 & 185 \\
& total solver time (s) & 2.33 & 130.6 & 310.0& 353.7& 1388.2&1801.0 &3896.9 \\
& linear solver time (s) & 1.86 & 113.3 &279.5 &269.9 & 995.6& 1268.0& 2590.7\\ \hline \multirow{3}{*}{\shortstack{Schur/Plasmo.jl\\ 30 threads}}& iterations & 44 & 94 & 100& 120& 139& 134& 178\\ & total solver time (s) & 22.3 & 60.7 & 89.6 & 211.0 & 910.1 & 1184.3 & 2436.1 \\ & linear solver time (s) & 21.6 & 58.2 & 80.2 & 190.8 & 728.0 &1003.3 & 1915.6 \\ \hline \multirow{3}{*}{\shortstack{PardisoMKL/Plasmo.jl\\ 30 threads}}& iterations & 47 & 94 &98 &123 &145 &144 &177 \\ & total solver time (s) & 4.48 & 10.7 &27.0 &90.0 &967.2 &1763.7 &6512.9 \\ & linear solver time (s) & 3.68 & 7.30 &16.5 &61.4 &746.4 &1504.8 &6016.5 \\ \hline
\end{tabular}
\label{tab:solution_times} \end{table}
\begin{table}[h]
\caption{Solution times per iteration with different problem sizes for {\tt MadNLP.jl} equipped with solvers MA57, Schur, and PardisoMKL. The problem was formulated as either a {\tt Plasmo.jl} OptiGraph or a {\tt JuMP.jl} model as indicated}
\centering
\begin{tabular}{c rrrrrrrr}
\hline
\hline
& & \multicolumn{7}{c}{Problem Size (Thousands of Variables)} \\
Solver & Value per Iteration & 12 & 57 & 114 & 285 & 853 & 1,138 & 1,707 \\
\hline
\hline
\multirow{2}{*}{\shortstack{MA57/Plasmo.jl\\ 1 thread}}& total solver time (s) & 0.05 & 1.26 & 3.02 & 2.97 & 9.37&12.62 &21.04 \\
& linear solver time (s) & 0.04& 1.18 &2.78 &2.26 &6.49 &8.50 &13.26 \\
\hline
\multirow{2}{*}{\shortstack{MA57/Plasmo.jl\\ 30 threads}}& total solver time (s) & 0.05 &1.31 &3.11 &2.42 &7.80 &10.17 &16.90\\
& linear solver time (s) & 0.04&1.28 &3.00 &2.30 &6.52 &8.59 &13.19 \\
\hline
\multirow{2}{*}{\shortstack{MA57/JuMP.jl\\ 30 threads}}& total solver time (s) &0.05 &1.37 &3.10 &2.88 &9.51 &12.42 &21.06\\
& linear solver time (s) &0.04 &1.19 &2.79 &2.19 &6.82 &8.75 &14.00 \\
\hline
\multirow{2}{*}{\shortstack{Schur/Plasmo.jl\\ 30 threads}}& total solver time (s) & 0.51& 0.65& 0.90&1.76 &6.54 &8.84 &13.69\\
& linear solver time (s) & 0.49& 0.62 &0.80 &1.59 &5.24 &7.49 &10.76 \\
\hline
\multirow{2}{*}{\shortstack{PardisoMKL/Plasmo.jl\\ 30 threads}}& total solver time (s) &0.10 &0.11 &0.27 &0.73 &6.67 &12.22 &36.74\\
& linear solver time (s) & 0.08& 0.08 &0.17 &0.50 &5.15 &10.42 &33.94 \\
\hline \hline
\end{tabular}
\label{tab:solution_times_pit} \end{table}
The results in Figure \ref{fig:sol_times} and Tables \ref{tab:solution_times} and \ref{tab:solution_times_pit} provide comparisons between different solver options. At small problem sizes ($\sim$ 10,000 variables), {\tt MadNLP.jl} with MA57 outperformed {\tt MadNLP.jl} with Schur or PardisoMKL based on total time per iteration. However, solver times saw the more drastic differences as the number of variables increased. MA57 with access to one thread took the longest time to run per iteration from about 100,000 to 1.1 million variables, but it was surpassed by PardisoMKL when run with 1.7 million variables. {\tt MadNLP.jl} operating with MA57 with access to 30 threads was 20\% faster per iteration at 1.7 million variables than when {\tt MadNLP.jl} had access to only one thread. The Schur solver did better than any run of MA57 after the problem was scaled beyond $\sim$ 10,000-50,000 variables. For the largest problem sizes, the Schur solver was 19\% faster per iteration than {\tt MadNLP.jl} operating with MA57 and 30 threads, and it was 35\% faster per iteration than {\tt MadNLP.jl} operating with MA57 and one thread . PardisoMKL also performed well at lower problem sizes ($\le$ $\sim$600,000 variables), but showed a substantial increase in solver time as problem sizes got very large, ultimately taking 2.7 times the time per iteration compared to the Schur solver at 1.7 million variables. \\
Notably, there were significant differences between the total time per iteration at the largest problem size for {\tt MadNLP.jl} operating in parallel or serial on either {\tt Plasmo.jl}'s OptiGraph or {\tt JuMP.jl}'s model. Solving the {\tt JuMP.jl} model with 30 threads took the same amount of time as solving a {\tt Plasmo.jl} OptiGraph with 1 thread while operating with MA57. This is because {\tt MadNLP.jl} does not parallelize the {\tt JuMP.jl} model since it is not constructed as a graph. However, when solving a {\tt Plasmo.jl} OptiGraph with 30 threads rather than 1 thread, {\tt MadNLP.jl} operating with MA57 had a reduction of more than four seconds in total solver time per iteration even though the linear solver times per iteration were virtually identical. This is because, while MA57 has limited parallelization capability (evidenced by the linear solver results), {\tt MadNLP.jl} is able to parallelize function and derivative evaluations outside of the linear solver to reduce total solution time. This highlights that non-trivial improvements in computational time can be achieved by exploiting structure outside the linear solver. \\
These results highlight some of the benefits of graph-structured optimization and the capabilities of {\tt MadNLP.jl}. First, graph-structured optimization and {\tt MadNLP.jl} allow for function and derivative evaluations to be run in parallel, even when the linear solver itself has limited parallelization abilities. In addition, graph-structured optimization provides an efficient method for implementing decomposition schemes such as Schur decomposition that can be run in parallel. While the Schur solver was equipped with MA57 as a linear solver, the linear solver time per iteration was less than MA57 alone beyond problem sizes of $\sim$ 10,000-50,000 variables. Thus graph-structured optimization has the ability to reduce solution times for complex problems. {\tt MadNLP.jl} takes advantage of this ability and efficiently implements these concepts.
\section{Conclusions and Future Work}
We highlighted many of the benefits of graph-structured optimization, including visualization capabilities, partitioning schemes, the parallelization of function and derivative evaluations, and decomposition schemes to reduce solver time. These advantages are utilized by {\tt Plasmo.jl} and {\tt MadNLP.jl} to model and solve graph-structured optimization problems. We gave an illustrative example of the visualization and partitioning capabilities of {\tt Plasmo.jl}, and we applied {\tt MadNLP.jl} to solve a case study of a stochastic natural gas pipeline using multiple solvers available within {\tt MadNLP.jl}. The results of this case study suggest that significant time can be saved by parallelizing function and derivative evaluations and using decomposition schemes to parallelize the linear solver. \\
While graph-based modeling offers many benefits, this paradigm can be expanded in a number of ways. We are interested in increasing the ability of {\tt MadNLP.jl} to operate with subgraphs without aggregating some subgraphs into nodes prior to solution. In addition, we are interested in additional decomposition schemes that could also be applied to graph-structured optimization. Additional work can be done to test the capabilities and applications of Schwarz decomposition, particularly its ability to operate as an iterative solver.
\section*{Acknowledgments} This work was partially supported by the U.S. Department of Energy under grant DE-SC0014114. We also acknowledge the contributions of Jordan Jalving for his recommendations for modeling and interfacing with {\tt Plasmo.jl}.
\end{document} | arXiv |
\begin{document}
\begin{abstract} A Lie algebra is called {\em nonsoliton} if it does not admit a soliton inner product. We demonstrate that the subset of nonsoliton Lie algebras in the moduli space of indecomposable $n$-dimensional $\ensuremath{\mathbb N}$-graded nilpotent Lie algebras is discrete if and only if $n \le 7.$
\end{abstract}
\maketitle
\section{Introduction}\label{introduction}
A Lie algebra may be endowed with infinitely many different inner products. Among these, soliton inner products are considered preferred inner products. An inner product $Q$ on a nilpotent Lie algebra $\ensuremath{\mathfrak{g}} $ is called {\em soliton} if the Ricci endomorphism $\operatorname{Ric}$ of $\ensuremath{\mathfrak{g}} $ defined by $Q$ differs from a derivation of $\ensuremath{\mathfrak{g}} $ by a scalar multiple of the identity map on $\ensuremath{\mathfrak{g}} .$ (See Section \ref{metric Lie algebras} for a precise definition of the map $\operatorname{Ric}$). We will call a Lie algebra {\em soliton} if it admits a soliton inner product and we will call it {\em nonsoliton} if it does not admit a soliton inner product.
Soliton inner products on nilpotent Lie algebras are called {\em
nilsoliton}. The study of nilsoliton inner products for nilpotent Lie algebras originated in the analysis of Einstein solvmanifolds (\cite{lauret01b}). Indeed, deep results of Heber and Lauret allow one to reduce the study of Einstein inner products on solvable Lie algebras to the study of soliton inner products on nilradicals (\cite{heberinv}, \cite{lauret01b}, \cite{lauret-standard}). Of independent interest, a soliton inner product on a nilpotent Lie algebra defines a metric on the corresponding simply connected nilpotent Lie group that is soliton in the sense that the Ricci flow moves the metric by diffeomorphisms and rescaling (\cite{lauret01b}). And, outside of the category of homogeneous spaces, soliton inner products are of use in a purely algebraic context in that they supply extra structure for algebraic computations and may give canonical presentations of Lie algebras (See Example 3.11 of \cite{payne-index}).
When they exist, nilsoliton inner products are unique up to scaling (\cite{lauret01b}). If a nilpotent Lie algebra admits a soliton inner product, then it is $\ensuremath{\mathbb N}$-graded. As not all nilpotent Lie algebras are $\ensuremath{\mathbb N}$-graded, not all nilpotent Lie algebras admit nilsoliton inner products. One can find continuous families of nonsoliton nilpotent Lie algebras by finding continuous families of characteristically nilpotent Lie algebras. (A Lie algebra is {\em characteristically nilpotent} if its derivation algebra is nilpotent.) Such families exist in dimensions seven and higher (see \cite{khakimdjanov02}). As the direct sum of nilpotent Lie algebras is soliton if and only if each summand is soliton (\cite{jablonski-09}, \cite{nikolayevsky-preEinstein}), we will restrict our attention to indecomposable nilpotent Lie algebras. We will study the subset of nonsoliton Lie algebras in the moduli space of all indecomposable $\ensuremath{\mathbb N}$-graded nilpotent Lie algebras of fixed dimension. In particular, we are interested in determining when the set of nonsoliton Lie algebras is discrete in this moduli space.
In dimensions $6$ and lower, the situation is well-understood: the moduli space of nilpotent Lie algebras is discrete (\cite{degraaf-07}), and all nilpotent Lie algebras of dimension $6$ and lower admit soliton inner products (\cite{lauret02}, \cite{will03}). In dimension $7,$ the moduli space of real nilpotent Lie algebras consists of a finite number of discrete points and a finite number of continuous families of nonisomorphic nilpotent Lie algebras (\cite{seeley-93}, \cite{gong-98}). Nikolayevsky proved that if two real nilpotent Lie algebras have the same complexification, then either both are soliton or both are nonsoliton (\cite{nikolayevsky-preEinstein}). This result was also proved independently by M.\ Jablonski using different methods (\cite{jablonski-08b}). Using this result about complex forms, along with Carles's classification of complex nilpotent Lie algebras of dimension 7 (\cite{carles-96}, \cite{magnin-online}), Culma determined precisely which $7$-dimensional complex nilpotent Lie algebras have real forms that admit nilsoliton inner products (\cite{culma1}, \cite{culma2}). Culma found that among the continuous families of nonsoliton nilpotent Lie algebras in dimension $7,$ none of them are $\ensuremath{\mathbb N}$-graded. It follows that the subset of nonsoliton Lie algebras in the moduli space of indecomposable $7$-dimensional $\ensuremath{\mathbb N}$-graded nilpotent Lie algebras is discrete.
Arroyo determined precisely which $\ensuremath{\mathbb N}$-graded filiform nilpotent Lie algebras of dimension 8 admit soliton inner products (\cite{arroyo-11}). She found although there are continuous families of solitons in that moduli space, there are precisely four isolated nonsoliton Lie algebras.
Eberlein and Nikolayevsky showed independently that except for in two cases, soliton Lie algebras are dense in the moduli space of two-step nilpotent Lie algebras (\cite{eberlein-08}, \cite{nikolayevsky-preEinstein}). (Note that all two-step nilpotent Lie algebras are $\ensuremath{\mathbb N}$-graded by a derivation that equals the identity on a complement to the center and that is twice the identity on the center.) Jablonski showed that the soliton Lie algebras are dense in the remaining two cases-- when the nilpotent Lie algebra is of type $(2k+1,2)$ or $(2k+1,(\begin{smallmatrix} 2k+1 \\ 2 \end{smallmatrix})-2)$ (\cite{jablonski-thesis}). \begin{thm} [\cite{eberlein-08}, \cite{nikolayevsky-preEinstein}, \cite{jablonski-thesis}] The set of soliton Lie algebras is dense in the moduli space of two-step nilpotent Lie algebras.
\end{thm} Given the classification results we have described, and the theorem just stated, one might wonder if the set of nonsoliton Lie algebras is always discrete in the moduli space of two-step $n$-dimensional nonsoliton $\ensuremath{\mathbb N}$-graded nilpotent Lie algebras. The answer is no. The first continuous families of nonsoliton $\ensuremath{\mathbb N}$-graded nilpotent Lie algebras were found by C.\ Will. \begin{thm}[\cite{will-08}]
The moduli space of indecomposable $9$-dimensional
two-step nilpotent Lie algebras contains two one-parameter families
of nonsoliton nilpotent Lie algebras. \end{thm}
Jablonski defined a general method for constructing families of two-step nilpotent Lie algebras called concatenation. He used the concatenation construction to define continuous families of irreducible nonsoliton two-step nilpotent Lie algebras in infinitely many dimensions. \begin{thm}[\cite{jablonski-09}]
For $n \ge 23,$ the moduli space of indecomposable $n$-dimensional
two-step nilpotent Lie algebras contains a continuous family of
nonsoliton Lie algebras. \end{thm}
There are two key issues involved in the results of Jablonski and Will. First of all, nonsoliton nilpotent Lie algebras are quite rare, and two-step nilpotent Lie algebras are not classified in dimensions 10 and higher. Therefore, finding examples of nonsoliton nilpotent Lie algebras, or curves of them, requires a thorough understanding of the structure of nilpotent Lie algebras and how that structure relates to the existence of a soliton inner product. Second, in contrast to the semisimple case, there are few fine algebraic invariants that allow one to distinguish nonisomorphic nilpotent Lie algebras, so it is a significant task to show that the curves of nonsoliton nilpotent Lie algebras are mutually nonisomorphic. Will used the Pfaffian defined by Scheuneman in \cite{scheuneman-67} to distinguish the Lie algebras in her families, while Jablonski used geometric invariant theory.
Our main result is that the moduli space of indecomposable $\ensuremath{\mathbb N}$-graded $n$-dimensional nonsoliton nilpotent Lie algebras is not discrete if $n \ge 8:$ \begin{thm}\label{main thm}
The moduli space of indecomposable $n$-dimensional nonsoliton
$\ensuremath{\mathbb N}$-graded nilpotent Lie algebras contains a one-parameter
family of nonsolitons if $n \ge 8.$ \end{thm}
As a corollary we can say exactly when the nonsoliton Lie algebras are isolated in the moduli space: \begin{cor} The set of nonsoliton Lie algebras is discrete in the moduli space of indecomposable $n$-dimensional nonsoliton
$\ensuremath{\mathbb N}$-graded nilpotent Lie algebras if and only if $n \le 7.$ \end{cor}
The families of nilpotent Lie algebras that we construct to prove the theorem are the first examples of continuous families of three-step nonsoliton nilpotent Lie algebras. It would be interesting to refine the result in Theorem \ref{main thm} by specializing it to the two-step case, determining in which dimensions nonsolitons are discrete in the moduli space of two-step nilpotent Lie algebras.
This manuscript is organized as follows. In Section \ref{preliminaries}, we review necessary background material related to nilpotent Lie algebras, inner products on Lie algebras, soliton inner products, and Nikolayevsky (pre-Einstein) derivations of nilpotent Lie algebras. In Section \ref{8 and 9}, we present two continuous families of indecomposable $\ensuremath{\mathbb N}$-graded nilpotent Lie algebras, one in dimension eight and one in dimension nine, and we prove that the Lie algebras in the families are nonsoliton. We also describe the derivation algebras of the Lie algebras in the families. In Section \ref{higher dim}, we use the $8$- and $9$-dimensional examples from Section \ref{8 and
9} to construct continuous families of indecomposable $\ensuremath{\mathbb N}$-graded nilpotent Lie algebras in dimensions $n \ge 10.$ We find Nikolayevsky derivations for these Lie algebras, and we prove that all of the Lie algebras in the families are nonsoliton. Last, we prove that for any dimension $n \ge 8,$ the Lie algebras in the family are all mutually nonisomorphic. In Section \ref{proof of main theorem}, we combine our results from Sections \ref{8 and 9} and \ref{higher dim} to prove the main result.
\section{Preliminaries}\label{preliminaries}
\subsection{Lie algebras}
The descending central series of a Lie algebra $\ensuremath{\mathfrak{g}} $ is defined by $\ensuremath{\mathfrak{g}} ^{(1)} = \ensuremath{\mathfrak{g}} $ and $\ensuremath{\mathfrak{g}} ^{(j + 1)} = [\ensuremath{\mathfrak{g}} , \ensuremath{\mathfrak{g}} ^{(j)}]$ for $j > 1.$ The Lie algebra $\ensuremath{\mathfrak{g}} $ is {\em nilpotent} if and only if there is an integer $r$ so that $\ensuremath{\mathfrak{g}} ^{(r)}$ is trivial. If $r$ is the smallest integer so that $\ensuremath{\mathfrak{g}} ^{(r + 1)}$ is trivial, then $\ensuremath{\mathfrak{g}} $ is said to be $r$-{\em step} nilpotent. An {\em $r$-step} nilpotent Lie algebra is said to of {\em type} $(n_1,n_2, \ldots, n_r)$ if $\dim (\ensuremath{\mathfrak{g}} ^{(j)}/ \ensuremath{\mathfrak{g}} ^{(j+1)}) = n_j$ for $j = 1, \ldots, r.$
Let $\operatorname{Der}(\ensuremath{\mathfrak{g}} )$ be the derivation algebra of $\ensuremath{\mathfrak{g}} .$ The algebra $\operatorname{Der}(\ensuremath{\mathfrak{g}} )$ has Levi decomposition $\ensuremath{\mathfrak{s}} \oplus (\ensuremath{\mathfrak{t}} _s \oplus \ensuremath{\mathfrak{t}} _c) \oplus \ensuremath{\mathfrak{n}} $ where $\ensuremath{\mathfrak{s}} $ is the semisimple Levi factor and the solvable radical $\operatorname{rad} (\operatorname{Der}(\ensuremath{\mathfrak{g}} )) = \ensuremath{\mathfrak{t}} \oplus \ensuremath{\mathfrak{n}} $ is the direct sum of its nilradical $\ensuremath{\mathfrak{n}} $ and a torus $\ensuremath{\mathfrak{t}} .$ The torus further decomposes as the sum $\ensuremath{\mathfrak{t}} = \ensuremath{\mathfrak{t}} _s \oplus \ensuremath{\mathfrak{t}} _c$ of an $\ensuremath{\mathbb R}$-split torus $\ensuremath{\mathfrak{t}} _s$ and a compact torus $\ensuremath{\mathfrak{t}} _c.$ The dimension of $\ensuremath{\mathfrak{t}} $ is called the {\em rank} of $\ensuremath{\mathfrak{g}} ,$ and the dimension of the $\ensuremath{\mathbb R}$-split torus $\ensuremath{\mathfrak{t}} _s$ is called the {\em real rank} of $\ensuremath{\mathfrak{g}} .$
A Lie algebra is {\em indecomposable} if it cannot be written as the direct sum of two nontrivial ideals.
\subsection{Metric Lie algebras and soliton inner
products}\label{metric Lie algebras}
A {\em metric Lie algebra} $(\ensuremath{\mathfrak{g}} ,Q)$ is a Lie algebra $\ensuremath{\mathfrak{g}} $ endowed with an inner product $Q.$ Associated to each metric Lie algebra is a unique homogeneous space $(G,g),$ where $G$ is the connected Lie group whose Lie algebra is $\ensuremath{\mathfrak{g}} ,$ and $g$ is the left invariant metric on $G$ such that that the restriction of $g$ to the tangent space $T_e G \cong \ensuremath{\mathfrak{g}} $ of $G$ at the identity coincides with $Q.$ The Ricci endomorphism $\operatorname{Ric}$ for the Riemannian manifold $(G,g),$ when restricted to $T_e G,$ is an endomorphism of $T_e G \cong \ensuremath{\mathfrak{g}} .$ We call $\operatorname{Ric}_e$ the Ricci endomorphism for the metric Lie algebra $(\ensuremath{\mathfrak{g}} ,Q)$ and we abuse notation to let $\operatorname{Ric}$ denote $\operatorname{Ric}_e.$
Let $(\ensuremath{\mathfrak{n}} ,Q)$ be a metric nilpotent Lie algebras with associated homogeneous space $(N,g).$
Then Ricci form $\mathsf{ric}_e$ for $(N,g)$ at the identity is the bilinear form on $T_e N \cong \ensuremath{\mathfrak{n}} $ given by \[ \mathsf{ric}(x,y) = -\frac{1}{2}\sum_{i=1}^n Q( [x,x_i], [y,x_i])+\frac{1}{4} \sum_{i,j = 1}^n Q( [x_i,x_j], x) Q( [x_i,x_j], y),\] for $x, y \in \ensuremath{\mathfrak{n}} ,$ and where $\{x_i\}_{i=1}^n$ is an orthonormal basis for $\ensuremath{\mathfrak{n}} .$
Then Ricci endomorphism $\operatorname{Ric}_e = \operatorname{Ric}$ at the identity is the endomorphism of $T_e N \cong \ensuremath{\mathfrak{n}} $ given by the condition that $Q(\operatorname{Ric}(x),y) = \mathsf{ric}(x,y),$ for all $x, y \in \ensuremath{\mathfrak{n}} .$
Let $(\ensuremath{\mathfrak{n}} _1,Q_1)$ and $(\ensuremath{\mathfrak{n}} _2,Q_2)$ be metric nilpotent Lie algebras with associated homogeneous spaces $(N_1,g_1)$ and $(N_2,g_2)$ respectively. A map $\psi : \ensuremath{\mathfrak{n}} _1 \to \ensuremath{\mathfrak{n}} _2$ naturally induces the map $\overline{\psi} : N_1 \to N_2,$ where $\overline \psi= \exp \circ \psi \circ \exp^{-1}.$ E.\ Wilson proved that $\overline{\psi}$ is an isometry if and only if $\psi$ is an isometric isomorphism (\cite{wilson82}); i.e. $\psi$ is an isomorphism and $Q_1(x,y)=Q_2(\psi(x),\psi(y))$ for all $x, y \in \ensuremath{\mathfrak{n}} _1.$
A metric Lie algebra $(\ensuremath{\mathfrak{g}} ,Q)$ is called {\em soliton} if its Ricci endomorphism $\operatorname{Ric} \in \operatorname{End}(\ensuremath{\mathfrak{g}} )$ differs from a scalar multiple of the identity map by a derivation; that is, there exists a $\beta \in \ensuremath{\mathbb R}$ called the {\em soliton
constant} so that $\hat D = \operatorname{Ric} - \beta \operatorname{Id}$ is a derivation. In the case that the Lie algebra is nilpotent, we call the inner product a {\em nilsoliton} inner product, we call $\beta$ the {\em nilsoliton constant} and we call the derivation $\hat D$ the {\em nilsoliton derivation}.
Let $\ensuremath{\mathfrak{g}} $ be a nonabelian Lie algebra with basis $\ensuremath{\mathcal{B}} .$ Let $\Lambda$ index the set of nonzero structure constants $\alpha_{ij}^k$ relative to $\ensuremath{\mathcal{B}} ,$ modulo skew-symmetry: \[ \Lambda = \{ (i,j,k) \, : \, \alpha_{ij}^k \ne 0, i < j \}. \] To each triple $(i,j,k) \in \Lambda,$ we associate the {\em root vector} $\ensuremath{\bm y} _{ij}^k = \ensuremath{\bm e} _i + \ensuremath{\bm e} _j - \ensuremath{\bm e} _k,$ where $\{\ensuremath{\bm e} _i\}_{i=1}^m$ is the standard basis for $\ensuremath{\mathbb R}^m.$ Let $\ensuremath{\bm y} _1, \ensuremath{\bm y} _2, \ldots, \ensuremath{\bm y} _m$ be an enumeration of the root vectors $\ensuremath{\bm y} _{ij}^k, (i,j,k) \in \Lambda,$ using some fixed ordering of $\ensuremath{\mathbb N}^3.$ Our convention is to order $\ensuremath{\mathbb N}^3$ so that for $i_1,j_1,k_1,i_2, j_2, k_2 \in \ensuremath{\mathbb N},$ \begin{itemize} \item{$(i_1,j_1,k_1) < (i_2,j_2,k_2)$ if $k_1 < k_2$} \item{$(i_1,j_1,k_1) < (i_2,j_2,k_1)$ if $i_1 < i_2$} \item{$(i_1,j_1,k_1) < (i_1,j_2,k_1)$ if $j_1 < j_2.$} \end{itemize} The {\em
Gram matrix} for $\ensuremath{\mathfrak{g}} $ with respect to $\ensuremath{\mathcal{B}} $ is the matrix $U = (u_{ij})$ whose entries are the inner products of the root vectors: $u_{ij} = \ensuremath{\bm y} _i \cdot \ensuremath{\bm y} _j.$ We will use the following theorem of Nikolayevsky to show that the nilpotent Lie algebras in our families do not admit soliton inner products.
\begin{thm}[Theorem 3, \cite{nikolayevsky-preEinstein}]\label{Uv} Let $\ensuremath{\mathfrak{n}} $ be a nonabelian nilpotent Lie algebra with basis $\ensuremath{\mathcal{B}} .$ Suppose that the Gram matrix $U$ for $\ensuremath{\mathfrak{n}} $ with respect to $\ensuremath{\mathcal{B}} $ has no entries of $2.$ Then $\ensuremath{\mathfrak{n}} $ admits a soliton inner product if and only if the matrix equation $U \ensuremath{\bm v} = [1]$ has a solution $\ensuremath{\bm v} $ with all positive entries. \end{thm} Note that we have replaced Nikolayevsky's hypothesis that the basis is a ``nice basis'' with an equivalent hypothesis on the Gram matrix $U$ (which depends only on the index set $\Lambda$).
\subsection{Nikolayevsky derivations}
A derivation $D^N$ of a Lie algebra $\ensuremath{\mathfrak{g}} $ is called a {\em
Nikolayevsky derivation} if it is semisimple with real eigenvalues and \begin{equation}\label{defn N der} \operatorname{trace}(D^N \circ F) = \operatorname{trace} (F) \end{equation} for all $F \in \operatorname{Der}(\ensuremath{\mathfrak{g}} ).$ Nikolayevsky defined such derivations and showed that they are unique up to automorphism. He called them {\em pre-Einstein derivations} because when the underlying Lie algebra is nilpotent, they are natural generalizations of the nilsoliton derivation used to define an Einstein solvable extension. Because they are purely algebraic objects of broader use, we prefer to call such a derivation a {\em Nikolayevsky derivation.} He also showed that if $\ensuremath{\mathfrak{g}} $ admits a soliton inner product, then the nilsoliton derivation is a scalar multiple of the Nikolayevsky derivation (\cite{nikolayevsky-preEinstein}). It follows from the proof of Theorem 1.1(a) of \cite{nikolayevsky-preEinstein} that $D^N$ is a Nikolayevsky derivation if and only if the condition in Equation \eqref{defn N der} holds for all $F$ in an $\ensuremath{\mathbb R}$-split torus $\ensuremath{\mathfrak{t}} ^s$ containing $D^N.$ Thus, it is elementary to find the Nikolayevsky derivations of a Lie algebra with real rank one:
\begin{prop}\label{rank 1 N der}[\cite{nikolayevsky-preEinstein}]
Let $\ensuremath{\mathfrak{g}} $ be a nilpotent Lie algebra with real rank
one. Let $D$ be a nontrivial semisimple derivation with real eigenvalues. Then \[ D^N = \frac{\operatorname{trace}(D)}{\operatorname{trace}(D^2)} \, D \] is the unique Nikolayevsky derivation for $\ensuremath{\mathfrak{g}} .$ \end{prop}
\subsection{Moduli spaces of soliton and nonsoliton nilpotent Lie
algebras}\label{moduli spaces}
We need to describe the structure of the moduli space of nilpotent Lie algebras of dimension $n.$ Let $\ensuremath{\mathbb R}^n$ be a real vector space of dimension $n$ with basis $\ensuremath{\mathcal{B}} = \{x_i\}_{i=1}^n.$ Suppose that $\ensuremath{\mathbb R}^n$ is endowed with a Lie bracket that defines a nilpotent Lie algebra structure on $\ensuremath{\mathbb R}^n.$ The Lie bracket is equivalent to a skew-symmetric vector-valued bilinear map $\mu$ in the vector space $V = \wedge^2 (\ensuremath{\mathbb R}^n)^\ast \otimes \ensuremath{\mathbb R}^n.$ The Jacobi identity and the nilpotency condition are polynomial constraints on the coefficients of $\mu$ in $V$ with respect to the basis $\{ x_i^\ast \wedge x_j^\ast \otimes x_k \, : \, 1 \le i < j \le n, 1 \le k \le n \},$ so we may identify each $\mu$ with an element $(\mu, \ensuremath{\mathcal{B}} )$ of an affine subvariety $X$ of $V.$ We let $\ensuremath{\mathfrak{n}} _\mu$ denote the corresponding nilpotent Lie algebra. The general linear group $GL_n(\ensuremath{\mathbb R})$ acts on $X$ by change of basis: for $\mu \in X,$ the element $g \cdot \mu$ of $X$ is defined by \[ (g \cdot \mu) (x,y) = g \mu(g^{-1}x,g^{-1} y), \] for $x, y \in \ensuremath{\mathbb R}^n.$ Two elements $\mu$ and $\nu$ of $X$ define isomorphic Lie algebras $\ensuremath{\mathfrak{n}} _\mu$ and $\ensuremath{\mathfrak{n}} _\nu$ if and only if $\mu$ and $\nu$ are in the same $GL_n(\ensuremath{\mathbb R})$ orbit. The quotient $\ensuremath{\mathcal{N}} _n$ of $X$ by this action is the moduli space of $n$-dimensional nilpotent Lie algebras. The equivalence class of $\mu$ is denoted by $\overline{\mu}.$ We endow $\ensuremath{\mathcal{N}} _n$ with the quotient topology.
The properties of whether a Lie algebra $\ensuremath{\mathfrak{n}} _\mu$ for $\mu \in X$ is $\ensuremath{\mathbb N}$-graded, and whether it is indecomposable are both invariant under the $GL_n(\ensuremath{\mathbb R})$ action. Hence we may define the moduli space $\widetilde \ensuremath{\mathcal{N}} _n$ of $\ensuremath{\mathbb N}$-graded, indecomposable nilpotent Lie algebras to be the set of elements $\overline \mu$ of $\ensuremath{\mathcal{N}} _n$ so that $\ensuremath{\mathfrak{n}} _\mu$ is $\ensuremath{\mathbb N}$-graded and indecomposable. We use the subspace topology for $\widetilde \ensuremath{\mathcal{N}} _n.$ The property of whether or not a Lie algebra $\ensuremath{\mathfrak{n}} _\mu$ admits a soliton inner product is also invariant under the $GL_n(\ensuremath{\mathbb R})$ action, so we may define the set $\text{Nonsol}(n) \subseteq \widetilde \ensuremath{\mathcal{N}} _n$ to be the set of all $\overline{\mu}$ in $\widetilde \ensuremath{\mathcal{N}} _n$ such that $\ensuremath{\mathfrak{n}} _\mu$ does not admit a soliton inner product.
\section{Continuous families of nonsoliton nilpotent Lie algebras in
dimensions 8 and 9}\label{8 and 9}
\subsection{A curve of nonsoliton nilpotent Lie algebras in dimension 8} \begin{defn}\label{dim 8} Let $s \in \ensuremath{\mathbb R},$ and let $\ensuremath{\mathcal{B}} =
\{x_i\}_{i=1}^8$ be a fixed basis for $\ensuremath{\mathbb R}^8.$
Define $\ensuremath{\mathfrak{n}} _s$ to be the
nilpotent Lie algebra with underlying vector space $\ensuremath{\mathbb R}^8$ whose
Lie algebra structure is determined by the bracket relations \begin{align} &[x_2,x_3]= e^{-s} x_4 & &[x_1,x_3]= e^s x_5 & &[x_1,x_2]=x_6 \notag \\ &[x_2,x_6]= e^s x_7 & &[x_3,x_4]= e^{-s} x_7 & &[x_1,x_6]= e^{-s} x_8 \label{m8}\\ &[x_2,x_4]= x_8 & &[x_3,x_5]= e^{s} x_8 & & \strut . \notag \end{align}
The Jacobi Identity may be checked by confirming that there are no distinct choices of $i, j$ and $k$ so that $[x_i,[x_j,x_k]]$ is nonzero. (That there are no such choices may also be deduced from Theorem 7 of \cite{payne-09b}).
It is not hard to verify that for all $s,$ the Lie algebra $\ensuremath{\mathfrak{n}} _s$ is three-step nilpotent of type $(3,3,2)$ and is indecomposable. For all $s,$ we may write the vector space $\ensuremath{\mathfrak{n}} _s$ as the direct sum $\ensuremath{\mathfrak{n}} _s = V_1 \oplus V_2 \oplus V_3,$ where $V_1, V_2$ and $V_3$ are the three steps \begin{align*} V_1 &= \operatorname{span} \{ x_1, x_2, x_3 \} \\ V_2 &= \operatorname{span} \{ x_4, x_5, x_6 \} \\ V_3 &= \operatorname{span} \{ x_7, x_8 \} . \end{align*} Define the derivation $D: \ensuremath{\mathfrak{n}} _s \to \ensuremath{\mathfrak{n}} _s$ by \begin{equation*}
D(x) = k x, \quad \text{ if $x \in V_k,$ for $k=1,2,3.$ } \end{equation*} The eigenspaces $V_1, V_2, V_3$ for $D$ define an $\ensuremath{\mathbb N}$-grading of $\ensuremath{\mathfrak{n}} _s.$ \end{defn}
Now we describe the derivation algebra of a typical nilpotent Lie algebra in the family defined in Definition \ref{dim 8}.
\begin{prop}\label{der alg 8} Let $\ensuremath{\mathfrak{n}} _s$ be the nilpotent Lie algebra
as defined in Definition \ref{dim 8}, for any fixed $s$ in $\ensuremath{\mathbb R}.$
Then the derivation algebra $\operatorname{Der}(\ensuremath{\mathfrak{n}} _s)$ of $\ensuremath{\mathfrak{n}} _s$ is a $16$-dimensional solvable algebra with real
rank one. The derivation algebra decomposes as $\operatorname{Der}(\ensuremath{\mathfrak{n}} _s) = \ensuremath{\mathbb R}
\, D + \ensuremath{\mathfrak{m}} ,$ where $D$ is the derivation $D$ defined above and
$\ensuremath{\mathfrak{m}} $ is the nilradical. The derivation $D^N = \frac{5}{11}D$ is a Nikolayevsky derivation of $\ensuremath{\mathfrak{n}} _s.$ \end{prop}
\begin{proof} The derivation algebra of $\ensuremath{\mathfrak{n}} _s$ is a subspace of
$\operatorname{End}(\ensuremath{\mathfrak{n}} _s).$ The subspace may be described by a system of $8^3$
linear equations in $8^2$ unknowns, where the coefficients of the
linear equations depend on the structure constants for $\ensuremath{\mathfrak{n}} _s.$
(See Section 1.9 of \cite{degraaf-book}.) The structure constants
for $\ensuremath{\mathfrak{n}} _s$ depend on the parameter $s.$ Using Matlab to solve
this system symbolically, we found that for any $s,$ the solution
space to the linear system is $16$-dimensional and is spanned by the
derivation $D$ and $15$ nilpotent derivations that span the
nilpotent subalgebra $\ensuremath{\mathfrak{m}} .$ Hence any semisimple derivation of
$\ensuremath{\mathfrak{n}} _s$ is a scalar multiple of $D$ and the real rank of
$\ensuremath{\mathfrak{n}} _s$ is one. By Proposition \ref{rank 1 N der}, $D^N =
\frac{5}{11}D$ is a Nikolayevsky derivation of $\ensuremath{\mathfrak{n}} _s.$ \end{proof}
Now we show that none of the nilpotent Lie algebras in the family defined in Definition \ref{dim 8} are soliton.
\begin{thm}\label{dim 8 theorem} Suppose that $\ensuremath{\mathfrak{n}} _s$ is a nilpotent Lie algebra as defined in Definition \ref{dim 8}. Then $\ensuremath{\mathfrak{n}} _s$ does not admit a soliton inner product. \end{thm}
\begin{proof} Fix $s$ in $\ensuremath{\mathbb R}$ and suppose that $\ensuremath{\mathfrak{n}} _s$ admits a soliton inner product $Q.$ With respect to the basis $\ensuremath{\mathcal{B}} ,$ the Gram matrix $U$ is \begin{equation}\label{U 8 def}
U = \begin{bmatrix} 3 &1 &1 & 1 & 0 & 0 & 0 &1 \\ 1 &3 &1 & 0 & 1 & 1 & 0& 0\\ 1 &1 &3 & 0 & 0 & 0 & 1& 0 \\ 1 & 0 & 0 & 3& 1 &1 & 1 & 0 \\
0 & 1 & 0 & 1& 3 & 0 &1 & 1\\
0 & 1 & 0 & 1& 0& 3& 1& 1\\
0 &0 &1 & 1& 1& 1& 3 & 1\\
1 & 0 & 0 & 0& 1& 1&1 & 3 \end{bmatrix} .\end{equation}
The solution space to the linear equation $U \ensuremath{\bm v} = [1]$ is $\{ \ensuremath{\bm v} _0 + t \ensuremath{\bm v} _1 \, : \, t \in \ensuremath{\mathbb R} \},$ where \begin{align*}
\ensuremath{\bm v} _0 &= \smallfrac{1}{11} (1,1,3,2,2,2,0,2)^T, \quad \text{and} \\ \ensuremath{\bm v} _1 &= (-1,1,0,1,-1,-1,0,1)^T. \end{align*} For all such solutions $\ensuremath{\bm v} = (v_i),$ $v_7 = 0.$ By Theorem \ref{Uv}, $\ensuremath{\mathfrak{n}} _s$ does not admit a soliton inner product. \end{proof}
We will show in Theorem \ref{nonisomorphic n} that no two nilpotent Lie algebras in the family defined in Definition \ref{dim 8} are isomorphic.
\subsection{A curve of nonsoliton nilpotent Lie algebras in dimension 9}
Now we define a one-parameter family of $9$-dimensional nilpotent Lie algebras, similar to the family of $8$-dimensional nilpotent Lie algebras defined in the previous section. \begin{defn}\label{dim 9} Let $\ensuremath{\mathcal{B}} = \{x_i\}_{i=1}^9$ be a basis for $\ensuremath{\mathbb R}^9.$ Let $s$ be a real number.
Let $\ensuremath{\mathfrak{n}} _{s}$ be the Lie algebra with underlying vector space $\ensuremath{\mathbb R}^9$ whose Lie
algebra structure is determined by the Lie bracket relations \begin{align*} &[x_2,x_3]= e^{4s} x_4 & &[x_1,x_3]= e^{-3s} x_5 & &[x_1,x_2]= e^{-s}x_6 \\ & [x_2,x_6]= e^{-4s} x_7 & &[x_3,x_4]= e^{4s} x_7 & &[x_1,x_6]= e^{4s} x_8 \\ &[x_2,x_4]= x_8 & &[x_3,x_5]= e^{-4s} x_8 & & [x_3,x_6] = e^{-s}x_9 \\ & & & [x_2,x_5] = e^s x_9. & \end{align*} The Jacobi Identity may be confirmed by direct computation, noting that the only time $[[x_i,x_j], x_k]$ is nontrivial for distinct $i,j,k$ is when $\{i,j,k\} = \{1,2,3\}.$ (The latter fact follows from Theorem 7 of \cite{payne-09b}.)
Each Lie algebra $\ensuremath{\mathfrak{n}} _s$ is three-step nilpotent of type $(3,3,3).$
We may write the vector space $\ensuremath{\mathbb R}^9 \cong \ensuremath{\mathfrak{n}} _s$ as the direct sum $\ensuremath{\mathbb R}^9 = V_1 \oplus V_2 \oplus V_3,$ where $V_1, V_2$ and $V_3$ are the three steps for any $\ensuremath{\mathfrak{n}} _s:$ \begin{align*} V_1 &= \operatorname{span} \{ x_1, x_2, x_3 \} \\ V_2 &= \operatorname{span} \{ x_4, x_5, x_6 \} \\ V_3 &= \operatorname{span} \{ x_7, x_8, x_9 \} . \end{align*} A derivation $D$ of $\ensuremath{\mathfrak{n}} _s$ is defined by \begin{equation*}
D(x) = k x, \quad \text{ if $x \in V_k,$ for $k=1,2,3.$ } \end{equation*} The eigenspaces $V_1, V_2, V_3$ for $D$ define an $\ensuremath{\mathbb N}$-grading of $\ensuremath{\mathfrak{n}} _s.$ \end{defn}
\begin{prop}\label{der alg 9} Let $s \in \ensuremath{\mathbb R},$ and let $\ensuremath{\mathfrak{n}} _s$ be as defined in Definition \ref{dim 9}.
The derivation algebra $\operatorname{Der}(\ensuremath{\mathfrak{n}} _s) = \ensuremath{\mathbb R} \, D + \ensuremath{\mathfrak{m}} $ of $\ensuremath{\mathfrak{n}} _s,$ is
$19$-dimensional and solvable, with $18$-dimensional nilradical $\ensuremath{\mathfrak{m}} .$ The real rank of $\ensuremath{\mathfrak{n}} _s$ is one, and $D^N = \frac{9}{14}D$ is a Nikolayevsky derivation of $\ensuremath{\mathfrak{n}} _s.$ \end{prop}
The proof of the proposition is analogous to that of Proposition \ref{der alg 8}, so we do not include it.
Now we show that none of the Lie algebras defined in Definition
\ref{dim 9} are soliton. \begin{thm}\label{dim 9 theorem} Let $s \in \ensuremath{\mathbb R},$ and let $\ensuremath{\mathfrak{n}} _s$ be a nilpotent Lie
algebra as defined in Definition \ref{dim 9}. Then $\ensuremath{\mathfrak{n}} _s$
does not admit a soliton inner product. \end{thm}
\begin{proof} The proof is the same as that of Theorem \ref{dim 8 theorem}, except that \begin{equation}\label{U9 def} U = \begin{bmatrix} 3 & 1 & 1& 1& 0 & 0 & 0 &1 &1 &1 \\ 1 & 3 & 1& 0 & 1& 1 & 0& 0& 1 & -1\\ 1 & 1 & 3& 0& 0& 0& 1& 0 & -1 & 1\\ 1 & 0 & 0& 3& 1& 1& 1& 0& 1 & 1\\ 0 & 1 & 0& 1& 3& 0& 1& 1& 1 & 0\\ 0 & 1 & 0& 1& 0& 3& 1& 1& 1 & 0\\ 0 & 0 & 1& 1& 1& 1& 3& 1& 0 & 1\\ 1 & 0 & 0& 0& 1& 1& 1& 3& 1 & 1\\ 1 & 1 & -1& 1& 1& 1& 0& 1& 3 & 1\\ 1 & -1 & 1& 1& 0& 0& 1& 1& 1 & 3\\ \end{bmatrix}
\end{equation} and the solution space to $U \ensuremath{\bm v} = [1]$ is $\{ \ensuremath{\bm v} _0 + s \ensuremath{\bm v} _1 + t\ensuremath{\bm v} _2 \, : \, s, t \in \ensuremath{\mathbb R} \},$ where \begin{align*} \ensuremath{\bm v} _0 &=
\smallfrac{1}{161}(5,25,39,18,28,28,0,18,16,30)^T. \\
\ensuremath{\bm v} _1 & = (-1,1,0,1,-1,-1,0,1,0,0)^T \\
\ensuremath{\bm v} _2 & = (0,1,-1,0,0,0,0,0,-1,1)^T \end{align*} All vectors $\ensuremath{\bm v} = (v_i)$ in the solution space have $v_7 = 0.$ \end{proof}
We will show in Theorem \ref{nonisomorphic n} of Section \ref{higher dim} that if $\ensuremath{\mathfrak{n}} _s$ and $\ensuremath{\mathfrak{n}} _t$ are nilpotent Lie algebras
in the family defined in Definition \ref{dim 9}, $\ensuremath{\mathfrak{n}} _s$ and
$\ensuremath{\mathfrak{n}} _t$ are isomorphic if and only if $s=t.$
\section{Constructions of curves of nonsoliton nilpotent Lie algebras
in higher dimensions}\label{higher dim}
\subsection{Examples in dimensions $n \ge 10$}
We describe how we construct the higher-dimensional examples from the $8$- and $9$- dimensional ones already defined. \begin{defn}\label{dim n+1} In each even dimension $8+2k,$ where $k \in \ensuremath{\mathbb N},$ a family of Lie algebras $\ensuremath{\mathfrak{n}} _s^{8 + 2k}$ is defined as follows. For each $s \in \ensuremath{\mathbb R},$ let $\ensuremath{\mathfrak{n}} _s$ be the $8$-dimensional nilpotent Lie algebra defined in Definition \ref{dim 8}. For $k \ge 1,$ the $(8+2k)$-dimensional nilpotent Lie algebra $\ensuremath{\mathfrak{n}} _s^{8 + 2k}$ is represented with respect to a basis $\{ x_i \}_{i=1}^8 \cup \{ y_i \}_{i=1}^{2k}$ so that the bracket relations defining the Lie algebra structure are those listed for $\ensuremath{\mathfrak{n}} _s$ in Equation \eqref{m8} of Definition \ref{dim 8}, along with the $k$ additional generating bracket relations \begin{equation*}\label{new brackets 1}
[y_i, y_{2k-i}] = x_8, \quad \text{for $i=1, \ldots,
k.$}\end{equation*} If necessary, we may let $x_{m+i} = y_i$ for $i=1,\ldots, 2k$ to define an ordering the basis $\ensuremath{\mathcal{B}} = \{x_i\}_{i=1}^m \cup \{y_i\}_{2k} = \{x_i\}_{i=1}^{m + 2k}$ in accord with the subscripts on the $x_i$'s. When $k=0,$ we let $\ensuremath{\mathfrak{n}} _s^{8 + 2k} = \ensuremath{\mathfrak{n}} _s.$
In odd dimensions $9+2k,$ the Lie algebras $\ensuremath{\mathfrak{n}} _s^{9 + 2k}$ are defined similarly. For any $s \in \ensuremath{\mathbb R},$ let $\ensuremath{\mathfrak{n}} _s$ be the $9$-dimensional nilpotent Lie algebra defined in Definition \ref{dim 8}. For $k \ge 1,$ the $(9+2k)$-dimensional nilpotent Lie algebra $\ensuremath{\mathfrak{n}} _s^{9 + 2k}$ is represented with respect to the basis $\{ x_i \}_{i=1}^9 \cup \{ y_i \}_{i=1}^{2k}$ with the bracket relations for $\ensuremath{\mathfrak{n}} _s$ in Definition \ref{dim 9}, along with the additional bracket relations \begin{equation*}\label{new brackets 2}
[y_i, y_{2k-i}] = x_9, \quad \text{for $i=1, \ldots,
k.$}\end{equation*} When $k=0,$ we let $\ensuremath{\mathfrak{n}} _s^{9 + 2k} = \ensuremath{\mathfrak{n}} _s.$
One may confirm without too much effort that the Jacobi Identity holds for all of these product structures. For any Lie algebra $\ensuremath{\mathfrak{n}} _s^{8 + 2k}$ or $\ensuremath{\mathfrak{n}} _s^{9 + 2k},$ with $k \in \ensuremath{\mathbb N},$ the only time a double bracket $[[x_i,x_j], x_k]$ vanishes for distinct $i,j,k$ is when $\{i,j,k\} = \{1,2,3\}.$ (This follows from Theorem 7 of \cite{payne-09b}.) One may also verify that for all $s \in \ensuremath{\mathbb R}$ and $k \ge 0,$ the Lie algebra $\ensuremath{\mathfrak{n}} _s^{8 + 2k}$ is three-step nilpotent of type $(2k + 3, 3 ,2),$ and the Lie algebra $\ensuremath{\mathfrak{n}} _s^{9 + 2k}$ is three-step nilpotent of type $(2k + 3, 3 ,3).$
Now fix $\ensuremath{\mathfrak{n}} _s^{m + 2k},$ where $m = 8$ or $9,$ and $k \in \ensuremath{\mathbb N}.$ Define the subspaces $V_2, V_3, V_4$ and $V_6$ by \begin{align}\label{def grading} \begin{split} V_2 &= \operatorname{span} \{ x_1, x_2, x_3\} \\ V_3 &= \operatorname{span} \{ y_1, \ldots, y_{2k}\} \\ V_4 &= \operatorname{span} \{ x_4, x_5, x_6 \} \\ V_6 &= \operatorname{span} \{ x_7, \ldots, x_m \}. \end{split} \end{align} When $k=0,$ we let $V_2, V_4,$ and $V_6$ be as defined above, and we let $V_3 = \{0\}.$ Then $\ensuremath{\mathfrak{n}} _s^{m + 2k} = V_2 \oplus V_3 \oplus V_4 \oplus V_6$ for all $k \ge 0.$
For $k \ge 0,$ define the derivation $D: \ensuremath{\mathfrak{n}} _s^{m + 2k}\to \ensuremath{\mathfrak{n}} _s^{m + 2k}$ by \begin{equation}\label{D-gen}
D(x) = k x, \quad \text{ if $x \in V_k,$ for $k=2,3, 4, 6.$ } \end{equation} Because $D$ is a derivation, for $m=8$ or $9,$ and $k \ge 0,$ the eigenspaces $V_i$ define an $\ensuremath{\mathbb N}$-grading of the Lie algebra $\ensuremath{\mathfrak{n}} _s^{m + 2k}:$ $[V_i,V_j] \subseteq V_{i + j},$ where $V_k = 0$ when $k \not \in \{2,3,4,6\}.$ \end{defn}
We will need the following lemma later.
\begin{lemma}\label{der lemma} Let $\ensuremath{\mathfrak{g}} $ be a Lie algebra, and let $\ensuremath{\mathfrak{i}} $ and $\ensuremath{\mathfrak{j}} $ be ideals in $\ensuremath{\mathfrak{g}} $ such that $\ensuremath{\mathfrak{g}} $ is the sum (not necessarily a direct sum) of $\ensuremath{\mathfrak{i}} $ and
$\ensuremath{\mathfrak{j}} ,$ and $[\ensuremath{\mathfrak{i}} ,\ensuremath{\mathfrak{j}} ] = 0.$
Let $\pi:\ensuremath{\mathfrak{g}} \to \ensuremath{\mathfrak{g}} $ denote a projection map from $\ensuremath{\mathfrak{g}} $ to $\ensuremath{\mathfrak{i}} ;$ i.e., an endomorphism such that
$\pi|_{\ensuremath{\mathfrak{i}} } = Id|_{\ensuremath{\mathfrak{i}} }$ and $\pi(\ensuremath{\mathfrak{g}} ) = \ensuremath{\mathfrak{i}} $. Let $D$ be a derivation of $\ensuremath{\mathfrak{g}} .$
Then the restriction of $\pi_1 \circ D$ to $\ensuremath{\mathfrak{i}} $ is a derivation of $\ensuremath{\mathfrak{i}} .$
Conversely, if $D_1: \ensuremath{\mathfrak{i}} \to \ensuremath{\mathfrak{i}} $ is a derivation of $\ensuremath{\mathfrak{i}} ,$ $D_2: \ensuremath{\mathfrak{j}} \to \ensuremath{\mathfrak{j}} $ is a derivation of $\ensuremath{\mathfrak{j}} ,$ and $D_1(z) = D_2(z)$ for all $z \in \ensuremath{\mathfrak{i}} \cap \ensuremath{\mathfrak{j}} ,$ then \[ D(z) = \begin{cases} D_1(z) & z \in \ensuremath{\mathfrak{i}} \\ D_2(z) & z \in \ensuremath{\mathfrak{j}} \end{cases} \] is a derivation of $\ensuremath{\mathfrak{g}} .$
The solvable radical of $\operatorname{Der}(\ensuremath{\mathfrak{g}} )$ contains the solvable radical of $\operatorname{Der}(\ensuremath{\mathfrak{i}} ).$ \end{lemma}
\begin{proof} There exists a basis $\{x_i\}_{i=1}^{m} \cup \{ y_j \}_{j=1}^{d}$ for $\ensuremath{\mathfrak{g}} $ such that \[ \ensuremath{\mathfrak{i}} = \operatorname{span} \{x_i\}_{i=1}^{m} \qquad \text{and} \qquad
\ensuremath{\mathfrak{j}} < \operatorname{span} \{ y_j \}_{j=1}^{d}+ \ensuremath{\mathfrak{z}} , \] and the projection $\pi: \ensuremath{\mathfrak{g}} \to \ensuremath{\mathfrak{i}} $ from $\ensuremath{\mathfrak{g}} $ to $\ensuremath{\mathfrak{i}} $ is given by \begin{align*} \pi(x_i) &= x_i, \quad \text{for $i=1, \ldots, m,$ and} \\
\pi(y_j) &= 0 \quad \text{for $j=1, \ldots, d.$}\end{align*}
Because $D$ is a derivation, for all $i,j = 1, \ldots, m,$ \begin{equation}\label{def derivation} D([x_i,x_j]) = [D(x_i), x_j] + [x_i, D(x_j)]. \end{equation} As $[y_l, x_j] = 0$ for all $l = 1, \ldots, d$ and $j=1, \ldots, m,$ we get \begin{equation}\label{der} D([x_i, x_j]) = [(\pi_1 \circ D)(x_i) ,x_j] + [x_i, (\pi_1 \circ D)(x_j)] \end{equation} for all $i,j = 1, \ldots, m.$ The vectors on the right side of Equation \eqref{def derivation} are in $\ensuremath{\mathfrak{i}} ,$ hence the vector on the left side is also in $\ensuremath{\mathfrak{i}} ,$ so we have \[ (\pi \circ D)([x_i, x_j]) = [(\pi \circ D)(x_i) ,x_j] + [x_i, (\pi \circ D)(x_j)] \] for all vectors $x_i$ and $x_j$ in the basis $\{x_i\}_{i=1}^m$ of $\ensuremath{\mathfrak{i}} .$ Hence, the restriction of $\pi \circ D$ to $\ensuremath{\mathfrak{i}} $ is a derivation of $\ensuremath{\mathfrak{i}} .$
To prove the converse, note that Equation \eqref{der} holds for all $x_i \in \ensuremath{\mathfrak{i}} $ and $x_j \in \ensuremath{\mathfrak{j}} $ due to the fact that $[\ensuremath{\mathfrak{i}} ,\ensuremath{\mathfrak{j}} ] = 0,$ while it holds if either both $x_i$ and $x_j$ are in $\ensuremath{\mathfrak{i}} $ or both $x_i$ and $x_j$ are in $\ensuremath{\mathfrak{j}} $ because $D_1$ and $D_2$ are derivations of $\ensuremath{\mathfrak{i}} $ and $\ensuremath{\mathfrak{j}} $ respectively. \end{proof}
The next proposition describes the Nikolayevsky derivations of the Lie algebras defined in Definition \ref{dim n+1}. \begin{prop}\label{der alg- n}
Let $m = 8$ or $9,$ and let $k \ge 0.$ For any $s \in \ensuremath{\mathbb R},$ let $\ensuremath{\mathfrak{n}} _s^{m + 2k}$
an $(m + 2k)$-dimensional nilpotent Lie
algebra as defined in Definition \ref{dim n+1}. The derivation $D^N = \frac{m + k -3}{6m + 3k -26} \, D,$ where $D$ is defined in Equation \eqref{D-gen}, is a Nikolayevsky derivation of $\ensuremath{\mathfrak{n}} _s^{m +
2k}.$ \end{prop}
\begin{proof} Fix $s,$ and let $\ensuremath{\mathfrak{n}} _s^{m + 2k}$ be
an $(m + 2k)$-dimensional nilpotent Lie
algebra as defined in Definition \ref{dim n+1}. Then the subspace $\ensuremath{\mathfrak{m}} =\operatorname{span} \{x_i\}_{i=1}^m$ is an ideal. If $m=8,$ the ideal $\ensuremath{\mathfrak{m}} $ is isomorphic to the Lie algebra $\ensuremath{\mathfrak{n}} _s$ defined in Definition \ref{dim 8}, and if $m=9,$ then $\ensuremath{\mathfrak{m}} $ is isomorphic to the Lie algebra $\ensuremath{\mathfrak{n}} _s$ defined in Definition \ref{dim 9}. Let $\ensuremath{\mathfrak{h}} = \operatorname{span} (\{x_m\} \cup \{y_j\}_{j=1}^{2k}).$ The subspace $\ensuremath{\mathfrak{h}} $ is an ideal isomorphic to the Heisenberg algebra of dimension $2k+1.$
Define the derivation $D_0$ of $\ensuremath{\mathfrak{m}} $ by \begin{equation}\label{defD0} D_0(w) = \begin{cases}
2\lambda w & w \in V_2 = \operatorname{span} \{x_i\}_{i=1}^3 \\ 3 \lambda w & w \in V_3 = \operatorname{span} \{y_j\}_{j=1}^{2k} \\ 4\lambda w & w \in V_4 = \operatorname{span} \{x_i\}_{i=4}^6\\ 6\lambda w & w \in V_6 = \operatorname{span} \{x_i\}_{i=7}^m \end{cases}, \end{equation} where $\lambda =\frac{m + k -3}{6m + 3k -26} .$
We want to show that $D_0$ is a Nikolayevsky derivation by showing that \[ \operatorname{trace} (D_0 \circ F) = \operatorname{trace}(F)\] for all $F$ in $\operatorname{Der}(\ensuremath{\mathfrak{n}} _s^{m + 2k}).$
Let $\pi_\ensuremath{\mathfrak{m}} : \ensuremath{\mathfrak{n}} _s^{m + 2k} \to \ensuremath{\mathfrak{m}} $ and $\pi_\ensuremath{\mathfrak{h}} : \ensuremath{\mathfrak{n}} _s^{m + 2k} \to \ensuremath{\mathfrak{h}} $ be the projection maps defined by the basis $\operatorname{span} \{x_i\}_{i=1}^m \cup \{ y_j\}_{j=1}^{2k}.$
By Lemma \ref{der lemma}, the restriction $\pi_\ensuremath{\mathfrak{m}} \circ F|_{\ensuremath{\mathfrak{m}} }$ of $F$ to $\ensuremath{\mathfrak{m}} $ is a derivation of $\ensuremath{\mathfrak{m}} .$ By Proposition \ref{der alg 8} (if $m = 8$) or Proposition \ref{der alg 9} (if $m = 9$),
the restriction of $F$ to $\ensuremath{\mathfrak{m}} $ is a nonzero scalar multiple of the restriction of $D_0$ to $\ensuremath{\mathfrak{m}} .$
Similarly, the restriction $\pi_\ensuremath{\mathfrak{h}} \circ F|_{\ensuremath{\mathfrak{h}} }$ of $F$ to $\ensuremath{\mathfrak{h}} $ is a derivation $F_1$ of
$\ensuremath{\mathfrak{h}} .$
Thus, we may write $F$ as the sum of derivations $F = \hat F + (F -
\hat F),$ where \begin{align*}
\hat F(w) &=
\begin{cases} \pi_\ensuremath{\mathfrak{m}} \circ F|_{\ensuremath{\mathfrak{m}} }(w) & w \in \ensuremath{\mathfrak{m}} = \operatorname{span} \{ x_1, \ldots, x_{m} \}\\
\pi_\ensuremath{\mathfrak{h}} \circ F|_{\ensuremath{\mathfrak{h}} } (w) & w \in \operatorname{span} \{ y_j \}_{j=1}^{2k}, \end{cases} \\ &= \begin{cases} cD_0(w) & w \in \ensuremath{\mathfrak{m}} = \operatorname{span} \{ x_1, \ldots, x_{m} \}\\
F_1(w) & w \in \operatorname{span} \{ y_j \}_{j=1}^{2k}, \end{cases} \end{align*} for some $c \in \ensuremath{\mathbb R},$ and $F_1 : \ensuremath{\mathfrak{h}} \to \ensuremath{\mathfrak{h}} $ is a derivation of the ideal $\ensuremath{\mathfrak{h}} .$
The derivation $\hat F$ fixes the ideals $\ensuremath{\mathfrak{m}} $ and $\ensuremath{\mathfrak{h}} ,$ and is block diagonal when represented with respect to the basis. The restriction of $\hat F$ to $\ensuremath{\mathfrak{h}} $ is the derivation $F_1$ of $\ensuremath{\mathfrak{h}} .$ The derivation $F - \hat F$ maps $\ensuremath{\mathfrak{m}} = \operatorname{span} \{ x_1, \ldots, x_{m} \}$ into $\operatorname{span} \{ y_j \}_{j=1}^{2k}$ and it maps $\operatorname{span} \{ y_j \}_{j=1}^{2k}$ into $\ensuremath{\mathfrak{m}} = \operatorname{span} \{ x_1, \ldots, x_{m} \},$
and when represented with respect to the basis $\ensuremath{\mathcal{B}} $ in block form has 0 blocks along the diagonal.
Using the definition of $D_0$ we see that $\operatorname{trace}(D_0 \circ (F - \hat F)) = 0;$ hence, \[ \operatorname{trace} (D_0 \circ \hat F) = \operatorname{trace}(D_0 \circ F).\] In addition, $\operatorname{trace} \hat F = \operatorname{trace} F.$ Thus, in order to show that $\operatorname{trace} (D_0 \circ F) = \operatorname{trace}(F)$ for all derivations $F,$ it suffices to show that $\operatorname{trace} (D_0 \circ \hat F) = \operatorname{trace}(\hat F)$ for all derivations $F.$
We will compute $\operatorname{trace}(D_0 \circ \hat F)$ and $\operatorname{trace}(\hat F)$ directly. But first we will prove the following claim: $k \cdot \operatorname{trace} \hat F|_{\ensuremath{\mathbb R} x_m} = \operatorname{trace} \hat F|_{V_3}.$ It is not hard to confirm the basic fact that the derivation $D^N_{\ensuremath{\mathfrak{h}} }: \ensuremath{\mathfrak{h}} \to \ensuremath{\mathfrak{h}} $ defined by \[
D^N_{\ensuremath{\mathfrak{h}} }(v) = \begin{cases} \frac{k+1}{k+2} \, v & v \in V_3 = \operatorname{span}
\{ y_l \}_{l=1}^{2k} \\ 2 \, \frac{k+1}{k+2}\, v & v \in \ensuremath{\mathbb R} x_m \end{cases} \] is a Nikolayevsky derivation for the ideal $\ensuremath{\mathfrak{h}} \cong \ensuremath{\mathfrak{h}} _{2k
+1}.$ By the defining property of the Nikolayevsky derivation $D^N_{\ensuremath{\mathfrak{h}} }$ for $\ensuremath{\mathfrak{h}} ,$ \[
\operatorname{trace}(D^N_{\ensuremath{\mathfrak{h}} } \circ \hat F|_{\ensuremath{\mathfrak{h}} }) = \operatorname{trace}(\hat F|_{\ensuremath{\mathfrak{h}} }), \] which becomes \[ \left( \frac{k+1}{k+2} \right) \left(
\operatorname{trace}(\hat F|_{V_3}) + 2 \operatorname{trace}(\hat F|_{\ensuremath{\mathbb R} x_m}) \right)=
\operatorname{trace}(\hat F|_{V_3}) + \operatorname{trace}(\hat F|_{\ensuremath{\mathbb R} x_m}). \] After simple arithmetic manipulations we get the desired equality
$k \cdot \operatorname{trace} \hat F|_{\ensuremath{\mathbb R} x_8} = \operatorname{trace} \hat F|_{V_3}.$ Then
\begin{equation}\label{trace is}
\operatorname{trace} (\hat F|_{V_3}) = k \cdot \operatorname{trace} \hat F|_{\ensuremath{\mathbb R} x_8} =
k \cdot \operatorname{trace} (c D_0)|_{\ensuremath{\mathbb R} x_8} = 6c\lambda k.\end{equation}
We return to the computation of $\operatorname{trace}(D_0 \circ \hat F)$ and $\operatorname{trace}(\hat F),$ using the definitions of $D_0$ and $\hat F,$ finding \begin{align*}
\operatorname{trace}(D_0 \circ \hat F) &= \operatorname{trace} \left((D_0 \circ \hat F)|_\ensuremath{\mathfrak{m}} \right) +
\operatorname{trace} \left((D_0 \circ \hat F)|_{V_3}\right)\\ &=
c\operatorname{trace} \left(D_0^2|_\ensuremath{\mathfrak{m}} \right) +
\operatorname{trace} \left((3 \lambda \operatorname{Id}_{V_3} \circ \hat F)|_{V_3}\right), \end{align*} where $\operatorname{Id}_{V_3}$ denotes the identity map on $V_3.$ Continuing, using the definition of $D_0$ we get \begin{align*} \operatorname{trace}(D_0 \circ \hat F) &= c(3 \cdot 4\lambda^2 + 3 \cdot 16\lambda^2 + (m-6) \cdot 36 \lambda^2)
+ 3 \lambda \operatorname{trace} (\hat F|_{V_3}) \\ &= c \lambda^2 (36m - 156) + 3\lambda \cdot 6 c \lambda k \quad \text{from Equation \eqref{trace is}}\\ &= c \lambda (36m - 156 + 18k) \cdot \lambda,\\ &= 6\lambda c (6m + 3k - 26) \frac{m + k -3}{6m + 3k -26} \quad \text{by definition of $\lambda$} \\ &= 6\lambda c \, (m + k - 3), \end{align*} while parallelly, \begin{align*}
\operatorname{trace}(\hat F) &= \operatorname{trace} (\hat F|_\ensuremath{\mathfrak{m}} ) +
\operatorname{trace} (\hat F|_{V_3})\\ &= 3 \cdot 2c\lambda + 3 \cdot 4c\lambda + (m-6) \cdot 6 c\lambda
+ \operatorname{trace} (\hat F|_{V_3}) \\ &= (6c\lambda m -18c\lambda)+ 6c\lambda k \\ &= 6\lambda c \, (m + k - 3). \end{align*}
Thus, $D_0$ is a Nikolayevsky derivation for $\ensuremath{\mathfrak{n}} _s^{m + 2k}$ as claimed. \end{proof}
Now we prove a technical lemma about isomorphisms of the algebras which we have defined. \begin{lemma}\label{technical lemma} Let $m = 8$ or $9,$ let $s\in \ensuremath{\mathbb R},$ and let $k \ge 0.$ Let $\ensuremath{\mathfrak{n}} _s^{m + 2k}$
be a $(m + 2k)$-dimensional nilpotent Lie
algebra as defined in Definition \ref{dim n+1}, and let $V_3, V_4$ and
$V_6$ be as defined in Equation \eqref{def grading}. Suppose that $\phi:
\ensuremath{\mathfrak{n}} _s^{m + 2k} \to \ensuremath{\mathfrak{n}} _s^{m + 2k}$ is an isomorphism. Then \begin{enumerate} \item{If $m=8,$ then $\phi$ maps the subspaces $\ensuremath{\mathbb R} x_1 \oplus V_3 \oplus V_4 \oplus V_6$ and $\ensuremath{\mathbb R} x_5 \oplus V_3 \oplus V_6$ to themselves. }\label{dim 8 isomorphism} \item{If $m=9,$ then $\phi$ maps the subspaces $\ensuremath{\mathbb R} x_1 \oplus V_4 \oplus V_6$ and $\ensuremath{\mathbb R} x_4 \oplus \ensuremath{\mathbb R} x_5 \oplus V_4 \oplus V_6$ to themselves.}\label{dim 9 isomorphism} \end{enumerate} \end{lemma}
\begin{proof}
The two subspaces $\ensuremath{\mathbb R} \, x_1 \oplus V_3 \oplus V_4 \oplus V_6$ and $\ensuremath{\mathbb R} x_4 \oplus \ensuremath{\mathbb R} x_5 \oplus V_3 \oplus V_6$ are fixed by isomorphisms because each may be uniquely characterized by algebraic properties that are preserved under isomorphisms.
We claim that when $m=8,$ the subset \[ S = \{ b_1 x_1 + v \, : \, b_1 \ne 0, v \in V_3 \oplus
V_4 \oplus V_6 \}\] is the set of all elements $x$ such that $\operatorname{ad}_{x}$ has rank 3, where $\operatorname{ad}_x$ is the adjoint map for either $\ensuremath{\mathfrak{n}} _s^{m + 2k}$ or $\ensuremath{\mathfrak{n}} _t^{m + 2k}.$
For example, when $k=1,$ if $x= \sum_{i=1}^8 b_i x_i + c_1 y_1 + c_2 y_2,$ then with respect to the usual basis $\ensuremath{\mathcal{B}} ,$ the adjoint map $\operatorname{ad}_x$ for $\ensuremath{\mathfrak{n}} _s^{10}$ is represented by the matrix \[ [\operatorname{ad}_x]_\ensuremath{\mathcal{B}} = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -e^{-s} b_3 & e^{-s}b_2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -e^sb_3 & 0 & e^s b_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -b_2 & b_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 &-e^s b_6 & -e^{-s} b_4 & e^{-s} b_3 & 0 & e^s b_2 & 0 & 0 & 0 & 0 \\ -e^{-s} b_6 & -b_4 & -e^s b_5 & b_2 & e^s b_3 & e^{-s} b_1 & 0 & 0 & -c_2 & c_1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}.\] The rank of the submatrix \[ M_1 = \begin{bmatrix} 0 & -e^{-s} b_3 & e^{-s}b_2 \\ -e^sb_3 & 0 & e^s b_1 \\ -b_2 & b_1 & 0 \end{bmatrix} \] is two if and only if $(b_1,b_2,b_3) \ne (0,0,0),$ and is zero otherwise.
Therefore if the rank of $\operatorname{ad}_x$ is
$3,$ then $(b_1,b_2,b_3) \ne (0,0,0).$ But if $b_2 \ne 0$ or $b_3
\ne 0,$ then the minor \[\begin{bmatrix} e^{-s} b_3 & 0 & e^s b_2 \\ b_2 & e^s b_3 & e^{-s} b_1 \end{bmatrix}\] has rank two. The block form of the matrix then forces the rank of the larger matrix to be at least four, a contradiction. Hence $b_2 = b_3 = 0,$ and $x \in S.$ Conversely, if $b_1 \ne 0$ and $b_2 = b_3 = 0,$ then rows 5, 6 and 8 form a basis for the row space of the matrix.
Since isomorphisms preserve the rank of $\operatorname{ad}_x,$ $S$ is invariant under isomorphisms. By continuity, an isomorphism fixes the closure of $S,$ $\ensuremath{\mathbb R}_1 x_1 \oplus V_3 \oplus V_4 \oplus V_6.$
The subspace $\ensuremath{\mathbb R} x_5 \oplus V_4 \oplus V_6$ can be characterized algebraically as the closure of the set of nonzero elements $x$ in the commutator ideal $V_4 \oplus V_6$ such that the rank of $\operatorname{ad}_x$ is 1. This is seen by letting $b_1, b_2, b_3, c_1$ and $c_2$ equal zero in the matrix representing $\operatorname{ad}_x.$
Thus we have shown that $\ensuremath{\mathbb R} x_1 \oplus V_3 \oplus V_4 \oplus V_6$ and $\ensuremath{\mathbb R} x_5 \oplus V_3 \oplus V_6$ are preserved by isomorphisms, establishing Part \eqref{dim 8 isomorphism} of the lemma in the case that $m=10.$ The same arguments apply in higher even dimensions $8 + 2k > 10.$
Now suppose the $m=9.$ Let \[ S = \{ b_1 x_1 + v \, : \, b_1 \ne 0, v \in V_4 \oplus V_6 \}. \] We assert that $S$ is the set of all elements $x$ such that $\operatorname{ad}_{x}$ has rank 3, and $x$ is not in the centralizer of the commutator ideal. The commutator
ideal is $V_4 \oplus V_6$ and its centralizer is \[ \ensuremath{\mathfrak{z}} (V_4 \oplus V_6) = V_3 \oplus V_4 \oplus V_6.\]
If $x= \sum_{i=1}^9 b_i x_i + c_1 y_1 + c_2 y_2,$ then adjoint map $\operatorname{ad}_x$ for $\ensuremath{\mathfrak{n}} _s^{11}$ is represented by the matrix \setcounter{MaxMatrixCols}{12} \[ \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -e^{4s} b_3 & e^{4s}b_2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -e^{-3s}b_3 & 0 & e^{-3s} b_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -e^{-a}b_2 & e^{-a}b_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 &-e^{4s} b_6 & -e^{4s} b_4 & e^{4s} b_3 & 0 & e^{-4s} b_2 & 0 & 0 & 0 & 0 & 0 \\ -e^{4s} b_6 & -b_4 & -e^{-4s} b_5 & b_2 & e^{-4s} b_3 & e^{4s} b_1 & 0 & 0 & 0 &0 & 0 \\ 0 & -e^sb_5 & -e^{-s}b_6 & 0 & e^sb_2 & e^{-s} b_3 & 0 & 0 & 0 & -c_2 & c_1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} .\] The rank of the submatrix
\[ \begin{bmatrix} 0 & -e^{4s} b_3 & e^{4s}b_2 \\ -e^{-3s}b_3 & 0 & e^{-3s} b_1 \\ -e^{-a}b_2 & e^{-a}b_1 & 0 \end{bmatrix} \] is two if and only if $(b_1,b_2,b_3) \ne 0$ and is zero otherwise.
Assume that $x$ is not in $\ensuremath{\mathfrak{z}} (V_4 \oplus V_6),$ and that
the rank of $\operatorname{ad}_x$ is three. Since $x$ is not in the centralizer of the commutator, $(b_1,b_2,b_3) \ne (0,0,0).$ If $b_2 \ne 0$ or $b_3 \ne 0,$ then the minor \[ \begin{bmatrix} e^{4s} b_3 & 0 & e^{-4s} b_2 \\ b_2 & e^{-4s} b_3 & e^{4s} b_1 \\ 0 & e^sb_2 & e^{-s} b_3 \end{bmatrix} .\]
has rank 2 or more, forcing the larger matrix to have rank greater than
three, a contradiction. Therefore $b_2 = b_3 = 0.$ Substituting these values into the larger matrix, we see that rows 5, 6 and 8 are independent, and row 9 will not being in their span unless $c_1 = c_2 =0.$ Hence, $x \in S.$
Conversely, if $x \in S,$ then $b_1 \ne 0,$ $b_2 = b_3 = 0,$ and $c_1 = c_2 = 0.$ Since $b_1 \ne 0,$ the vector $x$ is not in the centralizer of the commutator ideal. After substituting the zero values into the large matrix, we see that rows 7 and 9 are in the span of the independent rows 5, 6 and 8. Hence the
rank of $\operatorname{ad}_x$ is 3.
Thus we have shown that $S$ can be characterized as the set of all $x$ such that $\operatorname{ad}_x$ is not in the centralizer of the commutator ideal and $\operatorname{ad}_{x}$ has rank 3. Therefore $\phi(S) = S,$ and $\phi$ preserves the closure $\ensuremath{\mathbb R} x_1 \oplus V_4 \oplus V_6.$
We have just shown that an isometry $\phi$ preserves $W = \ensuremath{\mathbb R} x_1 \oplus V_4 \oplus V_6.$ Therefore $\phi$ will also map the centralizer of $W,$ \[ \ensuremath{\mathfrak{z}} (W) = \{ x \in \ensuremath{\mathfrak{n}} _s^{m + 2k} \, : \, [x,y] = 0 \enskip \text{for all $y \in \ensuremath{\mathbb R} x_1 \oplus V_4 \oplus V_6$} \}\] of $W$ for $\ensuremath{\mathfrak{n}} _s^{m+2k}$ to the centralizer of $\phi(W)=W$
for $\ensuremath{\mathfrak{n}} _t^{m+2k}.$ But \[ \ensuremath{\mathfrak{z}} (W) = \ensuremath{\mathbb R} x_4 \oplus\ensuremath{\mathbb R} x_5 \oplus V_3 \oplus V_6. \] Therefore, $\phi$ preserves $\ensuremath{\mathbb R} x_4 \oplus \ensuremath{\mathbb R} x_5 \oplus V_3 \oplus V_6$ as claimed.
This we have shown that Part \eqref{dim 9 isomorphism} holds in dimension $11.$ The same arguments apply in odd dimensions $9 + 2k$ greater than 11. \end{proof}
Now we are ready to show that for fixed $m$ and $k,$ no distinct two Lie algebras in the family $\ensuremath{\mathfrak{n}} _s^{m + 2k}, s \in \ensuremath{\mathbb R}$ are isomorphic.
\begin{thm}\label{nonisomorphic n}
Let $m = 8$ or $9,$ let $s, t \in \ensuremath{\mathbb R},$ and let $k \ge 0.$ Let $\ensuremath{\mathfrak{n}} _s^{m + 2k}$
and $\ensuremath{\mathfrak{n}} _t^{m + 2k}$ be two $(m + 2k)$-dimensional nilpotent Lie
algebras as defined in Definition \ref{dim n+1}. Then $\ensuremath{\mathfrak{n}} _s^{m
+ 2k}$ and $\ensuremath{\mathfrak{n}} _t^{m + 2k}$ are isomorphic if and only if $s = t.$ \end{thm}
\begin{proof} Suppose that $\phi : \ensuremath{\mathfrak{n}} _s^{m + 2k}\to \ensuremath{\mathfrak{n}} _t^{m + 2k}$ is an isomorphism. We view $\ensuremath{\mathfrak{n}} _s^{m + 2k} $ and $\ensuremath{\mathfrak{n}} _t^{m + 2k} $ as the same vector space endowed with different Lie brackets.
Let $V_i$ denote the eigenspace for the derivation $D$ with eigenvalue $i$ as in Equation \eqref{def grading}. Recall that eigenspaces $V_i,$ where $i=2,3,4,6,$ define an $\ensuremath{\mathbb N}$-grading of $\ensuremath{\mathfrak{n}} _s^{m+2k}:$ $[V_i,V_j] \subseteq V_{i + j},$ where $V_l = 0$ when $l \in \ensuremath{\mathbb N} \setminus \{2,3,4,6\},$ and $V_3 = \{0\}$ when $k=0.$ In particular, we know that \begin{gather*}
[V_3 \oplus V_4 \oplus V_6, \ensuremath{\mathfrak{n}} _t^{m +2k}] \subseteq V_6, \\
[V_3 \oplus V_4 \oplus V_6, V_4 \oplus V_6] = \{0\}, \end{gather*} and that $V_6$ is central.
By Lemma \ref{technical lemma}, Part \eqref{dim 8 isomorphism}, the isomorphism $\phi$ maps the subspace $\ensuremath{\mathbb R} x_1$ into the subspace $\ensuremath{\mathbb R} x_1 \oplus V_3 \oplus V_4 \oplus V_6.$ Therefore we may write \[ \phi(x_1) = a_{11} x_1 + v_1 \] for some vector $v_1 \in V_3 \oplus V_4 \oplus V_6$ and some $a_{11} \in \ensuremath{\mathbb R}.$
As $\phi$ is an isomorphism, $a_{11} \ne 0;$ were $a_{11}$ to vanish, $\phi(x_1)$
would be in the centralizer of the commutator while $x_1$ was not. We write \begin{align} \begin{split} \phi(x_2) & = a_{12} x_1 + a_{22} x_2 + a_{32} x_3 + v_2\\ \phi(x_3) & = a_{13} x_1 + a_{23} x_2 + a_{33} x_3 + v_3 \end{split} \end{align} for scalars $a_{12}, a_{22}, a_{32}, a_{13}, a_{23}$ and $a_{33},$ and vectors $v_2$ and $v_3$ in $V_3 \oplus V_4 \oplus V_6.$
To complete the proof, we will consider the cases $m=8$ and $m = 9$ separately. First suppose that $m=8.$ The first defining relation $[x_2,x_3] = e^{-s} x_4$ yields \begin{align}\label{DR1-n} \begin{split} \phi(x_4) &= e^s [\phi(x_2), \phi(x_3)] \\
&= e^s [a_{12} x_1 + a_{22} x_2 + a_{32} x_3, a_{13} x_1 + a_{23} x_2 +
a_{33} x_3] + v_4 \\ &= e^{s-t} ( a_{22} a_{33} - a_{32} a_{23}) x_4 + e^{s+t} (a_{12} a_{33} - a_{32} a_{13} ) x_5 \\ & \qquad \qquad + e^s (a_{12} a_{23} - a_{22} a_{13} ) x_6 + v_4, \end{split} \end{align} where $v_4$ is a vector in the center $V_6.$
The second relation $[x_1,x_3] = e^{s} x_5$ gives us \begin{align}\label{DR2-n} \begin{split} \phi(x_5) &= e^{-s} [\phi(x_1), \phi(x_3)] + v_5 \\
&= e^{-s} [a_{11} x_1, a_{13} x_1 + a_{23} x_2 + a_{33} x_3] + v_5\\ &= e^{-s+t} a_{11}a_{33} x_5 + e^{-s} a_{11}a_{23} x_6 + v_5 \end{split} \end{align} for some vector $v_5 \in V_6.$ Hence $\phi(x_5) \in V_4 \oplus V_6.$ By Lemma \ref{technical lemma}, Part \eqref{dim 8 isomorphism}, the subspace $\ensuremath{\mathbb R} x_5$ is mapped into $\ensuremath{\mathbb R} x_5 \oplus V_3 \oplus V_6$ by $\phi.$ Hence, the $x_6$ coefficient in Equation \eqref{DR2-n} is zero.
From the fact that $a_{11} \ne 0,$ we deduce that $a_{23} = 0.$ Now we have \begin{equation}\label{DR2b-n} \phi(x_5) = e^{-s+t} a_{11}a_{33} x_5 + v_5. \end{equation}
Next we get \begin{align}\label{DR3-n} \begin{split} \phi(x_6) &= [\phi(x_1),\phi(x_2)]\\
&= [a_{11} x_1, a_{12} x_1 + a_{22} x_2 + a_{32} x_3 ] + v_6\\ &= e^t a_{11}a_{32} x_5 + a_{11}a_{22} x_6+ v_6 \end{split} \end{align} for some vector $v_6 \in V_6,$ so $\phi(x_6) \in V_4 \oplus V_6.$
The bracket relation $[x_2,x_6] =e^s x_7$ implies that \begin{align} \label{DR4-n} \begin{split} \phi(x_7) &= e^{-s}[\phi(x_2),\phi(x_6)] \\ &= e^{-s}[a_{12} x_1 + a_{22} x_2 + a_{32} x_3, e^t a_{11}a_{32} x_5 + a_{11}a_{22} x_6] \\ &= e^{-s+t} a_{11}a_{22}^2 x_7 + a_{11}(e^{-s-t} a_{12}a_{22} + e^{-s+2t} a_{32}^2 )x_8. \end{split} \end{align} We use the relation $[x_3,x_4]=e^{-s}x_7,$ substituting $a_{23}=0$ when it occurs, to get \begin{align}\label{DR5-n} \begin{split} \phi(x_7) &= e^{s} [\phi(x_3),\phi(x_4)] \\ &= e^{s} [a_{13} x_1 + a_{23}x_2 + a_{33} x_3 + v_3,e^{s-t} a_{22} a_{33} x_4 \\ & \qquad + e^{s+t} (a_{12} a_{33} - a_{32} a_{13} ) x_5 - e^s a_{22} a_{13} x_6 + v_4] \\ &= e^{2s-2t} a_{33}^2 a_{22} x_7 \\ & \quad + \big( -e^{2s-t} a_{13}^2 a_{22} +e^{2s+2t} a_{33} (a_{12}a_{33} - a_{32} a_{13}) \big) x_8. \end{split} \end{align}
The bracket relation $[x_1,x_6]=e^{-s}x_8$ gives \begin{align}\label{DR8-a-n} \begin{split} \phi(x_8) &= e^{s} [a_{11}x_1, e^t a_{11}a_{32} x_5 + a_{11}a_{22} x_6] \\
&= e^{s-t} a_{11}^2 a_{22} x_8, \end{split} \end{align}
and from $[x_3,x_5]=e^s x_8$ we find \begin{align}\label{DR8-b-n} \begin{split} \phi(x_8) &= e^{-s} [a_{13} x_1 + a_{23}x_2+ a_{33} x_3, e^{-s+t} a_{11}a_{33} x_5]\\ &= e^{-2s+2t} a_{33}^2 a_{11} x_8. \end{split} \end{align} We know that $\phi(x_8) \ne 0,$ so Equations \eqref{DR8-a-n} and \eqref{DR8-b-n} tell us that $a_{22} \ne 0$ and $a_{33} \ne 0.$
Equating coefficients of $x_7$ in Equations \eqref{DR4-n} and \eqref{DR5-n} gives \begin{equation}\label{first expr-n}
e^{-s+t} a_{11}a_{22} =
e^{2s-2t} a_{33}^2 ,\end{equation} and equating $x_8$ coefficients in Equations \eqref{DR8-a-n} and \eqref{DR8-b-n} gives \begin{equation}\label{second expr-n} e^{s-t} a_{11} a_{22} = e^{-2s+2t} a_{33}^2. \end{equation}
Thus, \[ e^{3s - 3t} a_{33}^2 = a_{11}a_{22} = e^{-3s + 3t} a_{33}^2\] from Equations \eqref{first expr-n} and \eqref{second expr-n}. Because $a_{33}$ is nonzero, we may conclude that $s = t$ as desired.
Now suppose that $m=9.$ The bracket relation $[x_2,x_3] = e^{4s} x_4$ implies that \begin{align}\label{x4-n} \begin{split} \phi(x_4) = e^{4t-4s}& (a_{22} a_{33} - a_{32} a_{23}) x_4 + e^{-4s-3t}(a_{12}a_{33}-a_{32}a_{13}) x_5 \\ & + e^{-4s-t}(a_{12}a_{23}-a_{22}a_{13}) x_6 + v_4, \end{split} \end{align} where $v_4 \in V_6.$ We use $[x_1,x_3] = e^{-3s} x_5$ to obtain that \begin{equation*} \phi(x_5) = e^{3s-3t}a_{11}a_{33} x_5 + e^{3s-t}a_{11}a_{23} x_6 + v_5 \end{equation*} for some $v_5 \in V_6.$ By Lemma \ref{technical lemma}, we know that $\phi(x_5)$ is in the invariant subspace $\ensuremath{\mathbb R} x_5 \oplus\ensuremath{\mathbb R} x_5 \oplus V_4 \oplus V_6,$ so the coefficient of $x_6$ is zero. Hence $a_{23} = 0.$
The bracket relation $[x_1,x_2] = e^{-s}x_6$ yields \begin{equation}\label{x6-n} \phi(x_6) = e^{s-3t}a_{11}a_{32} x_5 + e^{s-t}a_{11}a_{22} x_6 + v_6 \end{equation} where $v_6 \in V_6,$ and the bracket relation $[x_1,x_6] = e^{4s} x_8$ gives us \[ \phi(x_8) = e^{-3s+3t} a_{11}^2 a_{22} x_8.\] Because $\phi(x_8) \ne 0,$ we see that $a_{22} \ne 0.$
The bracket relation $[x_2,x_6] = e^{-4s} x_7$ becomes \begin{align}\label{x7-n} \begin{split} \phi(x_7) =
e^{5s-5t}a_{11} &a_{22}^2x_7 + a_{11}(e^{5s+3t} a_{22}a_{12} + e^{5s-7t} a_{32}^2) x_8 \\ &+ 2 e^{5s-2t}a_{11}a_{22} a_{32} x_9. \end{split} \end{align} The bracket relation $[x_3,x_4]=e^{4s} x_7$ and that $a_{23}=0$ gives \begin{align}\label{x7-2-n} \begin{split} \phi(x_7) & = e^{-8s+8t} a_{22} a_{33}^2 x_7 \\ & \quad + \left( -e^{-8s+3t} a_{22}a_{13}^2 + e^{-8s-7t}
a_{33}(a_{12}a_{33}-a_{32}a_{13}) \right) x_8 \\ & \qquad - e^{-8s-2t} a_{22} a_{13} a_{33} x_9. \end{split} \end{align}
Setting the $x_7$ coefficients from Equations \eqref{x7-n} and \eqref{x7-2-n} equal, we get \begin{equation}\label{above-n}
a_{11} a_{22} = e^{13t-13s} a_{33}^2. \end{equation}
The bracket relation $[x_3,x_5] = e^{-4s} x_8$ and the fact that $a_{23}=0$ give \[ \phi(x_8) = e^{7s - 7t} a_{33}^2 a_{11} x_8.\] This means that $a_{33} \ne 0.$
Equating the $x_8$ coefficients in this expression and the previous expression for $\phi(x_8),$ we get \[ e^{-3s+3t} a_{11}^2 a_{22} = e^{7s - 7t} a_{33}^2 a_{11}, \] so \[ a_{11} a_{22} = e^{10s-10t} a_{33}^2 .\] This together with Equation \eqref{above-n} gives $e^{13t-13s} =e^{10s - 10t} .$ Hence, $s=t.$ \end{proof}
\begin{thm}\label{nonsoliton - n} Let $m =8$ or $9,$ and let $k \in \ensuremath{\mathbb N}.$ Let $\ensuremath{\mathfrak{n}} _s^{m + 2k}, s \in \ensuremath{\mathbb R}$
be the one-parameter family of nilpotent $(m + 2k)$-dimensional nilpotent Lie
algebras defined in Definition \ref{dim n+1}. None of the Lie
algebras in the family are soliton. \end{thm}
\begin{proof}
Let $x_{m+i} = y_i$ for $i=1,\ldots, 2k.$ Then the index set $\Lambda$ with respect to the basis $\ensuremath{\mathcal{B}} = \{x_i\}_{i=1}^m \cup \{y_j\}_{j=1}^{2k} = \{x_i\}_{i=1}^{m+2k}$ is $\Lambda_\ensuremath{\mathfrak{m}} \cup \Lambda_\ensuremath{\mathfrak{h}} ,$ where $\Lambda_\ensuremath{\mathfrak{m}} $ is the index set for $\ensuremath{\mathfrak{m}} $ with respect to the basis $\{x_i\}_{i=1}^m,$ and \[\Lambda_\ensuremath{\mathfrak{h}} = \{ (m+ i, m+ 2k - i, 8) \, : \, i=1, \ldots, k \}.\] The set $\Lambda$ is ordered as described in Section \ref{metric Lie
algebras}. In this ordering, if $(i_1,j_1,k_1) \in \Lambda_\ensuremath{\mathfrak{m}} $ and $(i_2,j_2,k_2) \in \Lambda_\ensuremath{\mathfrak{h}} ,$ then $(i_1,j_1,k_1) < (i_2,j_2,k_2).$
First we consider the case that $m=8.$ Let the family $\ensuremath{\mathfrak{n}} _s^{8 + 2k}, s \in \ensuremath{\mathbb R}$ of nilpotent Lie algebras
be as defined in Definition \ref{dim n+1}.
We will do a proof by contradiction, so we suppose that $\ensuremath{\mathfrak{n}} _s^{8 + 2k}$ admits a soliton inner product.
Let $U$ denote the Gram matrix for $\ensuremath{\mathfrak{n}} _s^{i + 2k}$ with respect to the basis $\ensuremath{\mathcal{B}} .$ For $a, b \in \ensuremath{\mathbb N},$ let $[0]_{a \times b}$ denote the $a \times b$ matrix with
all entries zero, and let $[1]_{a \times b}$ denote the $a \times b$
matrix with all entries one, and let $I_a$ denote the $a \times a$
identity matrix. The matrix $U$ has block form \[ U = \begin{bmatrix}
U_{11} & U_{12} & [0]_{5 \times k} \\
U_{21} & U_{22} & [1]_{3 \times k} \\
[0]_{k \times 5} & [1]_{k \times 3} & 3 I_k \end{bmatrix}, \] where \[ U_8= \begin{bmatrix} U_{11} & U_{12} \\ U_{21} & U_{22} \end{bmatrix} \] is the matrix $U$ in Equation \eqref{U 8 def}, broken into blocks $U_{11}, U_{12}, U_{21}, U_{22}$ of sizes $5 \times 5, 5 \times 3, 3 \times 5,$ and $3 \times 3$ respectively.
By Theorem \ref{Uv} there exists a solution $\ensuremath{\bm v} $ to $U\ensuremath{\bm v} = [1]$ with all positive entries. We may write $\ensuremath{\bm v} $ as \[ \ensuremath{\bm v} = \begin{bmatrix} \ensuremath{\bm v} _1 \\ \ensuremath{\bm v} _2 \\ \ensuremath{\bm v} _3 \end{bmatrix}, \]
where $\ensuremath{\bm v} _1 = (v_i)_{i=1}^5$ is $5 \times 1,$ $\ensuremath{\bm v} _2 =
(v_i)_{i=6}^8$ is $3 \times 1$ and $\ensuremath{\bm v} _3 = (v_i)_{i=9}^{8+k}$ is $k \times 1.$ Multiplying $U\ensuremath{\bm v} $ in block form gives \begin{align*}
U_{11} \ensuremath{\bm v} _1 + U_{12} \ensuremath{\bm v} _2 &= [1]_{5 \times 1} \\
U_{21} \ensuremath{\bm v} _1 + U_{22} \ensuremath{\bm v} _2 + [1]_{3 \times k} \ensuremath{\bm v} _3 &= [1]_{3
\times 1} \\
[1]_{k \times 3} \ensuremath{\bm v} _2 + 3 \ensuremath{\bm v} _3 &= [1]_{k
\times 1}. \end{align*} Substituting $[1]_{3 \times k} \ensuremath{\bm v} _3 = ( \sum_{i=9}^{8 + k} v_i) [1]_{3 \times 1}$ and $ [1]_{k \times 3} \ensuremath{\bm v} _2 =(\sum_{i=6}^{8} v_i)[1]_{k \times 1}$ into the above yields the equivalent system \begin{align}
U_{11} \ensuremath{\bm v} _1 + U_{12} \ensuremath{\bm v} _2 &= [1]_{5 \times 1} \label{U1}\\
U_{21} \ensuremath{\bm v} _1 + U_{22} \ensuremath{\bm v} _2 &= \left(1 - \sum_{i=9}^{8 + k} v_i
\right) [1]_{3
\times 1} \label{U2} \\ 3 \ensuremath{\bm v} _3 &= \left(1 - \sum_{i=6}^{8} v_i
\right)[1]_{k \times 1} \label{U3}. \end{align}
Equation \eqref{U3} implies that $\ensuremath{\bm v} _3 = c \, [1]_{k \times 1},$ where \begin{equation}\label{defn c} c = \frac{1}{3} \left(1 - \sum_{i=6}^{8} v_i
\right)> 0.\end{equation} It follows that \[\sum_{i=9}^{8 + k} v_i = k c = \frac{k}{3} \left(1 - \sum_{i=6}^{8} v_i
\right).
\] Now we bound the coefficient \[ a = 1- \sum_{i=9}^{8 + k} v_i\] of $[1]_{3 \times 1}$ in Equation \eqref{U2}. The matrices $U_{21}$ and $U_{22}$ are nontrivial and have no negative entries, and the entries of the vectors $\ensuremath{\bm v} _1$ and $\ensuremath{\bm v} _2$ are positive. Hence, Equation \eqref{U2} forces $a$ to be positive.
On the other hand, \begin{align*}
a = 1- \sum_{i=9}^{8 + k} v_i&= 1 - k c \\ &= 1 - \frac{k}{3} \left(1 - \sum_{i=6}^{8} v_i
\right) \\ &= \frac{3-k}{3} + \frac{k}{3} \sum_{i=6}^{8} v_i . \end{align*} The inequality in Equation \eqref{defn c} implies that $\sum_{i=6}^{8} v_i < 1$ , so \[ a < \frac{3-k}{3} + \frac{k}{3} = 1. \] Thus, we have shown that $a$ lies in the interval $(0,1).$
Returning to Equations \eqref{U1}-\eqref{U3}, we have that \[ \ensuremath{\bm x} = \begin{bmatrix} \ensuremath{\bm v} _1 \\ \ensuremath{\bm v} _2 \end{bmatrix} \] is a solution to the matrix equation \[ U_8 \, \ensuremath{\bm x} = \begin{bmatrix}
U_{11} & U_{12} \\
U_{21} & U_{22} \end{bmatrix} \, \ensuremath{\bm x} = \begin{bmatrix}
[1]_{5 \times 1} \\
[a]_{k \times 1} \end{bmatrix} . \]
Now this involves only the matrix $U_8,$ which is given explicitly in Equation \eqref{U 8 def}. Solving symbolically using Matlab,
we find that the general solution to the matrix equation is $\ensuremath{\bm v} = \ensuremath{\bm v} _0 + t \ensuremath{\bm v} _1,$ where \[ \ensuremath{\bm v} _0 = \smallfrac{1}{33}(7a + 2, 3-6a, 13-4a, 11-11a, 2a+10, 13a-1, 11a-11,0)^T\] and \[ \ensuremath{\bm v} _1 = (-1,1,0,1,-1,-1,0,1)^T. \] For all such solutions $\ensuremath{\bm v} = (v_i),$ the component $v_7$ is $\frac{a-1}{3}.$ We assumed that $v_7 > 0,$ so then $a - 1>0,$ a contradiction to the fact that $a \in (0,1).$
Therefore, for all $s \in \ensuremath{\mathbb R},$ and all $k \in \ensuremath{\mathfrak{n}} ,$ the Lie algebra $\ensuremath{\mathfrak{n}} _s^{8 + 2k}$ is not soliton, as claimed.
Now suppose that $m=9.$ Let $\ensuremath{\mathfrak{n}} _s^{9 + 2k}$ be one of the Lie algebras in the one-parameter family of nilpotent Lie algebras
defined in Definition \ref{dim n+1}. Suppose that $\ensuremath{\mathfrak{n}} _s^{9 +
2k}$ admits a nilsoliton inner product.
Let $U$ be the Gram matrix for $\ensuremath{\mathfrak{n}} _s^{9 + 2k}$ with respect to the basis $\{x_i\}_{i=1}^m \cup \{y_j\}_{j=1}^{2k} = \{x_i\}_{i=1}^{9 + 2k}.$ By examining the index set $\Lambda,$ we see that the matrix $U$ has is of form \[ U = \begin{bmatrix}
U_{11} & U_{12} & [0]_{8 \times k} \\
U_{21} & U_{22} & [1]_{2 \times k} \\
[0]_{8 \times k} & [1]_{2 \times k} & 3 I_k \\ \end{bmatrix}, \] where \[ U_9= \begin{bmatrix} U_{11} & U_{12} \\ U_{21} & U_{22} \end{bmatrix} \] is the $10 \times 10$ matrix $U$ in Equation \eqref{U9 def}, and the blocks $U_{11}, U_{12}, U_{21}$ and $U_{22}$ are size $8 \times 8, 8 \times 2, 2 \times 8,$ and $2 \times 2$ respectively.
By Theorem \ref{Uv}, the matrix equation $U\ensuremath{\bm v} = [1]$ has a solution $\ensuremath{\bm v} $ with all positive entries. We write $\ensuremath{\bm v} $ as \[ \ensuremath{\bm v} = \begin{bmatrix} \ensuremath{\bm v} _1 \\ \ensuremath{\bm v} _2 \\ \ensuremath{\bm v} _3 \end{bmatrix}, \]
where $\ensuremath{\bm v} _1$ is $8 \times 1,$ $\ensuremath{\bm v} _2$ is $2 \times 1$ and $\ensuremath{\bm v} _3$ is $k \times 1.$ Then \begin{align*}
U_{11} \ensuremath{\bm v} _1 + U_{12} \ensuremath{\bm v} _2 &= [1]_{8 \times 1} \\
U_{21} \ensuremath{\bm v} _1 + U_{22} \ensuremath{\bm v} _2 + [1]_{2 \times k} \ensuremath{\bm v} _3 &= [1]_{2 \times 1}. \end{align*} As $[1]_{2 \times k} \ensuremath{\bm v} _3 = ( \sum_{i=11}^{10 + k} v_i) [1]_{2 \times 1},$ we can rewrite this system as \begin{align}
U_{11} \ensuremath{\bm v} _1 + U_{12} \ensuremath{\bm v} _2 &= [1]_{8 \times 1} \label{U1b}\\
U_{21} \ensuremath{\bm v} _1 + U_{22} \ensuremath{\bm v} _2 &= \left(1 - \sum_{i=11}^{10 + k} v_i
\right) [1]_{2
\times 1}. \label{U2b} \end{align}
Solving Equation \eqref{U1b} for $[ \begin{smallmatrix} \ensuremath{\bm v} _1
\\ \ensuremath{\bm v} _2 \end{smallmatrix}],$ we get \[ \begin{bmatrix} \ensuremath{\bm v} _1
\\ \ensuremath{\bm v} _2 \end{bmatrix} = \ensuremath{\bm v} _0 + t_1 \ensuremath{\bm w} _1 + t_2 \ensuremath{\bm w} _2 + t_3\ensuremath{\bm w} _3, \] where \begin{align*} \ensuremath{\bm v} _0 & = \smallfrac{1}{11}(3,-1,3,0,4,4, 0,0,0,0)^T \\ \ensuremath{\bm w} _1 &= (-1,1,0,1,-1,-1,0,1,0,0)^T \\ \ensuremath{\bm w} _2 &= (-5,-2,6,0,-3,-3,0,0,11,0)^T \\ \ensuremath{\bm w} _3 &= (-5,9,-5,0,-3,-3,0,0,0,11)^T. \end{align*} But none of these solutions have all positive entries, a contradiction to Theorem \ref{Uv}. Therefore, for all $s \in \ensuremath{\mathbb R},$ the Lie algebra $\ensuremath{\mathfrak{n}} _s^{9 + 2k}$ does not admit a nilsoliton inner product. \end{proof}
\section{Proof of main theorem}\label{proof of main theorem}
Now we prove Theorem \ref{main thm}.
\begin{proof} Let $\widetilde \ensuremath{\mathcal{N}} _n$ denote the moduli space of
$\ensuremath{\mathbb N}$-graded, indecomposable nilpotent Lie algebras as described
in Section \ref{moduli spaces}. Let $\text{Nonsol}(n) \subseteq \widetilde \ensuremath{\mathcal{N}} _n$ be the set of all $\overline{\mu}$ in $\widetilde \ensuremath{\mathcal{N}} _n$ such that $\ensuremath{\mathfrak{n}} _\mu$ does not admit a soliton inner product as described
in Section \ref{moduli spaces}.
We know from the results of Lauret, Will and Culma described in the introduction (\cite{lauret02}, \cite{will03}, \cite{culma1}, \cite{culma2}) that the set of nonsoliton Lie algebras in $\widetilde \ensuremath{\mathcal{N}} _n$ is discrete when $n \le 7.$
By Theorem \ref{dim 8 theorem}, none of the $8$-dimensional Lie algebras defined in Definition \ref{dim 8} are soliton. By Theorem \ref{dim 9 theorem} none of the $9$-dimensional Lie algebras defined in Definition \ref{dim 9} are soliton. Theorem \ref{nonsoliton - n} implies that none of the Lie algebras in dimensions $n \ge 10$ defined in Definition \ref{dim n+1} are soliton.
By Theorem \ref{nonisomorphic n}, no two of the Lie algebras $\ensuremath{\mathfrak{n}} _s^n$ and $\ensuremath{\mathfrak{n}} _t^n$ defined in dimension $n \ge 8$ are isomorphic.
Therefore, in each dimension $n \ge 8,$ the mapping $\gamma: \ensuremath{\mathbb R} \to \widetilde \ensuremath{\mathcal{N}} _n$ mapping $s \in \ensuremath{\mathbb R}$ to the equivalence class of the nilpotent Lie algebra $\ensuremath{\mathfrak{n}} _s^n$ is one-to-one, with image in $\text{Nonsol}(n).$
Hence, the set $\text{Nonsol}(n)$ consisting of nonsoliton Lie algebras (modulo equivalence under isomorphism) in $\widetilde \ensuremath{\mathcal{N}} _n$ is not discrete when $n \ge 10.$ \end{proof}
\noindent{\it Acknowledgments.} We are grateful to Raz Stowe for illuminating discussions, and to Mike Jablonski for helpful comments and useful suggestions.
\end{document} | arXiv |
List of complex analysis topics
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| Wikipedia |
Changes in the Rate of Sea Level Rise
Willis Eschenbach / May 22, 2018
Guest Post by Willis Eschenbach
There's been some discussion of the rate of sea level rise lately, so I thought I'd take a look at some underlying data.
I started with a 2016 paper by the modern master of failed serial doomcasting, James Hansen. It has the frightening title of "Ice melt, sea level rise and superstorms: evidence from paleoclimate data, climate modeling, and modern observations that 2°C global warming could be dangerous" … yikes! Be very afraid!
In Figure 29 of that paper, Hansen claims to show that sea level rise has been accelerating, from 0.6 mm/year from 1900 to 1930, to 1.4 mm/year from 1930 to 1992, and 2.6 mm/year from 1993 to 2015.
Now, as is far too common with this charming fellow, James Hansen is playing fast and loose with the facts. First, he's taken the data of Church and White from 1900 to 1992 and multiplied it by 0.78. This has the effect of flattening the record and thus reducing the prior sea level trends … which of course makes it seem like there is more acceleration than might actually exist.
Next, he has cherry-picked the Church and White (C&W) data shown in blue. The C&W data actually goes from 1860 to 2009, and Hansen and his merry band have chopped off both the early and the late part of the data.
Finally, post-1992 he has spliced the satellite data (with a trend which differs from Hansen's specially flattened tide gauge data) on to the end of the tide gauge data. They are measuring different things, and thus cannot be directly compared. This is the reason for the "knuckle" in Hansen's Figure 29 at the year 1993.
In any case, as those who know me are aware, I prefer to go to the original data. I don't believe anyone until I've run the numbers myself … and this is another example of why I do that. As my beloved grandmother used to say, "You can believe half of what you read, a quarter of what you hear … and an eighth of what you say" … and Hansen's claims seemed unbelievable.
In this instance I went to the Church & White (2011) paper cited by Hansen above, entitled Sea-Level Rise from the Late 19th to the Early 21st Century. (I guess C&W didn't get the memo about how scientific papers now require terrifying titles.) I digitized the C&W Figure 5 and analyzed it. This is their Figure 5:
(Notice that because this data has not been subjected to the special Hansen flattening, the trends of the tide gauge and the satellite altimeter data are similar … but I digress.)
In particular, I wanted to look at the trends. Since Hansen had used a 31-year trend from 1900-1930, I looked at the same length trends. Here are all of the trailing 31-year trends, indexed by the final year of each trend, including of course the 1900-1930 trend referenced by Hansen et al.
I'm sure you can see the problem with making any general statements about whether or not there is any acceleration in the rate of sea level rise during the last hundred years or so …
You can also see why Hansen cherry-picked the 1900-1930 trend as his data to try to show acceleration … because if he'd used 1930-1960 instead, there wouldn't be any acceleration to show.
Here's my conclusion in all of this. Until we can say why the rate of sea level rise:
• decelerated from the start of the C&W record until 1930
• accelerated rapidly until 1960
• decelerated for the next ten years
• stayed about the same from 1970 to 2000
• and then started accelerating again,
… until that time, I say that making just about any statement about sea level acceleration is premature. However, one thing is clear:
There is no simple relationship between CO2 levels and the rate of sea level rise …
My best to all on a lovely spring day. Fog in the morning, sun in the afternoon, and now a foggy night. When the fog rolls in like this in the evening, on nights like tonight it sometimes traps the sound of the foghorn on the Bodega Bay breakwater six miles (ten km.) away, and carries that mournful wail up the hill to draw my mind away, away to the eternal sea …
As Always: When you comment, please quote the exact words you are discussing so that we can all understand your exact subject. Misunderstandings are the bane of the intarwebs—please avoid them by being crystal clear about the topic of your comment.
Data: The digitized C&W data is below:
Year Sea Level (mm)
1860 -189.26
1877 -157.7
1933 -95.73
1934 -99.6
1981 -3.64
May 22, 2018 in Bad science.
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185 thoughts on "Changes in the Rate of Sea Level Rise"
DeLoss McKnight says:
Wouldn't you expect to see a little acceleration of SLR coming out of the Little Ice Age? I would also blame to a lesser degree all the ground water pumped out for irrigation would have a slight impact as well.
Javert Chip says:
The oceans are huge (and I mean HUGE) heat-sinks. Even with alleged significant atmospheric warming, it will take a LONG time to impact the oceans.
RAH says:
Oh, I thought I had read that the oceans had been impacted significantly in more than one way when massive amounts of cold water were released due to the failure of ice dams.
4 Eyes says:
My thoughts too Javert and that is why the accelerations and decelerations in the rate of sea level rise suggest either no dependency on CO2, which has been rising steadily, or a general lack of accuracy or reliability of the data.
…and yet, one rain storm in one small part of Australia lower sea level……..bullcrap
TallDave (@TallDave7) says:
Causal relationships between systems with masses 300x different usually run from the more massive to the less. I would imagine CO2 is less than a rounding error for the oceans, except perhaps on millennial scales or larger.
"…it will take a LONG time to impact the oceans."
Apparently it takes 800 years.
There was a dramatic acceleration at the end of the Little Ice Age, pretty well as you'd expect. Up to about 1850 there was little or no sea level rise that may have lasted several centuries. But at about 1850 (the end of the LIA) the modern sea level rise started at 1 or 2 mm/year. Since 1850 the rate has been remarkably constant.
It's a perfect hockey stick – except that the blade started in 1850! Correct me if I'm wrong, but I don't think SUV's or mass air travel had really taken off in 1850.
ferdberple says:
The is a better correlation between sea level rise and me peeing in the ocean than there is to sea level rise and CO2.
Sea level has been rising not since the end of the LIA, but since its depths during the Maunder Minimum, c. AD 1695.
"Wouldn't you expect to see a little acceleration of SLR coming out of the Little Ice Age?"
Since sea level rise is a artifact of temp….I would expect to see it flatten out because temps have been flat
tomwys1 says:
Virtually all tide gauges – worldwide – show linear trends and inconsequential (de)acceleration; take your pick, affected by local tectonics. In places that are tectonically inert, moving neither up nor down, Sea-Level is rising between 1mm and 1.4mm per year. Slipshavn (Nyborg) Denmark is a good example. All the rest (& Hansen et al is among them) is pandered drivel, by people wallowing in grants, and accepting prizes from unknown foundations and governments desperate to raise revenue. Sad, very sad!!!
Hansen's Figure 29 is a bad joke and I'm thankfully glad that Willis zeroed in on it!!!
Martin Hovland says:
Yes, Hansen har made the HOCKEYSHTICK of sealevel change….It's just one of his many jokes, I presume.
Bill5150 says:
Yes tomwys1, its a sick sad world. The water levels at Fort Denison in Sydney show similar and no acceleration. I am living in hope that Tim Flannery walks out onto the road in front of my car one day. .
J Mac says:
That's a solid 'take down and pin', Willis.
pdtillman says:
Second to Jmac (and everyone else). Hard to believe Hansen expected to be taken seriously. Oh wait, this was just political agitprop…..
high treason says:
The entire notion that picking one variable (CO2) out of the 50 odd variables to make it chief culprit and trying to make it fit the hypothesis is pseudoscience. Manipulating data to make it fit with some hypothesis is NOT science, it is fraud.
The persistent fraud will discredit science across the board.
HUE MAURICE says:
Depuis des millions d'années il y a des milliards de km³ d'eaux douces (venus des pluies, des fleuves & des rivières) qui se sont déversés dans les mers & océans… SANS QU'ELLES OU ILS NE MONTENT !!! Çà alors ! Tout simplement parce que l'eau s'infiltre continuellement dans les planchers océaniques et maritimes vers le magma où cette soupe toxique (les poissons chient dans la mer !) y est chauffée/bouillie et remonte donc (comme dans une cafetière électrique) vers les sources (chaudes ou froides suivant l'altitude) et vers les nappes phréatique qu'elle remplit.
For millions of years there are billions of km³ of fresh water that are poured into the seas & oceans … WITHOUT WHERE THEY DO NOT UP !!! That's it! Quite simply because the water is continuously infiltrating the ocean and sea floors towards the magma where this poisonous soup (the fish shit in the sea!) It is heated / boiled and goes back up (as in a coffee maker) to the sources (hot or cold depending on the altitude) and the groundwater that it fills.
co2isnotevil says:
Hansen has his fingerprints on a lot of what's wrong with climate science. He was the first to misapply Bode's feedback analysis to the climate, is largely responsible for the fake legitimacy of homogenization and has produced numerous predictions of scenarios that failed to materialize and none that did. Given this guys history of buffoonery, why would anyone take anything he says seriously?
the great climate prophet's time is coming. When we hit the minimum this September his prediction that the Arctic would be ice free in the summer will be falsified. Then in a few more years when all of the West Side Hwy remains above SL he will be shown to be wrong again. Has he made any other predictions for which he can be held accountable during our life times?
Rich Davis says:
Because, and I say this much more in sadness than in anger, the public
1) never really grasps the details of the claims to fully grasp that they failed to materialize, and
2) has a collective attention span of about 30 seconds, so is off to the next sex scandal, etc., on the MSM narrative, and
3) Climate Change (TM) constantly evolves to ensure its non-falsifiability.
Just as politicians have an advantage from name recognition ("all publicity is good publicity"), Hansen is a "world-renowned climate scientist". I believe that's his official title in the stylebook isn't it?
Joe Born says:
Could you be a little more specific about "first to misapply Bode's feedback analysis to the climate"?
I'm no Hansen fan, and I don't for a moment believe the climate system exhibits much net positive feedback. Apart from the improbably high feedback parameters, though, I've yet to see anyone explain how "Bode's" feedback analysis is wrong as applied to the climate system.
In other words, you don't mean there's something wrong with , do you?
sailboarder says:
Just the way it is applied, resulting in a massive miss on part of climate scientists. (see Moncton's posts)
sailboarder:
You may well be right about its being applied incorrectly. However, I'd caution against relying on Lord Monckton as authority for that proposition.
In a recent series of posts on this site he completely misconstrued the equation he represented as based on Bode's chapter 3. And he seemed to confuse what in some circles are known as "small-signal" quantities with the "large-signal" quantities they imply.
Although he's been going on about feedback for years, he still flubs the most-basic concepts. So at least on that topic you're better off looking elsewhere.
There are 2 preconditions for applying Bode's LINEAR feedback analysis set out in the first 2 paragraphs of his book. This book is still the definitive reference on electronic feedback amplifiers, despite being written nearly a century ago. This book was the only feedback related reference in either Hansen's or Schlesinger's papers on climate system feedback. These papers comprised the primary theoretical justification for a sensitivity high enough to establish the IPCC and UNFCCC.
The first missing precondition is strict linearity between the input and output and there's absolutely no physics that supports a linear relationship between W/m^2 and degrees K as the input and output of a linear system. The units of lambda0 are K per W/m^2 and f is W/m^2 per degree K, where these units themselves are meaningless as are the Cn constants quantifying the many 'feedback' terms. Another consequence of linearity is that the absolute and incremental gains (sensitivity) must be the same and the incremental application of climate feedback applied to sensitivity estimations is itself non sequitur.
Second is the requirement for an implicit, internal source of Joules to power the gain. This can't be the forcing input (the Sun) since this is already accounted for as the 'signal' input to the system. It would be like plugging a signal source into both the source input and the power cord of an audio amplifier.
A Bode feedback amplifier measures the sum of the input and feedback to determine how much output to deliver from its implicit power supply. In the climate system, the source of the output Joules are the input Joules (not an implicit power supply) and this COE constraint has been ignored because Bode ignores it owing to his 2 preconditions.
The equation you cited is pure garbage and has absolutely no correspondence to anything having to do with the climate. There's so much wrong with how Hansen and then Schlesinger applied feedback analysis to the climate, it would be amusing that anyone actually bought this garbage if the consequences of the IPCC's oppressive agenda based on this weren't so dire.
https://wattsupwiththat.com/2016/09/07/how-climate-feedback-is-fubar/
Regarding Monkton's feedback analysis, he's basically correct. The flaws in his analysis are mostly due to him trying to use language and terms that are consistent with how consensus climate science applies them, rather than how Bode applies them.
Regarding the difference between small signal and large signal, there's no difference in a linear amplifier, as the gain is constant and independent of the input. When the signal gets so large that the output starts to clip, the amplifier is no longer operating in its linear range and Bode's analysis no longer applies.
The only distinction to be made is between the small signal or AC gain and the DC gain setting the operating point, where the DC gain can be different from the AC gain. For example, adding a resistor to the emitter leg of a transistor amplifier sets and stabilizes the DC gain with negative feedback, but adding a bypass capacitor across it that has a low impedance at the operating frequencies and the AC gain at those frequencies will be much higher as the negative feedback gets reduced. This distinction doesn't apply to the climate system, as again, it depends on an internal source of Joules to power the gain.
co2isnotevil:
Thanks for your input. But I'm having trouble getting my mind around your reasoning.
I do recognize that the circuits Bode had in mind employed power supplies not reflected in the math, but to me that means the math can be applied to something that doesn't have a power supply. And those circuits were vacuum-tube circuits, whose relationships of, e.g., plate current to grid-to-cathode voltage were highly non-linear. So he was obviously linearizing if he made a linearity assumption. And that's presumably what Hansen did.
Let's remind ourselves of what linearization involves. An output is a not-necessarily linear function of an intermediate quantity . in turn is the sum of (1) the (output-independent) input and (2) feedback that's a not-necessarily linear function of the output .
Note that no power supplies were employed in building that equation.
In these contexts it is the temperature-independent portion of 's argument that is normally thought of as the system's input. But the and functions aren't usually so obliging as to permit us analytically to obtain from that general large-signal equation a closed-form relationship between the input and the output. That is, we can't in general find a function analytically such that . Even when specific instances of those functions do admit of analytic solution, moreover, the resultant is not usually linear.
But linear math would make analysis easier, so analysts often "linearize" the system by concentrating on small changes from a known equilibrium state to other (in this context, equilibrium) states. In the equilibrium context we're dealing with here, linearization involves differentiating the foregoing large-signal equation to obtain:
where and are the first derivatives of and . The derivatives are then evaluated at some known equilibrium state , where may include a temperature-independent constituent such that . If we additionally replace the infinitesimal differentials with finite differences, we obtain the following approximation:
After making the substitutions and and converting approximation to equality, the response perturbation can be isolated to obtain the form I described above:
(Here we use instead of because we've used already, for the feedback function rather than the feedback coefficient.)
If you've followed that derivation you'll see that nothing in it was based on an external power supply. And, yes, the equation is linear, but it's the linearization of a non-linear equation.
So I don't see how power supplies or nonlinearity make the equation invalid, so long as the forcing and temperature increments are not large.
The non linearity of a vacuum tube (or transistor for that matter) is irrelevant to Bode's analysis because a DC bias moves the device into the linear portion of its transfer curve. The non linearity is only present in a small region where the device is just starting to conduct. Once a sufficient current is flowing through the device, the response of its output to the input signal becomes linear until the output runs out of power supply and it starts to clip. Bode's analysis assumes strictly linear systems, not linear approximations of non linear relationships, besides, the assumption of approximate linearity between W/m^2 and degrees K is far from valid over the range of temperatures found on the planet.
A misunderstanding arises by conflating the DC operating point with the steady state average gain of the system in order to establish a false distinction between the absolute gain and the incremental gain. BTW, gain is the proper term, not sensitivity, which per Bode has a completely different meaning. Also, per Bode, gain is dimensionless and the dimensionless small signal gain required to amplify the 240 W/m^2 of incident solar energy into the 385 W/m^2 of surface emissions at its average temperature is about 1.6. This is not the DC gain, but the average small signal gain which is the same as the incremental small signal gain where each W/m^2 of input results in 1.6 W/m^2 of surface output. There is no DC gain or DC bias because there is no DC power supply. The 240 W/m^2 of input is the small signal input to a passive (i.e. no internal source of energy) climate system.
What this means is that the incremental gain applied to the next W/m^2 is also 1.6 which means that the next W/m^2 increases surface emissions by 1.6 W/m^2, corresponding to a temperature increase from about 287K to 287.3K which is far from the 0.4-1.2 C claimed to arise from 1 W/m^2 of incremental solar input and corresponds to an effect from the 3.7 W/m^2 of EQUIVALENT forcing said to arise by doubling CO2 concentrations of about 1.1C.
The idea that the next W/m^2 can increase surface emissions by 4.3 W/m^2 corresponding to the nominal 0.8C increase claimed is absurd beyond reason. Based on how Bode's analysis was applied, the only possible source to replenish the additional 3.3 W/m^2 of surface emissions is the implicit, internal source of Joules that's not actually present in the system. The reason of course is that COE dictates that all Joules contribute equally to the work producing a result. In fact, Joules are the units of work and Watts are just a rate of Joules, which integrated over time becomes work.
Linearizing the system is the correct thing to do, but the proper linearization is between forcing and surface emissions, which is already quite linear, and not between forcing and the surface temperature. Note that degrees K and W/m^2 interchangeably represent the same thing via the SB LAW, the only difference is the units. This is the fundamental error that's making climate science so hard to get right. The problem with linearizing the relationship between W/m^2 and temperature is that this infers a constant derivative (i.e. gain/sensitivity). Since the relationship actually goes as T^4, the derivative goes as 1/T^3, thus the incremental gain where the output is expressed as a temperature is far from being a constant.
Forcing and surface emissions have the same units (W/m^2). Feedback must also have the same units as the input. The feedback fraction is defined as the dimensionless fraction of the output added back to the input and can be between -1 and 1, thus when the output has the same units as the input, the feedback term also has the proper units and can be summed with the input. The bottom line is that you can't add degrees K to W/m^2 and the Cn constants that convert one to the other have no physical justification and again are only claimed to be 'approximately linear'.
Why go through all this effort of coercing a linear relationship between degrees K and W/m^2 when a nearly perfect linear relationship already exists between W/m^2 of forcing and the equivalent W/m^2 of surface emissions? The are only 2 possible reasons. One is incompetence and the other is to confuse and deceive.
Let me try to explain this from another angle.
Analyzing passive circuits of inductors, capacitors, transformers and resistors in terms of how they behave in the time domain requires solving complex sets of simultaneous differential equations. If instead, we apply the Laplace transform to convert between the time domain and the frequency domain, the analysis is simplified to the linear application of Ohms Law on resistance that has both real and imaginary parts. Afterwards, we can apply the Laplace transform in reverse to produce time domain results.
The SB Law is the bidirectional transformation function that converts between the non linear domain of degrees K and the linear domain of energy. In fact, the T^4 component of the SB LAW is responsible for most of the stated non linearity!
The correct way to analyse the climate is to convert temperatures into W/m^2 using the SB LAW, perform linear calculations on W/m^2 and convert the resulting W/m^2 into a temperature using SB in reverse. To calculate a delta T, do this before and after the delta F and subtract the resulting T's from each other.
The reason consensus climate science rejects this (or more precisely fails to consider it), is because it produces an unambiguous answer that's not the answer they need, hence all the obfuscation from excess complexity, adding the wiggle room necessary to accommodate what the laws of physics can not.
The correct way to analyse the climate is to convert temperatures into W/m^2 using the SB LAW, perform linear calculations on W/m^2 and convert the resulting W/m^2 into a temperature using SB in reverse.
I think I agree with that as an abstract proposition, but I don't see why you think they're not doing it. In contrast, you seem to think Lord Monkton's right, and he adds temperatures where he should be adding forcings. There seems to be a disconnect.
Regarding Monkton's feedback analysis, he's basically correct.
Actually, he got it wildly wrong, as you can see if you work through what in his fourth post he introduced as the "standard, mainstream" equation that is "universal in all dynamical systems except climate":
In form, this equation can indeed be found in Bode, Network Analysis and Feedback Amplifier Design. But Lord Monckton completely misconstrued it.
The problem is that he switched the symbol's meaning when he introduced the "standard, normal" equation. He had previously used to represent the sum of a constant reference portion and a variable portion: a variable. (See, for instance, his first post's paragraph beginning with "The error.") In his proposed equation, though, he used it to represent only the constant, reference-value portion . So the equation's form makes it seem that the system input is a constant. Meanwhile, he represented the actual input's variable component in a highly unorthodox fashion. Specifically, he represented it in a variation of the gain parameter : .
Instead of causing the output to increase by increasing the input, that is, he did so by increasing the gain. He compounded the unconventional parameter dependence by making the feedback coefficient depend on and, worse, on the output : .
The overall result is that for every positive value of there's an input value for which the output blows up. That's not linear, and it's not what Bode intended.
I'm afraid I didn't find in your responses a compelling argument for the proposition that the equation I asked about is "pure garbage." To me your argument seems based on misunderstanding the greenhouse effect—and, for that matter, Bode's technological milieu. (Although there are some vacuum-tube operating regimes that are less nonlinear than others, for example, the whole reason for negative feedback is to combat those devices' nonlinearities.)
Nonetheless, I thank you for taking the time to respond.
Jim Gorman says:
Joe, Let me put it into simpler terms. Without an external power supply you can only get an output equal to the input. If you try to assume feedback will increase the output when there is no external power, where does the increased power come from? Continue down that road and you will invent perpetual motion as you are creating energy from nowhere!
" … but I don't see why you think they're not doing it. In contrast, you seem to think Lord Monkton's right,"
Consensus climate science is not doing this because if they did, they would get a sensitivity less then the lower limit claimed in IPCC reports and the debate would have ended decades ago.
The IPCC calculates the sensitivity to doubling CO2 as the 3.7 W/m^2 of EQUIVALENT forcing that arises by doubling CO2 times the sensitivity factor, which they cite, without foundation, as 0.8C +/- 0.4C per W/m^2 (3.7*0.8 = 3C). They then curve fit arbitrary values of lambda0 and f so that deltaT/deltaF is the 0.8C they need it to be. They really need to choose arbitrary values of lambda0 and f such that deltaT/deltaF is the 0.3C that fits the skeptics estimates of the ECS. This still won't make the equation correct, as the upper limit on the actual sensitivity is highly temperature dependent and given by 1/(4eoT^3), where e is the ratio between the average planet emissions and average surface emissions, o is the SB constant and T is the average surface temperature in degrees K. The lower limit is given by the same equation, except with e=1.
The problem with the derivation of the equation you cited is assuming the relationship between T and P is linear is to assume that dT/dP is constant, and it's not. This ends up decoupling the magnitude of dT/dP from the constraints of T^4/P allowing dT/dP to be arbitrarily high. After all, 0.8C per W/m^2 of forcing sounds plausible enough, while 4.3 W/m^2 of incremental surface emissions per W/m^2 of forcing is an obvious violation of COE, even as both represent the same thing. Feedback then becomes the only possible source replenishing the additional 3.3 W/m^2 of emissions and any system where the feedback exceeds the forcing is unconditionally unstable. Relative to the climate, the T^4/P constraint is about 1.6 W/m^2 of surface emissions per W/m^2 of forcing, or about 600mw of 'feedback' per W/m^2 of forcing.
Climate science doesn't understand that the exponent in T^4 is immutable from first principles and no feedback or anything else can change this, moreover; it's the non linearity consequential to this exponent that they try fudge away, when there's a far easier way.
My point still stands. Why make gross assumptions and invent a bunch of complexity to linearize something intrinsically non linear, when a simple transformation makes the relevant relationships trivially linear?
Regarding Monkton's analysis, conceptually, he's right, but there are errors in his analysis. As I mentioned, these seem to be the result of attempting to be consistent with the bad assumptions and subverted terminology used by consensus climate science. For example, the terms forcing, feedback and sensitivity are all defined in Bode's book, but as applied to the climate, they have retained the same meaning while being subverted to represent something so different that the original meanings are no longer relevant. More specifically, the mu term in Bode is also referred to as the open loop gain, is a dimensionless constant and is why Monkton went with the form of a ratio of temperatures, as ratios are the only way to arrive at dimensionless values and he was stuck with using temperatures as the inputs and outputs, rather than power densities which are already intrinsically linearly related to each other. Additionally, the beta term is the fraction of the output returned as feedback and is also a dimensionless constant and has the same issues. What he did was to shoehorn the climate feedback model into the Bode model in a way more consistent with Bode, and as you noticed, it didn't make much sense and if once you understand Bode, it makes even less sense the way it has been framed by the consensus climate science.
Jim Gorman:
Without an external power supply you can only get an output equal to the input.
A transformer, for example, needs no external power source for its output voltage to exceed its input voltage.
A voltage-doubler circuit requires no external power source for its output voltage to exceed its input voltage.
More to the point, in our equations the input is forcing and the output is temperature. How great a temperature is greater than how much forcing?
The power source is the sun. The temperature is a measure of how much of that energy from the sun has accumulated before the power out to space equals the power out to the sun.
Knowing what the inputs and outputs are would make your opinion on whether one can exceed the other more informed.
Greg F says:
Without an external power supply you can only get an output equal to the input. If you try to assume feedback will increase the output when there is no external power, where does the increased power come from?
Joe Born responded:
No Joe. You're wrong. Jim asked "where does the increased power come from?" There is no increase in power with a transformer. Double the voltage and you half the current. Nor is there an increase in power with a voltage doubler.
The Bode feedback model requires a amplifier. Amplifiers by definition are an active element that requires an external power source.
3.2 Elementary Theory of Feedback Circuits
In its simplest form, a feedback amplifier can be regarded as a combination of an ordinary amplifier, or u circuit, and a passive network, or B circuit, by means of which a portion of the output of the u circuit can be returned to its input.
The Bode model simply doesn't apply to the "climate feedback" which has no additional energy sources. A proper model for the "climate feedback" can only contain passive elements. Use of the Bode model is grossly inappropriate.
Yes, you're exactly correct. You seem to understand Bode and are similarly amused at how anyone can think a transformer is an active amplifier per Bode. You'll probably also get why this passive network of the climate makes the most sense.
Model the atmosphere as a transmission line between the surface and space that's well matched to free space at TOA, but mismatched at the boundary with the surface. The source of the 'feedback' power is surface emissions reflected back to the surface by the impedance mismatch.
co2isnotevil,
I was going to respond a couple of times today but you kept beating me to it. Damn you!
A transmission line model makes a lot of sense. Never considered that angle. I would add thermal capacitance for the energy that is absorbed by the surface. Having adequate data to construct a model is another story.
What I don't get is how anybody (with at least high school physics) could not see the "climate feedback" model as 100% passive. Calling it "feedback" … I just don't know what to say. Words have meaning. Obfuscation of the technical meaning of the word is just so wrong.
Greg F
You are confusing the mathematics with the application. Bode applied the mathematics to an amplifier, which is indeed an active-element. Hansen applied the mathematics to the earth, which indeed is a passive element. But the math requires no active element.
What carbon dioxide does is increase the atmosphere's optical depth, which increases the average altitude of the radiators that space receives earth-sourced radiation from, which, lapse rate being what it is, increases the difference between the surface temperature and the effective radiation temperature. For a given surface temperature, that is, it reduces the effective radiation temperature and thus causes an imbalance between the incoming radiation, which hasn't changed, and the outgoing radiation, which has fallen.
If we ignore lapse-rate changes, then the surface temperature has to increase by the necessary emission-altitude temperature increase. (This means that the change in surface radiation exceeds the forcing change, a fact that seems to bother co2isnotevil). Finally, the resultant surface-temperature increase causes more water vapor, and, according to Hansen's hypothesis (which I question), that causes more forcing. That additional, temperature-dependent forcing is what's referred to as feedback, since the output quantity, temperature, is causing a change in the input quantity, forcing.
Note that all of this happens without any external power source: on average, the power into the system equals the power out of the system, with the exception that the earth may occasionally change its temperature by temporarily emitting more or less energy than it absorbs.
If you can't explain why the feedback equation doesn't describe this, then your argument for its being "pure garbage" is not compelling.
You are confusing the mathematics with the application. Bode applied the mathematics to an amplifier, which is indeed an active-element. Hansen applied the mathematics to the earth, which indeed is a passive element.
You are still wrong. Bode applied the mathematics to a system that is composed of an active (μ) and passive (β) element. Hansen incorrectly applied the mathematics where there is no active element.
But the math requires no active element.
It most certainly does require an active element.
Ah, yes, the "Yes, it is, too" argument. Very effective.
Serious discussion, please. I'm not interested in a high-school debate. If you want to participate in an adult conversation, please identify what in the equation that requires an active element. Yes, Bode used to characterize his active element, but there's no reason why it can't be used to characterize a passive one.
Here, let me help:
For the sake of simplicity, let's say . (The actual function is more complicated, but I'm suggesting an easy function for discussion purposes.) That gives surface temperature as a function of the total forcing that caused it. (Again, that's not precisely correct, but it doesn't matter for present purposes.) All passive. No outside power sources involved.
The total forcing is the sum of temperature-independent forcing , which is caused by things like human-generated CO2, and temperature-dependent forcing , which is caused by, e.g. water vapor: Presumably , like , is highly nonlinear. If we adopt and , we get:
Same equation as Bode's, but no active elements involved.
There, I've laid it all out there for you. All you have to do is identify where it's wrong and give a logical reason for your belief. That's how adult discussions work.
You have identified the feedback fubar. In fact, Bode's math does require an active element. The active element is implied by the mu term which is why you don't perceive it and why Hansen missed it. This is stated in the first paragraph of Bode's book where he specified his simplifying assumptions. Mu is the open loop gain (gain with no feedback) and you can't have any open loop gain if there's no IMPLICIT power supply. Mu is transconductance which is a dimensionless ratio that represents the amplification of a small voltage into a high impedance on the input into a relatively larger current change on the output driving a lower impedance. If you don't recognize this as implicit power gain, then you definitely don't understand the math of feedback systems.
Feedback systems can be tricky and there's no hope of understanding how to analyze them until you thoroughly understand the implications of the simplifying assumptions. This is what led Hansen and all who followed him astray. Amateurs should not apply analysis they don't understand.
You also seem to be missing the distinction between voltage gain and power gain. While Bode's math is quantifying voltage gain, it only works when power gain is implied. The reason is impedance. In Bode's basic analysis, the input impedance is assumed to be infinite while the output impedance is assumed to be zero. The implication of this is that the input forcing plus feedback is measured by the input of an amplifier to determine how much power to deliver to the output from its implicit supply. The input impedance of the climate system is zero and the input plus forcing is consumed to produce the output power. The later is not covered by Bode's math and requires an additional COE constraint that has never been applied to the climate system.
Here's how power gain works relative to impedance. If the input impedance is 1K ohms and the output load is 100 ohms, even a voltage gain of unity has a power gain of 10.
Yes, Bode used \mu to characterize his active element, but there's no reason why it can't be used to characterize a passive one.
co2isnotevil has made a more than adequate response. As he stated μ is the open loop gain. A passive component cannot have gain. Gain requires additional energy. Why is that so hard to understand?
I mis-stated transconductance and mu. Mu is a dimensionless ratio of voltage gain calculated from the transconductance (which has units of 1/ohms) and the output impedance of the active element.
Transconductance is the change in output current divided by the change in input voltage. As long as the output impedance is less then the input impedance, which is always the case for vacuum tubes, the potential power gain is greater than 1.
Note that if no load is present on the output of the amplifier, there is no power gain and there is no current drawn from the implicit power supply (except for bias currents). Maximum power gain occurs when the load impedance the amplifier is driving is equal to the output impedance of the amplifier.
BTW, Bode does talk about how to adjust the model for finite input and output impedances, but this comes much later in his book and was clearly not accounted for by Hansen who stuck with the idealized model and its many assumptions.
Getting back to the adult conversation, there are several errors in the derivation you presented.
First is the assumption that GHG 'forcing' is on an equal footing with solar forcing, while only the incoming solar energy conforms to Bode's definition of forcing. This is the result of the IPCC's ambiguous definition and usage of forcing. When they say that doubling CO2 results in 3.7 W/m^2 of 'forcing' what they actually mean is that doubling CO2 is EQUIVALENT to increasing the solar forcing by 3.7 W/m^2 while keeping CO2 concentrations constant.
The ambiguity in the definition of forcing arises from its quantification as a delta flux up/down at TOT/TOA. This assumes that an instantaneous W/m^2 more post albedo solar energy from the SUN has the same influence as an instantaneous 1 W/m^2 decrease in the size of the transparent window (i.e. increased GHG concentrations). The problem is that in the steady state, all of the incremental solar forcing affects the surface temperature, while only about 1/2 of the incremental atmospheric absorption does the same, as the other half of this absorbed energy eventually exits into space rather than being returned to the surface. This brings up another failure in pedantic climate science where it's generally assumed that all of the energy absorbed by the atmosphere eventually ends up being returned back to the surface as 'feedback'.
The other error is a failure to account for COE relative to feedback power. Since the presumed gain element has an input impedance close to zero, feedback power is consumed by the input of the gain block and is no longer available as output power until it passes through the gain block once more. Your analysis counts this power twice, once as feedback power ultimately passing through the gain block to affect the temperature and again as directly contributing to replacing emissions consequential to the output temperature. Because of the malformed gain block whose input is forcing and whose output is temperature, you fail to notice that you must account for COE between the input and output. COE between the input and output doesn't apply to a Bode linear amplifier, again owing to the implicit power supply that provides all the output Joules required. Only COE between the output power and the implicit power supply actually matters.
I'm afraid you're making a lot of assertions with no reasoning to back them up. Just saying that a separate power supplied is implied by doesn't make it so. Just saying that transconductance is dimensionless doesn't stop it from being expressed in siemens.
Gain is nothing more than the ratio of output to input. It's math. It isn't exclusive to amplifiers. An amplifier's input impedance is usually high, but it doesn't have to be; people who know this stuff tell me that the input impedances of the common-base bipolar-transistor amplifiers used in some radio-frequency circuits aren't particularly high. And an amplifier without a load may exhibit high voltage gain even though the power in exceeds the power out.
Gain doesn't imply a power supply. Many antennas have plenty of gain with no power supply.
And "the input impedance of the climate system is zero and the input plus forcing is consumed to produce the output power" sounds to me like gobbledygook. In the first place, you haven't said what quantities you think are analogous to the voltage and current whose ratio is impedance. Secondly, forcing isn't something that's "consumed."
I appreciate your efforts, but you've fallen far short of demonstrating that the equation I asked about is "pure garbage."
"Just saying that a separate power supplied is implied by \mu doesn't make it so. "
I'm not saying this, but Bode did and as far as I'm concerned, Bode is the definitive reference for the math describing linear feedback amplifiers. Also, I corrected my mis-statement about the dimensionality of the transconductance, which has units of 1/ohms. The dimensionless gain, mu, is the transconductance times the output impedance. And yes, mu must be dimensionless per Bode, as this is the ratio of the output change divided by the input change, where both are expressed in the same units. The only one who took liberties with assumptions without backing them up was Hansen, Schlesinger and those who echo the broken feedback analysis. Just because Hansen claims something definitely doesn't make it true.
The 'power gain' of an antenna is just a convenience for analysis as the power exiting the antenna is not increased. Instead, it's focused on a smaller region, thus the energy density increases in some places, decreases in others and the total power remains the same.
Relative to the climate, amplifying 240 W/m^2 of incident solar energy into 390 W/m^2 of surface emissions requires power gain if you want to model this as a feedback amplifier. The point being that power gain by the climate system is impossible without an implicit power supply that's not the same as the forcing input. This being said, a Bode linear feedback amplifier is not an adequate model for the climate system. A passive transmission line between the surface and space (the 'gain' block) is a far better model. Alternatively, this can be quantified in the Z^-1 domain (discrete time) and the 'feedback' can be modelled as a delay element between surface emissions absorbed by the atmosphere and its return back to the surface and another between absorbed surface emissions and the fraction of them that ultimately leaves the planet.
Apparently, the COE argument went over your head, so let me try once more. You calculate feedback power in W/m^2 as some dimensionally scaled version of the output (surface) temperature. This means that the feedback power is already contributing to the output temperature, but since it's also being consumed by the gain block to produce the output power, this power is no longer available to contribute to the output temperature upon which the feedback power is predicated. In effect, you are counting this power twice, once as it contributes to T that the feedback is based on and again as the feedback passes through the gain block contributing further to the emissions corresponding to an increased T.
If you can't grasp how the input power is consumed by the climate feedback modelto produce the output power, then where is the output power coming from?
people who know this stuff tell me that the input impedances of the common-base bipolar-transistor amplifiers used in some radio-frequency circuits aren't particularly high.
They are by design due to transmission line effects. For RF they are typically 50 or 75 ohms. Network cables have a characteristic impedance of 110 ohms and network cards are all terminated with 110 ohms.
Many antennas have plenty of gain with no power supply.
The total radiated power doesn't change. The "gain" is merely a function of the directivity of the antenna relative to a reference antenna. The reference antenna is usually an ideal isotropic antenna (radiates in all directions equally). Cut the radiation pattern to a half sphere and relative to the ideal isotropic antenna you would have a 3 dB gain.
The input impedance of a common base amplifier is low because the input current is superimposed on the output current. Common base amplifiers do not amplify current, but do amplify voltage and this produces power gain. Common emitter amplifiers amplify both voltage and current. Emitter followers amplify current, but not voltage. Since power is voltage times current, all configurations exhibit power gain.
Even common emitter RF amplifiers have a low input impedance, but this is the consequence of some kind of transformer on the input. If a transformer has a 10:1 winding ratio, it will present a 100:1 impedance transformation. This is because a transformer multiplies voltage by the same factor it divides current, such that the voltage times the power is the same on the primary and secondary. Since impedance is voltage divided by current, increasing the voltage by N and decreasing the current by N increases the secondary impedance to N^2 times the primary impedance.
Oops, a small editing error.
"voltage times the power" was originally "voltage times current" that I changed to power and didn't delete all of the earlier phrase.
I'm not saying this, but Bode did and as far as I'm concerned, Bode is the definitive reference for the math describing linear feedback amplifiers. . . . The dimensionless gain, mu, is the transconductance times the output impedance. And yes, mu must be dimensionless per Bode, as this is the ratio of the output change divided by the input change, where both are expressed in the same units. The only one who took liberties with assumptions without backing them up was Hansen, Schlesinger and those who echo the broken feedback analysis.
I'm afraid our background differences are getting in the way here. I'm a retired lawyer, not a scientist, so I had to deal with a wide range of fact situations over the course of my professional life. Perhaps that has made me take a broader, more-conceptual view of the relevant disciplines. Among my clients feedback was used in electronic circuits, sure, but it was used it for other things as well. And in my experience people in those fields didn't slavishly follow Bode's nomenclature. On page 20 of his Control Systems Theory, for example, Olle Elgerd used gain for a quantity whose dimensions are newtons per radian. I'm sure the term has also been used for all manner of other dimensioned quantities.
If you can't grasp how the input power is consumed by the climate feedback model to produce the output power, then where is the output power coming from?
Who said anything about output power? In the equation we're dealing with the output is temperature, not power. Temperature is roughly a measure of how much energy has been received over time without being re-emitted. You seem to be going by how well Hansen's system matches Bode's, while I'm interested in whether the equation fits.
It's like the level of water that has entered a bathtub without leaving through a loosely plugged drain. If you increase the inflow, there's a temporary imbalance between the inflow and the outflow through the drain, but the resultant water-level increase causes the rate outflow to increase to reach a new equilibrium at the higher water level. If the resultant water-pressure increase pushes the plug in further, outflow decreases from what it was, so the water level increases still more: the outflow change is added to the inflow change.
If we knew enough about the bathtub system we could write an equation for the equilibrium relationship between the equilibrium water level and inflow that would take into account the effect that water-level has on the loosely plugged drain, the equation you'd have would be of the same one we've been dealing with. Inflow is the input, water level is the output, and gain is expressed in meters per liter per second.
Feedback outflow would be calculated from the output water level. If in your view this means that the outflow "is already contributing to the output" water level "but since it's also being consumed by the gain block to produce the output power, this power is no longer available to contribute to the output" water level "upon which the feedback power is predicated," then your view of feedback is fundamentally different from mine, and I am pessimistic about the prospects for convergence here.
But, again, thanks for the discussion.
"In the equation we're dealing with the output is temperature, not power."
And this is the fundamental reason for why consensus climate science is so screwed up. By focusing on temperature, rather than power, they conveniently ignore COE between the forcing and the temperature, thus the change in T is unconstrained by the energy requirements of the increased emissions consequential to that change. This is how they can arm wave a 0.8C increase in T that increases emissions by 4.3 W/m^2 and yet is the result of only 1 W/m^2 of new input to the system.
Where to you think the extra 3.3 W/m^2 of surface input in excess of the forcing is coming from in order to replace the energy that's being emitted away consequential to the presumed increase in temperature?
You can't decouple the temperature of a body from its emitted power where the emitted power is proportional to T^4. They are interchangeable metrics that represent the same exact state of a body and you can't have one without the other. Since energy can't be created or destroyed and all Joules are equivalent, the climate system must be linear in the energy domain, that is if one more Joule arrives, 1 one more must leave, or if each W/m^2 of forcing results in 1.6 W/m^2 of surface emissions, the next one must too and in no possible Universe can it be as large as the 4.3 W/m^2 required in order to be consistent with the climate sensitivity factor claimed by the IPCC. Note that the extra 0.6 W/m^2 of actual Earth 'feedback' is not new power as the consensus formulation allows, but power emitted by the surface in the recent past, delayed by GHG's and clouds in the atmosphere and subsequently returned to the surface. Note as well that 0.6 W/m^2 is less than the forcing of 1 W/m^2, thus the system is unconditionally stable, as it most certainly is, otherwise, we wouldn't even be here to worry about it.
The fundamental problem is that the language framing climate science is broken and we can't correct the science until we fix the language so the errors can actually be expressed. One of the biggest mistakes was to express sensitivity as a change in temperature due to a change in forcing, rather than as a change in emissions due to a change in forcing, where the deltaT is calculated by subtracting the SB equivalent temperatures of those emissions before and after the change in forcing. As best as I can tell, it was Schlesinger who formalized this as the sensitivity metric when he attempted to 'fix' the many mistakes in Hansen's feedback paper. He actually added more errors and made it far more convoluted.
"then your view of feedback is fundamentally different from mine"
Not necessarily. Your view and my view is also Bode's view of how a feedback amplifier works. The problem is that how a feedback amplifier works is irrelevant to how the climate system responds to change, thus Bode's math can't be legitimately used to model the climate. His view of feedback and gain (incorrectly referred to as a sensitivity) is based on 2 simplifying assumptions. One is strict linearity and the other is the existence of an implicit and infinite source of Joules to power the gain. While these assumptions are embodied by the climate feedback model, they have absolutely no correspondence to how the actual climate system responds to change.
The assumption of approximation of linearity does not conform to strict linearity. Solar power is the only true forcing input to the climate system and can not also be the implicit, infinite source of Joules powering Bode's ideal gain block. An implicit power supply does not mean connecting the power supply input to the forcing input, it means a power supply that's independent of the forcing.
CO2 'forcing' isn't real. Changing CO2 concentrations changes the system, not the stimulus (forcing). Changes to the system are said to be EQUIVALENT to changes in forcing keeping the system constant. What is called CO2 'forcing' is not forcing at all, but represents how much new forcing would be required to have the same effect as a change to system arising from a change in CO2 concentrations.
BTW, your tub model of feedback doesn't hold water.because once more, there's no active gain, although it's a better model of how the climate operates …
Co2 and Greg F ==>. You have done an excellent job of explaining. Are you guys EE's. That's my background. The suggestion of using transmission line theory will be important somewhere down the line. It answers my questions about delays and no extra energy being required.
Yes, I'm an EE (Cornell '78). My first exposure to feedback amplifiers when I was about 11 years old and received a 5-tube radio kit for my birthday. The output stage was a 2-tube feedback amplifier (actually 3 active elements, since the 12ax7 is a dual triode) and it came complete with a theory of operation which I kept reading until I understood it.
It's interesting how most EE's grasp the 'feedback fubar' intuitively, while many climate scientists are clueless.
Schlesinger doesn't, although he claims to have some kind of EE background and has asserted to me that he's the worlds leading expert on climate system feedback. After our exchanges, it seemed to me that he was either a failing EE student or was perpetrating a masterful deception.
Co2 ==> BSEE University of Kansas (1972), had my General Class ham radio license at 12. Made money in high school repairing tractor fender radios. The first ones were tubes and shook themselves apart!
By focusing on temperature, rather than power, they conveniently ignore COE between the forcing and the temperature, thus the change in T is unconstrained by the energy requirements of the increased emissions consequential to that change. This is how they can arm wave a 0.8C increase in T that increases emissions by 4.3 W/m^2 and yet is the result of only 1 W/m^2 of new input to the system.
In addition to looking at feedback differently, we lawyers also look at conservation of energy differently from the way you engineers do.
A lawyer proposes the following scenario. A 2-electron-volt photon enters a cavity. It bounces around inside the cavity, hitting the cavity's 1 m^2 bottom wall five times before leaving the cavity through an aperture 20 nanoseconds later. The lawyer would say energy has been conserved: 2 electron-volts entered, 2 electron-volts left, and none remains.
But that's the lawyer; he only knows about energy
The engineer, on the other hand, also knows about power and temperature. So he would instead count how many times a photon had hit the bottom wall's area in 20 nanoseconds and accuse that shyster of "arm-waving" whatever temperature half an electron-volt per nanosecond per square meter translates to "and yet is the result of only" a tenth of an electron-volt per nanosecond per square meter of input to the system.
As I say, our backgrounds make us look at things differently. I'm okay with that.
Joe Born ==> An engineer would ask at what ev level is that photon leaving. You assume the same as when it came in. Does a cue ball hit another ball with the same energy it started with after bouncing off the rail five times? In order for that to happen you must make several assumptions, like the walls are pure, ideal reflectors.
<blockquote An engineer would ask at what ev level is that photon leaving. You assume the same as when it came in. Does a cue ball hit another ball with the same energy it started with after bouncing off the rail five times? In order for that to happen you must make several assumptions, like the walls are pure, ideal reflectors.
Thank you for a perfect example of the engineer's focusing on irrelevant details at the expense of the relevant concept.
For the tediously literal-minded:
Commenter co2isnotevil thinks it violates conservation of energy for a forcing of 1 W/m^2 to cause a surface-temperature increase worth, say, 1.4 W/m^2. That is, it makes no sense to him that it would take 1.4 W/m^2 worth of surface-temperature increase to redress a top-of-the-atmosphere imbalance of only 1 W/m^2.
What I'm pointing out is that the surface 1.4 W/m^2 includes an increase in the amount of radiation circulating between the surface and the atmosphere, i.e., in the amount of energy retained on earth. That's what's analogous to the photon bouncing off the (yes, of course they're ideally reflecting) walls.
It's irrelevant that, e.g., the photons emitted by the atmosphere to the surface aren't the same photons that the surface emitted to the atmosphere (or, incidentally, that there are actually more—but lower-energy—photons emitted to space than are received without reflection from the sun). Disputants like co2isnotevil and Christopher Monckton regularly raise irrelevancies like that. Maybe they do so because they don't recognize the irrelevance, or maybe they do it to frighten the natives.
In any event, what's important is that in comparing power from space with power from the surface co2isnotevil is multiple-counting at the surface; you can't infer from that difference that energy is being created. The relevant comparison is power from space with power to space. And the equation we're discussing deals with conditions that prevail when those quantities are equal.
That said, I suppose I should admit to being provocative in drawing the lawyer/engineer distinction. I've known engineers in my time whose intellects were incandescent, while my brothers at the bar often embarrass me vicariously. Still, in writing the statement I quoted above an engineer was making a mere debating point, not a relevant argument, and that's something engineers accuse us lawyers of doing.
A physicist on the other hand would consider that the energy of a photon is quantized, so either the photon leaves the box or it's energy was absorbed by a wall, heating it. You can't divide the energy of a photon into pieces unless that photon was absorbed by matter and then re-emitted in a combination of lower energy absorption/emissions bands. You should think about the consequences of this relative to 'thermalization'.
"co2isnotevil thinks it violates conservation of energy for a forcing of 1 W/m^2 to cause a surface-temperature increase worth, say, 1.4 W/m^2. "
No, this is not even close to what I think, although it would be true for the Moon or a planet without an atmosphere. What I've said is that each W/m^2 of solar forcing results in 1.6 W/m^2 of surface emissions. This is not a COE violation because the retained and reflected energy is limited to the forcing, which sets an absolute upper limit on the emissions sensitivity as 2 W/m^2 of surface emissions per W/m^2 of forcing, i.e. the 100% positive feedback case.
My point is that the 4.3 W/m^2 of emissions that the IPCC claims results from each W/m^2 of forcing is a violation of COE and that this violation arose because the internal, infinite power supply magically (implicitly) added this power to output of the model, yet the climate system has no implicit, internal, infinite power supply powering the increase from 1 W/m^2 up to 1.6 W/m^2, Instead, the 600 extra milliwatts of 'feedback' is the energy of prior surface emissions that was absorbed by GHG's or clouds and then re-emitted back to the surface at a later time. This is why 'feedback' is limited to an additional 1 W/m^2.
The only way to avoid this is if the open loop gain was greater than 1, however; this doesn't work in the degenerate case with no atmosphere where the surface would still be warmer than the power from the Sun can support.
BTW, assuming unit open loop gain was a 'mistake' in Hansen's feedback paper that was 'corrected' by Schlesinger by changing the output of the model from W/m^2, which surprisingly, Hansen had correct, to degrees K and then providing the illusion of non unit open loop gain by obfuscating the Stefan-Boltzmann Law that converts between power and temperature as the open loop gain. After a few weeks of back and forth with Schlesinger about a decade ago, I brought this up at which point he got angry and didn't want to communicate with me any more.
I thought it might be helpful to explain why the feedback power is limited to the forcing, because this is not the case with the Bode model owing to the implicit power supply.
Consider a gain block with an open loop gain of 1. 1 W/m^2 goes into it and 1 W/m^2 leaves. For a Bode linear feedback amplifier, 100% positive feedback would be unstable, even with unit open loop gain. 1 W/m^2 is fed back, making the input 2 W/m^2. Now 2 W/m^2 comes out, all is fed back and 3 W/m^2 goes in and so on and so forth. This is the so called 'runaway' condition. Per Bode's simplifying assumptions, the additional energy coming out of the gain block originates from the implicit and infinite power supply. The reason the gain block doesn't consume the input and feedback power is because it has an infinite input impedance and the input consumes no power itself.
What they want you to believe is that the feedback model is a theoretical abstraction of incremental behavior. But, what's really being modelling from a physical perspective is not incremental and is identifiable, thus removing this level of abstraction. however; this level of abstraction is required to provide the wiggle room to support a sensitivity as high as they need it to be. When the abstraction is constrained by what it's modelling, the wiggle room disappears.
The physical correspondences are that the output of the model is the average state of the surface which is comprised of 2 components, its average temperature and its average emissions, immutably related to each other through the Stefan-Boltzmann Law. The model also has a single input, which is solar forcing, while the combination of unit open loop gain and feedback is modelling the atmosphere which adds additional complexity by modulating the fraction of surface power leaving the planet as well as the fraction of solar power arriving at the surface.
The power leaving the gain block can be either feedback power or output power, but not both, because there's no implicit power supply within what they're modelling, instead, the output power of the gain block can only originate from its input, moreover; feedback to the surface originates from the atmosphere and only half of what enters the atmosphere can be fed back to the surface, since the remaining half ultimately exits into space.
If 100% of the surface emissions were absorbed by the atmosphere, half of this would be returned to the surface as feedback and the remaining half would be emitted out into space. This sets the maximum amount of feedback power to half of the surface emissions which is what it would be 100% of the surface emissions were absorbed. Whether this is 50% or 100% feedback really doesn't matter as this is the maximum possible that it can be. In any event, this sets the maximum ratio between surface emissions and the incident energy to 2 which sets the maximum feedback to half of 2 which is equal to the 1 of forcing input. Linearity in the power domain means that this applies to all average Joules equally, including the next one. The current ratio is 1.6 as I have pointed out before.
If all of the surface emissions were absorbed and we needed to replace 240 W/m^2 of incident energy, all must come from the atmosphere, thus the atmosphere must be absorbing 480 W/m^2, so that half can be emitted into space, corresponding to a surface temperature of 303K which is the absolute highest AVERAGE temperature for the planet, given the amount of incident solar energy. Of course, for the atmosphere to be absorbing that much, it would need to be 100% covered by dense clouds which would reflect away a lot more power significantly reducing the 240 W/m^2 of post albedo incident power. This is as bad as a runaway condition can get (BTW, the surface temperature of Venus has a different origin).
If none of the surface emissions were absorbed (the zero feedback case) then surface emissions would be 240 W/m^2 corresponding to a temperature of 255K. However; the albedo would also be reduced increasing the total incident solar energy, increasing the temperature. If we consider the albedo to be that of the Moon, which is made from the same stuff, the average surface temperature would be about 271K. The consensus considers the zero feedback case, or reference, 255K, rather than 271K it should be.
Okay, so you don't know enough physics for that to be a good analogy. (Frankly, despite my surname I don't know much quantum optics myself, but I'm pretty sure the photons reflected from a mirror are identical to the incident ones, and any absorption occurs only to the extent that the mirror isn't, as we assume, ideal. That is, most photons are in essence reflected, while, in the case of a non-ideal mirror, a few photons randomly chosen in accordance with the laws of quantum mechanics are absorbed) So let's drop that analogy.
I seem not to have taken your meaning last time, but your response appears to share much of the same misapprehension:
[T]he retained and reflected energy is limited to the forcing, which sets an absolute upper limit on the emissions sensitivity as 2 W/m^2 of surface emissions per W/m^2 of forcing, i.e. the 100% positive feedback case. . . .
[T]he climate system has no implicit, internal, infinite power supply powering the increase from 1 W/m^2 up to 1.6 W/m^2, Instead, the 600 extra milliwatts of 'feedback' is the energy of prior surface emissions that was absorbed by GHG's or clouds and then re-emitted back to the surface at a later time. This is why 'feedback' is limited to an additional 1 W/m^2.
I'm afraid that reflects too profound a misunderstanding of a couple relevant disciplines for this exercise to be worth continuing further. For the benefit of any lurkers, though, I'll review the issue.
Let's not forget that the question before the house is whether is the right equation to use for approximating climate response. Instead of talking about what Bode said or whether gain implies a separate power supply, let's look at the equation in operation. For purposes of explanation, we'll describe it as happening in discrete steps, although in reality it's a continuous process.
Step 1: The CO2 concentration so increases that the resultant emission-altitude increase reduces top-of-the-altitude temperature by 0.33 K and the resultant emissions by . Note that this involves no increased power input. The same amount of power comes in from the sun, but it has a harder time getting out. The value of the temperature change before the resultant radiation imbalance has had any effect is zero.
Step 2: The resultant radiation imbalance warms the surface until its temperature has increased by = 0.33 K, i.e., by the amount needed to raise the new emission altitude's temperature back to what the previous emission altitude's was. Again, no internal power supply has been used; the surface temperature increased because more of what was always coming in from the sun gets backed up before it goes out. Once that previous emission temperature has been restored there would be no imbalance if nothing further happened.
Step 3: But the increased temperature makes water-vapor concentration increase enough that the emission-temperature decrease caused by the resultant emission-altitude increase reduces emission to space by an additional . No internal power source is involved. Evaporation energy came from the sun.
Now, I do not believe that , i.e., the gain experienced in thus traversing the "loop," is in reality the 0.67 value we're using. But that's not the issue here. The issue is whether we're using the right equation, and we are. This would be the way to estimate the response if that value were indeed the approximate derivative of the feedback, i.e., of temperature-dependent forcing, as a function of temperature.
Step 4: The resultant radiation imbalance warms the surface until its temperature has increased by another = 0.22 K. Again, no internal power supply was needed.
Step 5: Again, the increased temperature makes water-vapor concentration increase. And, again, no internal power source is involved. Evaporation energy came from the sun. The emission-temperature decrease caused by the resultant emission-altitude increase reduces emission to space by an additional . That raises the total forcing to .
The preceding sequence so continues as to cause to approach , all without the aid of any internal, infinite power supply.
The explanation gave by co2isnotevil makes no sense to me, so I'll probably misrepresent it again. But he seems to think that the difference between a forcing and the resultant additional surface emission is feedback: that somehow a value equal to forcing is emitted from the surface, and a percentage equal to the loop gain returns to the surface from the surface. As we saw above, though, the additional power from the surface appeared before, say, the water vapor responded, i.e., before feedback.
In any event I've already wasted too much time on this, this I'll leave it at that.
TimTheToolMan says:
Joe writes
Step 4: The resultant radiation imbalance warms the surface until its temperature has increased by another \Delta T_{\mathrm{eq}\,2}=\lambda_0\Delta F_{\mathrm{dep}\,1} = 0.22 K.
All standard AGW theory. However the warming effect originates from the TOA because its fundamentally an accumulation of energy but the water vapor feedback doesn't impact the TOA imbalance directly because its concentrated at the surface. Its not at all obvious that more GHG at the surface impacts the imbalance.
"Let's not forget that the question before the house is whether \Delta T_\mathrm{eq}=\lambda_0(\Delta F_\mathrm{ind}+k\Delta T_\mathrm{eq}) is the right equation to use for approximating climate response."
This is about as valid as \Delta T_\mathrm{eq}=k\Delta F. Neither are valid because the 'constants' are temperature dependent and non linearly dependent as well. Solving, you end up with a \Delta T on the left and linearized temperature dependent coefficients on the right.
To see why this is broken, consider the function a = b^4. At b == 2, we can linearly approximate it as a = 8b, but this curve fitted approximation only works for b == 2 and has absolutely no predictive power for other values of b. Now, to take this a step further, approximate the sensitivity of b to changes in a as the derivative of b with respect to a. The actual 'sensitivity' becomes 1/4b^3, which at b=2 becomes 1/32 while the 'sensitivity' of the approximation is 1/8 and independent of b.
Replace a with F, b with T and throw in a few scaling coefficients and this is the smoke and mirrors used to approximate their insanely high sensitivity while maintaining sufficient confusion. This was combined with their misappropriation of Bode as the theoretical justification for a sensitivity as high as the IPCC needed to justify their formation.
The problem is that it T is implicitly approximated to be linear to F which makes the dT/dF (sensitivity) meaningless with respect to the actual relationship between T and F.
chaamjamal says:
Also this
https://ssrn.com/abstract=3023248
john lenon says:
Hawaii lava is filling up the oceannnnnnnnnnnnnnnnnnnnnn
On Tue, May 22, 2018 at 9:33 PM, Watts Up With That? wrote:
> Willis Eschenbach posted: "Guest Post by Willis Eschenbach There's been > some discussion of the rate of sea level rise lately, so I thought I'd take > a look at some underlying data. I started with a 2016 paper by the modern > master of failed serial doomcasting, James Hansen. It has th" >
Leo Smith says:
It's worse. Hawaii lava lost to the sea is reducing the size of Hawaii so its GONNA SINK
Don K says:
Of course it's gonna sink. The evidence is quite strong actually and suggests that the tops of Mauna Kea and Mauna Loa will be slipping beneath the waves in about 2,000,000 years.– Subsidence 2.6mm/yr https://pubs.geoscienceworld.org/gsa/geology/article-abstract/19/2/171/205284/crustal-subsidence-rate-off-hawaii-determined-from?redirectedFrom=fulltext
Actually, there's other evidence — the presence of a small amount of basalt above the water at French Frigate Shoal about 1000km NW of Honolulu — that suggests the rate of subsidence probably slows as the Hawaiian hot spot moves further South and East away from the volcanoes. So maybe the higher parts of the Island of Hawaii will be around for 10,000,000 or even 20,000,000 years. But it almost certainly is going to sink.eventually.
RWturner says:
It's not subsidence that sinks these seamounts, it's erosion (denudation) of the basalt because it is quite unstable subaerially exposed. The rate of denudation is fast at first when the magma chamber empties and no longer provides a force against gravity, and then it slows to the rate you see on the eastern islands.
Jan Christoffersen says:
RWTurner,
Basalt is quite resistant to erosion, chemical or physical. Basaltic ocean sea mounts sink eventually (without continuous volcanism) because they are built on relatively thin oceanic crust and mantle, which cannot support them.
joelobryan says:
The final sentence of the (way, way too long) Hansen 2016 abstract says, "We discuss observations and modeling studies needed to re- fute or clarify these assertions."
(My bold).
They threw everything from accelerating SLR, melting poles, slowing AMOC, to children dropping dead in the streets into this 2016 paper in a quest for study money. Obviously they were expecting Hilly to win. (Okay maybe not children dropping dead, but close)
Conclusion: carnival barking rentseeking at work.
Willis Eschenbach says:
Thanks, Joel. I missed that in the paper … my stomach wouldn't take it after while.
commieBob says:
The decision to multiply the rate by 0.78 wasn't arbitrary, it was taken (according to the graph's caption) from Hay et al. 2015. That paper (If I have the right one) states:
Several previous analyses of tide gauge records–employing different methods to accommodate the spatial sparsity and temporal incompleteness of the data and to constrain the geometry of long-term sea-level change–have concluded that GMSL rose over the twentieth century at a mean rate of 1.6 to 1.9 millimetres per year. Efforts to account for this rate by summing estimates of individual contributions from glacier and ice-sheet mass loss, ocean thermal expansion, and changes in land water storage fall significantly short in the period before 1990. The failure to close the budget of GMSL during this period has led to suggestions that several contributions may have been systematically underestimated. [bold mine] Probabilistic reanalysis of twentieth-century sea-level rise
If I can paraphrase what they're saying: "We can't explain the data so we changed the data."
That, Madame, is intellectual baby-talk, …
Christopher Monckton of Brenchley did a wonderful WUWT story about logical errors as applied to climate change. I'm not sure which particular logical error Hay et al. committed (perhaps a form of argument from ignorance) but it is surely not logical to change the data simply because you can't explain why the data shows what it does.
Hansen didn't pull the 0.78 multiplier out of the air but when one looks at where it came from it's still bunk.
What things could Hay et al. have ignored in their attempt to close the GMSL budget? How about sediment? How about the volcanoes beneath the Ross Ice Sheet?
"We couldn't explain the data so we changed it." Dear God in Heaven!
Hivemind says:
Or maybe their theory was just wrong ? If they seriously thought their data was wrong, why didn't they go back and collect good data? No reputable scientist would change good data to match a bad theory.
Would any "reputable scientist" stand for his host sneaking in and disabling the AC and opening the windows in the Chamber where he/she are going to present on what has been historically the hottest day of the year in Washington, DC to try and make a point without informing those present what had been done?
No, no, you're confused. They changed bad data into good "data" that fits their preordained double-plus-good law of nature.
It would be so much less effort just to write a computer program to spit out reams of "data", instead of risking measurement error and all the labor of collecting and compiling. Oh wait, they already did that. GCMs. But they have rigorously checked the models with …
the output from more models.
Hugs says:
Figure 29. Estimated sea level change (mm) since 1900. Data through 1992 are the tide-gauge record of Church and White (2011) with the change rate multiplied by 0.78, so as to yield a mean 1901–1990 change rate of 1.2 mm year−1
(Hay et al., 2015). The two estimates for the satellite era (1993–2015) are from Nerem et al. (2010, updated at http://sealevel.colorado.edu) and Watson et al. (2015)
So, they used Hay et al. for their trend, forced Church and White on top of that, and then grafted with measurements (or should I say estimates) done with a totally different method. Did I get it right?
I think it is a misrepresentation of data. The accuracy of Church and White is used with the trend of Hay et al. I'm not saying that it looks childish, because I'm not a qualified world-known iconic climate scientist, but I do raise eye-brows. But the whole thing is very complicated so not many are able to say how terrible that kind of graph really is.
In my opinion, Hansen does these graphs not for science, but for alarmists to use in blogs, newspapers and the worst, eventually in Wikipedia, where they appear as 'sourced' and from a 'trusted journal'.
Nick Stokes says:
Willis,
"I digitized the C&W Figure 5 and analyzed it. "
CSIRO has a rather complete zipfile of C&W data here. The file that corresponds to the numbers you listed is CSIRO_Recons_gmsl_yr_2011.txt, though it starts in 1880. Your numbers look pretty good.
paulski0 says:
Yes, though out of date as it only goes up to 2009. The update to 2013 is available here.
31-year trends look like this:
https://i1.wp.com/bonjourplanetearth.files.wordpress.com/2018/05/cw2011slr-trends.png
Clear acceleration beyond previous rates. Of course, an update to 2017 would be even clearer.
A C Osborn says:
Er, no, not much difference between 1920-30 and the later trend.
Funny how that chart looks nothing like Mr Eschenbach's.
Mark BLR says:
To AC Osborn's "Funny how that chart looks nothing like Mr Eschenbach's".
Try narrowing the "width" (/ adjusting the X-axis) of your viewing window …
paulski0 plotted the 2015 UPDATE to C&W 2011 (data to 2013), Willis digitised and plotted the ORIGINAL data (to 2009, published in 2011).
Checking the differences (2015 update – 2011 original), there is a CONSTANT -1.6mm offset from 1880 to 1989, followed by variable but ALMOST always negative values from 1990 to 2001 (the exception is 1994, with a delta of +1.1mm).
From 2000 the deltas are :
2000 : -2.1
2002 : 2.3
2006 : 6
"Funny" how it's the most recent data that needs the most "adjustments" …
A C Osborn,
The most recent trend (1983-2013) is about 35% higher than the highest in the 1920s/30s. If you consider that not much difference, there's not much point in whatever scale you're using. And again, that difference has only increased beyond 2013.
I'm using the actual official published data as per the link. By suggesting an important discrepancy you can only be implying there's something wrong with Willis' chart, since his is a digitization. There are some visual differences obviously – Willis' chart is more squashed, horizontally and vertically (to emphasise how strange your notion is of it showing "not much difference", recent trends are waaaay above the top of Willis' y-axis). My x-axis dates refer to start of trend rather than end of trend. And the official published data goes back to 1880 rather than 1860. But, as Nick Stokes says above, Willis' digitization looks fine.
There are some small differences in the more recent trends due to increased availability of tide gauge data for the 2015 update, so the final data point in Willis' graph (for 1979-2009) has a trend of 2.3mm/yr in the official update.
stevefitzpatrick says:
AC,
That is rate of rise plot, not a plot of sea level. The current 30 year trailing rate is the highest in the record.
Mark BLR,
"Funny" how it's the most recent data that needs the most "adjustments"
Nothing to do with adjustments. It's due to an increase in tide gauge data availability (and therefore, presumably, accuracy). Look at the uncertainty estimate in the original data for 2009, compare with 2009 uncertainty in the update. Substantially reduced. This is because of new data coming in providing better coverage.
All this boils down to, more than what the actual measurements are, but do we trust that the people in don't have confirmation bias affecting them?
"look, these station data show a lot of rise, are they included?"
"look, these station data show a drop lately, I guess that must be because of the earthquake/Moon/Evil spirits made the data bad. Let's exclude it to make sure it doesn't contaminate good data"
Basically, that should not happen. In practice, I have no means to make sure it does not happen.
Given that satellite data have been said to be much higher than tide gauge, motivation to find low-biasing tide gauges has been high.
paulski0 writes
Clear acceleration beyond previous rates.
It also shows about 40 years of decreasing acceleration, and considerable acceleration at a time when CO2 isn't claimed to be the major driver of climate.
Both those features need to be adequately explained before you can make claims about recent rates of acceleration.
indefatigablefrog says:
Now let's see the error bars!!! People are hiding the truth. Literally by hiding things from view.
Do you think that Church and White do not have the confidence intervals to go with that graph?
Of course they will have computed them. So why are they not on the graph?
Could it be that they don't want us to see the massive flaw that lies at the heart of basing global policy on short term estimations of sea level rise rate. The estimate for short term trends is completely swamped by the huge statistical uncertainty.
see here: http://www.realclimate.org/index.php/archives/2011/07/is-sea-level-rise-accelerating/
D. J. Hawkins says:
How does that work, as your x-axis stops in 1980?
Paulski and Nick Stokes, thanks for the links. It appears that they have "adjusted" only the recent data. As Mark BLR pointed out,
I can understand adjustments if say unknown data from earlier days were discovered … but why is the change only in the recent data? Not saying it is wrong, just saying it is suspicious.
Next, you say:
I fear you are mistaking increasing sea levels with an increase in acceleration. Acceleration is the SLOPE of the data in your (or my) graph … and the recent slope is little different from the 1930-1960 slope.
Finally, I say again:
Clyde Spencer says:
paulski0,
What was being discussed is whether or not there has been recent "acceleration" in sea level rise. What you have presented is essentially the upward 'velocity' over time. The slope of the curve represents the acceleration over time. The slope of the velocity during the 1920s is comparable to the recent slope. That is, the accelerations are comparable. Osborn is correct in pointing out that the accelerations are comparable. Currently, the 'velocity' of upward movement is high, but the rate of change of velocity is not unprecedented. But, inasmuch as we have seen decelerations in the last century, I don't think that we should get too excited about the recent trends.
" It appears that they have "adjusted" only the recent data."
There's been no recent temp increase…..if anything the recent data should show sea level rise slowing…or even stopped if it's an artifact of temp like they claim….it had to be adjusted to show it rising
…and no, I do not trust a single one of them….Trump is a threat to the entire swamp
daveburton says:
Those graphs of avg SLR rate vs. year are basically graphs of the first derivative of sea-level, after high-frequency variations are filtered out. That means acceleration of SLR is appears as a "rising" trend in the graph, and deceleration of SLR appears as a falling trend in the graph.
That makes the variations look more significant than they really are. The difference between 1 mm/yr SLR rate and 2 mm/yr SLR rate is just four inches of sea-level per century, which is too little for a person to notice in his own lifetime, without very careful measurements.
Additionally, those graphs are almost invariably created using measurements from mixes of locations which vary over time. That means that the variations in rate of SLR might have more to do with variations in where it is measured than with real changes in trend.
A plain graph of sea-level vs. date is a lot simpler:
Quick ref:
http://sealevel.info/acceleration_primer_big_green_text.png
http://sealevel.info/acceleration_primer.html
https://mobile.twitter.com/ncdave4life/status/999630918309556225
https://twitter.com/ncdave4life/status/999630918309556225
Then you aren't talking about acceleration in the same sense as is typically the case in mainstream science. What you're talking about is really variability in rate, which to a large extent is a consequence of natural forced and unforced climate variability. What's meant by SLR acceleration is a sustained increase in rate, due to sustained imbalances in TOA flux, glaciers and ice sheets.
There is no simple relationship between CO2 levels and the rate of sea level rise
In terms of the variability in SLR rate over the 20th Century, that's a straw man. No-one has said there should be. See Dangendorf et al. 2017 for a representation of what modeling including the expected effects of CO2 (and other known factors) produces (blue curves).
The simple relationship is the fact that the rate at present is much high than (has accelerated from) the rate of the 19th Century and the 20th Century, and will keep increasing.
Oops, missed the right thread. Also, 'sustained imbalances in TOA flux, glaciers and ice sheets' should really say 'increasing imbalances in TOA flux, glaciers and ice sheets'.
due to sustained imbalances in TOA flux, glaciers and ice sheets.
Why would you expect a sustained imbalance to cause acceleration? From an energy perspective a sustained imbalance causes the energy to be retained in the ocean for melting and thermal expansion at the same rate over time. To accelerate, you need an increasing imbalance.
paulski0 went on to correct himself
'increasing imbalances in TOA flux, glaciers and ice sheets'.
And this is far from a given. It could go either way. There is no reason to suspect SLR will accelerate.
Ah…here we are.
She is even dressed in green..
https://youtu.be/Gb2X2-Qrn-U
Sorry that link was borked.
https://youtu.be/yO6s_Abez8M
Wim Röst says:
Thank you, Willis, for going back to the original data and for showing us to what happened with that data. An important and impressing analysis.
But I have got one question. I was confused about the third graphic, the thick blue line that shows the Trailing Trend. I suppose 'Trailing Trend' is the same as 'Moving Average'. I ever learned to show the moving average for a period in the year in the middle (!) of the period, so the average for 1960 to 1990 is shown in the graphic for the year 1975. Instead I see you using 'Year of End of Trailing Trend' which shows the trend for the last year of the data. But now, as I see it, all averages for every 31 year period are shown 15.5 year to the right in the graphic. Which is confusing for me.
If 'Trailing Trend' is the same as 'Moving Average' I would be pleased to get a graphic with the moving averages put 15.5 years to the left: in the middle of the 31 year period. That enables me to read the average for a period in the middle of that period. Which is conform reality.
Thanks, Wim. The problem with doing it your way is that it is non-causal, that is to say, it includes events that happen AFTER the date we assign them to …
For example, we can determine the trend for the past 31 years ending today,1988-2018.
But we can't determine the trend from fifteen years prior to fifteen years from now, 2003-2033.
That's why I posted it as a trailing trend.
Finally, posting it as a trailing trend makes it perfectly clear why Hansen used the period 1900-1930 …
PS—There are both a centered moving average and a trailing moving average …
Hi Willis, thanks for the quick reply.
In my old fashioned geography education I was obliged to show the moving average for the middle year of the period. I never learned about trailing moving averages.
Using a paint program I changed the years at the bottom of the graphic (putting them around 15 years to the right which places the moving average in the middle of the period) with an interesting result. (I don't know how to post the graphic in a way it shows up in the comments, it is a .png format)
It shows that the sea level trends if averaged over 31 year periods, rise faster from around 1926 to around 1950 and restart their rise in the beginning of the nineties. Conform what we should expect when we look at 'raw' temperature data.
Of course the most interesting of all numbers stays the same: an average sea level rise over the whole period
of one and a half century which is around 1.5 mm per year. Nothing to be worried about.
Writing Observer says:
Wim, I learned the same thing for "the" moving average in school myself. Then I got out into the REAL world – financial services, in my case.
You want to know what the trend is NOW – not at some time in the past (15.5 years in this case).
Now, it is just possible (although I think it unlikely) that there is an ~15 year lag between an increase in temperature and an increase in the acceleration of sea level rise. Of course, the main problem is that there is no real correlation between CO2 and temperature – except with a very long lag period.
oppti says:
Sea level has often a periodic signal at many places. It is always educating to look at this .
https://tidesandcurrents.noaa.gov/sltrends/sltrends_station.shtml?id=9414290
afonzarelli says:
Willis, i think your grandmother said it all. As (the tv) fonzie once wrote on the bathroom wall, bull makes the world go round…
As a grandpa it's Grandchildren that make my world go round.
tty says:
In Fig 5 is the satellite figures corrected by removing the "Global Isostatic Adjustment" by 0.4 mm/yr, which is supposed to compensate for a hypothetical increase in the volume of the ocean basins?
This must be done to be able to compare with non-satellite data.
Thanks, Willis.
I've been presenting similar sea-level trend graphs in my past couple of books.
Example using 21-year trends:
https://bobtisdale.files.wordpress.com/2018/01/exhibit-02.png
Here is a plot of my favorite data point whenever "acceleration" is discussed.
http://www.psmsl.org/data/obtaining/rlr.monthly.plots/70_high.png
It is from Kungsholmsfort in southern Sweden. This is an old coastal fort built on solid Precambrian bedrock of the Baltic Shield, one of the most stable and tectonically quiet areas on Earth. The only complication is the isostatic rebound from the last glaciation 12,000 years ago which is very nearly linear. By a coincidence the tide gauge at Kungsholmsfort was situated (when built in 1886) almost exactly on the line where sea-level rise and isostatic rebound are similar at about 1.8 millimeter per year. As you can see from the diagram it is still on that line.
It is very difficult to declare panic when looking at that, but I don't think Swedish Meteorological and Hydrological Institute, for example, would not be doing exactly that.
Note that if you flatten the graph, then apply a curve fit, you will probably see a minimum in the middle and a slight change from decelerating sea level to accelerating. Not that it would wet you feet, nor mine, which are about 30,000 units higher.
I'm hoping I live long enough to see the outcome of this, but I'm not sure, anymore, that I'll outlive the panic.
Here's Kungholmsfort, Sweden, 1887 – 2015:
http://www.sealevel.info/MSL_graph.php?id=70&boxcar=1&boxwidth=3
Linear trend is 0.025 ±0.237 mm/yr.
Acceleration = 0.01218 ±0.01421 mm/yr².
There was a slight acceleration evident in the early 20th century. So here it is with regressions just using data since the 1920s:
http://www.sealevel.info/MSL_graph.php?id=70&boxcar=1&boxwidth=3&c_date=1930/1-2019/12
Linear trend = 0.443 ±0.458 mm/yr
Acceleration = 0.00948 ±0.04124 mm/yr²
http://sealevel.info/050-081_Kungholmsfort_Sweden_1930-2015_vs_CO2.png
One would think that values of houses in areas close to the sea would take a tumble — but no. I see that a property in Sandbanks, Dorset (in the UK) is asking 7 — 8 Million £ making it the highest priced area after London.
Houses are at the top of the beach.
Yep it's the same for Miami which is a poster child for SLR for alarmists and thus the press. No effect on the prices at all despite years of SLR hype.
fernandoleanme says:
If I were building a house near a south Florida beach I would build the first floor slab about four meters above ground level (the area under this slab would be used as a garage and boat storage space), make it two stories high, use cement blocks for the outer walls, put removable stainless steel bars outside the windows, use a well anchored Spanish tile roof, install a small 2KW generator in an elevated platform on the ground floor, and a 200W solar panel covered with bullet proof glass on the roof. That house ought to last at least 60-70 years.
Alan the Brit says:
Was it not one Albert Gore who after cashing in his ill-gotten gains from An Inconvenient Truth, spent some $4M on sea-front property, after telling everyone how fast sea-levels were going to rise?
That might just mean that those who have a lot of money and like the beach just aren't interested in following the CAGW "debate". In fact if I got lucky and came into enough money to buy a $10,000,000 beachside palace my give a $hit factor in CAGW would probably disappear
One would expect SLR to impact the River Thames where the houses of parliament are 'precariously' perched. It would therefore seem logical that, when the opportunity arose to 'jump ship' and abandon the the place before it's swamped, it would be taken.
Instead, they are spending billions refurbishing the place.
Yep and Boston completed the most expensive public works project up to that time digging tunnels for freeways called "The Big Dig".
The "Big Dig" is already under water. It was designed that way.
Expensive as hell, but it goes right *under* Boston harbor.
At ~11 inches/century that Boston has, SLR will not impact the Big Dig until about the year 4520.
Yes, it's below SL and if SL rises until it reaches the entrances to the tunnels it will be done for. And that was the point! If SL is supposed to rise as they have said it will, it would have been foolish to build it.
A problem with that Church & White data is that the mix of locations which they use when computing their "average" varies over time. Since different locations experience different sea-level trends, that can lead to apparent accelerations and decelerations which are actually just artifacts of the changing mix of stations.
I prefer to look at long, high-quality, sea-level measurement records for individual locations. Here are "thumbnails" of graphed, seasonally-adjusted, mean sea-level from over 150 GLOSS-LTT tide stations:
http://sealevel.info/154_thumbnails.png
To view larger versions, click here and then click on the individual thumbnail graphs:
http://www.sealevel.info/MSL_NOAA2010_thumbs.html
The most obvious thing about those graphs is that all the long, high-quality, continuous or near-continuous measurement records are remarkably linear.
Some locations saw a very slight acceleration in rate sometime between 1850 and 1930, which increased the rate of sea-level rise by at most +1½ mm/yr. (It is most evident at Brest, France.) But when CO2 level rose above 310 ppmv, sea-level rise acceleration ceased.
Obviously the sea-level trends vary from one location to another. In a few places sea-level is rising quickly (mostly because of land subsidence). In some places sea-level is falling (where the land rising faster than the ocean). But in most places sea-level is rising very, very slowly; the average is only about 1½ mm/year, or 6″ per century. (That number is so small that it is hardly significant, since it is often dwarfed by natural processes like sedimentation and erosion.)
The trends vary a lot, but there's one thing that every long, high-quality, sea-level measurement record has in common: the lack of any significant "acceleration" (increase in rate) within the last ≈90 years. The sea-level trends are about the same now, with CO2 above 400 ppmv, as nine decades ago, when CO2 was under 310 ppmv.
Since precise measurements began, CO2 level has risen every year for 59 consecutive years (315🠆407 ppmv), yet those CO2 increases caused no detectable increase in rate of sea-level rise. That is ironclad proof that CO2 emissions from fossil fuels, and rising CO2 levels, don't significantly affect sea-level.
Here are four graphs of sea-level rise juxtaposed with CO2 level. The top two graphs are of particularly high-quality measurement records from tectonically stable locations on opposite sides of the world, with typical trends (sea-level is rising about 1½ mm/year = 6 inches per century). The bottom two graphs are atypical: they show sea-level trends at the two locations where Nobel Prize Committees meet; sea-level is falling at both of those locations, due to "post-glacial rebound" (i.e., the land is rising). The sea-level trends are obviously very different, but they have one thing in common: just like at everywhere else, there's been no acceleration in sea-level rise in more than nine decades:
http://sealevel.info/Wismar_Honolulu_Oslo_Stockholm_vs_CO2_annot1.png
You can look up sea-level trends for other locations here:
http://www.sealevel.info/data.psp
Farmer Ch E retired says:
Thanks for this very complete look at global tide data. I reviewed the NOAA data early last year and it was obvious that sea level change is a local issue and is disconnected from atmospheric CO2 concentrations. Sea level change ranged from -5.79 ft/100 yr (Alaska) to +5.31 ft/100 yr (Louisiana). Nowhere did I observe a change in slope of sea level rise as is shown in 1930 and 1992 in Figure 29 above. In Steward, Alaska, sea level is gradually decreasing (post-glacial rebound) except for one day in 1964 when it rose 3 ft as a result of the Good Friday earthquake.
Agreed, Farmer Ch E retired.
I agree that there are serious problems with these composite sea-level trend graphs from the alarmist community.
For instance, for many years climate realists have been pointing out that the claimed 1.7 to 1.8 mm/year "average" rate of global mean sea-level rise from teams like Church & White must be too high, because most measurement sites show lower rates. For example:
http://www.john-daly.com/ges/msl-rept.htm
More recently, I calculated an average sea-level trend of under 1½ mm/year from the best tide gauge records. I think nearly everyone who studies sea-level realized that those 1.7 to 1.8 mm/yr numbers must be too high, but, as in the story of the Emperor's new clothes, the "team players" wouldn't say it aloud.
It is not widely noted that the team players had been fudging their numbers for years, to report higher trends. There are a lot of games that can be played when computing "global mean sea-level rise" from large numbers of tide gauges. Church & White like to use very large numbers of short-record gauges, compute empirical orthogonal functions to fit them, and adjust them in a variety of ways. It enables them to find "global sea level rise" trends which are quite different from simple averages of the rates at tectonically stable locations.
Here's a quote from the most famous sea-level paper of all, Church & White (2006), A 20th Century Acceleration in Global Sea-Level Rise:
"An additional spatially uniform field is included in the reconstruction to represent changes in GMSL. Omitting this field results in a much smaller rate of GMSL rise…"
The added "additional spatially uniform field" was obviously a fudge factor, to increase the reported rate of sea-level rise. But I wondered: why did they say "spatially?"
Surely, I thought, since they were reporting measured acceleration trends, the "additional field" must at least have been temporally uniform. So why did they use the word "spatially?" What other sort of non-uniformity could there be, besides spatial and temporal?
I emailed Drs. Church & White and asked them why they used the adjective "spatially." Was the "additional field" temporally uniform, I asked?
I've never learned what that "field" was, but to my amazement Dr. Church replied that it was not temporally uniform.
In 2009 they posted on their web site a new set of averaged sea-level data, from a different set of tide gauges. But they published no paper about it, and I wondered why not. So I duplicated their 2006 paper's analysis, using their new data, and not only did it, too, show slight deceleration after 1925, all the 20th century acceleration had gone away, too. Even for the full 20th century their data showed a slight (statistically insignificant) deceleration.
My guess is that the reason they wrote no paper about it was that the title would have had to have been something like this:
Church and White (2009), Never mind: no 20th century acceleration in global sea-level rise, after all.
Finally, in 2015, an "insider," Carling Hay, published a paper saying, right out loud, that the emperor had no clothes, and the claimed GMSL rates were too high. Suddenly the "accepted" rate of 20th century sea-level rise became 1.4 mm/year instead of 1.7 or 1.8 mm/year.
But that left the other team players with a dilemma: what to do with all that data they'd been using? Hansen's solution was simply to scale the old Church & White data by 0.78, to match the new received wisdom.
BTW, Tony Heller memorably called the splicing together of tide gauge data with satellite altimetry to create the appearance of acceleration the "IPCC sea level Nature trick," when the IPCC did it in AR4.
I think that was, perhaps, a little bit unfair. (Willis & I corresponded about this, and he thinks I'm letting them off too easy.) But at least Hansen and the IPCC did not hide what they did in their spliced-together sea-level graphs. Hansen even used contrasting colors in his graph, to make it clear what he did. They just don't seem to realize it's wrong.
In contrast, Jones, Mann, Bradley, Hughes, Briffa & Osborn, in their infamous WMO Report cover "hockey stick" temperature graph, used the same colors for the proxy data and the spliced-on real temperatures, and even rounded the splice points, to hide the splicing. Phil Jones and his pals clearly knew what they did was wrong, because they tried to hide it.
So, what Hansen (and the IPCC) did with their spliced sea-level graphs was merely scientific malpractice. What Jones, Mann et al did with paleo-temperature data was similar, but added intentional deception to the sin, making it much worse.
I agree, also, that the area around Seward, AK is the only place I know which really did see disastrously high sea-level rise… once.
https://www.sealevel.info/MSL_graph.php?id=seward&boxcar=1&boxwidth=3&c_date=1964/2-2019/12
http://sealevel.info/9455090_Seward_problem_solved_67pct.png
daveburton – thanks for the backstory – very interesting
HDHoese says:
I'll take the 3.3mm and raise it to 5.58. Based on the rest of the coast this looks like an anomaly. https://tidesandcurrents.noaa.gov/sltrends/sltrends_station.shtml?id=8774770
Which is based on this station https://tidesandcurrents.noaa.gov/stationhome.html?id=8774770
With this — "The bench marks are near the Yacht Basin. The tide gauge is on the southern-most pier in the Yacht Basin." There has been construction work on this basin in recent years and construction companies look at such data to justify projects.
Winds at this location are important for sea level, also runoff, which may explain flat line during 50s drought. Which leads to the question of quality control with bench marks. On the Louisiana coast, almost floating in places, buildings have to been very careful with substrate. Piling depths are dependent on weight, especially important when you are talking about mm.
Lars P. says:
Thank you for posting real data!
So many continuous tide gauge measurements show no acceleration.
How can the sea level rise accelerate and not affect these sites?
For this to happen it must be either all these sites started to raise in a (synchronised) accelerated way, or the sea is making a huge blob somewhere 🙂
Paul Homewood says:
Thanks Willis.
The IPCC said exactly the same in AR5:
It is very likely that the mean rate of global averaged sea level rise was 1.7 [1.5 to 1.9] mm/yr between 1901 and 2010 and 3.2 [2.8 to 3.6] mm/yr between 1993 and 2010. Tide gauge and satellite altimeter data are consistent regarding the higher rate during the latter period. It is likely that similarly high rates occurred between 1920 and 1950.,/b>
http://ar5-syr.ipcc.ch/topic_observedchanges.php#node11
We should also remember that the satellite figures are inflated by a GIA adjustment of 0.3mm/yr. While this may be valid as far as "ocean volume" is concerned, it is clearly not relevant for "sea levels" (The argument being that ocean floors have been sinking since the Ice Age)
As such, it should not be used when comapring satellite data with tidal gauge data (which measures sea levels at the coast)
Both satellite and tide gauge data are adjusted upward by ~0.3 mm per year in most studies, including Church and White.
son of mulder says:
Is the 60 year cycle peak to peak and trough to trough tied to anything to do with Ocean oscillations, in the Church and White 2011 chart.
I do not think so. I looked at Boston tide gauge data.
What I saw was a 74.4 year wave form tied directly to the orbit of the moon.
Specifically, the moon's apsidal precession, periodicity (8.0 years), and nodal precession, periodicity (18.6 years).
You would expect that these variations will be different in different locations.
For example, some areas get two high tides and two low tides per day. Some areas get only one. Some get one large high tide and one small high tide per day.
So it is with the various lunar precessions on longer time frames.
That doesn't conflict with the idea that the oceanic oscillations might be an average of all the stuff that falls into the categories you raise.
I think the AMO and PDO influence is small at most locations. (It certainly isn't as obvious as the shorter-cycle ENSO influence.)
However, quite a few authors claim to have detected evidence of such an influence:
Schlesinger, M. & Ramankutty, N. (1994), An oscillation in the global climate system of period 65-70 years. Nature, Vol. 367, pp. 723-726 (24 February 1994), doi:10.1038/367723a0
Douglas, B. (1995). Global sea level change: Determination and interpretation. Reviews of Geophysics 33(S1): doi:10.1029/95RG00355. issn: 8755-1209.
Douglas B (1997). Global Sea Rise: a Redetermination, Surveys in Geophysics, Vol. 18, No. 2-3 (1997), 279-292, doi:10.1023/A:1006544227856.
Excerpt: "It is well established that sea level trends obtained from tide gauge records shorter than about 50-60 years are corrupted by interdecadal sea level variation…"
Klyashtorin, L., and Lyubushin, A. (2007), Cyclic Climate Changes and Fish Productivity, VNIRO Publishing, 2007. 224 p. ISBN 978-5-85382-339-6.
Excerpt (p.8): "At the background of the secular linear trend, Global dT undergoes longperiod, up to 60-year long, fluctuations… Global dT detrending allows detection of 2.5 cycles of approximately 60-year Global dT fluctuations."
Jevrejeva, S., J. C. Moore, A. Grinsted, and P. L. Woodworth (2008), Recent global sea level acceleration started over 200 years ago? Geophys. Res. Lett., 35, L08715, doi:10.1029/2008GL033611.
Excerpt (p. 3): "The multi-decadal variability in global sea level for the past 300 years shows the same pattern as previously found in the climate system [Delworth and Mann, 2000], including a 60 – 70 years variability in sea surface temperature (SST) and sea level pressure (SLP).Similar 60-year cycles exist in early instrumental European records of air temperature (1761 – 1980) and longer paleoproxies from different locations around the world [Shabalova and Weber, 1998, 1999], suggesting a global pattern of 60-year variability. A global pattern of 60-year variability is supported by comparison of the GSL and North East Atlantic variability…"
Frolov, I., et al (2010), Climate Change in Eurasian Arctic Shelf Seas: Centennial Ice Cover Observations. Springer Science & Business Media, 2010. ISBN: 354085875X, 9783540858751. (The abstract notes an evident 60 year cycle, and Section 2.4 discusses it.)
In most individual tide gauge records there's no obvious AMO influence, but Murmansk looks like a possible exception:
http://sealevel.info/AMO_vs_murmansk_7.png
John in NZ says:
I went to the sea recently and it was raining.
No wonder the sea is rising.
Shanghai Dan says:
My bad. But in my defense, I did just finish a 32 ounce ice tea, so…
What sea-level rate graphs show is the 60-year oceanic-related climate periodicity and a small constant acceleration due to post-LIA warming that is unrelated to CO₂ forcing.
https://i.imgur.com/kI3WANh.png
Your last graph shows the same.
This is the same 60-year oscillation you defended doesn't exist.
https://wattsupwiththat.com/2014/04/25/the-elusive-60-year-sea-level-cycle/
I guess you just found it in the data. Congratulations. Better late than never.
Given the third graphic by Willis and graphic b of Javier:
The cyclical tendency in sea level rise seems to reflect a cyclical tendency of heat uptake by the oceans.
The atmosphere reacts on higher sea surface temperatures by showing a rise in temperature. As demonstrated by Ole Humlum, atmospheric temperatures follow sea surface temperatures with a delay of a couple of months.
The Cloud Hypothesis of Roy Spencer comes into mind: less clouds, more heat uptake by the oceans and warmer oceans are warming the atmosphere. All quite logic. A cyclical sea level rise could be seen as a proof for this hypothesis.
If warming happens this way, the next question is: what causes clouds to behave in a cyclic manner?
Charles May says:
Just like the AMO a 62 year cycle is present. It seems to be in a lot of places. I used Willis' data at the bottom of his post and analyzed it. I was quite surprised at what I found. I have looked at other sea rise data sets but I have never found one that had something close to a 1000 year cycle of something close to the 208 year cycle. I do have both in my analysis of the H4 data. The fit is simply outstanding.
https://1drv.ms/u/s!AkPliAI0REKhgZhsLcxMQ1u1fHxjTQ
The projection is shocking.
https://1drv.ms/u/s!AkPliAI0REKhgZhufrmr26bsbpcHGA
Here is how some of the waves fit into the projection.
https://1drv.ms/u/s!AkPliAI0REKhgZhtPJs_LwrMSPlRlQ
BTW, I did not check yet to see where the 874 year cycle peaks but I would not be surprised if it came close to 2135 you once identified.
The interesting part does not end there. I now have four datasets showing a nice drop in values before 2020.
Is that coincidence?
Here are the AMO data and its projection.
https://1drv.ms/u/s!AkPliAI0REKhgZhv7XILqEiNx6fW3Q
https://1drv.ms/u/s!AkPliAI0REKhgZhwuS0iX3jUSr9l6Q
That step change shocked me.
Here is the PDO. Instead of a 62 year cycle it shows an 82 year cycle. There is no step change like the AMO but we do have a nice drop in temperature before 2020 coming.
https://1drv.ms/u/s!AkPliAI0REKhgZhpTDctmQeAOHNCPQ
https://1drv.ms/u/s!AkPliAI0REKhgZhr_P-InTa4jNSCfA
I am going to be completely honest here. I am not sure what to make out of all this yet. I do have four datasets that are indicating a measurable drop in temperature before 2020. I am open to your suggestions.
Very interesting set of graphs. And even more interesting to see what it happens over the next three years. However, I do not think that sea level is going to start dropping significantly. It has been going up for quite a long time and didn't decrease during the long Gleissberg solar minimum of ~ 1900. Even if the rate goes to zero or slightly negative during a couple of decades that should not cause a significant drop in sea levels. One possibility is that the cycles are modulated by longer cycles. The 210-year de Vries solar cycle is strongly modulated by the 2500-year Bray solar cycle, so it only has a significant effect at ± 500 years of a Bray low.
Also I've read a couple of reports about a possible strong La Niña in the 2019-2020 time frame. That could produce some cooling and affect oceanic indexes. Other than that I find fascinating that AMO, solar, sea level rate, LOD and other indexes are synchronizing on a cooling phase at about the same time. It is unlikely to be by chance. Without the very strong El Niño of 2015-16 we could have been seeing signs of a moderate cooling already.
Thanks for taking the time to respond. I too am dubious of the change in predicted sea level rise but as you pointed out. But the fact that I have a number of datasets indicating a shift that seems nearly synchronized may the most significant discovery here.
On your mention of the Bray cycle, I have found it in my analysis of some of the longer proxy records but introducing it here with such a short time record is questionable. All of the cycles that I used in the sea level rise were found from the OFT. I made no manual additions.
BTW, I do analyze all four Nino regions and in my latest analysis of Nino region 3.4 it indicates a modest El Nino early in 2019. I am a little bothered by the fact that the last update of the daily record for all four regions was dated 04/06/2018. Normally, I look for these around the end or beginning of the month and here it past mid-May. I hope they are not being delayed because they are playing games with the data again. I just checked a few moments ago and there is sill no update.
Adrian Burgess says:
Thanks Willis for doing some proper 'peer review' that was obviously lacking before your keen eye!
I'm an interested CAGW (now intangibly described as 'Climate Change' [nobody likes change!]) observer, often denigrated as a 'denier' for denying 'irrefutable facts' I never seem to be presented with… I think in the climate world, 'facts' are synonymous with 'made up numbers based on opinions'.
Serge Wright says:
The graph provided by James Hansen is a pure case of climate fraud. How anyone can get away with publishig a graph showing an inflexion point of increased SLR from around 1990 that is not supported by a single tide global gauge, is indeed a great mystery.
Berényi Péter says:
http://www.psmsl.org/data/obtaining/rlr.annual.plots/12_high.png
It is a NEW YORK tide gauge (THE BATTERY, southern tip of Manhattan). There is no acceleration whatsoever since 1856. That's either because land is moving in sync with water or Hansen is silly. The former option is extremely unlikely, so…
co2islife says:
I just finished an article on this topic.
Multiple Sources Now Confirm; Climate Data "Adjustments" are Obvious Fravd to Anyone Choosing to Look
https://co2islife.wordpress.com/
J. Richard Wakefield says:
AGW Cultists like to defy the laws of physics. Any time this comes up and I show Tide & Current's graphs showing no acceleration, they claim those are individual locations, but GLOBALLY there is acceleration. So let me see if I have this correctly. 1000 individual cars are moving at a constant speed on the highway, but collectively they are all accelerating in speed. Got it.
James Snook says:
Hansen made a Faustian pact with the politicians controlling the purse strings at the moment he connived in arranging the over heating of the meeting room in Washington in which he made his first pitch to them on CAGW.
Since then, the politest thing that can be said about his activist 'science' is that It is prone to confirmation bias.
Kip Hansen says:
Nonsensical to use data that claims to be "global sea level" to the nearest 100th of a millimeter (as in "1860 -189.26") — even last years sea level can not be calculated to anything near that degree of accuracy or precision. Thus the whole exercise is just "sea level taxonomy" — fooling around with numbers and calculations — none of which adds to our knowledge or understanding of the real world.
Not Willis' fault — he is just playing their game — and a game is all it is.
In relation to "the game". 1) Just how do tidal gauges measure "sea level"? It seems that if the land is rising or falling (either above or below the water level), then all they measure is some relationship of water level to land, not necessarily "real sea level" above the center of the earth. 2) If they are only measuring a relative level at any location, how do you combine them and say "this is the global sea level"? Measuring different things with different instruments is just not amenable to averaging and obtaining a true answer. 3) We have hundreds of years of paintings and photos that should have unique physical characteristics (buildings, rocks, sea port heights, etc.) at various locations throughout the world to see how relative levels have changed over time. Am I the only one who wonders why a forensic database of the changes has not been established in order to check these so-called "scientific" projections against?
Paintings and photos are practically useless since they will only tell you what the state of the tide was at the time. Even in areas where there is no tide sea level varies 3-4 feet depending on wind and air pressure. Only by taking continuous measurements and averaging them over months or years is it possible to measure the true (relative) sea-level. And even then it takes decades before a trend is visible.
I don't think that is totally true. I am not talking about absolutely accurate measurements. But a painting from hundreds of years ago or a more recent photo can give relative differences. If a rock that has been visible for hundreds of years is now submerged one could make a conjecture about the reason. Likewise a building right at the oceans edge is now yards away, should generate some questions.
Tom Halla says:
James Hansen does seem to be trying to be the Harold Camping of climate predictions.
McPherson says:
When you have Ellison, a converted Muslim, being elected Senator in 2006, just prior to Obama being elected, , one gets the idea that the stage was being set for further action.
When Sovereignty is lost, there is no where for Law Abiding citizens to turn to for redress, as they no longer have elected leaders, only Appointed ones..Re: California.. Here, We are being governed by Environmental and Social NGOs, who have fought up all the Elected Officials, along with the Unions. An UnHoly alliance if there ever was one.
How are you? It would be fun to get together for a lunch or a talk. Love to all, Clark >
I have a graph showing the sea level rise in Sweden.
Between 1995 and 2016 the slope is 3 mm/year (20 Years)
There are 3 more periods showing this rate of rise:
1896-1926 (30 Years)
The average slope is 1,85 mm/year.
Yet the SMHI(gov) are confident to say -We see an acceleration to 3 mm/year due to global warming.
https://www.smhi.se/polopoly_fs/1.133603.1523542160!/image/Havsniv%C3%A5h%C3%B6jning.png_gen/derivatives/Original_1256px/image/Havsniv%C3%A5h%C3%B6jning.png
That is funny especially since the land is rising in Sweden since it got free of ice at the end of the ice age…
One can pick their favourite tide gauge from here:
http://www.psmsl.org/data/obtaining/
The graph above from SMHI is adjusted for land rise.
Larry in Texas says:
Lars, same is true for the Great Lakes region. Rising land due to glacier pressure now rebounding, lake levels lower. See my questions below.
Actually one pmsl data is enough to debunk the acceleration hypothesis.
How would this tide gauge show this continuous trend and be exempt from acceleration? Did the land started to accelerate the rise to counterbalance the accelerated sea level rise?
How on earth can this work if overall sea level rise is accelerating?
Menicholas says:
Just so…it takes very little to debunk erroneous assertions.
SMHI is extremely politicized, if anything it is worse than NOAA.
Richard C Savage says:
Beautifully accurate analysis, Willis. Today's Wall St Journal has letters in response to Fred Singer's previous article on CO2 and sea level rise – including one from Michael Mann. I've quoted your article in rebuttal, accurately I hope. Hope your analysis will sway some taken in by Hansen's ilk.
Just an out-of-curiosity question: how much is it possible to determine whether some of the sea level rise data is due to coastal subsidence as opposed to just a rise in sea level? I'm thinking about this because in parallel, there was concern about lake level lowering in the Great Lakes region some years ago, and it was apparently determined in that case that land adjacent to the Great Lakes was still rebounding from about 10,000 years of glacier pressure on top of that land. This meant the lake levels were not lowering, they were staying the same while the land was actually rising slowly. Is is possible the converse is true with ocean levels? Another example of course is the Tohoku earthquake next to Japan. Coastal land levels sunk about 3m after that quake. How much is the data affected by events like earthquakes in general?
Willis, your thoughts are welcome.
Nowadays it is possible by locating a sensitive GPS receiver nearby. This will measure the height of the ground in geocentric coordinates. However it takes a number of years to get good data.
Normally lake levels are not affected by glacial rebound, since both the land around and under the lake and the lake water rebound together. However if the lakes are very large the rebound will be greater at the northern end (which deglaciated later) and you get a phenomenon kalled "sjööverstjälpning" in Swedish,which translates as "lake overtipping", i e the water in the lake will tip over towards the southern end where the relative water level will rise while it sinks in the north. I would guess that this effect should be quite noticeable e. g. in Lake Michigan.
Cold in Wisconsin says:
Thanks to everyone for a very interesting discussion and to Willis for starting it off. What I did not see was the way in which Willis was able to detect or determine where the data from Hansen was derived from different data sets, adjusted, and then patched together? I love the forensic work, but was it listed in their Materials and Methods, or did you just figure it out by deduction?
A high school science student would get an F for combining data in this way. My brother in law is a high school science teacher, and believes strongly in AGW, and yet as I remind him, he would never allow one of his students to change their data to fit their hypothesis. I just cannot understand how rational educated adults accept this stuff as scientific or authoritative. He also cannot countenance the idea a scientist would act in any way unethically to advance their career. Yet he believes that every business person is a greedy schmuck.
I look forward to future neuroscience experiments that determine how and why our world views are so rigidly fixed at some early period such that facts become irrelevant. And what kind of psychic shock is needed to alter those thought patterns, as sometimes (but rarely) occurs. I wonder if studies on psychedelic drugs would convert any of these AGW believers in the same way as they can treat opioid addicts?
Cold in Wisconsin May 23, 2018 at 9:31 am
Thanks to everyone for a very interesting discussion and to Willis for starting it off. What I did not see was the way in which Willis was able to detect or determine where the data from Hansen was derived from different data sets, adjusted, and then patched together?
Thanks, Cold. It's all listed in the caption under his Figure 29, shown in the head post.
Steve Zell says:
In Eschenbach's Figure 5, there does seem to be a "knee" (inflection point?) in the curve at around 1930, but little acceleration (or deceleration) since then. If we use -105.16 mm in 1930 and +52.43 mm in 2009, then the average rate of sea level rise (per Eschenbach) would be (52.43 + 105.16) / (2009 – 1930) = 1.99 mm/year, or about 7.85 inches per century.
Raise your hand if you think it's possible to build an 8-inch high seawall around your coastal city in the next 100 years!
I'm also wondering–how did Church & White (or those whose data they quoted) measure sea level to the nearest hundredth of a millimeter as far back as 1860, when the tide normally rises and falls twice a day by over a full meter in some places? Shouldn't the data be rounded to the nearest full millimeter to account for imprecision of the measurements?
Willis said that he digitized the chart, so it's the precision of his program only.
Did you know that the maximum height of tide in the Bay of Fundy (Nova Scotia) is 16 meters?
sorry, 16,000.00 mm
Tidal gauges are just that. They measure the tides. When you subtract the tides, what is left is sea-level change. Most of this is due to weather. Over a sufficiently long time this will average out.
Also if you look at fig 5 you will see that the uncertainty is much larger in the early part of the diagram when there were fewer stations.
And don't underestimate nineteenth century instrument makers. Here is a swedish tidal gauge from 1886:
https://upload.wikimedia.org/wikipedia/commons/f/f9/Mareograph.jpg
Sea level is measured continuously by a float and mechanically transferred to a pencil which draws a curve on a paper roll driven by the clock while a second pencil, driven by the same clock, marks each full hour.
M Courtney says:
Hypothesis:
The change in trend since the LIA is all manmade.
1) Improved irrigation and hydro power plants (like the Boulder dam) became prevalent at the start of the 20th century. This slowed the loss of water from the land to the sea and allowed vegetation to tie up more moisture. The sea level rise slowed.
2) Nuclear tests after WW2 caused small areas of the sea to hold far more thermal energy than before. Ths caused some thermal expansion but also allowed local warm water pockets to survive long enough to circulate round the world. This increased Arctic ice loss. The sea levekl rise spikes.
3) Early 1960s, Khruschev and JFK ban nuclear tests. The change in trend disappears. The rest of the graph is just measurement noise.
Nuclear testing is utterly insignificant in this context
Yeah, probably.
But I've found a correlation.
That's more than enough for climatology.
bill hunter says:
Nice article Willis! One small clarification. Hansen attributes the multiplying the Church and White data by .78 to Hay, et al 2015. In this paper the Church and White data is adjusted in order to close the sea level rise budget from various inputs (heat expansion, ice melt, and land water storage changes, etc.). So there may be some rationale existing for the adjustments since these factors are presumably are being considered in more recent papers. However, it should bring attention to adjustments in general as years ago surface temperature record trends were adjusted down in part because of low levels of recorded sea level change not to speak of tossing some early Argo data on ocean warming trends due to sea level change. Having those adjustments then lead to adjusting sea level change downwards because of a lower warming trend speaks to creeping bias in the whole program.
Changing actual data to conform with theory is an extremely dubious practice. Nobody knows what the ice melt in Antarctica was in 1901 for example (or in 1950 for that matter). If you have sea-level measurements that don't fit your modelled sea-level budget, I submit that there is probably something wrong with your budget. And how likely is it that tide gauges all over the world were consistently wrong by a constant 22% over a 90-year period? And apparently this problem then disappeared just as satellite measurements started.
thingadonta says:
moving averages is a very average argument.
I am unmoved by that assertion.
Church paper itself states:
The linear trend from 1900 to 2009 is 1.7 ± 0.2 mm year−1 and from 1961 to 2009 is 1.9 ± 0.4 mm year−1. Note the error estimates.
Willis…my man!
Keeping it real and taking names!
Marlo Lewis says:
Willis, pardon my ignorance, but I don't understand your digitized C&W data. What does it mean to say that sea level in 1860 is 189.26 mm? Or that in 1960 it was 40.06 mm? Surely sea levels were higher in 1960 than 1860. So maybe the numbers refer to annual sea level rise–but obviously they don't, otherwise Al Gore's 20 foot wall of water would have washed over us long ago.
Marlo, the only stupid questions on my planet are the ones you don't ask. Those always come back to bite you in the differential housing … you've simply missed the minus (-) sign. In 1860 it was -189.26 mm. All of these numbers are anomalies around a specified zero point.
Stan Robertson says:
Is there some reason why one would not just run a standard F-test to see if the addition of a quadratic term in the expression for sea level would be significant?
The reason is the presence of autocorrelation and a high Hurst exponent. You can't use standard statistics on highly autocorrelated datasets.
prcgoard says:
Having been interested in this topic since John Daly's time, I am perplexed as to why the papers of the expert, Nils-Axel Mörner are completely ignored? The paper: "Glacial Isostasy: Regional—Not Global" has a detailed account of the Fennoscandian region's glacial rebound and the Netherland's sinking with the location of the 'hinge.' [http://dx.doi.org/10.4236/ijg.2015.66045]
Other of his papers also discuss the problems of tide gauge and satellite data. The above paper has references to many of his papers. If not located, ask him for a copy.
Church & White made a curious/odd selection of tide gauge sites leading to questionable conclusions, especially when they tack on the satellite data, which in turn supposedly relates to a 'spheroidal surface,' not necessarily the ocean surface. My guess is that the "additional spatially uniform field" mentioned above, is to make the satellite data 'match' the greater-than-a-global average rise of about 1.4 mm/yr, of the USA coastal data (around 3 mm/yr) due to slight sinking of the USA part of the continent as the Canadian part rises. Was this to be parochially correct?
Church & White conveniently ignored their "home" long-term tide gauge, Fort Denison in Sydney Harbour with a rate of rise <1 mm/yr.
As to short-term changes in rate of sea level rise, when ocean basins are subject to cyclical lunar tide changes, atmospheric pressure changes, prevailing wind flow and storm surges it is not surprising that such fluctuations are seen. Thermal expansion is a doubtful factor as changes in continental shelf gauges are no different to those on stable islands surrounded by deep ocean.
prcgoard May 24, 2018 at 10:49 pm
Having been interested in this topic since John Daly's time, I am perplexed as to why the papers of the expert, Nils-Axel Mörner are completely ignored?
I ignore him because he claims over and over that the sea level is not rising anywhere on the globe … which contradicts the tide gauges everywhere.
I also ignore him because he claims he is able to determine historical sea level rises in Fiji from the cuts made by waves in the tropical islands. Having lived in Fiji, I know that is simply not possible at the level of precision and accuracy that he is claiming.
I've met him, he's a charming guy, but he truly doesn't understand sea level.
There's a discussion of his other … well … eccentric claims here … read'm and weep. | CommonCrawl |
Icosidodecahedron
In geometry, an icosidodecahedron is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.
Icosidodecahedron
(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 32, E = 60, V = 30 (χ = 2)
Faces by sides20{3}+12{5}
Conway notationaD
Schläfli symbolsr{5,3}
t1{5,3}
Wythoff symbol2 | 3 5
Coxeter diagram
Symmetry groupIh, H3, [5,3], (*532), order 120
Rotation groupI, [5,3]+, (532), order 60
Dihedral angle142.62°
$\cos ^{-1}\left(-{\sqrt {{\frac {1}{15}}\left(5+2{\sqrt {5}}\right)}}\right)$
ReferencesU24, C28, W12
PropertiesSemiregular convex quasiregular
Colored faces
3.5.3.5
(Vertex figure)
Rhombic triacontahedron
(dual polyhedron)
Net
Geometry
An icosidodecahedron has icosahedral symmetry, and its first stellation is the compound of a dodecahedron and its dual icosahedron, with the vertices of the icosidodecahedron located at the midpoints of the edges of either.
Its dual polyhedron is the rhombic triacontahedron. An icosidodecahedron can be split along any of six planes to form a pair of pentagonal rotundae, which belong among the Johnson solids.
The icosidodecahedron can be considered a pentagonal gyrobirotunda, as a combination of two rotundae (compare pentagonal orthobirotunda, one of the Johnson solids). In this form its symmetry is D5d, [10,2+], (2*5), order 20.
The wire-frame figure of the icosidodecahedron consists of six flat regular decagons, meeting in pairs at each of the 30 vertices.
The icosidodecahedron has 6 central decagons. Projected into a sphere, they define 6 great circles. Buckminster Fuller used these 6 great circles, along with 15 and 10 others in two other polyhedra to define his 31 great circles of the spherical icosahedron.
Cartesian coordinates
Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the even permutations of:[1]
• (0, 0, ±φ)
• (±1/2, ±φ/2, ±φ2/2)
where φ is the golden ratio, 1 + √5/2.
The long radius (center to vertex) of the icosidodecahedron is in the golden ratio to its edge length; thus its radius is φ if its edge length is 1, and its edge length is 1/φ if its radius is 1. Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional icosidodecahedron, and the two-dimensional decagon. (The icosidodecahedron is the equatorial cross section of the 600-cell, and the decagon is the equatorial cross section of the icosidodecahedron.) These radially golden polytopes can be constructed, with their radii, from golden triangles which meet at the center, each contributing two radii and an edge.
Orthogonal projections
The icosidodecahedron has four special orthogonal projections, centered on a vertex, an edge, a triangular face, and a pentagonal face. The last two correspond to the A2 and H2 Coxeter planes.
Orthogonal projections
Centered by Vertex Edge Face
Triangle
Face
Pentagon
Solid
Wireframe
Projective
symmetry
[2] [2] [6] [10]
Dual
Surface area and volume
The surface area A and the volume V of the icosidodecahedron of edge length a are:
${\begin{aligned}A&=\left(5{\sqrt {3}}+3{\sqrt {5}}{\sqrt {3+4\varphi }}\right)a^{2}&&=\left(5{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}&&\approx 29.3059828a^{2}\\V&={\frac {14+17\varphi }{3}}a^{3}&&={\frac {45+17{\sqrt {5}}}{6}}a^{3}&&\approx 13.8355259a^{3}.\end{aligned}}$
Spherical tiling
The 60 edges form 6 decagons corresponding to great circles in the spherical tiling.
The icosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Pentagon-centered
Triangle-centered
Orthographic projection Stereographic projections
Orthographic projections
2-fold, 3-fold and 5-fold symmetry axes
Related polytopes
The icosidodecahedron is a rectified dodecahedron and also a rectified icosahedron, existing as the full-edge truncation between these regular solids.
The icosidodecahedron contains 12 pentagons of the dodecahedron and 20 triangles of the icosahedron:
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
{5,3} t{5,3} r{5,3} t{3,5} {3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3.3.3.3.3 V3.4.5.4 V4.6.10 V3.3.3.3.5
The icosidodecahedron exists in a sequence of symmetries of quasiregular polyhedra and tilings with vertex configurations (3.n)2, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With orbifold notation symmetry of *n32 all of these tilings are wythoff construction within a fundamental domain of symmetry, with generator points at the right angle corner of the domain.[2][3]
*n32 orbifold symmetries of quasiregular tilings: (3.n)2
Construction
Spherical Euclidean Hyperbolic
*332 *432 *532 *632 *732 *832... *∞32
Quasiregular
figures
Vertex (3.3)2 (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.8)2 (3.∞)2
*5n2 symmetry mutations of quasiregular tilings: (5.n)2
Symmetry
*5n2
[n,5]
Spherical Hyperbolic Paracompact Noncompact
*352
[3,5]
*452
[4,5]
*552
[5,5]
*652
[6,5]
*752
[7,5]
*852
[8,5]...
*∞52
[∞,5]
[ni,5]
Figures
Config. (5.3)2 (5.4)2 (5.5)2 (5.6)2 (5.7)2 (5.8)2 (5.∞)2 (5.ni)2
Rhombic
figures
Config. V(5.3)2 V(5.4)2 V(5.5)2 V(5.6)2 V(5.7)2 V(5.8)2 V(5.∞)2 V(5.∞)2
Dissection
The icosidodecahedron is related to the Johnson solid called a pentagonal orthobirotunda created by two pentagonal rotundae connected as mirror images. The icosidodecahedron can therefore be called a pentagonal gyrobirotunda with the gyration between top and bottom halves.
(Dissection)
Icosidodecahedron
(pentagonal gyrobirotunda)
Pentagonal orthobirotunda
Pentagonal rotunda
Related polyhedra
The truncated cube can be turned into an icosidodecahedron by dividing the octagons into two pentagons and two triangles. It has pyritohedral symmetry.
Eight uniform star polyhedra share the same vertex arrangement. Of these, two also share the same edge arrangement: the small icosihemidodecahedron (having the triangular faces in common), and the small dodecahemidodecahedron (having the pentagonal faces in common). The vertex arrangement is also shared with the compounds of five octahedra and of five tetrahemihexahedra.
Icosidodecahedron
Small icosihemidodecahedron
Small dodecahemidodecahedron
Great icosidodecahedron
Great dodecahemidodecahedron
Great icosihemidodecahedron
Dodecadodecahedron
Small dodecahemicosahedron
Great dodecahemicosahedron
Compound of five octahedra
Compound of five tetrahemihexahedra
Related polychora
In four-dimensional geometry the icosidodecahedron appears in the regular 600-cell as the equatorial slice that belongs to the vertex-first passage of the 600-cell through 3D space. In other words: the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed hypersphere from a pair of opposite vertices, are the vertices of an icosidodecahedron. The wire frame figure of the 600-cell consists of 72 flat regular decagons. Six of these are the equatorial decagons to a pair of opposite vertices. They are precisely the six decagons which form the wire frame figure of the icosidodecahedron.
If a 600-cell is stereographically projected to 3-space about any vertex and all points are normalised, the geodesics upon which edges fall comprise the icosidodecahedron's barycentric subdivision.
Icosidodecahedral graph
Icosidodecahedral graph
5-fold symmetry Schlegel diagram
Vertices30
Edges60
Automorphisms120
PropertiesQuartic graph, Hamiltonian, regular
Table of graphs and parameters
In the mathematical field of graph theory, a icosidodecahedral graph is the graph of vertices and edges of the icosidodecahedron, one of the Archimedean solids. It has 30 vertices and 60 edges, and is a quartic graph Archimedean graph.[4]
Icosidodecahedra in nature
The Hoberman sphere is an icosidodecahedron.
Icosidodecahedra can be found in all eukaryotic cells, including human cells, as Sec13/31 COPII coat-protein formations. [5]
Trivia
In Star Trek Universe, the Vulcan game of logic Kal-Toh has the goal to create a holographic icosidodecahedron.
See also
• Cuboctahedron
• Great truncated icosidodecahedron
• Icosahedron
• Rhombicosidodecahedron
• Truncated icosidodecahedron
Notes
1. Weisstein, Eric W. "Icosahedral group". MathWorld.
2. Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
3. Two Dimensional symmetry Mutations by Daniel Huson
4. Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
5. Russell, Christopher; Stagg, Scott (11 February 2010). "New Insights into the Structural Mechanisms of the COPII Coat". Traffic. 11 (3): 303–310. doi:10.1111/j.1600-0854.2009.01026.x. PMID 20070605.
References
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
• Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.
External links
• Eric W. Weisstein, Icosidodecahedron (Archimedean solid) at MathWorld.
• Klitzing, Richard. "3D convex uniform polyhedra o3x5o - id".
• Editable printable net of an icosidodecahedron with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
Archimedean solids
Tetrahedron
(Seed)
Tetrahedron
(Dual)
Cube
(Seed)
Octahedron
(Dual)
Dodecahedron
(Seed)
Icosahedron
(Dual)
Truncated tetrahedron
(Truncate)
Truncated tetrahedron
(Zip)
Truncated cube
(Truncate)
Truncated octahedron
(Zip)
Truncated dodecahedron
(Truncate)
Truncated icosahedron
(Zip)
Tetratetrahedron
(Ambo)
Cuboctahedron
(Ambo)
Icosidodecahedron
(Ambo)
Rhombitetratetrahedron
(Expand)
Truncated tetratetrahedron
(Bevel)
Rhombicuboctahedron
(Expand)
Truncated cuboctahedron
(Bevel)
Rhombicosidodecahedron
(Expand)
Truncated icosidodecahedron
(Bevel)
Snub tetrahedron
(Snub)
Snub cube
(Snub)
Snub dodecahedron
(Snub)
Catalan duals
Tetrahedron
(Dual)
Tetrahedron
(Seed)
Octahedron
(Dual)
Cube
(Seed)
Icosahedron
(Dual)
Dodecahedron
(Seed)
Triakis tetrahedron
(Needle)
Triakis tetrahedron
(Kis)
Triakis octahedron
(Needle)
Tetrakis hexahedron
(Kis)
Triakis icosahedron
(Needle)
Pentakis dodecahedron
(Kis)
Rhombic hexahedron
(Join)
Rhombic dodecahedron
(Join)
Rhombic triacontahedron
(Join)
Deltoidal dodecahedron
(Ortho)
Disdyakis hexahedron
(Meta)
Deltoidal icositetrahedron
(Ortho)
Disdyakis dodecahedron
(Meta)
Deltoidal hexecontahedron
(Ortho)
Disdyakis triacontahedron
(Meta)
Pentagonal dodecahedron
(Gyro)
Pentagonal icositetrahedron
(Gyro)
Pentagonal hexecontahedron
(Gyro)
Convex polyhedra
Platonic solids (regular)
• tetrahedron
• cube
• octahedron
• dodecahedron
• icosahedron
Archimedean solids
(semiregular or uniform)
• truncated tetrahedron
• cuboctahedron
• truncated cube
• truncated octahedron
• rhombicuboctahedron
• truncated cuboctahedron
• snub cube
• icosidodecahedron
• truncated dodecahedron
• truncated icosahedron
• rhombicosidodecahedron
• truncated icosidodecahedron
• snub dodecahedron
Catalan solids
(duals of Archimedean)
• triakis tetrahedron
• rhombic dodecahedron
• triakis octahedron
• tetrakis hexahedron
• deltoidal icositetrahedron
• disdyakis dodecahedron
• pentagonal icositetrahedron
• rhombic triacontahedron
• triakis icosahedron
• pentakis dodecahedron
• deltoidal hexecontahedron
• disdyakis triacontahedron
• pentagonal hexecontahedron
Dihedral regular
• dihedron
• hosohedron
Dihedral uniform
• prisms
• antiprisms
duals:
• bipyramids
• trapezohedra
Dihedral others
• pyramids
• truncated trapezohedra
• gyroelongated bipyramid
• cupola
• bicupola
• frustum
• bifrustum
• rotunda
• birotunda
• prismatoid
• scutoid
Degenerate polyhedra are in italics.
| Wikipedia |
One-half of one-seventh of $T$ equals one-third of one-fifth of 90. What is the value of $T$?
From the problem, we write the equation \[\frac{1}{2}\cdot\frac{1}{7}\cdot T=\frac{1}{3}\cdot\frac{1}{5}\cdot90.\]Simplifying, we have \begin{align*}
\frac{1}{14}\cdot T&=\frac{1}{15}\cdot90 \quad \implies \\
\frac{1}{14} \cdot T &=6 \quad \implies \\
T &= \boxed{84}.
\end{align*} | Math Dataset |
Monitoring fluid intake by commercially available smart water bottles
The weight of water
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Rachel Cohen1,2,
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Scientific Reports volume 12, Article number: 4402 (2022) Cite this article
Fluid intake is important to prevent dehydration and reduce recurrent kidney stones. There has been a trend in recent years to develop tools to monitor fluid intake using "smart" products such as smart bottles. Several commercial smart bottles are available, mainly targeting health-conscious adults. To the best of our knowledge, these bottles have not been validated in the literature. This study compares four commercially available smart bottles in terms of both performance and functionality. These bottles are the H2OPal, HidrateSpark Steel, HidrateSpark 3, and Thermos Smart Lid. One hundred intake events for each bottle were recorded and analyzed versus ground truth obtained from a high-resolution weight scale. The H2OPal had the lowest Mean Percent Error (MPE) and was able to balance out errors throughout multiple sips. The HidrateSpark 3 provided the most consistent and reliable results, with the lowest per sip error. The MPE values for HidrateSpark bottles were further improved using linear regression, as they had more consistent individual error values. The Thermos Smart Lid provides the lowest accuracy, as the sensors do not extend through the entire bottle, leading to many missed recordings.
Dehydration is a very serious issue as it can lead to adverse complications, including confusion, falls, hospitalization, and death. Fluid intake balance is important especially among the elderly and people with underlying conditions affecting fluid regulation. High fluid intake is recommended for patients at risk of recurring stone formation. Therefore, monitoring fluid intake is a useful way to determine if adequate fluid has been consumed1,2. There are many reports in the literature of attempts to create systems or devices that can aide in tracking and managing fluid intake3. Unfortunately, most of these studies have not resulted in commercially available products. The bottles that are available in the market mainly target recreational athletes or health-conscious adults hoping to increase their hydration3. In this paper, we aim to determine if common commercially available water bottles are a viable solution for researchers and patients alike. We compare four commercial water bottles in terms of performance and functionality. These bottles are the HidrateSpark 34, HidrateSpark Steel5, H2O Pal6 and the Thermos Smart Lid7, as shown in Fig. 1. These were selected as they were the only four popular bottles that were (1) available to purchase in Canada and (2) had sip volume data that could be accessed via a mobile app.
Images of analyzed commercial bottles: (a) HidrateSpark 34, (b) HidrateSpark Steel5, (c) H2OPal6, (d) Thermos Smart Lid7. The dashed red boxes show the location of the sensors.
Out of the aforementioned bottles, only a previous version of the HidrateSpark has been validated in research8. This study found that over a 24 h fluid intake period, the HidrateSpark bottle was accurate in measuring total intake within 3% error8. The HidrateSpark has also been used in clinical studies to monitor intake volume in kidney stone patients9. Since then, HidrateSpark has developed new bottles with different sensors. The H2OPal has been used in other research studies to track and promote fluid intake, but there is no specific study to validate its performance2,10. Plecher et al. compared several commercial bottles in terms of functionality for seniors and available information online, however they did not perform any validation on their accuracy11.
All four commercial bottles include a free proprietary application to display and store intake events that are transferred via Bluetooth. The HidrateSpark 3 and the Thermos Smart Lid have the sensors down the middle of the bottle, likely using capacitive sensors, whereas the HidrateSpark Steel and H2Opal have sensors in the base, using load or pressure sensors. The sensor locations are shown in Fig. 1 by the dashed red boxes. In the Thermos Smart Lid, the sensor does not reach the bottom of the container.
Each bottle was tested in two phases: (1) a controlled sip volume phase and (2) a free-living phase. In both phases, the results recorded by the bottle (obtained from the products' mobile apps used on Android 11) were compared to the ground truth obtained using a 5 kg weight scale (Starfrit Electronic Kitchen Scale 93756). All bottles were calibrated before data collection using the apps. In Phase 1, sip sizes ranging from 10 to 100 mL in increments of 10 mL were measured in a random order, five times each—for a total of 50 measurements per bottle. These events were not actual drinking events by a human but were poured out so the volume of each sip could be better controlled. In this phase, the bottles were recalibrated if the sip error was larger than 50 mL and were re-paired if the app lost Bluetooth connection with the bottle. In the free-living phase, a single user drank from the bottles freely during the day, taking varying sip sizes of their choice. This phase also consisted of 50 sips over time, but not all in succession. Therefore, each bottle had a dataset of 100 measurements in total.
To determine summative liquid intake and ensure proper daily hydration, it is more important to have an accurate volume intake detection throughout the whole day (24 h), rather than of each sip. However, to determine just-in-time intervention prompts, there is a need for each sip to have a low error, as done in the study by Conroy et al.2. If a sip is not recorded or is poorly recorded, it is crucial that the bottle can balance out the volume in the next recordings. Therefore, the error (measured volume − actual volume) was adjusted manually. For example, assume that the subject drinks 10 mL and the bottle reports 0 mL, but then subsequently the subject drinks 20 mL and the bottle reports 30 mL total, then the adjusted error is 0 mL.
Table 1 lists the various performance metrics for each bottle considering both phases (100 sips). The Mean Percentage Error (MPE) per sip, the Mean Absolute Error (MAE) per sip and the Cumulative MPE are calculated as follows:
$$\text{Sip} \; \text{MPE}=\frac{1}{n}\sum_{i=1}^{n}\frac{({S}_{act}^{i}- {S}_{est}^{i})}{{S}_{act}^{i}}\times 100$$
$$\text{Sip} \; \text{MAE}=\frac{1}{n}\sum_{i=1}^{n}\left|{S}_{act}^{i}- {S}_{est}^{i}\right|$$
$$\text{Cumulative} \; \text{MPE}=\frac{1}{n}\sum_{k=1}^{n}\frac{({C}_{act}^{k}- {C}_{est}^{k})}{{C}_{act}^{k}}\times 100$$
where \({S}_{act}^{i}\) and \({S}_{est}^{i}\) are the actual and estimated intake volume for \({i}_{th}\) sip, respectively and \(n\) is the total number of sips. \({C}_{act}^{k}\) and \({C}_{est}^{k}\) represent the cumulative intake volume from the last \(k\) sips. The Sip MPE looks at the percent error for each individual sip and the Cumulative MPE looks at the total percent error over time. According to the results in Table 1, the H2OPal has the minimum number of missed recordings, the lowest Sip MPE and the lowest Cumulative MPE. When determining the total intake over a period of time, the Mean Error is preferred as a comparative metric over the Mean Absolute Error (MAE). Because it accounts for the bottle's ability to recover a poor measurement over time when recording the subsequent measurements. The sip MAE has also been included for applications where the accuracy of each sip is important, as it calculates the absolute error of each sip. The Cumulative MPE also measures how well the measurements balance out over the entire phase and does not penalize individual sips. Another observation was that 3 out of 4 bottles underestimated the volume intake per sip shown in Table 1 with negative numbers.
Table 1 Performance data for each commercial bottle.
The R-square Pearson correlation coefficients for all bottles are also shown in Table 1. The HidrateSpark 3 provided the highest correlation coefficient. Although the HidrateSpark 3 had some missed recordings, the majority of those were small sips (< 40 mL), so that it did not affect the correlation coefficient as heavily. The H2OPal and HidrateSpark Steel both had high correlations with r = 0.88, where the Thermos SmartLid had the lowest correlation (r = 0.75).
The Bland–Altman plots in Fig. 2 also confirmed that the HidrateSpark 3 had the smallest Limits of Agreements (LoA) compared to the other three bottles. The LoA analyzes the extent to which the actual and measured values agree. In addition, almost all measurements were in the range of LoAs confirming that this bottle provides consistent results, as shown in Fig. 2c. However, most of the values are below zero, meaning that generally the sip sizes are being underestimated. The same is true for the HidrateSpark Steel in Fig. 2b, where most of the error values are negative. Therefore, these two bottles provided the highest MPEs and Cumulative MPEs compared to H2Opal and Thermos Smart Lid where the errors were distributed above and below 0 as seen in Fig. 2a,d.
Bland–Altman plots for (a) H2OPal, (b) HidrateSpark Steel, (c) HidrateSpark 3, and (d) Thermos Smart Lid. The dotted lines represent the confidence intervals around the mean, calculated from the standard deviation in Table 1.
The HidrateSpark Steel and H2OPal had similar standard deviations of 20.04 mL and 21.41 mL, respectively. Figure 2a,b also demonstrated that the HidrateSpark Steel's values bounced consistently around the mean but generally stayed within the LoA region, while the H2Opal had more values outside of the LoA region. The Thermos Smart Lid had the largest standard deviation of 35.42 mL and more than 10% of measurements were outside of the LoA region shown in Fig. 2d. This bottle provided the minimum Sip Mean Error and relatively small Cumulative MPE, despite having the highest number of missed recordings and largest standard deviation. The Thermos SmartLid had many missed recordings because the sensor straw does not extend to the bottom of the container, causing missed recordings when the liquid contents are below the sensor stick (around 80 mL). This should lead to underestimating the fluid intake; however, the Thermos is the only bottle that had a positive MPE and Sip Mean Error, meaning that the bottle is overestimating the fluid intake. Therefore, the reason the Thermos had a very low mean sip error is because nearly each measurement from the bottle is a large overestimation. When these overestimations are averaged including the many missed sips that are not recorded at all (or "underestimated") the mean result balances out. When excluding the missed recordings from the calculation, the Sip Mean Error became + 10.38 mL, confirming that there is a large overestimation of individual sips. Though this may appear to be positive, in reality this bottle is inaccurate in individual sips estimation and unreliable as it misses many drinking events. Additionally, as seen in Fig. 2d, the Thermos SmartLid appears to have an increased error as the sip size increases.
In summary, the H2OPal is the most accurate at estimating sips over time and was the most reliable to measure most of the recordings. The Thermos Smart Lid was the least accurate and missed more sips than the other bottles. The HidrateSpark 3 bottles had more consistent error values, however underestimated the majority of sips leading to poor performance throughout time.
The results suggest that the bottles may have a certain offset that could be compensated using a calibration algorithm. This is especially true for the HidrateSpark bottles that have small standard deviations of errors and always underestimate individual sips. The Least-Square (LS) method was used with Phase 1 data while excluding any missed recordings to obtain the offset and gain values. The obtained equation was used on the measured sip intakes of Phase 2 to calculate the actual value and determine the error after calibration. Table 2 shows that the calibration improved the Sip Mean Error for both HidrateSpark bottles, but not the H2OPal or Thermos Smart Lid.
Table 2 Sip mean error before and after calibration for phase 2.
Bottle liquid level comparison
In Phase 1 to complete all the measurements, each bottle was refilled multiple times so it is possible that the calculated MAE is impacted by the filled level of the bottle. To determine this, each bottle was divided into three liquid levels as high, medium, and low, based on the total volume of each bottle. For the measurements in Phase 1, a one-way ANOVA test was performed to determine if the liquid level had a significant difference on the absolute error. For the HidrateSpark 3 and Steel, there was no significant differences in the error in each of the three categories. For the H2OPal and the Thermos bottles, there was a borderline significant difference (p < 0.05) using the Welsh test for unequal variance. Subsequently, a multiple comparison Tukey HSD test was performed on these two bottles. In both cases, the significant difference was between the "high" level and the "low" level categories. For the H2OPal, the "high" category had the largest mean and standard deviation, meaning that there is a higher error when the bottle is more filled. However, in the Thermos, the "low" category had the highest error. This is likely because the sensor does not extend to the bottom of the bottle.
Real world vs simulated phases
A two tailed t-test was conducted to compare the Phase 1 and Phase 2 errors for each bottle. For all bottles, we achieved p > 0.05 meaning that the two populations were not significantly different. However, it was observed that the number of missed recordings was much greater in Phase 2 for both HidrateSpark bottles. For the H2OPal, the number of missed recordings were almost equal (2 vs 3), and for the Thermos SmartLid there were fewer missed recordings in free living scenario (6 vs 10). Since the HidrateSpark bottles were both improved after calibration, a t-test was also conducted after-calibration. For the HidrateSpark 3, the Phase 1 and 2 errors were borderline significantly different (p = 0.046). This is more likely due to the larger number of missed recordings in Phase 2 compared to Phase 1.
Usability analysis and limitation
This section provides insight on the usability of the bottles and their apps, as well as additional functionality information. Although the accuracy of the bottles is important, the usability factors are also of interest when choosing a bottle.
App performance and functionality
The HidrateSpark 3 and HidrateSpark Steel are equipped with LEDs that blink to remind the users to drink if they are not on track to meet their goal, or to blink a certain number of times a day (set by the user). They can also be set to blink every time the user drinks. The H2OPal and Thermos Smart Lid do not have any visual feedback to remind the user to drink. However, all purchased bottles have mobile notifications to remind the users to drink via the mobile app. The number of notifications per day can be customized in the HidrateSpark and H2OPal app.
The HidrateSpark 3 and Steel use a linear trend to guide the user when to drink, and give an hourly suggested target the user should aim for to reach the goal by the end of the day. The H2OPal and Thermos Smart Lid only provide one total daily target. In all bottles, if the device is not connected to the app by Bluetooth, the data is stored locally and synchs once paired.
None of these four bottles focuses on elderly hydration. Additionally, the formulae the bottles use to determine the daily intake goal were not available, so it is difficult to determine if they are appropriate for seniors. The bottles are mostly large, heavy and are not tailored to seniors. The use of the mobile app may also not be ideal for seniors, though it could be useful for researchers to collect data, remotely.
Hardware and software limitations
All bottles could not determine if the liquid was consumed, discarded or spilled. All bottles also needed to be placed down on a surface after each sip to record the intake, accurately. This means that it is possibile to miss drinks if the bottle is not placed down, especially if it is refilled.
Another limitation is that the devices needed to be re-paired regularly with the app to synchronize the data. The Thermos needed to be re-paired every time the app is opened, and the HidrateSpark bottles often had difficulty finding the Bluetooth connection. The H2OPal was the easiest to re-pair with the app if the connection was interrupted. All bottles were calibrated before the testing began, and all had to be recalibrated at least once during the process. The HidrateSpark bottles and H2OPal had to be emptied and filled fully to calibrate.
All the bottles did not have the option to download or save data long term. Additionally, none were able to be accessed via an API.
The HidrateSpark 3 and H2OPal use replaceable lithium-ion batteries and the HidrateSpark Steel and Thermos SmartLid use rechargeable batteries. The rechargeable batteries should last up to 2 weeks on a full charge, as stated by the manufacturers, however, the Thermos SmartLid had to be charged almost every week when using it frequently. This is a limitation as many people will not remember to charge the bottles regularly.
Additional factors
There are various factors that impact the selection of a smart bottle especially when the users are elderly. The heaviness and bulkiness of the bottle is an important factor, as it needs to be easy to use for frail older adults. As mentioned, these bottles are not tailored to older adults. The price and the amount liquid each bottle can contain is also another factor. Table 3 shows the height, weight, liquid volume and price of each bottle. The Thermos Smart Lid is the cheapest and lightest, as it is made entirely of lighter plastic. It can also hold the most amount of liquid compared to other three bottles. Conversely, the H2OPal is the tallest, heaviest and most expensive among the study bottles.
Table 3 The height, weight, volume and price of the bottles.
Conclusion and future works
Commercially available smart bottles are very useful to researchers as there is no need to prototype a new device. Though there are many smart water bottles available, the most prevalent issue was that the data or raw signals were not accessible to the users, and only some had the results displayed in a mobile app. The development of a widely available smart bottle with high accuracy and completely accessible data is needed, especially one tailored to older adults. Of the four bottles tested, out of the box the H2OPal had the lowest Sip MPE, Cumulative MPE, and number of missed recordings. The HidrateSpark 3 had the highest linearity and smallest standard deviation and lowest MAE. The HidrateSpark Steel and HidrateSpark 3 can be simply calibrated manually to decrease the Sip Mean Error using LS method. For a more accurate sip recording, the HidrateSpark 3 is the preferred bottle, and for more consistent measurements over a period of time, the H2OPal is preferred. The Thermos SmartLid had the most unreliable performance with the most missed number of sips and a large over estimation of individual sips.
This study is not without limitations. In a real-world scenario, many users will drink from other vessels, especially for hot liquids, store bought beverages and alcohol. Future work should evaluate how the form factor of each bottle might affect the error to guide smart water bottle design.
Rule, A. D., Lieske, J. C. & Pais, V. M. Jr. Management of kidney stones in 2020. JAMA 323, 1961–1962. https://doi.org/10.1001/jama.2020.0662 (2020).
Conroy, D. E., West, A. B., Brunke-Reese, D., Thomaz, E. & Streeper, N. M. Just-in-time adaptive intervention to promote fluid consumption in patients with kidney stones. Health Psychol. 39, 1062 (2020).
Cohen, R., Fernie, G. & Roshan Fekr, A. Fluid intake monitoring systems for the elderly: A review of the literature. Nutrients 13, 2092. https://doi.org/10.3390/nu13062092 (2021).
Inc, H. HidrateSpark 3 Smart Water Bottle & Free Hydration Tracker App—Black https://hidratespark.com/products/black-hidrate-spark-3. Accessed 21 April 2021.
HidrateSpark STEEL Insulated Stainless Steel Smart Water Bottle & App—Hidrate Inc. https://hidratespark.com/products/hidratespark-steel. Accessed 21 April 2021.
H2Opal. https://www.h2opal.com/products/h2o-pal. Accessed 9 November 2020.
Thermos® Connected Hydration Bottle with Smart Lid. https://www.thermos.com/smartlid. Accessed 9 November 2020.
Borofsky, M. S., Dauw, C. A., York, N., Terry, C. & Lingeman, J. E. Accuracy of daily fluid intake measurements using a "smart" water bottle. Urolithiasis 46, 343–348. https://doi.org/10.1007/s00240-017-1006-x (2018).
Bernard, J., Song, L., Henderson, B. & Tasian, G. E. Association between daily water intake and 24-hour urine volume among adolescents with kidney stones. Urology 140, 150–154. https://doi.org/10.1016/j.urology.2020.01.024 (2020).
Fallmann, S., Psychoula, I., Chen, L., Chen, F., Doyle, J., Triboan, D. Reality and perception: Activity monitoring and data collection within a real-world smart home. in Proceedings of the 2017 IEEE SmartWorld, Ubiquitous Intelligence & Computing, Advanced & Trusted Computed, Scalable Computing & Communications, Cloud & Big Data Computing, Internet of People and Smart City Innovation (SmartWorld/SCALCOM/UIC/ATC/CBDCom/IOP/SCI), 1–6 (IEEE, 2017).
Plecher, D. A. et al. Interactive drinking gadget for the elderly and alzheimer patients. In Proceedings of the Human Aspects of IT for the Aged Population. Social Media, Games and Assistive Environments (eds Zhou, J. & Salvendy, G.) 444–463 (Springer International Publishing, 2019).
This work was supported by a Canadian Institutes of Health Research (CIHR) foundation Grant (FDN-148450). Dr Fernie receives this funding as the Creaghan Family Chair in Prevention and Healthcare Technologies.
The Kite Research Institute, Toronto Rehabilitation Institute - University Health Network, Toronto, Canada
Rachel Cohen, Geoff Fernie & Atena Roshan Fekr
Institute of Biomedical Engineering, University of Toronto, Toronto, Canada
Department of Surgery, University of Toronto, Toronto, Canada
Geoff Fernie
Rachel Cohen
Atena Roshan Fekr
Conceptualization—R.C.; methodology—R.C., A.R.; writing—original draft preparation—R.C., A.R.; writing—review and editing, G.F., A.R.; supervision—A.R., G.F. All authors have read and agreed to the published version of the manuscript.
Correspondence to Rachel Cohen.
Cohen, R., Fernie, G. & Roshan Fekr, A. Monitoring fluid intake by commercially available smart water bottles. Sci Rep 12, 4402 (2022). https://doi.org/10.1038/s41598-022-08335-5
Fluid intake recommendations in urolithiasis and general advice to patients without metabolic risk factors
Murat Can Kiremit
Abubekir Boyuk
Kremena Petkova
World Journal of Urology (2023) | CommonCrawl |
\begin{definition}[Definition:Tarski's Geometry]
Tarski's geometry is an axiomatic treatment of geometry.
Unless specified otherwise, {{ProofWiki}} will use the term Tarski's geometry to be a formal systematic treatment of geometry containing only:
:$(1):\quad$ The language and axioms of first-order logic, and the disciplines preceding it
:$(2):\quad$ The undefined terms of Tarski's Geometry
:$(3):\quad$ Tarski's Axioms of Geometry.
\end{definition} | ProofWiki |
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